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import abc
import logging
import numpy as np
import torch
import torch.nn.functional as F
from torch import nn as nn
import rlkit.torch.pytorch_util as ptu
from rlkit.policies.base import ExplorationPolicy
from rlkit.torch.core import torch_ify, elem_or_tuple_to_numpy
from rlkit.torch.distributions import (
Delta, TanhNormal, MultivariateDiagonalNormal, GaussianMixture, GaussianMixtureFull,
)
from rlkit.torch.networks import Mlp, CNN, GAT
from rlkit.torch.networks.basic import MultiInputSequential
from rlkit.torch.networks.stochastic.distribution_generator import (
DistributionGenerator
)
class TorchStochasticPolicy(
DistributionGenerator,
ExplorationPolicy, metaclass=abc.ABCMeta
):
def get_action(self, obs_np, ):
actions = self.get_actions(obs_np[None])
return actions[0, :], {}
def get_actions(self, obs_np, ):
dist = self._get_dist_from_np(obs_np)
actions = dist.sample()
return elem_or_tuple_to_numpy(actions)
def _get_dist_from_np(self, *args, **kwargs):
torch_args = tuple(torch_ify(x) for x in args)
torch_kwargs = {k: torch_ify(v) for k, v in kwargs.items()}
dist = self(*torch_args, **torch_kwargs)
return dist
class TorchGATStochasticPolicy(
DistributionGenerator,
ExplorationPolicy, metaclass=abc.ABCMeta
):
def get_action(self, obs_np, ):
actions = self.get_actions(obs_np)
return actions[0, :], {}
def get_actions(self, obs_np, ):
dist = self._get_dist_from_np(obs_np)
actions = dist.sample()
return elem_or_tuple_to_numpy(actions)
def _get_dist_from_np(self, *args, **kwargs):
torch_args = tuple(torch_ify(x) for x in args)
torch_kwargs = {k: torch_ify(v) for k, v in kwargs.items()}
dist = self(*torch_args, **torch_kwargs)
return dist
class PolicyFromDistributionGenerator(
MultiInputSequential,
TorchStochasticPolicy,
):
"""
Usage:
```
distribution_generator = FancyGenerativeModel()
policy = PolicyFromBatchDistributionModule(distribution_generator)
```
"""
pass
class MakeDeterministic(TorchStochasticPolicy):
def __init__(
self,
action_distribution_generator: DistributionGenerator,
):
super().__init__()
self._action_distribution_generator = action_distribution_generator
def forward(self, *args, **kwargs):
dist = self._action_distribution_generator.forward(*args, **kwargs)
return Delta(dist.mle_estimate())
class MakeGATDeterministic(TorchGATStochasticPolicy):
def __init__(
self,
action_distribution_generator: DistributionGenerator,
):
super().__init__()
self._action_distribution_generator = action_distribution_generator
def forward(self, *args, **kwargs):
dist = self._action_distribution_generator.forward(*args, **kwargs)
return Delta(dist.mle_estimate())
|
|
import numpy as np
def final_score(y_pred, threshold=0.5):
y_mae = np.where(y_pred[0] < threshold, 0, np.ceil(y_pred[0])).astype(int)
y_mse = np.where(y_pred[1] < threshold, 0, np.ceil(y_pred[1])).astype(int)
y_pred_avg = np.average([y_mae, y_mse], axis=0)
y_pred_avg = np.around(y_pred_avg).astype(int)
return y_pred_avg
|
|
# -*- coding: utf-8 -*-
"""Demo156_Outliers_Detection_LocalOutlierFactor.ipynb
## Outliers
An outlier is an observation that lies outside the overall pattern of a distribution __[Moore and McCabe, 1999]__.
- Outliers can either be treated special completely ignored
- E.g., Fraudulant transactions are outliers, but since we want to avoid them, they must be paid special attention
- If we think that the outliers are errors, we should remove them
## Which of the ML models care about Outliers?
Affected models:
- AdaBoost
- Linear models
- Linear regression
- Neural Networks (if the number is high)
- Logistic regression
- KMeans
- Heirarchical Clustering
- PCA
Unaffected models:
- Decision trees
- Naive bayes
- SVMs
- Random forest
- Gradient boosted trees
- K-Nearest Neighbors
### Identification
- Extreme Value Analysis
- IQR = 75th quantile - 25th quantile
- Upper boundary = 75th quantile + (IQR * 1.5)
- Lower boundary = 25th quantile - (IQR * 1.5)
- Upper Extreme boundary = 75th quantile + (IQR * 3)
- Lower Extreme boundary = 25th quantile - (IQR * 3)
"""
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np
from google.colab import drive
drive.mount('/content/gdrive')
data = pd.read_csv("gdrive/My Drive/Colab Notebooks/FeatureEngineering/train_date.csv")
data.head()
# Outliers according to the quantiles + 1.5 IQR
sns.catplot(x="Survived", y="Fare", kind="box", data=data)
sns.despine(left=False, right=False, top=False)
data['Fare'].describe()
# Get outliers
IQR = data['Fare'].quantile(0.75) - data['Fare'].quantile(0.25)
ub = data['Fare'].quantile(0.75) + (IQR * 3)
lb = data['Fare'].quantile(0.25) - (IQR * 3)
data[(data['Fare']>ub) | (data['Fare']<lb)].groupby('Survived')['Fare'].count()
# First let's plot the histogram to get an idea of the distribution
fig = data.Age.hist(bins=50, color='green')
fig.set_title('Distribution')
fig.set_xlabel('X')
fig.set_ylabel('#')
# Get outliers
IQR = data['Age'].quantile(0.75) - data['Age'].quantile(0.25)
ub = data['Age'].mean() + data['Age'].std()
lb = data['Age'].mean() - data['Age'].std()
data[(data['Age']>ub) | (data['Age']<lb)].groupby('Survived')['Age'].count()
from sklearn.metrics import roc_auc_score
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score
data = data.drop(['Date'], axis=1)
data.columns
data[['Fare','Age']].isnull().mean()
X_train, X_test, y_train, y_test = train_test_split(data[['Age', 'Fare']].fillna(0),
data['Survived'],
test_size=0.2)
X_train.shape, X_test.shape
# We will cap the values of outliers
data_processed = data.copy()
_temp = np.ceil(data['Age'].mean() + data['Age'].std())
data_processed.loc[data_processed.Age >= _temp, 'Age'] = _temp
IQR = data['Fare'].quantile(0.75) - data['Fare'].quantile(0.25)
_temp = np.ceil(data['Fare'].quantile(0.75) + (IQR * 3))
data_processed.loc[data_processed.Fare > _temp, 'Fare'] = _temp
X_train_processed, X_test_processed, y_train_processed, y_test_processed = train_test_split(
data_processed[['Age', 'Fare']].fillna(0),
data_processed['Survived'],
test_size=0.2)
from sklearn.neighbors import LocalOutlierFactor
df_outliers = data.copy()
df_outliers = df_outliers.fillna(0)
column_name = 'Fare'
obj = LocalOutlierFactor()
_temp = obj.fit_predict(df_outliers[[column_name]])
print(np.unique(_temp, return_counts=True))
central = df_outliers[_temp==1][column_name].mean()
max_val = df_outliers[_temp==1][column_name].max()
min_val = df_outliers[_temp==1][column_name].min()
df_outliers.loc[_temp==-1,[column_name]] = df_outliers.loc[_temp==-1,[column_name]].apply(lambda x: [max_val if y > central else y for y in x])
df_outliers.loc[_temp==-1,[column_name]] = df_outliers.loc[_temp==-1,[column_name]].apply(lambda x: [min_val if y < central else y for y in x])
print(data.shape)
print(df_outliers.shape)
column_name = 'Age'
obj = LocalOutlierFactor()
_temp = obj.fit_predict(df_outliers[[column_name]])
print(np.unique(_temp, return_counts=True))
central = df_outliers[_temp==1][column_name].mean()
max_val = df_outliers[_temp==1][column_name].max()
min_val = df_outliers[_temp==1][column_name].min()
df_outliers.loc[_temp==-1,[column_name]] = df_outliers.loc[_temp==-1,[column_name]].apply(lambda x: [max_val if y > central else y for y in x])
df_outliers.loc[_temp==-1,[column_name]] = df_outliers.loc[_temp==-1,[column_name]].apply(lambda x: [min_val if y < central else y for y in x])
print(data.shape)
print(df_outliers.shape)
X_train_outliers, X_test_outliers, y_train_outliers, y_test_outliers = train_test_split(
df_outliers[['Age', 'Fare']],
df_outliers['Survived'],
test_size=0.2)
print(X_train.shape)
print(X_train_processed.shape)
print(X_train_outliers.shape)
print(X_test.shape)
print(X_test_processed.shape)
print(X_test_outliers.shape)
from sklearn.linear_model import LogisticRegression
classifier = LogisticRegression()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.linear_model import RidgeClassifierCV
classifier = RidgeClassifierCV()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.linear_model import RidgeClassifierCV
classifier = RidgeClassifierCV()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.svm import SVC
classifier = SVC()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.neural_network import MLPClassifier
classifier = MLPClassifier()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.svm import LinearSVC
classifier = LinearSVC()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.ensemble import RandomForestClassifier
classifier = RandomForestClassifier()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.tree import DecisionTreeClassifier
classifier = DecisionTreeClassifier()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.ensemble import GradientBoostingClassifier
classifier = GradientBoostingClassifier()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.linear_model import SGDClassifier
classifier = SGDClassifier()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.linear_model import Perceptron
classifier = Perceptron()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.naive_bayes import GaussianNB
classifier = GaussianNB()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
from sklearn.neighbors import KNeighborsClassifier
classifier = KNeighborsClassifier()
classifier.fit(X_train,y_train)
y_pred = classifier.predict(X_test)
y_pred = np.round(y_pred).flatten()
print(accuracy_score(y_test, y_pred))
classifier.fit(X_train_processed,y_train_processed)
y_pred_processed = classifier.predict(X_test_processed)
y_pred_processed = np.round(y_pred_processed).flatten()
print(accuracy_score(y_test_processed, y_pred_processed))
classifier.fit(X_train_outliers,y_train_outliers)
y_pred_outliers = classifier.predict(X_test_outliers)
y_pred_outliers = np.round(y_pred_outliers).flatten()
print(accuracy_score(y_test_outliers, y_pred_outliers))
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######################################################################
###
### Decision tree - Titanic
### The purpose of this dataset is to predict which people
### are more likely to survive after the collision with the iceberg.
### The dataset contains 13 variables and 1309 observations.
### The dataset is ordered by the variable X.
### https://www.guru99.com/r-decision-trees.html#4
###
#######################################################################
### Variable Definition Key
### survival Survival 0 = No, 1 = Yes
### pclass Ticket class 1 = 1st, 2 = 2nd, 3 = 3rd
### sex Sex
### Age Age in years
### sibsp # of siblings / spouses aboard the Titanic
### parch # of parents / children aboard the Titanic
### ticket Ticket number
### fare Passenger fare
### cabin Cabin number
### embarked Port of Embarkation C = Cherbourg, Q = Queenstown, S = Southampton
### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ###
rm(list=ls())
### Libraries
library(tree)
library(dplyr)
library(rpart.plot)
library(rpart)
library(ROCR)
library(randomForest)
library(caret)
library(e1071)
# Questa funzione installa una librerie che servirà per i random forest
#options(repos='http://cran.rstudio.org')
#have.packages <- installed.packages()
#cran.packages <- c('devtools','plotrix','randomForest','tree')
#to.install <- setdiff(cran.packages, have.packages[,1])
#if(length(to.install)>0) install.packages(to.install)
#library(devtools)
#if(!('reprtree' %in% installed.packages())){
# install_github('araastat/reprtree')
#}
#for(p in c(cran.packages, 'reprtree')) eval(substitute(library(pkg), list(pkg=p)))
library(reprtree)
### ### ### ### ### ###
set.seed(678)
DIR = "/Users/gianlucamastrantonio/Desktop/statistico/lavori/lezioni/DataSpace/2018/Lez7/"
Dataset = read.csv(paste(DIR,"titanic.csv",sep=""))
head(Dataset)
tail(Dataset)
# Puliamo il dataset
#
DatasetClean <- Dataset %>%
select(-c(home.dest, cabin, name, X, ticket)) %>%
#Convert to factor level
mutate(pclass = factor(pclass, levels = c(1, 2, 3), labels = c('Upper', 'Middle', 'Lower')),
survived = factor(survived, levels = c(0, 1), labels = c('No', 'Yes'))) %>%
na.omit()
glimpse(DatasetClean)
summary(DatasetClean)
# dividiamo il campioni in train and test
shuffle_index = sample(1:nrow(DatasetClean))
head(shuffle_index)
DatasetShuffled = DatasetClean[shuffle_index,]
### Creiamo una funzione per fare lo split
create_train_test = function(data, size = 0.8) {
#
n_row = nrow(data)
total_row = round(size*n_row)
train_sample = 1: total_row
ret = list(Train = data[train_sample,],Test=data[-train_sample,])
return(ret)
}
App = create_train_test(DatasetShuffled, 0.8)
DataTrain = App$Train
DataTest = App$Test
dim(DataTrain)
dim(DataTest)
rm(App)
# controlliamo che la randomizzazione abbia funzionato
prop.table(table(DataTrain$survived))
prop.table(table(DataTest$survived))
# automatizziamo il processo
# versione 1
PropTable = function(x)
{
prop.table(table(x))
}
apply(DataTrain[,c(1,2,3,5,6,8)],2,PropTable)
apply(DataTest[,c(1,2,3,5,6,8)],2,PropTable)
# versione 2
IndexVar = c(1,2,3,5,6,8)
for(i in IndexVar)
{
print(colnames(DataTest)[i])
print(prop.table(table(DataTrain[,i])))
print(prop.table(table(DataTest[,i])))
}
# il modello
fit = rpart(survived~., data = DataTrain, method = 'class')
rpart.plot(fit, extra = 1,type=4)
rpart.plot(fit, extra = 106,type=4)
rpart.plot(fit, extra = 100,type=4, under=T)
#?rpart.plot per vedere i vari tipi grafici
rpart.rules(fit)
# facciamo previsione (classe)
Pred = predict(fit, DataTest, type = 'class')
plot(Pred)
# Controlliamo la previsione
Tab = table(DataTest$survived, Pred)
Tab
Accuracy = sum(diag(Tab))/sum(Tab)
# curva roc
# stimiamo le probs
PredProb = predict(fit, DataTest)
summary(PredProb)
pred = prediction(PredProb[,2], DataTest$survived)
perf = performance(pred,"tpr","fpr")
plot(perf,colorize=TRUE)
abline(a=0,b=1)
## vadiamo meglio come funziona la curca ROC
i = 5
a = perf@alpha.values[[1]][i]
fp = perf@x.values[[1]][i]
tp = perf@y.values[[1]][i]
glm.pred=rep("Down",252)
X = ifelse(PredProb[,2]>=a,"Yes","No")
table(X,DataTest$survived)
57/(37+51)
13/(108+13)
# tuning degli hyperparametri
# creaiamo uan funzione che mostre l'accuratezza
accuracy_tune = function(fit) {
predict_unseen = predict(fit, DataTest, type = 'class')
table_mat = table(DataTest$survived, predict_unseen)
accuracy_Test = sum(diag(table_mat)) / sum(table_mat)
accuracy_Test
}
# per fare il tuning dobbiamo settare
# rpart.control(minsplit = 20, minbucket = round(20/3), maxdepth = 30)
# tra i molti parametri abbiamo
# -minsplit: Set the minimum number of observations in the node before the algorithm perform a split
# -minbucket: Set the minimum number of observations in the final note i.e. the leaf
# -maxdepth: Set the maximum depth of any node of the final tree. The root node is treated a depth 0
# facciamo untest e vediamo come migliorare l'accuratezza
control = rpart.control(minsplit = 4,
minbucket = round(5 / 3),
maxdepth = 3,
cp = 0)
tune_fit <- rpart(survived~., data = DataTrain, method = 'class', control = control)
accuracy_tune(tune_fit)
# e vediamo la nuova roc
PredProbTune = predict(tune_fit, DataTest)
summary(PredProbTune)
predTune = prediction(PredProbTune[,2], DataTest$survived)
perfTune = performance(predTune,"tpr","fpr")
plot(perfTune,colorize=TRUE)
abline(a=0,b=1)
plot(perf, add=T,colorize=TRUE, lty=2)
### ### ### ### ### ### ### ### ### ### ###
### Random forest
### ### ### ### ### ### ### ### ### ### ###
# modello base
RF = randomForest(survived~. , ntree=10, mtry=1,data = DataTrain)
plot(RF)
RF
varImpPlot(RF)
# alcuni parametri
# ntree: number of trees in the forest
# mtry: Number of candidates draw to feed the algorithm. By default, it is the square of the number of columns.
# maxnodes: Set the maximum amount of terminal nodes in the forest
# importance=TRUE: Whether independent variables importance in the random forest be assessed
## plottiamo l'albero 2
reprtree:::plot.getTree(RF,k=2)
# Cerchiamo il miglior set di parametri
# troviamo mtry
tuneGrid <- expand.grid(mtry = c(1: 10))
trControl <- trainControl(method = "cv",
number = 10,
search = "grid")
rf_mtry <- train(survived~.,
data = DataTrain,
method = "rf",
metric = "Accuracy",
tuneGrid = tuneGrid,
trControl = trControl,
importance = TRUE,
nodesize = 14,
ntree = 300)
rf_mtry
best_mtry <- rf_mtry$bestTune$mtry
# cerchiamo il miglior maxnode
store_maxnode <- list()
tuneGrid <- expand.grid(.mtry = best_mtry)
for (maxnodes in seq(6,30,by=3)) {
#set.seed(1234)
rf_maxnode <- train(survived~.,
data = DataTrain,
method = "rf",
metric = "Accuracy",
tuneGrid = tuneGrid,
trControl = trControl,
importance = TRUE,
nodesize = 14,
maxnodes = maxnodes,
ntree = 300)
current_iteration <- toString(maxnodes)
store_maxnode[[current_iteration]] <- rf_maxnode
}
results_mtry <- resamples(store_maxnode)
summary(results_mtry)
# cerchiamo il miglior ntree
store_maxtrees <- list()
for (ntree in c(250, 300, 350, 400, 450, 500, 550, 600, 800, 1000, 2000)) {
set.seed(5678)
rf_maxtrees <- train(survived~.,
data = DataTrain,
method = "rf",
metric = "Accuracy",
tuneGrid = tuneGrid,
trControl = trControl,
importance = TRUE,
nodesize = 14,
maxnodes = 24,
ntree = ntree)
key <- toString(ntree)
store_maxtrees[[key]] <- rf_maxtrees
}
results_tree <- resamples(store_maxtrees)
summary(results_tree)
# best model
fit_rf <- train(survived~.,
DataTrain,
method = "rf",
metric = "Accuracy",
tuneGrid = tuneGrid,
trControl = trControl,
importance = TRUE,
nodesize = 14,
ntree = 800,
maxnodes = 24)
# stimiamo il modello con random Forest
RFtot = randomForest(survived~. , ntree=800, maxnodes = 24,nodesize = 14,mtry=best_mtry,data = DataTrain)
varImp(RFtot)
varImpPlot(RFtot)
## plottiamo l'albero 2
reprtree:::plot.getTree(RFtot,k=2)
# facciamo previsione
prediction <-predict(RFtot, DataTest)
# possiamo vedere la matrice di confusione
confusionMatrix(prediction, DataTest$survived)
# curva ROC
PredProbRF = predict(fit, DataTest, type="prob")
summary(PredProbRF)
predRF = prediction(PredProbRF[,2], DataTest$survived)
perfRF = performance(pred,"tpr","fpr")
plot(perfRF,colorize=TRUE)
abline(a=0,b=1)
plot(perfTune,colorize=TRUE,add=T)
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1706.07434
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\section{Caveats, discussion and conclusions}
\label{sec:discussion}
We have shown that the \emph{relative} role that environment is observed to have on the \HI fractions of satellites \citep{brown17} can be recovered within a semi-analytic model of galaxy formation with a self-consistent evolution of disc structure, namely {\sc Dark Sage} \citep{stevens16}. However, this model also predicts the mean \HI fractions of satellites as a whole to be too low. In the current design of the model, these two results are inherently tied together. One possible explanation is that the model is missing a physical process that would systematically (and exclusively) raise the gas content of satellites, irrespective of their environment.
One process hypothesised to help explain why low-mass galaxies in denser environments are observed to have brighter X-ray haloes \citep{mulchaey10} is `confinement pressure'. The idea here is that the temperature differential between the hot gas of a satellite and the hotter intracluster medium (ICM) creates an inward pressure, thus helping that satellite to retain its hot gas and dampening the effect of hot-gas stripping. However, it has been shown with both a simple analytic model and the results of hydrodynamic simulations that confinement pressure only has a strong dampening effect on hot-gas stripping once a satellite reaches pericentre, at which point most of its hot gas has been removed already \citep{bahe12}. The satellite galaxies in the {\sc Dark Sage} results without hot-gas stripping presented in this paper were also prescribed to maintain any reheated gas from feedback within their own hot reservoir (Section \ref{ssec:hot}). In effect, this run of the model shows an extreme for what confinement pressure could achieve.
More recently, using a cosmological hydrodynamic simulation focussed on a galaxy cluster, \citet*{quilis17} have suggested that the hot gas of satellite galaxies can be replenished by accretion from the ICM. This helps to counter-act the stripping of hot gas, where cold-gas stripping is shown by \citeauthor{quilis17} to have a greater net impact on these galaxies. Complementary results from the EAGLE simulations have explicitly shown that the gas accretion rates through a fixed 30-kpc aperture around satellite galaxies still show a strong dependence on environment \citep{voort17}. Note, though, that gas accretion rates through a fixed aperture can be wildly different to accretion rates onto a (sub)halo \citep{stevens14}.
Unlike central galaxies, which accrete hot gas cosmologically, satellites cannot acquire hot gas from an external source within the current {\sc Dark Sage} model. This feature remains a shared standard for semi-analytic models in general \citep[see the latest versions of other models, e.g.][]{gargiulo15,henriques15,lacey16}. If subhaloes were able to accrete hot gas from the ICM within a semi-analytic framework, this should systematically raise \HI fractions and lower quiescent fractions. Both of these outcomes would be favourable for {\sc Dark Sage}, potentially offering a solution (at least partially) for simultaneously recovering the absolute and relative impacts of environmental stripping processes.
Another possible explanation for the discrepancy in the mean \HI fractions of satellites in the observations and model lies with the group-finding algorithm applied to the observational data. \citet{campbell15} studied how well several group-finding algorithms performed in correctly identifying satellites and centrals by applying them to a mock catalogue from simulated data. The authors found that the \citet{yang05,yang07} technique returns a population of satellites with a purity of $\sim \! 0.6$--$0.7$ for all haloes with $M_{\rm vir} \! > \! 10^{12}\,{\rm M}_{\odot}$ \citep[see fig.~6 of][]{campbell15}. This means it is possible that 30--40 per cent of galaxies classed as satellites in our observational sample (Section \ref{sec:obs}) might actually be centrals. This could more than account for the difference in the fraction of satellites in the observed and model samples (38.2 and 29.7 per cent, respectively -- Section \ref{ssec:satcen_full}). While repeating the efforts of \citeauthor{campbell15} for {\sc Dark Sage} is beyond the scope of this paper, we can easily test what the model results would be after contaminating the satellite population with misidentified centrals by taking a weighted mean of the true central and satellite galaxy properties. Using respective weights of 0.6 and 0.4 for satellites and centrals gives a mean \HI fraction as a function of stellar mass that is consistent with a satellite population with a purity of 0.6. We have plotted this for {\sc Dark Sage} in Fig.~\ref{fig:contam}$a$, where we have also included central-galaxy results after applying a purity of 0.8. The contaminated satellite population meets the relation for observed satellites to a remarkable degree. While this could offer an easy out for the too-gas-poor nature of {\sc Dark Sage} satellites, cross contamination of satellites and centrals leads to a reduction in the relative gas fractions of satellites in bins of halo masses, as seen in Fig.~\ref{fig:contam}$c$. This would, therefore, simply trade one good result for a different one. Either way, the treatment of satellites in the semi-analytic model still requires further attention before it can reproduce all the observational results we have shown.
\begin{figure*}
\includegraphics[width=0.34\linewidth]{HIfrac_contamination.pdf}
\includegraphics[width=0.34\linewidth]{Quiescent_contamination.pdf}
\caption{Full-model results for {\sc Dark Sage} after cross-contaminating the central and satellite populations. We have invoked a purity, $P$, of 0.6 and 0.8 for satellites and centrals, respectively, for all stellar and halo masses. Observational data remain as measured. Panels $a$, $b$, $c$, and $d$ should be compared with Figs.~\ref{fig:satcen}, \ref{fig:quiescent}, \ref{fig:satenv}, and \ref{fig:quiescent_env}, respectively.}
\label{fig:contam}
\end{figure*}
Our results with {\sc Dark Sage} suggest the effects of hot- and cold-gas stripping on galaxies are almost entirely separable. Cold-gas stripping removes the highest-$j$ gas in the disc, which is at low surface density and is dominated in mass by H\,{\sc i}. This generates a difference in the mean \HI fractions between central and satellite galaxies at fixed stellar mass (Fig.~\ref{fig:satcen}), and nicely recovers observed trends in the relative \HI fractions of satellites in different environments (probed by halo mass -- Fig.~\ref{fig:satenv}). Molecular gas, which exists almost exclusively at the centres of these galaxies' discs, is almost unaffected by this process. Recent CO observations of 3 satellite galaxies in the Virgo cluster by \citet{lee17} show that, while the morphology of molecular gas can be disturbed by ram pressure, there is no evidence that ram-pressure stripping affects the total molecular gas content. For {\sc Dark Sage}, only low-$m_*$ galaxies in high-mass haloes have their molecular gas and star formation activity directly impacted by ram-pressure stripping (Fig.~\ref{fig:quiescent_env}).
On the other hand, hot-gas stripping shuts down galaxy accretion, thereby affecting the evolution of gas across the entire disc. The replenishment of molecular gas in the galaxy is then suppressed, causing the satellites to become quiescent after consuming their available H$_2$ in star formation. This is a major contributor behind the observed difference in quiescent fractions between satellites and centrals at fixed stellar mass (Fig.~\ref{fig:quiescent}). Based on the relative colours, sizes, and masses of centrals and satellites, observational evidence supports a picture where hot-gas stripping is the responsible mechanism for satellite quenching \citep{bosch08}. While showing improvement relative to earlier results from semi-analytic models, {\sc Dark Sage} still overproduces the fraction of quiescent satellites in the range $10^{10} \! \lesssim \! m_*/{\rm M}_{\odot} \! \lesssim \! 10^{11}$. Cross-contamination of centrals and satellites in the observational data from the group finder could also be playing a role here, mind; after cross-contaminating the {\sc Dark Sage} galaxies, we find the excess of quiescent satellites at mid masses goes away (Fig.~\ref{fig:contam}$b$). Again though, this reduces the relative strength of environment on quiescence in the model (Fig.~\ref{fig:contam}$d$).
We note that \citet{thesis} has shown that the stellar half-mass radii of disc-dominated {\sc Dark Sage} galaxies are systematically lower than the half-light radii of galaxies in the GAMA\footnote{Galaxy And Mass Assembly} survey by $\lesssim \! 0.1$\,dex \citep[cf.][]{lange16}. Galaxies that never suffered an instability in their existence, however, meet the observed relation quite precisely. Instabilities drive mass inwards and angular momentum outwards in {\sc Dark Sage} discs \citep{stevens16}. Because the model lacks a detailed consideration of radial dispersion support at the centres of discs, their surface density profiles grow an exaggerated cusp. This leads to smaller stellar sizes and higher molecular-to-atomic hydrogen mass ratios, especially at the discs' centres. Because the integrated \HI content of the model galaxies is calibrated to meet observations \citep{brown15}, the average radius of an \HI element will be greater than in real galaxies \citep[see fig.~4 of][]{stevens16}. This does not prevent the nominal \HI radii of the galaxies (where $\Sigma_{\mathrm{H}\,\LARGE\textsc{i}} \! = \! 1\,{\rm M}_{\odot}\,{\rm pc}^{-2}$) from agreeing with observations for fixed \HI mass, however (see Lutz et al.~in preparation). We compare the molecular-to-atomic ratio as a function of stellar mass for {\sc Dark Sage} galaxies against data presented by \citet{boselli14b} from the \emph{Herschel} Reference Survey \citep{herschel,boselli14a} in Fig.~\ref{fig:h2frac}. Indeed, the ratios of the model galaxies are systematically higher than the observational data, regardless of whether the conversion between CO luminosity and H$_2$ mass is taken as variable or a constant. While the \citeauthor{boselli14b} data are not volume-limited and only account for 74 late-type galaxies with detections in both \HI and CO, they are designed to be representative of average galaxies in the local Universe. With that in mind, we encourage that while the information in Fig.~\ref{fig:h2frac} is important to consider in interpreting our results, it is unlikely to be what drives them. Future work in improving how the centres of {\sc Dark Sage} discs are modelled, as well as larger H$_2$ surveys, will help to clarify our results.
\begin{figure}
\includegraphics[width=\linewidth]{H2HIfrac_one.pdf}
\caption{Molecular-to-atomic hydrogen mass ratio as a function of stellar mass for galaxies at $z=0$ with $m_{\rm H_2} \! \geq \! 10^{8.6}\,{\rm M}_{\odot}$, $m_{\rm H_2} \! \geq \! 0.01\,m_{\mathrm{H}\,\LARGE\textsc{i}}$, and bulge-to-total ratios $\! < \! 0.3$ (these include both centrals and satellites). These cuts are to approximately match the completeness limit and morphology of the compared, observed galaxies from \citet{boselli14b}. The $x$ and $y$ positions of the points are log of each of the mean stellar mass and mean molecular-to-atomic ratio, respectively, for four bins. Square and starred points have been manually shifted to the right by 0.03 and 0.06 dex, respectively, for the sake of clarity. These present the same actual data, where squares employ a luminosity-dependent conversion factor between CO emission and H$_2$ mass, and starred points use a constant conversion factor. The error bars give the standard deviation in $\log_{10}(m_{\rm H_2}/m_{\mathrm{H}\,\LARGE\textsc{i}})$ within each bin. {\sc Dark Sage} data are for the full model with the $f_{\rm H_2}(Z)$ prescription, using (sub)haloes with $N_{\rm p,max} \! \geq \! 100$.}
\label{fig:h2frac}
\end{figure}
We should be optimistic about soon converging on a detailed description of how environmental and secular processes affect the gas content and star formation activity of galaxies in a cohesive manner that explains our observations.
Already, data from MUSE\footnote{Multi Unit Spectroscopic Explorer} \citep{muse} have been used to show the optical effects of ram pressure in exquisite detail for a small sample of galaxies \citep[e.g.][]{muse1,gasp1}, which provide clues for this picture.
Upcoming surveys such as the ASKAP\footnote{Australian Square Kilometre Array Pathfinder} \HI All-Sky Survey \citep[also known as {\sc Wallaby};][]{wallaby} will complement these with large-number statistics of the \HI content, structure, and kinematics of environmentally influenced satellite galaxies.
On the theory side, where detailed cosmological hydrodynamic simulations have been limited in their volume \citep[$\sim\!10^6\,{\rm Mpc}^3$ -- e.g.][]{dubois14,illustris,khandai15,schaye15}, follow-up simulations with the same well-tested physical prescriptions are now targeting Local Group analogues and massive clusters, providing more opportunity to learn about the impact of stripping processes \citep[][respectively]{sawala16,bahe17}.
Based on the level of agreement with observations we have presented in this paper, and without any concerns regarding volume or computational efficiency, semi-analytic models are bound to continue to play an important role in this quest as well.
\section{Introduction}
\label{sec:intro}
It is widely accepted that the seeds of galaxies are born out of the gravitational fragmentation of gas within dark-matter-dominated overdensities in the Universe \citep[\'{a} la][]{white78}, more commonly referred to as `haloes'. Our modern theoretical consensus is that galaxies then continue to accrete gas cosmologically, either directly through cold streams or via their hot circumgalactic medium, where ionised gas must cool to a predominantly neutral phase before settling into the galactic disc \citep[see][]{rees77,white91,birnboim03,keres05,dekel06,benson11,stevens17}. In addition, the hierarchical nature of galaxy growth means galaxies can acquire large volumes of gas at once through mergers. As the principal component of gas in galaxies, and because cold gas is the raw material for forming new stars, neutral atomic hydrogen (H\,{\sc i}) is one of the most important ingredients in the formation and evolution of galaxies. Therefore, understanding the ensemble of internal and external astrophysical mechanisms that regulate \HI content and star formation activity is critical if we are to realise a coherent picture of galaxy evolution.
Internally, beyond the direct consumption of gas in the formation of stars and the accretion of black holes, the evolution of galaxies' gas reservoirs is consequently dictated by feedback from stellar evolution and active galactic nuclei. This feedback manifests in the form of energetic winds that eject and/or heat gas reserves, as evidenced and detailed by a variety of observations \citep[e.g.][]{sharp10,feruglio10,anderson15,cicone15,nielsen16} and simulations \citep*[e.g.][]{springel03,brook11,costa15,taylor15,bower17}. Instabilities and turbulence within the interstellar medium also help regulate the distribution of gas and its ability to form stars \citep*[see, e.g.,][]{federrath15,stevens16}. While these secular processes all impact galaxies' gas fractions, it has become increasingly clear that external effects, driven by environment, are of comparable importance.
The role of environment in the suppression of \HI was first demonstrated by observations showing cluster galaxies to be gas-poor compared to the field \citep{giovanelli85,solanes01}. This was exemplified in exquisite detail by the VLA\footnote{Very Large Array} Imaging of Virgo in Atomic gas survey \citep[VIVA;][]{chung09}, which demonstrated the importance of the highest density environments in shutting down star formation via strong gas depletion mechanisms. Other observational \citep[e.g.][]{cortese11,catinella13} and theoretical \citep[e.g.][]{mccarthy08,rafie15,marasco16} efforts support this picture, with both camps generally describing external processes that are distinguishable based upon the time-scales over which they act: (i) those that act swiftly and directly upon the cold gas to remove it from the galaxy via an interaction of the interstellar and intracluster (or intragroup) media \citep[$\sim$10s of Myr, i.e. ram-pressure stripping;][]{gunn72}, or (ii) those that regulate the rate at which gas is able to accrete onto the galaxy from its dark-matter halo over more lengthy time-scales \citep*[i.e.~strangulation, where galaxies consume their available gas for star formation in $\gtrsim$1 Gyr;][]{larson80}. Using this distinction, \citet{brown17} provide strong evidence that the gas loss in massive haloes ($M_{\rm vir} > 10^{13}\,{\rm M}_{\odot}$) is considerably faster than the subsequent quenching of star formation.
To zeroth order, environment can be thought of as a dichotomy between `central' and `satellite' galaxies. These terms are most relevant for numerical simulations, where a central galaxy belongs to the most massive subhalo of a halo \citep[see, e.g.,][]{springel01,onions12}. All remaining subhaloes host satellite galaxies. Under this definition, a central can be both isolated or have any number of satellites associated with it. By definition, only satellite galaxies are prone to gas-stripping processes. For observational data to be interpreted in this framework, group-finding algorithms are typically employed to arrange galaxies into haloes. These use a known halo mass function from simulations with abundance matching to assign halo masses, where the brightest or most massive galaxy is labelled the central and the remainder are tagged as satellites \citep[see, e.g.,][]{huchra82,tucker00,yang05,campbell15}. The strength of environmental stripping processes depends primarily on the parent halo mass, within which the satellite resides, as evidenced by their \HI fractions \citep{brown17}.
In recent years, single-dish surveys have begun to provide \HI measurements for tens of thousands of galaxies. For example, the \HI Parkes All Sky Survey \citep[HIPASS;][]{meyer04} scanned $\sim$75 per cent of the sky (30,000 deg$^2$) and detected 21-cm emission in $\sim$5000 nearby galaxies. Its successor, the Arecibo Legacy Fast ALFA\footnote{Arecibo L-band Feed Array} survey \citep[ALFALFA;][]{giovanelli05} went even further, and the latest data release ($\alpha$.70) provides a census of \HI gas for $\sim$25\,000 galaxies over $\sim$7000 deg$^2$ of sky \citep[for a presentation of the earlier $\alpha$.40 release, see][]{haynes11}. In addition, the stacking of undetected \HI sources based on optical position and redshift has become be a major tool for pushing 21-cm surveys beyond their nominal sensitivity limit, providing representative, statistical studies of \HI as a function of galaxy properties and environment in the local Universe \citep{fabello11a,fabello11b,fabello12,brown15,brown17}.
In their paper, \citet{brown17} compare their average gas content scaling relations with predictions from the cosmological hydrodynamic simulations of \citet{dave13} and the {\sc galform} semi-analytic model \citep{gp14}. They find that both sets of predictions are too gas-poor at fixed stellar mass and specific star formation rate (sSFR), although there is qualitative reproduction of ram-pressure stripping in the group and cluster regime. \citet{marasco16} also investigate the role of environment in dictating the \HI content of galaxies within the EAGLE\footnote{Evolution and Assembly of GaLaxies and their Environments} suite of cosmological hydrodynamic simulations \citep{schaye15}. They use the simulations to successfully reproduce the findings of \citet{fabello12} and \citet{catinella13}, demonstrating that EAGLE galaxies in more massive haloes have lower gas content at fixed stellar mass. However, rather than a continuous trend of \HI depletion as a function of environment \citep[e.g.][]{stark16,brown17}, their work finds the role of environment to be more binary, determining whether or not galaxies have any \HI at all. Higher-resolution runs of these simulations have found the \HI fractions to increase \citep{bahe16,marasco16,crain17}.
Despite significant progress, large uncertainties remain concerning where and how precisely, in terms of epoch and environment, secular and environmental processes influence galaxy gas reservoirs. We therefore use the {\sc Dark Sage} semi-analytic model \citep{stevens16} to investigate the contributions of various evolutionary and environmental processes in determining the gas fractions and star formation activity of both central and satellite galaxies, relating this to their stellar masses and specific star formation rates. Because semi-analytic models describe the phenomenology of hydrodynamical effects without directly modelling hydrodynamics at all, all gas-stripping processes are manually prescribed. The advantage this gives is that we can learn about the relative roles each effect has on the gas content of galaxies by turning the effects on and off. {\sc Dark Sage} is especially well equipped, as the one-dimensional disc structure of galaxies is self-consistently evolved in the model, naturally leading to a spatial dependence of environmental effects within galaxies.
This paper is structured as follows. We describe the main aspects of the {\sc Dark Sage} semi-analytic model in Section \ref{sec:sam}, including new features we have introduced for this work. In Section \ref{sec:obs}, we describe the observational data with which our results are closely compared. We examine the \HI fractions and quiescence (lack of star formation) of galaxies within the central--satellite dichotomy in Section \ref{sec:satcen} by comparing a set of {\sc Dark Sage} model variants against our observed data. The role of environment in shaping the \HI fractions and quiescence of satellite galaxies is then studied in greater detail in Section \ref{sec:satenv}. We discuss the context of our results with recent literature and offer concluding remarks in Section \ref{sec:discussion}.
\section{The semi-analytic model}
\label{sec:sam}
The {\sc Dark Sage} semi-analytic model \citep{stevens16} is a heavily modified version of the models developed by \citet{croton06,sage}. Whereas most semi-analytic models evolve the integrated properties of singular baryonic reservoirs, {\sc Dark Sage} evolves the one-dimensional structure of galactic discs in annuli of fixed specific angular momentum \citep[similar to][]{stringer07,dutton09}, where $j \! = \! r \, v_{\rm circ}$. Evolutionary and environmental processes thus affect each galaxy on local scales. We refer the reader to section 3 of \citet{stevens16} for full details, but briefly outline the key features of the model here.
\begin{itemize}
\item \emph{Cooling:} Hot gas cools onto galaxies following the method introduced by \citet{white91}, using the cooling tables of \citet{sutherland93}. The cooling gas is assumed to have an exponential surface density profile as a function of $j$ and to immediately localise itself with material in the disc of the same $j$.
\item \emph{Star formation, evolution, and feedback:} Passive star formation goes as $\Sigma_{\rm SFR} \! \propto \! \Sigma_{\rm H_2}$, where 43\% of star-forming gas is approximated as instantly recycled \citep[cf.][]{cole00}. Every star formation episode results in supernova feedback, which heats gas out of the disc. The mass of gas heated in an annulus is directly proportional to the mass of stars formed and inversely proportional to the local gas surface density. The latter approximates supernova energy as being more readily dissipated in denser gaseous regions \citep[this model follows][]{fu10,fu13}. Excess energy from supernovae can eject gas out of the halo.
\item \emph{Instabilities:} A two-component Toomre $Q$ value \citep{toomre64,romeo11} for each annulus is routinely calculated. If $Q<1$, a starburst in that annulus can occur, and any remaining unstable mass is transferred to the two adjacent annuli in proportion such that angular momentum is conserved. Sinking unstable stars in the innermost annulus are transferred to a fully dispersion-dominated bulge component, while gas can be accreted onto the central black hole.
\item \emph{Mergers:} For minor mergers, the specific angular momentum of the merging satellite is measured \emph{relative to the central galaxy} to determine where in the disc the satellite's cold gas ends up, while its stars are transferred to the central's bulge. For major mergers, the gas disc vectors are summed to define the new plane onto which both gas discs are projected, while both stellar discs are destroyed and form a bulge. Starbursts occur in the annuli where gas originated from both systems, based on a variation of \citet*{somerville01}. Black holes are directly fed gas during mergers phenomenologically \citep[following][]{kauffmann00}.
\item \emph{Active galactic nuclei:} Radio mode accretion of hot gas onto a central black hole and subsequent feedback occurs at each time-step, whereby hot gas is prevented from cooling \citep{sage}. Quasar mode feedback is triggered by mergers and instabilities. The model steps out annulus by annulus, determines whether the quasar energy is sufficient to strip the cumulative gas up to that annulus, and if so, transfers all the gas in that annulus to the hot reservoir. The remaining energy can eject gas out of the halo entirely.
\end{itemize}
By default, whenever gas is reheated by feedback, it is transferred to the hot and/or ejected reservoirs associated with the \emph{central} galaxy within the halo, even if the gas originated from a satellite (but see Section \ref{ssec:hot}). While the choice of whether reheated gas remains in a satellite's hot reservoir or not can have some effect on the gas properties of model galaxies \citep[e.g., as raised by][]{font08,lagos14}, this effect is entirely negligible for the results we present for {\sc Dark Sage}.
{\sc Dark Sage} has been run on the standard merger trees of the Millennium simulation \citep{millennium}. The simulation included $2160^3$ particles of mass $1.18 \! \times \! 10^9\, {\rm M}_{\odot}$ in a periodic box with a comoving length of 685\,Mpc. This assumes $h\!=\!0.73$, $\Omega_M\!=\!0.25$, $\Omega_\Lambda\!=\!0.75$, $\Omega_b\!=\!0.045$, and $\sigma_8\!=\!0.9$ \citep{wmap1}. The merger trees carry 64 snapshots. As in {\sc sage} \citep{sage}, galaxies in {\sc Dark Sage} are evolved on 10 sub-time-steps between each snapshot.
Galaxy catalogues built from {\sc Dark Sage} are available through the Theoretical Astrophysical Observatory \citep{tao}.\footnote{\url{https://tao.asvo.org.au}} Here, photometric properties and light-cones for {\sc Dark Sage} galaxies can be constructed. The codebase itself has now also been made publicly available.\footnote{\url{https://github.com/arhstevens/DarkSage}}
\subsection{Updates to the model}
For the purposes of this work, we have updated some of the default prescriptions in {\sc Dark Sage} and added a few optional variants to the code as well. These were added to improve the average gas content of the entire galaxy population at low stellar masses (see Section \ref{ssec:recal}) and to increase our ability to investigate the impact of environment on gas content (see below).
\subsubsection{Atomic and molecular hydrogen}
\label{ssec:hih2}
The amount of gas in the form of \HI and H$_2$ in each annulus of each {\sc Dark Sage} galaxy is regularly recalculated. In \citet{stevens16}, this was based on the mid-plane pressure of the disc \citep{blitz04}, and closely followed the prescription of \citet{fu10}. While we include that prescription in this work, we have now opted for a metallicity-based prescription as our default, where
\begin{subequations}
\label{eq:RH2}
\begin{equation}
R_{{\rm H}_2}(r) \equiv \frac{\Sigma_{{\rm H}_2}(r)}{\Sigma_{\rm H\,\textsc{i}}(r)} = \left[ \left( 1 - \frac{3\,s(r)}{4+s(r)}\right)^{-1} - 1 \right]^{-1}~,
\end{equation}
\begin{equation}
s(r) \equiv \frac{\ln\left(1+0.6\chi(r) +0.01\chi^2(r) \right)}{0.6 \tau(r)}~,
\end{equation}
\begin{equation}
\chi(r) \equiv \frac{3.1}{4.1} \left[ 1 + 3.1 \left(\frac{Z(r)}{{\rm Z}_{\odot}}\right)^{0.365} \right]~,
\end{equation}
\begin{equation}
\tau(r) \equiv 0.66\, \frac{Z(r)}{{\rm Z}_{\odot}} \frac{c_f(r)\, \Sigma_{\rm gas}(r)}{{\rm M}_{\odot}\, {\rm pc}^{-2}}~,
\end{equation}
\begin{equation}
c_f(r) \equiv \phi \left(\frac{Z(r)}{{\rm Z}_{\odot}}\right)^{-\gamma}~,
\end{equation}
\end{subequations}
where $\Sigma(r)$ is the local surface density of the subscripted quantity [where $\Sigma_{\rm gas}(r)$ is for \emph{all} gas in the disc], $Z(r)$ is the local surface density ratio of gaseous metals to all gas, and $c_f(r)$ is the local clumping factor. This is based on the work of \citet{mckee10} and follows the implementation of \citet{fu13}. These equations originate from modelling spherical gas clouds subject to a photoionizing ultra-violet background, where metallicity represents the dust content of those clouds, which shield the cloud from the ionizing photons, allowing a greater fraction of the gas to be in molecular form \citep*[for further details, see][]{krumholz08,krumholz09}. Here, we assume a solar metallicity of ${\rm Z}_{\odot} \! = \! 0.02$ and treat $\phi$ and $\gamma$ as free parameters, for which we find values of 3.0 and 0.3, respectively, after recalibrating the model (Section \ref{ssec:recal}).
Note that $R_{\rm H_2}(r)$ can only be calculated by equation (\ref{eq:RH2}) provided $s(r) \! < \! 2$. This requires gas to have non-zero metallicity. In {\sc Dark Sage}, gas begins with zero metallicity. If passive star formation from H$_2$ were the only means of forming stars in the model, {then we would have a paradoxical chicken--egg scenario with neither chickens nor eggs,} and hence star formation would never ignite with this method. This is resolved by instabilities and mergers, which both induce starbursts independent of any H$_2$ (for full details, see \citealt{stevens16}). In other words, once a galaxy has experienced an instability or merger, which tends to happen quickly, it can form stars regularly from H$_2$ via equation (\ref{eq:RH2}).
In this paper, we show results from {\sc Dark Sage} using both the metallicity- and pressure-based prescriptions for determining the ratio of H$_2$ (and H\,{\sc i}) in gas disc annuli. We denote these as $f_{\rm H_2}(Z)$ and $f_{\rm H_2}(P)$, respectively, where $f_{\rm H_2}$ is formally defined as the mass ratio of H$_2$ to \emph{all} gas within an annulus, as per equation 12 of \citet{stevens16}, and hence has a one-to-one mapping with $R_{\rm H_2}$.
By changing the prescription for $f_{\rm H_2}$, the relative contributions of each star formation channel to the final stellar populations at $z\!=\!0$ are altered for galaxies of mass $m_* \! \lesssim \! 10^{10}\,{\rm M}_{\odot}$. This is demonstrated in Fig.~\ref{fig:sfchannels}. Compared to $f_{\rm H_2}(P)$, switching to $f_{\rm H_2}(Z)$ reduces the contribution of the passive H$_2$ channel of star formation, and increases the contributions from both instability- and merger-driven starbursts. Regardless of the $f_{\rm H_2}$ prescription though, the stars in galaxies of mass $m_* \! \lesssim \! 10^9\,{\rm M}_{\odot}$ are predominantly born out of the H$_2$-dependent passive channel of star formation, whereas the stars in galaxies of mass $m_* \! \gtrsim \! 10^{9.5}\,{\rm M}_{\odot}$ are mainly born in instability-driven starbursts. This has important consequences for some of the runs of the model we present in this paper, which we come back to in Section \ref{ssec:evolution}.
\begin{figure}
\includegraphics[width=\linewidth]{SFchannels_paper.pdf}
\caption{Fraction of stars formed through the three possible channels in {\sc Dark Sage} as a function of galaxy stellar mass at $z\!=\!0$. This accounts for both in-situ and ex-situ star formation. Galaxies are grouped in 0.1-dex-wide bins. Solid curves use the metallicity-based prescription for determining the ratio of atomic and molecular hydrogen in gas disc annuli. Dashed curves use the mid-plane pressure prescription. Both include the full model's physics otherwise.}
\label{fig:sfchannels}
\end{figure}
\subsubsection{Ram-pressure stripping of cold gas}
\label{ssec:rps}
As was the case in \citet{stevens16}, the cold gas in each annulus of each satellite galaxy is subject to ram-pressure stripping from the hot gas associated with the central (most massive) subhalo, based on the landmark work of \citet{gunn72}:
\begin{equation}
\rho_{\rm hot, cen}(R_{\rm sat})\, v_{\rm sat}^2 \geq 2 \pi G\, \Sigma_{\rm gas}(r) \left[\Sigma_{\rm gas}(r) + \Sigma_*(r) \right]~,
\label{eq:rps}
\end{equation}
where $\rho_{\rm hot, cen}(R_{\rm sat})$ is the local density of the intracluster medium (or the circumgalactic medium of the central) at the position of the satellite and $v_{\rm sat}$ is the relative speed of the satellite to the central. If this inequality is met for an annulus, all gas in that annulus is transferred to the hot reservoir of the central.
However, equation (\ref{eq:rps}) does not provide a criterion for \emph{when} a disc will feel ram pressure; it simply tells us \emph{if} it is vulnerable to ram pressure how much gas will be stripped. In order for a disc to be vulnerable, the satellite's hot gas on its leading side must first be stripped away. While the stripping of a satellite's hot gas would be asymmetric in reality, {\sc Dark Sage} has built-in symmetry assumptions about matter in (sub)haloes (like most semi-analytic models). Bearing all of this in mind, we have added a condition to {\sc Dark Sage} that equation (\ref{eq:rps}) is only applied to satellites if the total baryonic mass (cold gas + stars) of the galaxy exceeds the hot-gas mass of the subhalo. While this phenomenological condition misses a lot of the complex physics surrounding ram-pressure stripping, it provides a simple means of ensuring that some of a satellite's protective hot gas must be lost before cold gas in the disc is susceptible to stripping. This is more important for massive satellites \citep[see, e.g., fig.~2 of][]{sage}.
\subsubsection{Stripping of hot gas in subhaloes}
\label{ssec:hot}
As an alternative to hot gas being stripped in proportion to dark matter \citep[see][]{sage}, which we maintain as the default for {\sc Dark Sage}, we have now included an option for a ram-pressure-stripping-like prescription of the hot gas in subhaloes \citep[\`{a} la][]{font08,mccarthy08}. Specifically, we assume equation 6 of \citet{mccarthy08} describes the gravitational restoring force per unit area on the hot gas of the satellite, which is assumed to be distributed in a singular isothermal sphere. We then find the radius $R$ from the satellite's centre where
\begin{equation}
\rho_{\rm hot, cen}(R_{\rm sat})\, v_{\rm sat}^2 = \frac{G\,M_{\rm sat}(<\!R)\,m_{\rm hot,sat}}{8\,R_{\rm vir,sat}\,R^3}~,
\label{eq:rps_hot}
\end{equation}
which equates ram pressure with the gravitational restoring force density (where subscript `sat' implies the subhalo the satellite is associated with, and $M_{\rm sat}$ accounts for all matter). Any hot gas in the subhalo external to this radius is assumed to be stripped over the course of the simulation snapshot interval, meaning one tenth of that mass is stripped in a sub-time-step on which the galaxies are evolved. For the sake of simplicity, after each stripping calculation, the remaining hot gas is assumed to redistribute itself into a singular isothermal sphere again, such that equation (\ref{eq:rps_hot}) is always applicable. If left unchecked, this artificial redistribution of hot gas to larger radii would lead to continually stronger stripping on each consecutive sub-time-step. To counteract this, the stripped hot-gas mass at a given sub-time-step is not allowed to exceed that of the first sub-time-step within the same snapshot interval. Once the next snapshot is reached, the virial radius of the subhalo will have reduced, and hence so will the extent to which the hot gas is assumed to be distributed.
We also include the option of not stripping hot gas at all. With this option, we also return reheated gas from feedback to the hot component of the subhalo, rather than that of the main halo, which would otherwise be standard.
\subsection{Recalibration}
\label{ssec:recal}
Before investigating how environmental and evolutionary processes can affect the gas fractions of galaxies, we first need to ensure the average gas fraction of {\sc Dark Sage} galaxies is representative of the real Universe. This was always a constraint used in calibrating the model \citep[using the data from][]{brown15}, but there was still an overall deficit in the gas content of galaxies at low stellar mass \citep[see appendix A of][]{stevens16}. This has now been alleviated through a combination of the updates to the code above and opting for the \HI fraction of galaxies to become the constraint of greatest emphasis. We further exclude any subhaloes that were never composed of at least 100 particles in their history, ensuring that we use well-resolved galaxies in calibrating the model. This is now in much finer agreement with \citet{brown15}, as presented in Fig.~\ref{fig:hifrac}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{HIfrac_Mstar_paper.pdf}
\caption{\HI fraction of {\sc Dark Sage} galaxies at $z\!=\!0$ as constrained against the observations of \citet{brown15}. The points connected by solid lines give the logarithm of the mean along each axis for five bins, for each of the observed and model galaxies. The dashed curve and shaded region give the median \HI fraction and 16$^{\rm th}$--84$^{\rm th}$ percentile range of the model galaxies, respectively.}
\label{fig:hifrac}
\end{figure}
The other $z\!=\!0$ constraints of the model have been maintained as well. These include the stellar, H\,{\sc i}, and H$_2$ mass functions, the black hole--bulge mass relation, the stellar mass--gas metallicity relation, and the Baryonic Tully--Fisher relation. See section 3 and appendix A of \citet[][and references therein]{stevens16} for further details.
We have only modified one parameter from the original version of {\sc Dark Sage}; we have reduced the passive star formation efficiency from H$_2$ by a factor of 3 to $\epsilon_{\rm SF} \! = \! 1.3 \! \times \! 10^{-4}\,{\rm Myr}^{-1}$ (cf.~table 1 of \citealt{stevens16}). This means a greater fraction of stars are formed through disc instabilities. Specifically, with the old parameter value, the dominant star formation channel transitioned from passive H$_2$ to instabilities at a factor of $\sim$10 higher in stellar mass versus what is seen now in Fig.~\ref{fig:sfchannels}. This has a minimal effect on the stellar mass function, but it is important for when stars form in a galaxy, and for their total \HI and H$_2$ content. This parameter change has been implemented for \emph{all} runs of the model presented in this paper.
\subsection{Model galaxy sample}
\label{ssec:sample}
Throughout this paper, we study galaxies from {\sc Dark Sage} at $z\!=\!0$, and only include those with $10^9 \! < \! m_*/{\rm M}_{\odot} \! \leq \! 10^{11.5}$, matching our sample of observed galaxies, described in Section \ref{sec:obs}. In principle, we could have used the $z\!=\!0.041$ snapshot from Millennium, which would have more accurately matched the redshift range for the observations. However, the evolution of galaxies is sufficiently low at $z \! \lesssim \! 0.05$ such that our results would have no noticeable changes.
We note that while the \HI and stellar mass measurements we present for {\sc Dark Sage} galaxies are instantaneous quantities for the $z\!=\!0$ snapshot, star formation rates are average quantities over the period since the previous snapshot. The time-scale for the star formation rates is hence $\sim$260\,Myr. While this is an order-of-magnitude difference in time-scale from the H$\alpha$ measurements for the observed galaxies (cf.~Section \ref{sec:obs} of this paper and section 6 of \citealt{brown17}), it is not a cause of significant concern. \citet{benson12} has shown that improving the temporal resolution of a simulation above Millennium's (64 snapshots) will only result in a $\lesssim$5 per cent change in the universal average SFR from the {\sc galacticus} semi-analytic model \citep{galacticus} at any time. The same is true for the stellar mass function at $z \! \lesssim \! 4$ \citep[see fig.~11 of][]{benson12}. \citet[][see appendix B]{thesis} showed similar results for {\sc sage} \citep{sage} using the GiggleZ simulation suite \citep{gigglez}. Using the same simulations, we have confirmed that any sSFR-related results for {\sc Dark Sage} are negligibly affected by temporal resolution, whether we increase it by a factor of 4 or decrease it by a factor of 2 (not shown here).
\section{Observational data}
\label{sec:obs}
This section describes the observational data used in this paper. Galaxies are selected according to their stellar mass ($m_* \! \geq \! 10^9 \, {\rm M}_{\odot}$) and redshift ($0.02 \! \leq \! z \! \leq \! 0.05$) from the overlap in volume between the Sloan Digital Sky Survey Data Release 7 \citep[SDSS DR7;][]{abazajian09} and the ALFALFA survey \citep{giovanelli05}. This yields a representative parent sample of 30,695 galaxies that is defined in Section 2 of \citet{brown15}. We refer the reader to that paper for a more detailed description.
\HI 21-cm spectra are extracted from ALFALFA data cubes using the SDSS DR7 coordinates and redshift. This provides spectra for every galaxy in the sample, regardless of whether it is a formal \HI detection or not.
The MPA-JHU SDSS DR7 catalogue\footnote{\url{http://www.mpa-garching.mpg.de/SDSS/DR7/} and improved stellar masses from \url{http://home.strw.leidenuniv.nl/~jarle/SDSS/}}
provides stellar mass and star formation rate (SFR) estimates for the entire sample. Masses are constrained following the methodology of \citet{kauffmann03} using stellar absorption lines and fits to the optical photometry. The catalogue follows \citet{brinchmann04} in deriving global SFRs by using fits to emission H$\alpha$ line fluxes and, where no or low-signal-to-noise lines are detected, the 4000-\AA~break strength. We note these probe completely different time-scales, but, as raised in Section \ref{ssec:sample}, both can be safely compared with our model results.
Halo masses are assigned using the SDSS DR7 group catalogue,\footnote{\url{http://gax.shao.ac.cn/data/Group.html}} detailed in \citet{yang07}. We refer the reader to their paper for a full description. Briefly, the authors use the friends-of-friends, iterative group-finding algorithm of \citet{yang05} to identify galaxy groups with SDSS DR7. Groups are then assigned a halo mass via the abundance matching technique, whereby individual group luminosity or stellar mass is ranked and then matched to the halo mass function of \citet{warren06}. For this work, we use the halo masses estimated using the stellar mass rank order.
For a full breakdown of the optical and \HI data, see sections 2.1 and 2.2 of \citet{brown15}. The sample's stellar mass, SFR, and halo mass estimates are described in more detail in section 2 of \citet{brown17}. The galaxy and halo properties presented in those papers assumed a \citet{kroupa01} initial mass function (IMF) and a dimensionless Hubble parameter $h\!=\!0.7$. In order to be consistent with the model galaxies we compare against (see Section \ref{sec:sam}), all quantities have been converted to assume $h\!=\!0.73$ and a \citet{chabrier03} IMF. We have assumed $0.66\, m_{\rm *, Chabrier}\!=\!0.61\, m_{\rm *, Kroupa}$ and $0.67\, {\rm SFR}_{\rm Chabrier}\!=\!0.63\, {\rm SFR}_{\rm Kroupa}$ \citep[as in][]{madau14}.
\subsection{\HI spectral stacking}
\label{ssec:stacking}
As outlined in the Section \ref{sec:intro}, \HI spectral stacking improves the statistical size of studies way beyond what is currently possible using only detections, probing the gas content of galaxies across the entire gas-rich to -poor regime. The stacking technique in this paper is based upon the technique developed by \citet{fabello11a} and described fully in section 3 of \citet{brown15}. To briefly summarise, we shift the \HI spectra of a given ensemble of galaxies to their rest frame velocity and co-add them regardless of their detection status, weighting each by the inverse of their corresponding stellar mass. Doing so yields a weighted average stacked profile and, therefore, the average \HI fraction, $\langle m_{\mathrm{H}\,\LARGE\textsc{i}} / m_* \rangle$, for the galaxies in each stack.
Errors on the stacking results are estimated using a statistical {\it delete-a-group jackknife} routine whereby a random 20 per cent of spectra in a given stack are iteratively discarded without replacement, each time re-estimating the average gas fraction. This provides five separate estimates of gas fraction with each spectra discarded once. Following equation 3 in \citet{brown15}, the error is then calculated as the standard deviation of these five average gas fraction measurements multiplied by a $\sqrt{N-1}$ factor, where $N\!=\!5$.
\section*{Acknowledgements}
We thank Claudia Lagos for comments on this paper, as well as Luca Cortese, Violeta Gonzalez-Perez, and Manodeep Sinha for helpful discussion regarding this work. ARHS also thanks Darren Croton for continued academic support during the writing of this manuscript.
The {\sc Dark Sage} codebase is publicly available at \url{https://github.com/arhstevens/DarkSage}. This was developed from the \textsc{sage} codebase, which is also publicly available at \url{https://github.com/darrencroton/sage}. Catalogues from both models are available at \url{https://tao.asvo.org.au/tao/}.
\bibliographystyle{mnras}
\input{bibliography}
\end{document}
\section{Satellites versus centrals}
\label{sec:satcen}
In this section, we break both the observed and model galaxies into centrals and satellites, and study the relative differences in their \HI fractions and quiescence. For the purposes of this paper, a central is the most massive galaxy of a halo, including isolated galaxies that are the sole occupant of a halo. Satellites constitute all other galaxies. We perform multiple runs of {\sc Dark Sage} with various environmental and evolutionary processes switched on and off to understand the effect each has on the gas content of galaxies. In each panel of Fig.~\ref{fig:satcen}, we present the mean \HI fraction of the observed galaxies for fixed values of stellar mass, comparing to one of the runs of {\sc Dark Sage}. Similarly, in Fig.~\ref{fig:satcen_ssfr}, we assess \HI fraction as a function of specific star formation rate (sSFR). It is important that we consider this, as \citet{brown15} showed that near-ultraviolet$-r$ band colour, which is a proxy for sSFR, is more tightly related to \HI fraction than stellar mass is. For {\sc Dark Sage}, we use bins of width 0.2 dex for Figs.~\ref{fig:satcen}--\ref{fig:satenv}. We plot the log of the mean value along each axis for each bin, for both observed and model galaxies.
In drawing conclusions from comparisons of the \HI fractions of observed and model galaxies, one needs to be mindful of how the relative \HI and H$_2$ content in the model galaxies is determined. Ideally, we would simultaneously examine the results of {\sc Dark Sage} with the mean H$_2$ fractions of observed galaxies. Unfortunately, only limited samples of galaxies have their H$_2$ content inferred from CO observations \citep*[e.g.][]{young95,leroy09,saintonge11,boselli14a}, and no H$_2$-blind survey analogous to ALFALFA currently exists (although ongoing surveys such as xCOLDGASS will help alleviate this -- Saintonge et al. in preparation). There is, however, a wealth of evidence, drawn primarily from these surveys, that closely connects the H$_2$ in these galaxies to their star formation activity \citep[e.g.][]{kennicutt98,rownd99,bigiel08,saintonge11b,saintonge16,boselli14b}. We can, therefore, use the quiescent fraction of galaxies in observations and models as a first order indication of the relative levels of H$_2$. We present the quiescent fractions of central and satellite galaxies for the various runs of {\sc Dark Sage}, as a function of stellar mass, alongside the same quantity for our observed sample, in Fig.~\ref{fig:quiescent}. For our purposes, we consider a quiescent galaxy to be one with ${\rm sSFR}\!<\!10^{-11}\,{\rm yr}^{-1}$.
\begin{figure*}
\includegraphics[width=\linewidth]{HIfrac_satcen.pdf}
\caption{Mean atomic hydrogen gas fraction of galaxies as function of mean stellar mass (for pre-determined stellar mass bins) at $z\!=\!0$, split into satellites and centrals. Each panel includes the same observational data, and shows a different version of {\sc Dark Sage}, with a variation of one process in each case as labelled. Error on the mean from jackknifing are given for the observational data (Section \ref{ssec:stacking}). The left-hand panels represent the complete model with the relative \HI and H$_2$ content of each galaxy annulus calculated via a metallicity- or pressure-based prescription ($f_{\rm H_2}(Z)$ and $f_{\rm H_2}(P)$, respectively; see Section \ref{ssec:hih2}). The three upper right panels consider variations on environmental processes that directly affect satellites. These assume the metallicity-based prescription for calculating $f_{\rm H_2}$. The three lower right panels consider variations on evolutionary processes of galaxies and assume the pressure-based prescription for $f_{\rm H_2}$. Dashed curves consider {\sc Dark Sage} galaxies for all (sub)haloes in the Millennium simulation merger trees (which are composed of at least 20 particles), whereas solid curves only consider (sub)haloes that included 100 or more particles at some point in their history.}
\label{fig:satcen}
\end{figure*}
\begin{figure*}
\includegraphics[width=\linewidth]{HIfrac_satcen_ssfr.pdf}
\caption{As for Fig.~\ref{fig:satcen}, but now mean \HI fraction as a function of mean specific star formation rate (for pre-determined sSFR bins).}
\label{fig:satcen_ssfr}
\end{figure*}
\begin{figure*}
\includegraphics[width=\linewidth]{Quiescent_satcen.pdf}
\caption{Quiescent fraction (${\rm sSFR}\!<\!10^{-11}\,{\rm yr}^{-1}$) of central and satellite galaxies as a function of stellar mass. Solid curves only account for (sub)haloes with a minimum historical maximum number of particles of 100, whereas dashed curves include the full sample of {\sc Dark Sage} galaxies. Error bars on the data are Poissonian in the vertical ($\sqrt{N_{\rm quiescent}} / N_{\rm total}$ for each bin) and indicate the width of the bins in the horizontal. The precise horizontal position of each data point is the mean stellar mass within that bin. For comparison, panel $b$ also includes the quiescent fraction of galaxies from the original version of {\sc sage} \citep[][dotted curves]{sage}.}
\label{fig:quiescent}
\end{figure*}
\subsection{Observations and the full model}
\label{ssec:satcen_full}
As seen in all panels of Figs.~\ref{fig:satcen} \& \ref{fig:satcen_ssfr}, the difference in the mean \HI fraction between the observed centrals and satellites, whether at fixed stellar mass or specific star formation rate, is $\lesssim$\,0.2\,dex. Both at $m_* \! \lesssim \! 10^{9.5}\,{\rm M}_{\odot}$ and ${\rm sSFR} \! \lesssim \! 10^{-1.5}\,{\rm Gyr}^{-1}$, there is little to separate centrals from satellites. At higher stellar masses, satellites have less \HI than centrals, suggesting that a significant population of satellites are subject to phenomena (stripping, strangulation, etc.) that deplete their gas reservoirs \citep[\'{a} la][]{gunn72,larson80}.
At high sSFR, satellites have greater \HI fractions than centrals. Two effects are important for this comparison. First is the rate at which gas depletion takes place in satellites. If depletion processes are slow-acting, the sSFR and \HI fraction will decrease for an individual galaxy at comparable rates. If they are of moderate speed or fast-acting, the \HI fraction will be observed to decrease before sSFR \citep*[see][]{balogh00,mccarthy08,vollmer12,brown17}. One might na\"{i}vely except then that at fixed sSFR, satellites would have lower \HI fraction than centrals, which would oppose what is observed. This expectation would only be legitimate if one were comparing equivalent populations of satellites and centrals (i.e.~of the same stellar mass in each bin), however. In Fig.~\ref{fig:ssfr_m}, we present the mean stellar mass of satellites and centrals for equivalent bins in sSFR as used in Fig.~\ref{fig:satcen_ssfr}. Here, we see that at fixed sSFR, centrals tend to have greater stellar mass. As such, they are biased towards lower gas fractions (cf.~Fig.~\ref{fig:satcen}).
\begin{figure}
\includegraphics[width=\linewidth]{sSFR_vMeanMass.pdf}
\caption{Stellar mass of observed and full-model {\sc Dark Sage} galaxies for bins of specific star formation rate, split into centrals and satellites. Curves and points give the means for the model and observations, respectively, while the error bars and shaded regions cover the 16$^{\rm th}$--84$^{\rm th}$ percentiles (for both variables for the observations). Only {\sc Dark Sage} galaxies in (sub)haloes that were composed of 100 or more particles at some time in their history are included here.}
\label{fig:ssfr_m}
\end{figure}
The complete model of {\sc Dark Sage} (regardless of the prescription for the breakdown of atomic and molecular hydrogen) shows a much greater separation in the \HI fractions of satellites and centrals, typically $\lesssim$\,0.65\,dex (panels $a$ \& $b$ of Figs.~\ref{fig:satcen} \& \ref{fig:satcen_ssfr}). Even though the model is tuned to meet the average \HI fraction of \emph{all} galaxies, this large separation is possible, as the satellites only make up a minority of the galaxies (see below), and because the contribution of the more-massive centrals dominates the average. As such, {\sc Dark Sage} is in good agreement with observations for the \HI fraction as a function of stellar mass for the centrals (only). This is largely independent of the choice of $f_{\rm H_2}$ prescription, although there is slightly better agreement with observations at the low-mass end when we employ the metallicity-based prescription (cf.~panels $a$ and $b$ of Fig.~\ref{fig:satcen}). In effect, the properties of the satellites in the model are unconstrained. \emph{All} results we present concerning satellites are hence predictions (or `\emph{post}dictions') of the model.
38.2 per cent of galaxies in the observational sample are classed as satellites, whereas the {\sc Dark Sage} Millennium sample has 29.7 per cent satellites. Some difference in these values is expected, due to the nature of group finding versus subhalo finding for the observed galaxies and simulated data, respectively. In principle, subhalo finding for simulations is a more precise means of identifying satellites \citep[although there is certainly some variation amongst codes -- see][]{onions12,knebe13b,behroozi15}. Projection effects make it possible for true centrals to be observationally classified as satellites with a group finder, but the converse is unlikely \citep[see, e.g.,][]{campbell15}. We return to this important point and its potential influence on our results in Section \ref{sec:discussion}.
The {\sc Dark Sage} galaxies also show a flattening in their mean \HI fraction at ${\rm sSFR} \! \gtrsim \! 10^{-1}\,{\rm Gyr}^{-1}$ (panels $a$ and $b$ of Fig.~\ref{fig:satcen_ssfr}), whereas the \HI fractions continue to increase in the observations at high sSFR, with the separation between satellites and centrals also increasing. A much smaller fraction of the model galaxies occupies these high sSFRs versus the observations, implying galaxies in the model form their first generation of stars too early, leaving them less star-forming at $z\!=\!0$ for an equivalent stellar mass. Indeed, fig.~A7 of \citet{stevens16} demonstrates that {\sc Dark Sage} galaxies are too star-forming at high redshift. Similar behaviour is exhibited in Fig.~\ref{fig:ssfr_m}, where the mean stellar mass of {\sc Dark Sage} galaxies as a function of sSFR begins to flatten at $\sim$$10^{-1}\,{\rm Gyr}^{-1}$, whereas the observed galaxies continue towards lower stellar masses. In the observations, satellites are also found to generally be of lower stellar mass for fixed sSFR versus centrals (Fig.~\ref{fig:ssfr_m}), highlighting why satellites have higher gas fractions at high sSFR (Fig.~\ref{fig:satcen_ssfr}); at low stellar masses, the gradient of mean gas fractions in satellites is steep (Fig.~\ref{fig:satcen}).
To close the loop on the connection between stellar mass, \HI fraction, and specific star formation rate, we compare the quiescent fraction of the full-model {\sc Dark Sage} galaxies against the observations in Fig.~\ref{fig:quiescent}$a$. Matching the observed quiescent fraction of galaxies as a function of stellar mass has proven challenging for semi-analytic models, especially for satellites \citep[see, e.g.,][]{font08,guo11,guo13,guo16,sage,luo16,henriques17}. For the centrals, {\sc Dark Sage} exhibits the common feature of overproducing low-mass quenched galaxies and underproducing high-mass quenched galaxies. For satellite galaxies, at least at low masses, the results are more promising. Only for $10^{10} \! \lesssim \! m_*/{\rm M}_{\odot} \! \lesssim \! 10^{11}$ is there an overabundance of quiescent satellites. Overall, the quiescent fraction of satellites is notably improved from recent results from both semi-analytic models and hydrodynamic simulations \citep[cf.][]{guo16}.
\subsection{Environmental processes}
In panels $c$--$e$ of Figs.~\ref{fig:satcen} \& \ref{fig:satcen_ssfr}, we present the mean \HI fraction of {\sc Dark Sage} galaxies after altering the prescriptions of gas stripping in subhaloes in three ways. The results in panel $c$ of these figures exclude ram-pressure stripping of cold gas entirely (but maintain all other aspects of the model, including hot-gas stripping). Conversely, those in panel $d$ exclude any consideration of hot-gas stripping (but maintain cold-gas stripping). Panel $e$ includes both forms of stripping, but uses the alternative ram-pressure prescription for hot gas (Section \ref{ssec:hot}). Similarly, panels $b$--$d$ of Fig.~\ref{fig:quiescent} present the quiescent fraction of galaxies for these model variants. In principle, despite these stripping processes only {\it directly} affecting satellites, the hierarchical nature of galaxy formation means that centrals could be causally affected as well. We do not find any note-worthy changes to the $z\!=\!0$ population of {\sc Dark Sage} central galaxies here, however.
\subsubsection{The overall effect of cold-gas stripping}
\label{ssec:satcen_cold}
In Fig.~\ref{fig:satcen}$c$, we see that, for {\sc Dark Sage} galaxies, without cold-gas stripping, there is little to separate the \HI fraction of satellites and centrals as a function of stellar mass, highlighting that this process is far more effective at reducing \HI fractions of satellites in the model than removal of hot gas. That said, at least for $m_* \! \lesssim \! 10^{10.2}\,{\rm M}_{\odot}$, the satellite \HI fractions are closer to the observational data than the complete, default model (cf.~panels $a$ and $c$ of Fig.~\ref{fig:satcen}).
In Fig.~\ref{fig:satcen_ssfr}$c$, we see that without cold-gas stripping, the model galaxies now display similar qualitative behaviour to the observed galaxies, in terms of the \HI fraction as a function of sSFR. That is, at low sSFR, the {\sc Dark Sage} satellites and centrals show the same \HI fraction on average, but for increasing sSFR, the satellites begin to show higher \HI fractions than the centrals. Although, with this run of the model, this splitting occurs at $\sim$1\,dex lower sSFR than the observations. Without any fast-acting processes to remove gas from satellites, there are two reasons for satellites showing higher gas fractions than centrals in Fig.~\ref{fig:satcen_ssfr}$c$. First, the bias of satellites having lower stellar mass than centrals for fixed sSFR, as shown in Fig.~\ref{fig:ssfr_m}, is still present in this run. Second, even if we were to consider only satellites and centrals of the same stellar mass for a given sSFR, the satellite would have a suppressed supply of fresh gas due to the stripping of hot gas from its subhalo. The satellite would then have to rely on its instantaneous cold-gas reservoir more heavily for the formation of stars, and hence would need to have a greater mass of \HI at an instant than an equivalent central.
Our results suggest the direct stripping of cold gas in galaxies should have a large impact on their \HI content (which we reaffirm in Section \ref{sec:satenv}), but that the current implementation in {\sc Dark Sage} is systematically too efficient at removing H\,{\sc i}. Potential solutions could include: (i) non-instantaneous stripping of the cold gas within the annuli, (ii) a more physically strict criterion for when cold gas stripping is allowed (rather than the current case that $m_{\rm cold} \! + \! m_* \! > \! m_{\rm hot}$), (iii) adding inclination effects to equation (\ref{eq:rps}), or (iv) the observed \HI masses include gas that the model would not consider `part of the galaxy' (which is hard to define, even when one has full three-dimensional information like in a hydrodynamic simulation -- see \citealt{stevens14}). However, each of these is unlikely to be satisfactory by itself, as this will have consequences for the relative \HI fractions of satellites in different environments (as the strength of stripping is dependent on local environment), which, as we show in Section \ref{sec:satenv}, is an area the model performs well in. We discuss this further in Section \ref{sec:discussion}.
Where cold-gas stripping has a negligible effect is on the quiescent fraction of galaxies (cf.~panels $a$ and $c$ of Fig.~\ref{fig:quiescent}). As indicated by equation (\ref{eq:rps}), ram-pressure stripping of cold gas is more dominant in the low-surface-density areas of satellites' discs, towards their outskirts. The hydrogen in annuli with low $\Sigma_{\rm gas}$ will also be predominantly atomic (regardless of the specific choice of $f_{\rm H_2}$ prescription -- see Section \ref{ssec:hih2} of this paper and section 3.4 of \citealt{stevens16}). Star-forming gas is predominantly molecular and hence deeper in the potential well. As a result, star-forming gas is less susceptible to ram-pressure stripping, and thus this process has little impact on the quiescent fraction of satellites.
We note that even without cold-gas stripping, our quiescent-fraction results are a significant improvement over {\sc sage} \citep[][from which {\sc Dark Sage} was developed]{sage} for both centrals and satellites, especially at low masses. To highlight this, we have included results from {\sc sage} in Fig.~\ref{fig:quiescent}$b$ (shown in panel $b$ as {\sc sage} does not include cold-gas stripping).
\subsubsection{The overall effect of hot-gas stripping}
\label{ssec:satcen_hot}
Fig.~\ref{fig:satcen}$d$ shows that when there is no given environmental mechanism to remove the hot-gas reservoir of subhaloes, the mean \HI fraction of {\sc Dark Sage} galaxies is remarkably close to observations for a given stellar mass. The maintenance of that reservoir allows gas to continue to cool onto satellite galaxies in the model for longer. Based on the severity of the difference seen in the model when cold-gas stripping is removed, it is safe to say the greatest reason removing hot-gas stripping from the model has an effect on the \HI fraction of galaxies is because hot gas shields the cold gas from being stripped (see Section \ref{ssec:rps}). This minimises the difference between the gas fractions of satellites and centrals of the same mass, which is favourable next to observational data.
Excluding hot-gas stripping does, however, lead to satellites having systematically lower \HI fractions than centrals as a function of sSFR, for all sSFR (Fig.~\ref{fig:satcen_ssfr}$d$). This is at odds with observations. Without hot-gas stripping, satellites can continue to accrete plenty of fresh gas, which, by design of the model, settles onto the disc in an approximately exponential form. Because cold-gas stripping is still present, the outer gas that cools is quickly removed, whereas the inner gas enhances star formation. Hot-gas stripping therefore affects the molecular-to-atomic gas ratio of satellites, where, without it, galaxies would have higher sSFR at fixed \HI fraction, as seen in Fig.~\ref{fig:satcen_ssfr}$d$.
The impact of hot-gas stripping on the star-forming gas content of satellites is highlighted further in Fig.~\ref{fig:quiescent}$d$. Without hot-gas stripping, the fraction of quiescent galaxies drops dramatically for satellites with $m_* \! \gtrsim 10^{10}\,{\rm M}_{\odot}$. This result would be unsurprising for any semi-analytic model \citep[see, e.g., the difference in gradual and instantaneous hot-gas stripping from][]{font08}, but the point here is that even when cold-gas stripping is included (which is not the case in many models), hot-gas stripping remains the dominant process for regulating the quiescent fraction of satellites. Moreover, we see that low-mass galaxies in {\sc Dark Sage} are not plagued by being over-quenched from hot-gas stripping, which has been a persistent issue with semi-analytic models in the past, as raised in Section \ref{ssec:satcen_full}.
Finally, in Figs.~\ref{fig:satcen}$e$, \ref{fig:satcen_ssfr}$e$, and \ref{fig:quiescent}$e$, we show the results of {\sc Dark Sage} with the alternative, ram-pressure-like prescription for hot gas (Section \ref{ssec:hot}). With this implementation, high-sSFR satellites begin to exhibit slightly higher \HI fractions than their central counter-parts, qualitatively more in line with the observations, whereas the opposite is true for high-mass galaxies. There is also a minimal increase in the fraction of quiescent satellites. By and large though, the two implementations of hot-gas stripping are relatively consistent when it comes to the mean differences in gas fractions of satellites and centrals.
\subsection{Evolutionary processes}
\label{ssec:evolution}
In this subsection, we assess processes that affect all galaxies (regardless of whether they are centrals or satellites). We focus on the effects of (i) disc instabilities, (ii) stellar feedback and evolution, and (iii) AGN feedback, which all appear to play some role in regulating the relations between \HI fraction, stellar mass, and sSFR. We present results from {\sc Dark Sage} with each of these features independently switched off in panels $f$--$h$ of Figs.~\ref{fig:satcen} \& \ref{fig:satcen_ssfr}.
It should be noted that unlike for the previous runs where we altered stripping processes, simply removing feedback or instabilities from the model has a significant impact on mass functions and scaling relations, which the model is usually constrained to meet. These runs have not been recalibrated; they are, therefore, an attempt to gauge what the \HI fractions of galaxies would be if each physical phenomenon did not exist, rather than attempting to recover realistic galaxies in the absence of said phenomena. This provides a more direct measure of how each process independently helps to shape the \HI fractions of galaxies.
We also note that, despite using the metallicity-based prescription for determining the fraction of gas in each annulus in the form of \HI and H$_2$ (equation \ref{eq:RH2}) as the new default for {\sc Dark Sage}, we have assumed the pressure-based prescription for the runs without feedback or instabilities. As noted in Section \ref{ssec:hih2}, passive star formation from H$_2$ will only ignite after initial metal enrichment, which happens when a galaxy first experiences a starburst from either an instability or merger. Without instabilities in the model then, only galaxies that experience a merger will have the passive H$_2$ star formation channel open to them. This channel of star formation is the dominant mechanism by which stars are formed in low-mass galaxies. Because the majority of galaxies in the Millennium simulation merger trees do not have a recorded merger, one would end up with a non-physically large number of star-less galaxies with large gas reservoirs. A similar problem arises when stellar evolution is removed from the model, as this removes metal enrichment entirely. Both cases are resolved by switching to the mid-plane pressure prescription for $f_{\rm H_2}$, which is unaffected by gas metallicity. For consistency, we also use this prescription for the run without AGN feedback (but this does not make a significant difference in this case).
\subsubsection{The effect of disc instabilities}
As shown in Fig.~\ref{fig:sfchannels}, most star formation in low-mass {\sc Dark Sage} systems comes from the passive H$_2$ channel, whereas the high-mass systems formed most of their stars through the instability channel. By eliminating any consideration of disc instabilities from the model, star formation, that would have otherwise more directly harnessed available \HI reservoirs, becomes progressively less efficient towards the high-mass end. As a result, come $z\!=\!0$, not only are there fewer galaxies of high stellar mass, but the gradient of the mean \HI fraction as a function of stellar mass has become shallower, which we show in Fig.~\ref{fig:satcen}$f$.
With less integrated star formation having occurred in the lead up to $z\!=\!0$ without instabilities in the model, galaxies are a lot more gas rich on average. Also, because instabilities are unresolved, most of the gas remains gravitationally unstable in dense clumps, and hence there is more H$_2$ than H\,{\sc i}. This means that the (instantaneous) star formation rates are actually higher at $z\!=\!0$ than they were for the full model. This is seen in Fig.~\ref{fig:satcen_ssfr}$f$, where the average \HI fraction for a given sSFR has decreased.
\citet{stevens16} showed that instabilities regulated the stellar specific angular momentum of {\sc Dark Sage} galaxies as a function of stellar mass. It is no coincidence that they also regulate the gas fractions of galaxies; recent analytic work by \citet{ob16} and results from hydrodynamic simulations \citep{lagos17} have shown that the gas fraction and specific angular momentum of a galaxy are inherently related. We will address this connection in more specific detail with {\sc Dark Sage} in future work.
\subsubsection{The effect of stellar feedback and evolution}
Without stellar feedback or stellar evolution, an equivalent amount of gas can be used to form a greater mass of stars. This is because the instantaneous recycling approximation no longer needs to be upheld, nor does a fraction of the cold gas in a star-forming annulus need to be reserved for being reheated out of the disc. Hence, the run of {\sc Dark Sage} without stellar feedback (or evolution) shows a deficit in the mean \HI fraction for fixed sSFR, as in Fig.~\ref{fig:satcen_ssfr}$g$.
Provided galaxies continue to accrete fresh gas, the lack of gas reheating raises the frequency of disc instabilities. There also tends to be greater cold-gas reservoirs in galaxies during mergers. These both not only mean that starbursts become stronger in the model, but also that black holes are able to accrete gas faster. Larger black holes lead to stronger AGN feedback, which shuts down fresh gas accretion at lower redshift. This leads to a deficit in the stellar mass function in the range $10^{10.5} \! \lesssim \! m_*/{\rm M}_{\odot} \! \lesssim \! 10^{11.5}$ and an excess of galaxies with lower masses. Resultantly, at $z\!=\!0$, the gas fractions of galaxies at lower masses are systematically reduced, as seen in Fig.~\ref{fig:satcen}$g$. In this situation, only galaxies that experience many mergers can accumulate enough cold baryons to grow to larger stellar masses. Because all mergers are more gas rich, the highest-mass galaxies have larger \HI fractions than in the full model. This is also seen in Fig.~\ref{fig:satcen}$g$.
\subsubsection{The effect of active galactic nuclei}
Without an AGN engine to regulate the rate at which galaxies accrete gas, all galaxies, but especially the high-mass ones, become extra gas-rich. This is seen in Fig.~\ref{fig:satcen}$h$. Where an AGN makes a smaller difference in {\sc Dark Sage} is with how gas fraction varies with sSFR. Fig.~\ref{fig:satcen_ssfr}$h$ shows that the most star-forming galaxies have almost unchanged \HI fractions when AGNs are removed from the picture. On the other hand, the more quiescent galaxies have lower \HI fractions for fixed sSFR in this case. This comes back to the extra accretion galaxies receive; much of this extra gas can quickly be consumed in star formation, leading to higher sSFRs (which are averaged over a finite time-scale) for a given instantaneous gas fraction. This is equivalent to getting lower instantaneous gas fractions for a given sSFR.
\section{Satellites in various environments}
\label{sec:satenv}
In this section, we investigate how the \HI fractions of satellite galaxies vary as a function of parent halo mass, which serves as a measure of environment. We consider three bins of halo mass: (i) $M_{\rm vir} \! < \! 10^{12}\,{\rm M}_{\odot}$, (ii) $10^{12} \! \leq \! M_{\rm vir}/{\rm M}_{\odot} \! < \! 10^{13.5}$, and (iii) $10^{13.5} \! \leq \! M_{\rm vir}/{\rm M}_{\odot} \! < \! 10^{15}$. For the {\sc Dark Sage} galaxies, virial masses of haloes are directly given from the Millennium simulation. As discussed in Section \ref{sec:obs}, the halo mass measurements for the observed galaxies follow the group-finding algorithm of \citet{yang07}.
There is an argument to be made that, for a fairer comparison, one should run the same group-finding algorithm on the {\sc Dark Sage} output as was used for the observations, and use those returned halo masses. We have tested that by rank ordering the stellar and virial masses of haloes, and reassigning halo masses by the ranked virial mass, any changes to our results are entirely negligible. Of course, this only mimics part of the method used in estimating the halo masses of the observed galaxies. Any further differences would come as a result of cross-contamination of centrals and satellite populations (as raised in Section \ref{ssec:satcen_full}). We have opted for the simpler, more digestible comparison for our main results (presenting the true model output with the observational data as they are), but we discuss the effect cross-contamination could have in Section \ref{sec:discussion}.
The main results of this section are presented in Fig.~\ref{fig:satenv} \& \ref{fig:quiescent_env}. In Fig.~\ref{fig:satenv}, instead of presenting the mean \HI fractions of satellite galaxies for the various halo mass bins on the $y$-axis (e.g., as in \citealt{brown17}), we present $\Delta \log_{10}(m_{\mathrm{H}\,\textsc{i}} / m_*)$, which is the \emph{separation} between the mean \HI fraction curve for that halo mass bin and the mean curve for all satellite galaxies (the curves and points in Figs.~\ref{fig:satcen} \& \ref{fig:satcen_ssfr}). This means we can directly compare how environment splits the \HI content of satellite galaxies in the observations and model without being concerned about any normalisation issues between the two (which have been discussed in Section \ref{sec:satcen}). Similarly, in Fig.~\ref{fig:quiescent_env}, rather than presenting the raw quiescent fraction of satellites in different environments, we show the \emph{difference} in those fractions from the \emph{overall} quiescent fraction of satellites (given in Fig.~\ref{fig:quiescent}).
\begin{figure*}
\includegraphics[width=\linewidth]{DeltaHIfrac_env.pdf}
\includegraphics[width=\linewidth]{DeltaHIfrac_env_ssfr.pdf}
\caption{Difference in mean atomic hydrogen gas fraction, as a function of stellar mass (top row) and specific star formation rate (bottom row), for \emph{satellite} galaxies at $z\!=\!0$ for bins of parent halo mass. Each panel right of the thick vertical line changes an environmental process in {\sc Dark Sage}, as labelled. Panel $d$ further recalculates \HI fractions in post-processing by redistributing matter into a new, exponential profile (see text for details). Solid curves only consider (sub)haloes composed of at least 100 particles at some point in their history, whereas dashed curves include the full sample of {\sc Dark Sage} galaxies. All panels include the same observational data, with errors on the means from jackknifing.}
\label{fig:satenv}
\end{figure*}
\begin{figure*}
\includegraphics[width=\linewidth]{Quiescent_satenv_diff.pdf}
\caption{Difference in the quiescent fraction of satellite galaxies within halo mass bins from the overall quiescent fraction of satellites. This is given as a function of stellar mass for three halo mass bins (see the legend of Fig.~\ref{fig:satenv}). Error bars on the data are Poissonian in the vertical and indicate the full width of the bins in the horizontal. The precise horizontal position of each data point is the mean stellar mass within that bin. Dotted curves in panel $b$ compare results from the original version of {\sc sage} \citep{sage}.}
\label{fig:quiescent_env}
\end{figure*}
\subsection{Full model versus observations}
In Fig.~\ref{fig:satenv}, we show how halo mass (environment) affects the \HI fraction of satellite galaxies in the full {\sc Dark Sage} model and the observations with stellar mass (panel $a$) and sSFR (panel $e$). For $m_* \! \gtrsim \! 10^{9.5}\,{\rm M}_{\odot}$, the observations and model are in remarkable agreement: better than 0.1\,dex for the mean \HI fraction in all halo mass bins. Only at lower stellar masses do we begin to see a discrepancy, where the model shows less environmental splitting than the observations. This is true regardless of whether we only consider subhaloes from the $z\!=\!0$ snapshot of Millennium with a historical maximum number of particles of $N_{\rm p,max} \! \geq \! 100$, or use all subhaloes with $\geq$20 particles. In Fig.~\ref{fig:quiescent_env}$a$, we show that {\sc Dark Sage} also recovers the \emph{relative} role environment plays on the quiescent fraction of satellites to an encouraging degree. In this case, results at low-$m_*$ are more in line with observations when we apply the cut of $N_{\rm p,max} \! \geq \! 100$.
An important consideration is whether the distribution of satellites in the various halo mass bins is consistent between the observations and model data. While environmental processes will not inherently care about the number of galaxies they act upon, the differences in \HI fractions and quiescent fraction of satellites in a given environment from the overall mean will depend on the contribution of satellites in that environment \emph{to} the overall mean. As such, we list the number of satellites in each halo mass bin for the observed and model samples in Table \ref{tab:Nsat}.
\begin{table*}
\centering
\begin{tabular}{l r r r} \hline
& \multicolumn{3}{c}{$N_{\rm sat}$}\\
$\log_{10}(M_{\rm vir}/{\rm M}_{\odot})$ & $<\,12$ & $\in [12,13.5)$ & $\in [13.5,15)$\\\hline
Observations & 1\,030 (11.9\%) & 3\,486 (40.2\%) & 4\,148 (47.9\%) \\
{\sc Dark Sage}, all & 194\,240 (13.3\%) & 704\,781 (48.4\%) & 544\,154 (37.4\%) \\
{\sc Dark Sage}, $N_{\rm p,max}\!\geq\!100$ & 155\,430 (12.9\%) & 585\,254 (48.6\%) & 453\,286 (37.6\%) \\\hline
\end{tabular}
\caption{Number of satellite galaxies in the various halo mass bins for both the observations and full model of {\sc Dark Sage} (which, apart from the cut of $m_* \! \geq \! 10^9\,{\rm M}_{\odot}$, comes from Millennium), for which results are shown in Figs.~\ref{fig:satenv} \& \ref{fig:quiescent_env}. $N_{\rm p,max}$ refers to the historical maximum number of particles that each subhalo was composed of in the Millennium merger trees.}
\label{tab:Nsat}
\end{table*}
Another important consistency check between the observations and models is the probability distribution of halo masses (within each halo mass bin). Because the group finder used to associate halo masses with observed galaxies drew from a halo mass function produced by $N$-body simulations whose cosmology is very similar to Millennium's \citep[cf.][]{millennium,warren06}, we should expect these to be consistent. However, because the observed sample of galaxies is limited in its volume and stellar mass range, differences in the distributions of halo masses can arise. We have tried extracting, as precisely as possible, a set of haloes from the Millennium simulation that matches the halo catalogue from the observations + group finder. While the smaller sample size introduces noise, no significant conclusions from our figures come about by making this selection (not shown here). This is in spite of the fact that the specific cases of the Coma and Hercules clusters have a large influence on the largest halo mass bin for the observational data.
\subsection{The relative impact of cold-gas stripping}
In Fig.~\ref{fig:satcen}, we demonstrated that cold-gas stripping in {\sc Dark Sage} was almost entirely responsible for the systematic separation in \HI fractions between satellite and central galaxies of the same stellar mass. Similarly, removing cold-gas stripping from the model almost eliminates any environmentally driven splitting in the \HI fractions of satellite galaxies. This is seen not only as a function of stellar mass in Fig.~\ref{fig:satenv}$b$, but also as a function of sSFR in Fig.~\ref{fig:satenv}$f$.
The results of {\sc Dark Sage} presented here are in contrast to those of the \citet{gp14} version of the {\sc galform} semi-analytic model presented alongside the same observations in \citet{brown17}. That is, despite the model being run on the same simulation \citep[Millennium;][]{millennium} and lacking a prescription for ram-pressure stripping of cold gas entirely, {\sc galform} still displayed \HI fraction splitting with halo mass for satellite galaxies. There are many differences between {\sc Dark Sage} and {\sc galform} in terms of the specifics of how galaxies are evolved \citep[for details on {\sc galform}, see][]{cole00,benson10,lacey16}. One vital aspect, which we consider here, is how the \HI and H$_{2}$ surface density profiles of discs are ultimately derived. We note that a modified set of merger trees \citep{jiang14} for Millennium is employed for {\sc galform}, but this algorithm is only expected to have a significant impact on semi-analytic galaxy properties for higher-resolution $N$-body simulations \citep[see][]{merson13,jiang14}.
The key feature of {\sc Dark Sage} is that the full two-phase angular-momentum structure of discs (and hence their radial structure also) is numerically evolved self-consistently. Similar to many other semi-analytic models \citep[cf.][]{hatton03,somerville08,guo11}, {\sc galform} only evolves the \emph{total} angular momentum of discs (gas and stars together), and then assumes that discs always carry an exponential surface density profile with radius. Although discs are seeded as approximately exponential in {\sc Dark Sage}, they develop strong cusps come $z\!=\!0$, partly caused by instability-induced radial flows of gas \citep{stevens16}. These cuspy gas discs are qualitatively in agreement with both observations and hydrodynamic simulations \citep[see][respectively]{bigiel12,stevens17}. A cuspy disc will have higher $R_{\rm H_2}(r)$ at small $r$ and relatively lower $R_{\rm H_2}(r)$ at large $r$ (e.g., see Equation \ref{eq:RH2}). Hence, a {\sc Dark Sage} galaxy will have more H$_2$ than a {\sc galform} galaxy with the same stellar mass, cold gas mass, and halo properties. Some processes that affect the \HI content of {\sc galform} galaxies might, therefore, affect the H$_2$ content of {\sc Dark Sage} galaxies as well/instead. We quantify the impact of this further in Section \ref{ssec:disc_modelling}.
Within {\sc Dark Sage}, cold-gas stripping only affects the star formation activity of low-$m_*$ satellites in the most massive haloes. This is evidenced by Fig.~\ref{fig:quiescent_env}$b$, where only in this regime is there a notable change to the relative quiescent fraction when cold-gas stripping is removed. The combination of these low-mass galaxies having shallow local potential wells and the high-mass haloes in which they reside inflicting strong ram pressure means that even the molecular gas in the satellites is susceptible to being directly stripped.
We also include results from the regular {\sc sage} model \citep{sage} in Fig.~\ref{fig:quiescent_env}$b$. The fact that {\sc sage} does not include cold-gas stripping, but is still able produce a decent result for the \emph{relative} role of environment on quiescent fraction, is testament to cold-gas stripping only being of secondary importance for satellites' quiescence. Differences seen between {\sc sage} and {\sc Dark Sage} here are the result of detailing the evolution of discs' structure.
\subsection{The relative impact of hot-gas stripping}
As discussed in Section \ref{ssec:satcen_hot}, any changes to the mean \HI fraction of satellite galaxies from removing hot-gas stripping from {\sc Dark Sage} were mostly caused by a reduction in the opportunity for cold-gas stripping to take place. As a result, the splitting by environment of satellites' \HI fractions as a function of stellar mass is also reduced from the full model, as shown in Fig.~\ref{fig:satenv}$c$. Despite the fact that removing hot gas from the model brought the difference in the mean \HI fractions of satellites and centrals more in line with observations (Fig.~\ref{fig:satcen}$c$), the splitting of satellites' \HI fractions by halo mass has become weaker than the observations for this run. This implies that while stripping processes in {\sc Dark Sage} are systematically too strong (Fig.~\ref{fig:satcen}$a$), they are largely of the correct \emph{relative} strength in haloes of different mass.
Hot-gas stripping plays a major role in regulating the quiescent fraction of satellites, especially at higher stellar masses (Section \ref{ssec:satcen_hot}). As shown in Fig.~\ref{fig:quiescent_env}$c$, removing hot-gas stripping also reduces the environmental splitting of the quiescent fraction of satellite galaxies with $m_* \! \lesssim \! 10^{10}\,{\rm M}_{\odot}$. Satellites of equivalent stellar mass (or subhalo mass at infall) experience greater depletion of their hot-gas reservoir when falling into more massive haloes. This is true regardless of whether we prescribe hot gas to be stripped at the same rate as dark matter or impose ram pressure on the hot gas (cf.~panels $a$ and $d$ of Fig.~\ref{fig:quiescent_env}). However, Fig.~\ref{fig:quiescent_env} also highlights that a \emph{relative} environmental dependence would still be present (and consistent with observations) for the quiescent fraction of satellites without any stripping processes. This is because the hot-gas reservoirs of satellites are not allowed to grow through cosmological accretion within {\sc Dark Sage} (we come back to this in Section \ref{sec:discussion}). As a result, the longer a galaxy exists as a satellite, the less gas it will have available to accrete, and thus the more quiescent it will become. Satellites in larger haloes have longer orbital and merging time-scales, and therefore will have greater opportunity to consume their gas and become quiescent.
\subsection{The impact of disc structure modelling}
\label{ssec:disc_modelling}
As an experiment in post-processing, we reset the specific angular momentum of the {\sc Dark Sage} discs so that $j_{\rm disc}\!=\!j_{\rm halo}$, and we redistribute both gas and stars in the disc such that they follow $\Sigma_{\rm gas}(r) \! \propto \! {\rm e}^{-r}$ with a common scale radius. This meets the disc model employed in the standard version of {\sc sage} \citep{sage}. We then recalculate the \HI and H$_{2}$ masses of each galaxy according to the $f_{\rm H_2}(P)$ prescription \citep[which matches the method of {\sc galform} -- see][]{lagos11}. We further recalculate their SFR based on their adjusted H$_{2}$ content, assuming $\Sigma_{\rm SFR}(r) \! = \! \epsilon_{\rm SF} \Sigma_{\rm H_2}(r)$, maintaing $\epsilon_{\rm SF} \! = \! 1.3 \! \times \! 10^{-4}\,{\rm Myr}^{-1}$. This allows us to estimate what differences we would find regarding the relative influence of environment on the \HI fractions of satellites if {\sc Dark Sage} did not include a detailed treatment of the angular momentum of discs and did not allow them to be cuspy. We present results for this for a run of the model that uses the ram-pressure prescription for hot-gas stripping, and excludes cold-gas stripping entirely. These choices give us the most directly comparable results to those from {\sc galform} version GP14+GRP \citep{gp14} published in figure 7 of \citet{brown17}. Our results are given in panels $d$ and $h$ of Fig.~\ref{fig:satenv}.
Fig.~\ref{fig:satenv}$d$ shows that an environmental splitting of \HI fractions as a function of stellar mass caused entirely by hot-gas stripping can be recovered for a more rudimentary model of galaxy discs. The strength of this splitting is still significantly less than what is observed in real galaxies. It is also less than half the strength seen in {\sc galform} \citep[cf.][]{brown17}. Meanwhile, the splitting with sSFR has become flat, now too weak at low sSFR and too strong at high sSFR versus observations (Fig.~\ref{fig:satenv}$h$). Of course, because the gas redistribution is done in post-processing, the evolution of the galaxies is not affected. It is an obvious statement that if aspects of galaxy evolution within one model were changed to match that of another model, the two models would end up giving very similar results. Such an exercise is hence not particularly informative. What the results of panels $d$ and $h$ of Fig.~\ref{fig:satenv} do tell us is that the method for determining $f_{\rm H_2}$ only has a limited potential contribution to the dominance of cold-gas stripping in driving the environmental dependence of satellites' \HI fractions in {\sc Dark Sage}. In other words, these results still favour ram-pressure stripping of cold gas to be the main mechanism for curbing the \HI content of satellite galaxies.
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1706.07418
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\section{Introduction}
Given
a graph $G=(V,E)$ and for each vertex $v \in V$ a subset $B(v)$ of the set $\{0,1,\ldots, d_G(v)\}$, where $d_G(v)$ denotes the degree of vertex $v$ in the graph $G$, a $B$-matching of $G$ is any set $F \subseteq E$ such that $d_F(v) \in B(v)$ for each vertex $v$, where $d_F(v)$ denotes the number of edges of $F$ incident to $v$. The general matching problem asks the existence of a $B$-matching in a given graph. A set $B(v)$ is said to have a {\em gap of length} $p$ if there exists a natural number $k \in B(v)$ such that $k+1, \ldots, k+p \notin B(v)$ and $k+p+1 \in B(v)$. Without any restrictions the general matching problem is NP-complete \cite{Lovasz}. However, for the case when no set $B(v)$ contains a gap of length greater than $1$, Lovasz \cite{Lovasz} developed a structural description and Cornuejols \cite{Cor} presented a polynomial time algorithm for finding a $B$-matching, if it exists. In this paper we consider a maximum/minimum size version of the general matching problem, in which we are interested in finding a $B$-matching having maximum (or minimum) number of edges.
{\bf Previous work} If $B(v)=\{0,1\}$ for each vertex $v$, then a $B$-matching is in fact a {\em matching}, i.e., a set of vertex-disjoint edges.
A {\em perfect matching} is a $B$-matching such that $B(v)=1$ for each vertex $v$. Given a function $b: V \rightarrow N$, a $b$-matching is any set
$F \subseteq E$ such that $d_F(v) \leq b(v)$ for each vertex $v$ and a perfect $b$-matching or a $b$-factor is any set
$F \subseteq E$ such that $d_F(v) = b(v)$ for each vertex $v$. If in addition to a function $b$ we are also given a function $a: V \rightarrow N$, then an $(a,b)$-matching is any set
$F \subseteq E$ such that $a(v) \leq d_F(v) \leq b(v)$ for each vertex $v$.
All these special cases of the general matching problem are well-solved, both in unweighted and weighted versions, see \cite{Sch} for a good survey for example.
In the {\em antifactor} problem for each vertex $v$ we have $|\{0,1,\ldots, d_G(v)\} \setminus B(v)| = 1$, meaning that for each vertex there is exactly one degree excluded from the set $B(v)$. Graphs that have an antifactor have been characterized by Lovasz in \cite{Lovasz1973}.
For the more general case when no set $B(v)$ contains a gap of length greater than $1$ Cornuejols \cite{Cor} in 1988 presented two solutions to the problem of finding such $B$-matching, if it exists. One uses a reduction to the edge-and-triangle partitioning problem, in which we are given a graph $G=(V,E)$ and a set $T$ of triangles (cycles of length $3$) of $G$ and are to decide if the set of vertices $V$ can be partitioned into sets of cardinality of $2$ and $3$ so that each set of cardinality $2$ is an edge of $E$ and each set of cardinality $3$ is a triangle of $T$. The other is based on an augmenting path approach applied in the modified graph $G'=(V \cup V', E')$ in which each edge $e$ of $G$ is split with two new vertices into three edges. For each new vertex $v'$ the set $B(v')$ is defined to be $\{1\}$ and we start from the set $F\subseteq E'$ such that all requirements regarding vertices of $G$ are satisfied, i.e., $d_F(v) \in B(v)$ for each vertex $v \in V$ and for each vertex $v' \in V'$ it is $d_F(v') \leq 1$. Next we aim to gradually augment $F$ so that it also satisfies the requirements regarding new vertices $V'$ and
$d_F(v')=1$ for each $v' \in V'$. In either case, the computed $B$-matching is not guaranteed to be of maximum or minimum cardinality. A good characterization of graphs that have a $B$-matching \cite{Sebo1993} was provided in 1993 by Seb\H{o} \cite{Sebo1993}.
A $B$-matching is said to be {\em uniform} if each $B(v)$ is either an interval, i.e., has the form $\{a(v), a(v)+1, \ldots, b(v)\}$ for some nonnegative integers $a(v) \leq b(v)$ or an interval intersected with either even or odd numbers, i.e., has the form $\{a(v), a(v)+2, \ldots, b(v)\}$ for two nonnegative integers $a(v) \leq b(v)$ such that $b(v)-a(v)$ is even. A maximum/minimum weight uniform $B$-matching problem was shown to be solvable in polynomial time by Szab{\'o} \cite{Szabo}. In the solution to the weighted uniform $B$-matching Szab{\'o} uses the following result of Pap \cite{Pap}. Let $\cal{F}$ be an arbitrary set of odd length cycles of graph $G$, where a single vertex
is considered a cycle of length $1$. A {\em perfect} $\cal{F}$-matching is any set of cycles and edges of $G$ such that each vertex belongs to exactly one edge or cycle and each cycles belongs to $\cal{F}$. Pap gave a polynomial time algorithm which minimizes a linear function over the convex hull of perfect $\cal{F}$-matchings.
{\bf Our results} We give the first polynomial time algorithm for the maximum/minimum $B$-matching for the case when no set $B(v)$ contains a gap of length greater than $1$.
We provide a structural result which states that given two $B$-matchings $M$ and $N$, their symmetric difference $M \oplus N= (M \setminus N) \cup (N \setminus M)$ can be decomposed into a set of {\em canonical paths}, a notion which we define precisely later and which plays an analogous role as that of an {\em alternating path} in the context of standard matchings. A path $P$ is alternating with respect to a matching $M$ if its edges alternate between edges of $M$ and edges not belonging to $M$.
Roughly speaking, a canonical path (with respect to a given $B$-matching $M$) consists of a meta-path, that is a sequence of alternating paths, and possibly some number of meta-cycles attached to the endpoints of this meta-path. A meta-cycle is a sequence of alternating paths such that the beginning of the first alternating path coincides with the end of the last alternating path in the sequence. After the application of a canonical path ${\cal P}$ to a $B$-matching $M$ we obtain another $B$-matching $M'$ such that only the parities of the degrees in $M$ and $M'$ of the endpoints of ${\cal P}$ are different.
Equipped with this structural result we show how finding a maximum/minimum $B$-matching can be reduced to a series of computations of a maximum/minimum weight
uniform $B$-matching. In fact we prove that in order to verify if a given $B$-matching $M$ has maximum/minimum weight it suffices to check if there exists a uniform $B$-matching of so called {\em neighbouring type} to $M$, whose weight is greater/smaller than that of $M$.
Additionally, we show a very simple reduction of a weighted uniform $B$-matching to a weighted $(a,b)$-matching, which yields a more efficient and simpler algorithm than the one by Szab{\'o}.
{\bf Related work} In deficiency problems the goal is to find matching that is as close as possible to given sets $B(v)$. Hell and Kirkpatrick \cite{HellKirkpatrick} gave an algorithm for finding a minimum deficiency $(a,b)$-matching among all $(0,b)$-matchings, where deficiency is measured as sum of $a(v)-d(v)$ over all vertices whose degree is not between $a(v)$ and $b(v)$. They also proved that for another measure of deficiency, namely number of vertices whose degree is outside $(a(v),b(v))$, the problem is NP-hard.
Another related problem is decomposing graph into $(a,b)$-matchings (a graph that can be decomposed into $(a,b)$-matchings is called $(a,b)$-factorable). Kano gave sufficient condition for graph to be $(2a,2b)$-factorable \cite{Kano}. Cai generalized this result for $(2a-1,2b)$, $(2a,2b+1)$ and $(2a-1,2b+1)$ -factorable graphs \cite{Cai1991}. Hilton and Wojciechowski showed another sufficient condition for $(r,r+1)$-factorization of graphs \cite{hilton2005}.
$(a,b)$-matchings were also studied in stable framework - Biro et al. proved that checking whether stable $(a,b)$-matching exists is NP-hard \cite{Biro}.
A special case of general matching problem is an extended global cardinality constraint problem (EGCC): given set of variables $X$, set of values $D$, a domain for each variable $D(x) \subseteq D$ and a cardinality set $K(d)$, for each $d \in D$, the goal is to find valuation of variables, such that the number of variables with value $d$ is in $K(d)$ \cite{Samer2008}. For empirical survey on EGCC see \cite{Nightingale}.
\section{Uniform $B$-matching}
In this section we show a reduction of a uniform $B$-matching to an $(a,b)$-matching.
Suppose the instance of a uniform $B$-matching involves a graph $G=(V,E)$ and for each vertex $v \in V$ a subset $B(v)$ of the set $\{0,1,\ldots, d_G(v)\}$.
We construct a graph $G'=(V , E \cup E')$ and functions $a,b: V \rightarrow N$ as follows.
If for a vertex $v$, the set $B(v)$ is an interval $\{c(v), c(v)+1, \ldots, d(v)\}$ for some nonnegative integers $c(v) \leq d(v)$, then we set $a(v)=c(v)$ and $b(v)=d(v)$. If for a vertex $v$ the set $B(v)$ has the form $\{c(v), c(v)+2, \ldots, d(v)\}$, i.e., $B(v)$ contains all odd numbers between $c(v)$ and $d(v)$, and $c(v)$ and $d(v)$ are also odd, or $B(v)$ contains all even numbers between $c(v)$ and $d(v)$, and $c(v)$ and $d(v)$ are even, then we add $\frac{d(v)-c(v)}{2}$ loops incident to $v$ and set $a(v)=b(v)=d(v)$. Each loop has weight $0$. Apart from this each edge $e \in E$ has the same weight in $G$ and $G'$. Thus $E'$ consists of some number of loops that are added to each vertex $v$
such that $B(v)$ is not an interval.
\begin{theorem}
\label{thm:uniform_matching}
There is a one-to-one correspondence between $B$-matchings of $G$ and $(a,b)$-matchings of $G'$.
A maximum weight $(a,b)$-matching of $G'$ yields a maximum weight $B$-matching of $G$.
\end{theorem}
Since $G'$ contains loops, it is not a simple graph. There also exists, however, a simple reduction from a uniform $B$-matching to an $(a,b)$-matching in a simple graph and even a reduction from a uniform $B$-matching to a perfect matching in a simple graph.
\section{Structure of general $B$-matchings}
Let us first recall and generalise some notions and facts from matching theory. In the case of matchings, it is often convenient to consider the symmetric difference of two $B$-matchings.
Given two $B$-matchings $M$ and $N$ the symmetric difference of $M$ and $N$, denoted as $M\oplus N$, is equal to $(M \setminus N) \cup (N \setminus M)$.
The symmetric difference $M\oplus N$ can be decomposed into a set of edge-disjoint alternating paths and alternating cycles, the definition of which is as follows.
\begin{definition}
Let $M$ be any $B$-matching of $G$. A sequence of edges $P=\\((v_1, v_2), (v_2, v_3), \ldots, (v_{2k-1}, v_{2k}), (v_{2k}, v_1))$
is said to be an {\em alternating cycle} (with respect to $M$) if
\begin{itemize}
\item for every $i$ such that $1 \leq i \leq k$ the edge $(v_{2i-1}, v_{2i})$ belongs to $M$,
\item $(v_{2k},v_1) \notin M$ and for every $i$ such that $1 \leq i \leq k-1, \ (v_{2i}, v_{2i+1}) \notin M$,
\item each edge of $G$ occurs in $P$ at most once,
\item vertices $v_1, \ldots, v_{2k}$ are not necessarily distinct.
\end{itemize}
An {\em alternating path} (with respect to $M$) is a sequence of edges $P=\\((v_1, v_2), (v_2, v_3), \ldots, (v_{k}, v_{k+1}))$ such that
\begin{itemize}
\item for every $i$ such that $1 \leq i \leq k-1$ exactly one of the edges $(v_{i}, v_{i+1}),(v_{i+1}, v_{i+2})$ belongs to $M$,
\item each edge of $G$ occurs in $P$ at most once,
\item vertices $v_1, \ldots, v_{k}$ are not necessarily distinct,
\item if $v_1=v_{k+1}$, then either both edges $(v_1, v_2)$ and $(v_k, v_1)$ are in $M$, or both are not in $M$.
\end{itemize}
Vertices $v_1$ and $v_{k+1}$ are called the {\em endpoints} of $P$ and edges $(v_1, v_2), (v_{k}, v_{k+1})$ the {\em ending edges} of $P$.
\end{definition}
Examples of alternating paths and cycles are shown in Figure \ref{fig:alternating_paths}.
\begin{figure}[t]
\input{alternating_path.tex}
\caption{}
\label{fig:alternating_paths}
\end{figure}
The decomposition of the symmetric difference of two $B$-matchings into alternating paths and cycles is not unique. Nevertheless we are interested in {\em maximal} decompositions, i.e., such ones that the concatenation of any two alternating paths from the decomposition
does not result in a new alternating path or cycle.
By {\em applying} an alternating path or cycle $P$ to a $B$-matching $M$ we mean the operation, whose result is $M \oplus P$. We can notice that given any alternating cycle $P$ with respect to a $B$-matching $M$, the set $M'=M \oplus P$ is also a $B$-matching, because $d_{M'}(v)=d_M(v)$ for each vertex $v$. However not for every alternating path $P$ with respect to a $B$-matching $M$, it is true that $M' =M \oplus P$ is also a $B$-matching. If $v_1, v_2$ are the endpoints of $P$, then $d_{M'}(v_1)\neq d_M(v_1)$ and $d_{M'}(v_2)=d_M(v_2)$ and it may happen that $d_{M'}(v_1) \notin B(v_1)$ or $d_{M'}(v_2) \notin B(v_2)$.
We observe the following.
\begin{fact}
Given two $B$-matchings $M$ and $N$. Let $D_-$ and $D_+$ denote the sets, respectively, $\{v \in V: d_N(v) < d_M(v)\}$ and $\{v \in V: d_N(v) > d_M(v)\}$ and let $D$ denote $D_- \cup D_+$.
Then any maximal decomposition of $M \oplus N$ has the property that each endpoint of an alternating path from the decomposition belongs to $D$. Also, every ending edge of an alternating path $P$ incident to a vertex $v$ in $D_-$ such that $v$ is an endpoint of $P$, belongs to $M$ and similarly, every ending edge of an alternating path $P$ incident to a vertex $v$ in $D_+$ such that $v$ is an endpoint of $P$, belongs to $N$.
\end{fact}
Since the application of an alternating path to a $B$-matching does not necessarily lead to a new $B$-matching, we need to introduce some generalisation of an alternating path that can be applied in the context of $B$-matchings in a similar way as an alternating path in the context of (standard) matchings.
From alternating paths of a maximal decomposition of the symmetric difference of two $B$-matchings $M$ and $N$ we build {\em meta-paths} and {\em meta-cycles}. Let $P(u,v)$ denote an alternating path with the endpoints $u$ and $v$ (note that $u,v \in D$). A meta-cycle $\cal{C}$ (of $M$ and $N$) is a sequence of alternating paths of the form
$(P(v_1, v_2), P(v_2, v_3), \ldots, P(v_k, v_1)$ such that vertices $v_1, \ldots, v_{k}$ are pairwise distinct. Analogously, a meta-path ${ \cal P}(v_1,v_{k+1})$ (of $M$ and $N$) is a sequence of alternating paths of the form
$(P(v_1, v_2), P(v_2, v_3), \ldots, P(v_k, v_{k+1}))$ such that vertices $v_1, \ldots, v_{k+1}$ are pairwise distinct. Let us note that a meta-cycle of $M$ and $N$ may consist of one alternating path of the form $P(v,v)$.
For a vertex $v$ and $k\in B(v)$ let $u_k(v)$ be a maximum element of $B(v)$, such that $B(v)\cap [k,u_k(v)]$ does not contain element of different parity than $k$. From that and because $B(v)$ has gap of length at most $1$ it follows that $B(v)\cap [v,u_k(v)] = \{k, k+2, k+4, \dots, u_k(v)\}$. Also, either $u_k(v) + 1 \in B(v)$ or $u_k(v)$ is a maximum element of $B(v)$, as otherwise we could increase $u_k(v)$. Similarly let us define $l_k(v)$ to be a minimum element of $B(v)$, such that $B(v)\cap [l_k(v),k]$ does not contain an element of different parity than $k$.
We define $B_k(v)$ to be $B(v)\cap [l_k(v), u_k(v)] = \{l_k(v), l_k(v)+2, \dots, k, \dots, u_k(v)\}$. Note that $\{B_k(v)\}_{k\in B(v)}$ is a partition of the set $B(v)$. For a $B$-matching $M$ we also define $B_M(v) = B_{d_{M(v)}}(v)$.
Given a $B$-matching $M$ we say that a $B$-matching $N$ is {\bf \em of the same uniform type} as $M$ if for every vertex $v$ it holds that $d_N(v) \in B_M(v)$.
A $B$-matching $N$ is said to be {\bf \em of neighbouring type} to a $B$-matching $M$ if there exists a set $W$ of at most two vertices such that $\forall w\in W d_N(w)\notin B_M(w)$ and $\forall v\notin W d_N(v) \in B_M(v)$ and:
\begin{itemize}
\item $|W| = 0$, or
\item $|W| = 2$ and for $w\in W$ $B_M(w)$ and $B_N(w)$ are adjacent, that is $\max(B_M(w)) + 1 = \min B_N(w)$ or $\max(B_N(w)) + 1 = \min B_M(w)$, or
\item $|W| = 1$ and for $w\in W$ there is $k$, such that $B_k(w)$ is adjacent to both $B_M(w)$ and $B_N(w)$.
\end{itemize}
In other words we allow two vertices to have degree outside of $B_M(v)$, but we limit how much they can deviate from that set.
We are now ready to give a definition of a {\bf \em canonical path} - a notion that is going to prove crucial in further analysis and which plays an analogous role as an alternating path in the context of matchings.
\begin{definition}
A canonical path ${\cal S}(v_1, v_k)$ (with respect to a $B$-matching $M$) consists of some number of meta-cycles ${C}_1, {C}_2, \ldots, {C}_p$ incident to vertex $v_1$, some number of meta-cycles ${C'}_1, {C'}_2, \ldots, {C'}_q$ incident to $v_k$ and
and in case $v_1 \neq v_k$ - of a meta-path $P(v_1,v_{k})$ such that the application of all meta-cycles ${C}_1, {C}_2, \ldots, {C}_p$, ${C'}_1, {C'}_2, \ldots, {C'}_q$ and the
path $P(v_1, v_k)$ to $M$ results in a $B$-matching of neighbouring type to $M$.
\end{definition}
We will often refer to the weight of a path - that is the effect it has on a $B$-matching $M$. More precisely $w_M(\S) = w(M\oplus \S) - w(M) = \sum_{e\in \S\setminus M} w(e) - \sum_{e\in \S\cap M} w(e)$. Note that if $\S_1$ and $\S_2$ are edge disjoint, then $w_M(\S_1) = w_{M\oplus \S_2}(\S_1)$. Usually we will write $w(\S)$ for $w_M(\S)$ when choice of $M$ is clear. Also when constructing new canonical paths we will use the notion of a fine vertex - we say that a vertex $v$ is {\em fine in $\S$} if the number of edges incident to $v$ in $M \oplus \S$ belongs to $B(v)$ and {\em wrong} otherwise. We say that an endpoint of $\S$ is fine (wrong) if it is fine (wrong) in $\S$.
In our algorithm we want to subsequently find and apply positive weight canonical paths until a $B$-matching is optimal.
Let us start by showing that it is necessary to consider canonical paths, that is that it may happen that matching is not optimal, but there is no meta-path or meta-cycle augmenting it (i.e. increasing its size). Consider an unweighted graph in Figure \ref{fig:ex_meta} and let $B(v) = \{0,1,3,5\}$, $B(u) = \{0,1\}$, $B(w) = \{0,2\}$ and $B(t) = \{0,2\}$. For every other vertex $x$ let $B(x) = \{1\}$. Then we cannot apply any of the meta-cycles incident to $v$, because the degree of $v$ would be $2$. On the other hand applying the meta-path decreases the size of the matching. So we need to apply both meta-cycles and meta-path at the same time (which together form a canonical path) to obtain a feasible $B$-matching of greater size.
\begin{figure}[t]
\centering
\input{canonical_path_ex.tex}
\caption{Example of matching, which is not optimal, but there is no meta-path or meta-cycle improving it.}
\label{fig:ex_meta}
\end{figure}
The definition of a canonical path is quite general, so we would like to restrict ourselves to a more limited notion. In the example above we saw that we cannot consider only minimal (with respect to inclusion) canonical paths. Therefore, we introduce another notion, similar to a minimal canonical path but taking into account the weight of a path.
\begin{definition}
We say that ${\S}$ is a basic (canonical) path if it is a canonical path and for no proper subset ${\S}'\subsetneq {\S}$ $\S'$ is a canonical path such that either $w({\S}') \geq w(\S)$ or $w(\S') > 0$.
\end{definition}
\begin{observation}
\label{obs:basic_path}
Let $M$ be a $B$-matching. If there exists a canonical path $\S$ w.r.t. $M$, then there exists a basic canonical path $\S'\subseteq \S$ w.r.t $M$.
\end{observation}
\begin{lemma}\label{decomp}
Let $M,N$ be two $B$-matchings. Then there exists a sequence ${\cal S}_1, {\cal S}_2, \ldots, {\cal S}_k$ and a set of alternating cycles
$C_1, C_2, \ldots, C_l$ that satisfy the following.
\begin{enumerate}
\item Let $M_0$ denote $M \oplus \bigcup_{i=1}^l C_i$. For each $i$ such that $0 < i \leq k$ ${\cal S}_i$ is a basic canonical path with respect to $M_{i-1}$ and $M_i = M_{i-1} \oplus {\cal S}_i$.
Also, $M_k=N$.
\item $M\oplus N = \bigcup_{i=1}^k {\cal S}_i \cup \bigcup_{i=1}^l C_i$, where every two elements of the set $\{\S_1, \ldots, \S_k, C_1, \ldots, C_l\}$ are edge-disjoint.
\end{enumerate}
\end{lemma}
\dowod
Let us consider some fixed maximal decomposition of $M \oplus N$.
Let $C_1, C_2, \ldots C_l$ denote all alternating cycles of this decomposition. By $M_0$ we denote $M \oplus \bigcup_{i=1}^l C_i$.
If $d_M(v)=d_N(v)$ for every vertex $v$, then $M \oplus N$ consists solely of alternating cycles $C_1, C_2, \ldots C_l$ and $M_0=N$ and we are done.
The maximal decomposition of $M_0 \oplus N$ consists only of alternating paths.
The {\em distance} of two $B$-matchings $M$ and $N$ denoted as $dist(M,N)$ is defined as $\sum_{v \in V} |d_N(v)-d_M(v)|$. In the distance of two $B$-matchings
it is enough to consider the vertices belonging to $D$, i.e., $dist(M,N)= \sum_{v \in D} |d_N(v)-d_M(v)|$.
Let $M_0$ and $N$ be two matchings such that the set $D$ corresponding to them is not empty, i.e. there exists a vertex $v$ such that $d_{M_0}(v)\neq d_N(v)$
and hence $dist(M_0,N) >0$.
We show how to construct some canonical path $\cal{S}$ with respect to $M_0$ such that the $B$-matching $M_1=M_0 \oplus \cal{S}$ satisfies: $D(M_1,N) \subseteq D(M_0,N)$, $D_{-}(M_1,N) \subseteq D_{-}(M_0,N), D_{+}(M_1,N) \subseteq D_{+}(M_0,N)$ and $dist(M_1,N) < dist(M_0,N)$.
We start from any alternating path $P$ that belongs to a maximal decomposition of $M_0 \oplus N$. $P$ may have two different endpoints or one endpoint. If $P$ is not a canonical path, then it means that after its application for at least one of its endpoints $v_1$ or $v_2$ it holds that $d_{M_0 \oplus P}(v_i) \notin B(v_i)$, where $i\in \{1,2\}$. We can notice that apart from this $P$ satisfies all the other
conditions of a canonical path. We are going to gradually extend $P$ so that we obtain $\cal{S}$ that is a canonical path. At each stage of the construction
the candidate $\cal{S}$ for a canonical path has all the properties of a canonical path except for the fact that for one or two of its endpoints
it holds that $d_{M_0 \oplus \cal{S}}(v_i) \notin B(v_i)$, where $i\in \{1,2\}$.
Note that in $\S$ both endpoints have degree one. If $v_i$ is not fine in $\S$ it means that $B(v)$ contains $d_{M_0}$ and $d_{M_0} + 2$, but it does not contain $d_{M_0}$. Then if we add another alternating path starting at $v_i$ it will not be an endpoint and its degree increases by 1, so its degree is in $B_M(v)$. This will be true at each step of our construction - if vertex $v$ is not an endpoint then its degree is in $B_M(v)$. Also another invariant is, that if there are two endpoints of $\S$ their degree will be odd, and if they join into one (so $v_1=v_2$) then their degree is even.
Assume then that we have some candidate path, which is not a canonical path, so $d_{M_0 \oplus \cal{S}}(v_1) \notin B(v_1)$. Since $N$ is a $B$-matching there exists an alternating path $P'$ in the maximal decomposition of $ (M_0 \oplus \cal{S}) \oplus N$ with one endpoint $v_1$. This path has the property that either $P$ and $\S$ both diminish the number of edges incident to $v_1$, or they both increase the number of edges incident to $M_0$, or our alternating paths would not be maximal. After adding $P$ to $\cal{S}$ the following things may happen:
\begin{enumerate}
\item $P$ has two different endpoints $v_1, v_3$. Then vertex $v_1$ is then fine in $\S \cup P$. If $v_3$ is not an endpoint of any alternating path belonging to $\cal{S}$, then $v_3$ is a new endpoint of $\cal{S} \cup P$ and either (i) $v_3$ is fine in $\S \cup P$ and we have decreased the number of wrong endpoints by one or (ii) $v_3$ is wrong in $M \oplus (\S \cup P)$ and the number of wrong endpoints of $\S \cup P$ is the same as the number of wrong endpoints of $\S$ and we continue the process treating $\S \cup P$ as the new candidate for a canonical path. If $v_3$ is an endpoint of some alternating path belonging to $\cal{S}$, then we have created
a new meta-cycle $\cal{C}$ incident to $v_3$. If $v_3$ is fine in $\S \cup C$ then we decreased the number of wrong endpoints. If $v_3$ is fine in $C$ then $C$ is a canonical path with respect to $M_0$. Otherwise it means that $d_{M_0}(v_3) + 2 \notin B(v_3)$, so $v_3$ must be the other endpoint of $\S$. Then we have only one wrong endpoint left, $v_3$, and we continue extending $\S$ from $v_3$. Note that now that two endpoints joined in $v_3$, we will move one of the endpoints to make $v_3$ fine, but the other will always be $v_3$.
\item $P$ has one endpoint $v_1$. If $v_1$ is fine in $\cal{S} \cup \cal{C}$, then we have decreased the number of wrong endpoints of a candidate for a canonical path. Otherwise if
$P$ is a canonical path we are done. The only case left is when $v_1$ is not fine but $d_{M_0}(v_1) + 2\notin B(v_1)$. This may only happen if both endpoints of $v_1$ are the same vertex and then we continue extending $\S$ with only one wrong endpoint left.
\end{enumerate}
That way we constructed a canonical path $\S$ of $M$. Therefore, by Observation \ref{obs:basic_path} it means that there is a basic canonical path $\S'$.
We can continue finding canonical paths in the same way, this time in $M_0\oplus \S'\oplus N$. Each such path decreases distance between $M$ and $N$, so thhat way we can decompose $M_0 \oplus N$ into a finite number of basic canonical paths.
\koniec
Now we are ready to state the key technical lemma.
\begin{restatable}{lemma}{technicallemma}\label{podst}
Let $M$ and $N$ be two $B$-matchings, such that $w(M) < w(N)$. Let $\Q$ be a basic canonical path of $M$ and $N$ and $\R$ a basic canonical path of $M\oplus \Q$ and $N$ such that $w(\Q) \leq 0$ and $w(R) > 0$. Then there exists a canonical path $\T$ of $M$ and $N$ such that $w(\T) > w(\Q)$.
\end{restatable}
We defer the proof of this lemma to Section \ref{sec:key_lemma} and now let us focus on its consequences.
\begin{theorem}
\label{thm:canonical_path}
If there exists a $B$-matching of greater weight than $M$, then there exists a $B$-matching of greater weight than $M$ that is of the same uniform type as $M$ or that is of neighbouring type to $M$.
\end{theorem}
\dowod
Suppose that there does not exist a $B$-matching $M'$ of the same uniform type as $M$ and with greater weight than $M$ but there exists a $B$-matching $N$ having greater weight than $M$.
By Lemma \ref{decomp} we know that there exists a sequence of basic canonical paths ${\cal S}_1, {\cal S}_2, \ldots, {\cal S}_k$ and a set of alternating cycles
$C_1, C_2, \ldots, C_l$ such that $M\oplus N = \bigcup_{i=1}^k {\cal S}_i \cup \bigcup_{i=1}^l C_i$. The weight of $N$ satisfies $w(N)=w(M)+ \sum_{i=1}^l w(C_i)+
\sum_{i=1}^k w({\cal S}_i)$. Since $w(N)>w(M)$ there exists some alternating cycle $C_i$ among the cycles $C_1, \ldots, C_l$ with positive weight or there exists
some canonical path ${\cal S}_i$ among the canonical paths ${\cal S}_1, {\cal S}_2, \ldots, {\cal S}_k$ with positive weight.
We may, however, observe, that if some alternating cycle $C_i$ has positive weight, then $M \oplus C_i$ is of the same uniform type as $M$ and has greater weight than $M$. As alternating cycles do not change degree of any vertex we may apply them after canonical paths. Therefore let $N' = M \oplus \bigcup_{i=1}^k {\S}_i$ and note that it is also a canonical path, as $\forall v d_N'(v) = d_N(v)$. Its weight, however, is greater than weight of $N$, because we skipped applying negative alternating cycles. Therefore we can assume that decomposition of $M\oplus N$ does not contain any alternating cycles.
By Lemma \ref{decomp} there is some sequence of basic canonical paths that forms a decomposition of $M \oplus N$, but it is not neceessarily unique. From all such sequences let us choose the one, such that ${\cal S}_1$ is a basic canonical path with respect to $M_0=M$ of maximum weight and $M_1 = M \oplus {\cal S}_1$. For each $i>1$, ${\cal S}_i$ is a basic canonical path with respect to $M_{i-1}$ of maximum weight and $M_i=M_{i-1} \oplus \S_i$. In other words with each sequence of canonical paths we associate a sequence of weights. We want to choose such sequence of basic canonical paths that sequence of weights is maximum w.r.t. lexicographical order.
Note that when choosing $\S_i$ of maximum weight we will always be able to complete the sequence of canonical paths, because $M_i$ is a $B$-matching so we can apply Lemma \ref{decomp}.
Some basic canonical path ${\cal S}_i$ must of course have positive weight. Let $i$ be the smallest such index. If $i=1$, then we are
done. Assume then, that $i>1$.
It means that $\S_i$ has positive weight and $w({\cal S}_{i-1}) \leq 0$ and ${\cal S}_{i-1}$. There exists then ${\cal S'}_i$ that is a basic canonical path with respect to $M_{i-1}$ with positive weight. But then, by Lemma \ref{podst} and Observation \ref{obs:basic_path}, there exists a basic canonical path ${\cal S'}_{i-1}$ with respect to $M_{i-2}$ such that $w(\S'_{i-1}) > w(\S_{i-1})$, and which contradicts the properties of our decomposition, because instead of adding ${\cal S}_{i-1}$, we could have done better and have added ${\cal S'}_{i-1}$.
Such argument cannot be applied only if the weight of ${\cal S}_1$ is already positive, which shows that the claim of the Theorem is correct. \koniec
\section{Algorithm for computing a maximum cardinality $B$-matching}
In this section we will show the algorithmic consequences of Theorem \ref{thm:canonical_path}, namely we will present a polynomial time algorithm for a maximum cardinality $B$-matching.
First, let us assume that we have some $B$-matching $M$. We want to be able to either verify that it is maximum or find a better $B$-matching.
According to Theorem \ref{thm:canonical_path}, $M$ is not maximum if and only if there is a larger $B$-matching $M'$ such that at most two vertices' degrees are not in $B_M(v)$. Therefore we can consider all possible sets of at most two vertices - whose degrees would not be restricted to $B_M(v)$. For the rest of vertices we allow them to have any degree in $B_M(v)$. This is an instance of a uniform $B$-matching, so we use Theorem \ref{thm:uniform_matching} to solve it.
This approach requires solving $O(n^2)$ instances of a maximum weight
uniform $B$-matching problem.
Now we can find maximum cardinality $B$-matching. We start by running Cornuejols' algorithm, which finds any $B$-matching or verifies that graph does not have a $B$-matching. Then we subsequently improve this matching until it is maximum. The size of maximum matching can be bounded by the number of edges in the graph, so the total complexity of the algorithm is strongly polynomial.
\begin{comment}
The key observation that allows a polynomial time computation of a maximum weight $B$-matching is that if a current $B$-matching $M$
is not of maximum weight and there does not exists a $B$-matching $M'$ of the same uniform type as $M$ and with greater weight than $M$, then
there exists a canonical path ${\cal S}$ with respect to $M$ such that $M \oplus {\cal S}$ has bigger weight than $M$. This observation
is proved below in Lemma (Theorem) \ref{}.
Based on this observation the approach towards finding a maximum weight $B$-matching is as follows. Using Cornuejols' algorithm we find some $B$-matching $M$ if it exists. Next we compute a maximum weight $B$-matching $M'$ of the same uniform type as $M$. Further we check if there exists a $B$-matching $N$ of neighbouring type to $M'$ with weight greater than that of $M'$. This requires solving $O(n^2)$ instances of a maximum weight
uniform $B$-matching problem. If there exists a $B$-matching $N$ of neighbouring type to $M'$ with weight greater than that of $M'$, we continue the process, otherwise we know that the current $B$-matching $M'$ is of maximum weight.
\end{comment}
\noindent \fbox{
\begin{minipage}[t]{0.956\textwidth}
\vspace{0.5cm}
{\bf \em \hspace{0.5cm} Algorithm Max $B$-Matching}
\vspace{0.5cm}
\begin{enumerate}
\item Using Cornuejols' algorithm find some $B$-matching $M$.
\item {\bf while} there exists a $B$-matching $M'$ of neighbouring type to $M$ with cardinality greater than that of $M$ {\bf do:}
\hspace{1cm} $M \leftarrow M'$
\item Output $M'$.
\end{enumerate}
\vspace{0.5cm}
\end{minipage}
}
\section{Structure and properties of a basic canonical path}
\label{sec:key_lemma}
In this Section we will prove Lemma \ref{podst}. Let us start with some notation we will use throughout this section. Each path and cycle in this section denotes a meta-path and a meta-cycle. Also we will use the relative notation of degrees. If $M$ is a $B$-matching and $v$ some vertex, then we will use the set $\{d - d_M(v) : \forall d \in B(v)\}$. Particularly, we will use $0$ to denote the current degree.
We say that a vertex $v$ is {\bf odd w.r.t. $M$} if $deg_{M}(v) +1 \in B(v)$ and {\bf even w.r.t. $M$} otherwise.
We will often omit $M$ and say that a vertex $v$ is odd (even) if it is odd (even) w.r.t. $M$. We will also say that a vertex $v$ is odd (even) w.r.t. $\S$ if it is odd (even) w.r.t. $M\oplus \S$.
For any canonical path $\S$ of $M$ we will also assume that if any vertex $v$ is in $D$, then it is in $D_+$. That is because the case when $v\in D_-$ is completely symmetrical, so we will avoid repeating each argument twice.
Now let us state the following observation which is a consequence of the definition of a $B$-matching of neighbouring type.
\begin{observation}
\label{obs:degrees_canonical_path}
Let $M$ be a $B$-matching and let $\S$ be a canonical path. Let $N=M\oplus \S$ and $v, v'$ be the endpoints of $\S$. Then:
\begin{enumerate}
\item For each vertex $u$ other than $v$ and $v'$, then $\{0, 2, \dots, d_{N\oplus M}(u)\} \subseteq B(u)$.
\item If $v$ and $v'$ are distinct, then for $u \in \{v,v'\}$ there is some $k\in \{0,2,\dots, d_{M\oplus N}(u) - 1\}$ such that $\{0,2,4,\dots, k, k+1, k+3, \dots, d_{M\oplus N}(u)\} \subseteq B(u)$.
\item If $v = v'$ then $\{0, 2, \dots, d_{M\oplus N}(v)\}\subseteq B(v)$ or there are $k_1 \in \{0, 2, \dots, d_{M\oplus N}(v)-2\}$ and $k_2 \in \{k_1+1, k_1+3,\dots, d_{M\oplus N}(v)-1\}$ such that $\{0,2, \dots, k_1, k_1 + 1, k_1+3, \dots, k_2, k_2+1, k_2+3, \dots, d_{M\oplus N}(v))\} \subseteq B_v$.
\end{enumerate}
\end{observation}
In the following lemmas we will derive some structure of basic canonical paths, which will be useful in proving Lemma \ref{podst}. We summarize these lemmas in Corollary \ref{cor:summary}.
\begin{lemma}
\label{lem:endpoints_type}
Let $\cal S$ be a basic path, such that its endpoints are distinct. Let $v,u$ be enpoints of $\cal S$. Then there is no $k\in \{1,2,\dots, d_{\cal S}(v)-2\}$ such that $v$ allows $\{k, k+1\}$.
\end{lemma}
\dowod
The proof will be by contradiction. Assume that there is a $k\in \{1,2,\dots, d_{\cal S}(v)-2\}$ such that $v$ allows $\{k,k+1\}$. Therefore by Observation \ref{obs:degrees_canonical_path} $v$ allows $\{0,2,\dots, k-2, k, k+1, k+3, \dots, d_{\cal S}(v)-2, d_{\cal S}(v)\}$. Let $m\in \{0,1,\dots,d_{\cal S}(v)-1\}$ be such that $u$ allows $\{m,m+1\}$. We will now construct a subset $\S'$ of $\S$ which is a canonical path and such that $w(\S') \geq \min(w(\S),0)$. Let us consider three cases:
\begin{enumerate}
\item $m = 0$. For any cycle $C$ in $\S$ $\S \setminus C$ is a canonical path. So if $\S$ contains a cycle $C$ such that $w(\S \setminus C) \geq \min(w(\S),0)$ $\S$ is not a basic path. Otherwise $\forall C w(C) > 0$. If there is a cycle incident to $v$ and not incident to $u$ then it is a canonical path with positive weight. If not then there must be a cycle $C$ incident to $u$ and $v$ (as $v$ is incident to at least one cycle). We split $C$ into two paths connecting $u$ and $v$ and we remove the one with smaller weight and the metapath connecting $u$ and $v$ (from definition of canonical path). We decreased degree of both endpoints by $2$ so it is a canonical path and weight of all cycles and remaining part of $C$ is positive
\item $m = d_{\S}(v) - 1$. If there is a cycle $C\in \S$ such that $w(C)\geq \min(w(\S),0)$ then it is a canonical path. Otherwise if there is a cycle $c$ incident only to $v$ then $\S \setminus C$ is a canonical path and $w(\S \setminus C) \geq w(\S)$. Finally if there is no such cycle we take a cycle $C$ incident to $u$ and $v$ and we split it into two paths. The path with greater weight with metapath connecting $u$ and $v$ forms a canonical path. As we removed some cycles, each of negative weight, and one part of $C$, which also has negative weight, it follows that resulting canonical path has weight greater than $w(\S)$.
\item $0 < m < d_{\S}(v) - 1$. We take any cycle $C$. It is a canonical path, so if it has positive weight it contradicts the assumption. Otherwise $\S \setminus C$ is also a canonical path and contradicts the assumption.
\end{enumerate}
\koniec
Lemma \ref{lem:endpoints_type} shows that if a canonical path $\S$ has distinct endpoints then its endpoint $v$ allows either $\{0,1,3,\dots, d_{\S}(v) - 2, d_{\S}(v)\}$ or $\{0,2,4,\dots, d_{\S}(v) - 3, d_{\S}(v) - 1, d_{\S}(v)\}$. The first case happens when $v$ is odd and the second case when it is even.
\begin{lemma}
\label{lem:cycle_both_endpoints}
Let $\S$ be a basic path with distinct endpoints. If $\S$ contains a cycle $C$ incident to both endpoints then one of those endpoints is odd and the other is even.
\end{lemma}
\dowod
The proof will be by contradiction. Let us assume that either both endpoints are odd or both are even. In the first case if $w(C) \leq 0$ then $\S \setminus C$ is a canonical path such that $w(\S\setminus C) \geq w(\S)$. Otherwise we split $C$ into two paths connecting endpoints of $\S$. As $C$ has positive weight, one of those paths also has positive weight and it is a canonical path. Case when both endpoints are even is similar.
\koniec
\begin{lemma}
\label{lem:cycle_degrees}
Let $\S$ be a basic path, such that its endpoints are the same vertex (so metapath from definition of canonical path is empty and $\S$ is a collection of cycles). Let $v$ be the endpoint of $\S$. Then $\S$ is either (a) a single meta-cycle or (b) $v$ allows degrees $\{0,1,3,\dots,d_{\S}(v) - 1, d_{\S}(v)\}$.
\end{lemma}
\dowod
Assume that $v$ allows an even degree $2k$, such that $0<k<\frac{d_{\S}(v)}{2}$. Then let $\S'$ be $k$ cycles of $\S$ of greatest weight. If all of these cycles have positive weight then $\S'$ has positive weight. Otherwise all excluded cycles have nonpositive weight, so $w(\S') > w(\S)$.
\koniec
\begin{lemma}
\label{lem:cycle_weight}
Let $\S$ be a basic path with distinct endpoints. Let $u,v$ be endpoints of $\S$. If $v$ is even then all cycles incident to $v$ but not to $u$ have nonpositive weight. If $v$ is odd then all cycles incident to $v$ but not to $u$ have positive weight.
\end{lemma}
\dowod
Let $v$ be an even endpoint, and $C$ a cycle incident only to $v$. Then if $w(C) > 0$ then $C$ is a canonical path of positive weight, which means that $\S$ is not a basic path.
Similarly if $v$ is an odd endpoint, then we can remove any incident cycles of nonpositive weight.
\koniec
\begin{comment}
\begin{corollary}
Let $\S$ be a basic path with distinct endpoints and let $C$ be its meta-cycle. If $w(C) > 0$ then it is incident to odd endpoint of $\S$ (and possibly also even endpoint). If $w(C) \leq 0$ then it is incident to even endpoint of $\S$ (and possibly also odd endpoint).
\end{corollary}
\end{comment}
We summarize those lemmas in the following Corollary. We will often implicitly refer to this Corollary in the proof of Lemma \ref{podst}.
\begin{corollary}
\label{cor:summary}
Let $\S$ be a basic canonical path with endpoints $u$ and $v$. Then:
\begin{itemize}
\item For every vertex $w$ which is not an endpoint of $\S$, $B(w)$ contains $0, 2, \dots, d_S(w)$;
\item If $u\neq v$ and $u$ is an odd endpoint, then $B(u)$ contains $0, 1, 3, \dots, d_S(u)$. If $u$ is an even endpoint, then $B(u)$ contains $0, 2, \dots, d_S(u)-1, d_S(u)$;
\item If $u=v$, then either $\S$ is a single meta-cycle and $B(u)$ contains $0,2$ or $B(u)$ contains $0,1,3,\dots, d_S(u)-1, d_S(u)$;
\item If $u \neq v$ and $u$ is odd, then any cycle incident only to $u$ is positive. If $u$ is even, then any cycle incident only to $u$ is non-positive;
\item If $u \neq v$ and $\S$ contains a cycle $C$ incident to both $u$ and $v$, then $u$ is odd and $v$ is even.
\end{itemize}
\end{corollary}
\begin{lemma}
\label{lem:cycle_weight_both_endpoints}
Let $\S$ be a basic path with distinct endpoints and let $v$ be its endpoint. If $w(\S) \leq 0$ then we can assume one of the following about $\S$ (but not both):
\begin{enumerate}
\item If $v$ is even then $v$ is incident to a cycle of nonpositive weight.
\item If $v$ is odd let $\C$ be cycles of $\S$ incident to $v$. Then $w(\C)\geq w(\S)$.
\end{enumerate}
Similarly if $w(\S) > 0$ and $v$ is odd let $\C$ be cycles of $\S$ incident to $v$. We can assume that $w(\C) > 0$. In particular, if $v$ is incident to any cycle, we can assume that it is incident to cycle of positive weight.
\end{lemma}
\dowod
Let us assume that $v$ is even. If there is a cycle in $\S$ incident only to $v$ (but not the other endpoint) then from Lemma \ref{lem:cycle_weight} it has nonpositive weight.
Therefore we assume that there is no cycle incident only to $v$ and let $\C$ be nonempty set of cycles in $\S$ incident to both endpoints (which means that the other endpoint is odd). If $\C$ contains cycle of nonpositive weight we are done. Otherwise all cycles in $\S$ have positive weight (as the other endpoint is odd), so the meta-path $P$ connecting endpoints of $\S$ has nonpositive weight. Then we can take any cycle of $\C$ and split it into two meta-paths $P_1$ and $P_2$ between endpoints of $\S$. Assume $w(P_1) \geq w(P_2)$. Then we make $P_1$ the meta-path of $\S$ and $P_2\cup P$ a meta-cycle of $P$ which has nonpositive weight.
The other cases are similar.
\koniec
Now we will prove Lemma \ref{podst}.
\technicallemma*
\dowod
To construct a canonical path $\T$ we will consider how $\Q$ and $\R$ interact with each other, that is what common vertices they have. Firstly let us notice that we can assume that $\Q$ and $\R$ do not have a common vertex $v$ that is not an endpoint of any of them. That is because $v$ allows degrees $0,2,4,\dots, d_{Q\cup R}(v)$. Therefore we can create $k := d_{Q\cup R}(v)/2$ new vertices $v_1, v_2, \dots, v_k$ and replace $v$ with a different vertex in each meta-path or meta-cycle containing $v$. Each of these vertices $v_i$ will allow degrees $\{0,2\}$, if it is an endpoint of some alternating path, or $\{0\}$ otherwise. Then any canonical path we will find in the new graph corresponds to some canonical path in the old graph.
The structure of the proof is as follows. First we will prove some auxillary lemmas. Then we will split the proof into a few cases depending on the structure of $\Q$ and $\R$. If both $\Q$ and $\R$ have two endpoints we use lemmas \ref{lematR1} and \ref{lematR}. In the second we assume that $\R$ contains at least two edge-disjoint paths between both endpoints of $\R$ (or equivalently that there is a cycle incident to both endpoints). If $\R$ has one endpoint we use Lemma \ref{lem:R_one_endpoint}. Finally if $\Q$ has one endpoint and $\R$ has two endpoints we use Lemma \ref{lem:Q_one_endpoint}.
We say that a path or cycle {\bf goes through} vertex $b$ if two edges of this cycle or path are incident to $b$.
\begin{lemma} \label{helpbas}
Let $\S \subseteq \R$ be a path with the endpoints $u$ and $v$ such that (i) $w(\S)>0$, (ii) both $u$ and $v$ belongs to $\Q$, (iii) $\S$ does not go through an even endpoint of $Q$. Then every path contained in $\Q$ between $u$ and $v$ that does not go through any even endpoint of $\Q$ has weight at least $w(\S)$ and thus positive.
\end{lemma}
\dowod Otherwise, we could replace such path with $\S$ and obtain a canonical path of greater weight than $\Q$.
\begin{lemma} \label{help}
Let $\S$ be a path $\P(c,d)$ of positive weight such that both $c$ and $d$ lies on $\Q$.
Then, the existence in the graph of any of the listed below implies the existence of a canonical path $\T$ w.r.t. $M$ such that $w(\T) > w(\Q)$:
\begin{enumerate}
\item $\S$ such that there exists a path $\P(c,d)$ contained in $\Q$ that does not go through any endpoint of $\Q$;
\item $\Q$ has two odd end-points $a$ and $b$ and $\S$ either contains a path $\P(a,b)$ or $\Q$ contains a path $\P(c,d)$ that contains
a path $\P(a,b)$
\end{enumerate}
\end{lemma}
\dowod If there exists a path $\P(c,d)$ contained in $\Q$ that does not go through any endpoint of $\Q$, then by Lemma \ref{helpbas} $\P(c,d)$ has positive weight and $\P(c,d) \cup \S$ form a positive cycle that goes only through even vertices and hence is a canonical path w.r.t $M$.
Suppose now that both $a$ and $b$ are odd. Thus $\Q$ contains exactly one path connecting $a$ and $b$.
Assume also that $\Q$
contains a path $\P'=\P(c,d)$ that contains
a path $\P(a,b)$. One endpoint of $\P'$, say $c$ must lie on a cycle $\C_1$ of $\Q$ incident to $a$ and the other - $d$ on a cycle $\C_2$ incident to $\Q$. This means that we can extract from $\C_1$ and $\C_2$ positive weight paths $\P_1=\P(a,c)$ and $\P_2=\P(b,d)$. Then $\S \cup \P_1 \cup \P_2$ is a positive canonical path w.r.t. $M$. Let us notice that this holds regardless of the fact if $\S$ goes through $a$ or $b$ or even both of them.
Suppose now that $\S$ contains a path $\P(a,b)$. Therefore $\S$ consists of paths: $\P_1=\P(c,a), \P_0=\P(a,b)$ and $\P_2=\P(b,d)$. If $w(\P_0)>0$, we are done. Otherwise, $w(\P_1) + w(\P_2) > 0$. By Lemma \ref{helpbas} this means that $w(\Q)$ contains two edge-disjoint paths $\P'_1=\P(a,c)$ and $\P'_2=\P(d,b)$ such that $w(\P'_1) + w(\P'_2)\geq w(\P_1) + w(\P_2) > 0$. Then $\P'_1 \cup \P'_2 \cup \S$ forms a positive canonical path w.r.t $M$. \koniec
We assume that $\Q$ has two endpoints $a$ and $b$ and $\R$ has two endpoints $c$ and $d$.
\begin{lemma}\label{cykl}
Let $\C \subset \R$ be a cycle with positive weight that contains at least one of the endpoints of $\Q$. Then there exists a canonical path of weight greater than $\Q$.
\end{lemma}
\dowod
\noindent {\bf Case: $\C$ contains no odd vertex.} $\C$ forms then a canonical path.
\noindent {\bf Case: $\C$ contains at least two odd vertices.} Suppose that $\C$ contains $k$ odd vertices. We then split $\C$ into $k$ paths with odd endpoints. At least one of these paths must have positive weight and forms a canonical path with positive weight.
\noindent {\bf Case: $\C$ contains exactly one odd vertex $c$ that belongs to $\R \setminus \Q$.} $\C$ must contain at least one even endpoint of $\Q$. We split $\C$ into three paths or two paths depending on whether $\C$ contains one or two even endpoints of $\Q$. We choose the path $\S$ with positive weight. If the endpoints of $\S$ are even endpoints of $\Q$, we are done - by Lemma \ref{help}. Otherwise one of the endpoints of $\S$ is $c$ and the other an even endpoint of $\Q$, let us call it $b$. By Lemma \ref{lem:cycle_weight_both_endpoints} $\Q$ contains a cycle $\C'$ going through $b$ that has non-positive weight. Also, if $\Q$ has two even endpoints $a$ and $b$, then $\C'$ does not go through $a$. Then $\Q \cup \S \setminus \C'$
forms a canonical path with the endpoints $a$ and $c$ and weight greater than that of $\Q$.
\noindent {\bf Case: $\C$ contains exactly one odd vertex $a$ that belongs to $\Q$.}
If $\C$ does not contain a vertex that is odd w.r.t. $Q$, we can see that $\Q \cup \C$ is of the same uniform type as $\Q$ and has bigger weight. Assume then that $\C$ contains a vertex that is odd w.r.t. $\Q$. Let us note that $\C$ cannot contain two vertices that are odd w.r.t. $\Q$ because by Lemma \ref{lem:cycle_both_endpoints} a basic canonical path with two endpoints does not contain a cycle that goes through both endpoints if both of them are odd or both of them are even.
Let us consider first the case when $a=c$ and $a$ is odd w.r.t. $\Q$. We remove from $\Q$ a path between $a$ and $b$ of minimum weight and each cycle incident to $b$ and not going through $a$ - the remaining part of $\Q$ has positive weight or smaller than that of $\Q$. It is so because each cycle contained in $\Q$ going through $a$ and not $b$ has positive weight, each cycle going through $b$ and not $a$ has non-positive weight and either each path between $a$ and $b$ has positive weight or at least one of them has non-positive weight. To thus modified $\Q$ we add $\C$ and obtain a canonical path $\Q'$ with one endpoint $a$ such that $deg_{\Q'}(a)=deg_{\Q}(a) +1$.
Now we assume that $\C$ contains a vertex $d \neq a$ that is odd w.r.t. $\Q$. If $\C$ goes through an even endpoint $b$ of $\Q$, we proceed as follows. By Lemma \ref{lem:cycle_weight_both_endpoints} $\Q$ contains a cycle $\C'$ with non-positive weight going through $b$. Thus $\Q \setminus \C' \cup \C$ forms a canonical path with the endpoints $a$ and $b$ and weight greater than $w(\Q)$. Next we examine the case when $\C$ does not go through any endpoint of $\Q$ different from $a$.
If $\Q$ contains a cycle $\C'$ with non-positive weight going through both $b$ and $d$, where $b$ is even then there exists a path $\P \subset \Q$ between $d$ and $b$ of non-positive weight and we build $\Q'= \Q \cup \C \setminus \P$, which is a canonical path with the endpoints $a$ and $d$ and weight greater than $w(\Q)$.
Otherwise, we build $\Q'$ as follows - we extract from $\Q$ a path $\S$ between $a$ and $d$ - note that $w(\S)>0$ by Lemma \ref{helpbas} as $\C$ contains a path between $a$ and $d$ of positive weight. Next we add every cycle contained in $\Q$ incident to $a$ but not the one containing $\S$ - each such cycle has positive weight. $\Q'$ also contains $\C$. The weight of $\Q'$ is clearly positive. It is also a canonical path with
the endpoints $a$ and $d$ because the degree of $a$ in $\Q'$ is odd and $deg_{\Q'}(d)=deg_{\Q}(d) +1$. To see that the degree of $d$ in $\Q'$ is as claimed let us notice that $d$ does not belong to any cycle contained in $\Q$ that goes through $b$ and with non-positive weight, which means that $d$ either lies on a path between $a$ and $b$ or on a cycle incident to $a$. Also, there cannot exist two edge-disjoint paths between $a$ and $b$ going through $d$ because then they would form two edge-disjoint cycles - one going through $a$ and $d$ and the other through $b$ and $d$. If the cycle going through $b$ and $d$ has positive weight, it forms a canonical path because it does not go through any odd vertex. \koniec
\begin{lemma} \label{L1}
Let $\C \subset \R$ be a cycle with positive weight such that it goes through $c$ and $c \in \Q$. Then there exists a canonical path $\T$ w.r.t. $M$ such that $w(\T) > w(\Q)$.
\end{lemma}
\dowod
The only case that requires explanation is when $\C$ does not contain any end-point of $\Q$. Other cases are covered by Lemma \ref{cykl} above.
Then $\C$ itself forms a canonical path because it does not go through any odd vertex. \koniec
\begin{lemma}\label{lematZ}
Let $Z \subseteq \R$ consist of a path between $c$ and $b$ and cycles incident to $c$ and be such that it does not go through any even end-point of $\Q$. Also, $w(Z)>0$, $b$ is even and $c$ is fine in $\Q \cup Z$.
If $Z$ goes through $d$, then $d$ is even.
Then there exists a canonical path w.r.t. $M$ with weight greater than that of $\Q$.
\end{lemma}
\dowod By Lemma \ref{lem:cycle_weight_both_endpoints} $\Q$ contains a cycle $\C$ incident to $b$ of non-positive weight.
Suppose first that $c$ belongs to $\C$. It means that $c \in \Q$ and thus by Lemmas \ref{cykl} and \ref{L1}, $c$ is either even w.r.t. $\Q$ or
$deg_{\Q \cup Z}(c)= deg_{\Q \cup \R}(c) = deg_{\Q}(c)+1$.
We extract from $\C$ a path $\P$ with the endpoints $b$ and $c$ and non-positive weight. We construct $\Q'=\Q \cup Z \setminus \P$. $\Q'$ is a canonical path with the endpoints $a$ and $b$ because the degrees of $c$ in $\Q'$ and in $\Q$ have the same parity
and the degree of $b$ is the same in $\Q'$ as in $\Q$. If $c$ does not belong to $\C$, we construct $\Q'=\Q \cup Z \setminus \C$. $\Q'$ is a canonical path with the endpoints $a$ and $c$. In both cases $w(\Q') > w(\Q)$. \koniec
\begin{lemma}\label{lematR1}
If $\R$ contains a path $\R_{max}$ between $c$ and $d$ and no cycle going through both $c$ and $d$, then there exists a canonical path of weight greater than $\Q$.
\end{lemma}
\dowod The general approach in this proof is the following. We start by considering a set $Z$ consisting of a path $R_{max}$ and every cycle $\C \subset \R$ that does not go through any endpoint of $\Q$. By Lemmas \ref{cykl} and \ref{L1} the weight of $Z$ is positive because every cycle $\C \subset \R$ that we have not included has non-positive weight. If both $c$ and $d$ is fine in $\Q \cup Z$, we either go to the second or the last case of this proof, or if $R_{max}$ goes through an even endpoint of $Q$, we split $Z$ into parts and apply Lemma \ref{lematZ}. Observe that $c$ is not fine in $\Q \cup Z$ iff $c$ is even w.r.t. $\Q$ (and thus also even) and some cycle $\C \subset \R$ goes through $c$ and an endpoint of $\Q$. This is because the degree of $c$ is odd in $\Q \cup Z$. Next we want to add parts of the non-selected cycles to $Z$ to make $c$ and $d$ fine in $\Q \cup Z$ or show directly that a given case implies that $w(\Q)$ is already positive.
Also in the proof we often assume that $\{a,b\}\cap \{c,d\}=\emptyset$, but the claim of the lemma also holds if some of the endpoints are the same.
In the rest of the proof by saying that an endpoint $v$ is fine we mean that $v$ is fine in $\Q \cup Z$. Let us also note that a vertex that is not fine must be even.
\noindent {\bf Case: (i) $d$ is not fine and some cycle $\C \subset \R$ incident to $d$ goes through an even endpoint $b$ of $\Q$ and (ii) $c$ is fine or no cycle $\C' \subset \R$ incident to $c$ goes through any even endpoint of $\Q$.}
We apply Lemma \ref{lematZ}.
\noindent {\bf Case: both $c$ and $d$ are not fine in $\Q \cup Z$.}
There exists then a cycle $\C_1 \subset \R$ incident to $c$ that goes through $a$ and a cycle $\C_2 \subset \R$ incident to $d$ that goes through $b$. We split $\C_1$ into two paths between $a$ and $c$ and choose the one with maximum weight - let us call it $\P_1$. Similarly, we split $\C_2$ into two paths between $b$ and $d$ and choose the one with maximum weight and call it $\P_2$.
We note that the path $\S=\R_{max} \cup \P_1 \cup \P_2$ has positive weight. (Every cycle $\C \subset \R$ incident to an endpoint of $\R$ has non-positive weight because it either goes through some endpoint of $\Q$ and then by Lemma \ref{cykl} it has non-positive weight or it goes only through even vertices and if such a cycle existed, it would form a canonical path with positive weight. Therefore $w(\R_{max}) \geq w(Z)>0$.)
If both $a$ and $b$ are even or both of them are odd, we are done, as either $\S$ forms a canonical path w.r.t. $M$ or by Lemma \ref{help} its existence implies the existence of a positive cycle going only through even vertices. The case when $a$ is odd and $b$ is even is covered above. Let us note that if $R_{max}$ goes through some endpoint(s) of $\Q$, we may also need to split $\S$.
\noindent {\bf Case: (i) $d$ is fine and belongs to a cycle $\C \subset \Q$ of non-positive weight that goes through an even endpoint $b$ of $\Q$ and (ii) $c$ is fine or no cycle $\C' \subset \R$ incident to $c$ goes through any even endpoint of $\Q$.}
Let us notice that regardless of the fact if $d$ is fine or not, if we reduce its degree from in $\Q \cup Z$ by one, it will be fine, i.e., $deg_{\Q \cup Z}(d)-1 \in B(d)$. The same holds for $c$.
If $\C$ does not go through $c$, we extract from $\C$ a path $\P$ with the endpoints $b$ and $d$ and non-positive weight. If $c$ is not fine, we add all cycles contained in $\R$ incident to $c$ to $Z$. Each such cycle goes through an odd endpoint $a$ of $\Q$.
As a result $c$ is fine and the weight of $Z$ remains positive. Next we construct $\Q'=\Q \cup Z \setminus \P$, which is a canonical path with the endpoints $a$ and $c$ and weight greater than $w(\Q)$.
If $\C$ goes through $c$, we extract from $\C$ a path $\P$ with the endpoints $b,d$ or $b,c$ or $c,d$ and non-positive weight. If $\P$ has endpoints $b,d$, we proceed as above. Otherwise we build $\Q'=\Q \cup Z \setminus \P$ and obtain a canonical path with the endpoints $a$ and $d$, or $a$ and $b$; and weight greater than $w(\Q)$.
\noindent {\bf Case: $c \notin \Q$, $c$ is fine and $d$ is not.}
There exists then a cycle $\C \subset \R$ incident to $d$ that contains some odd endpoint of $\Q$.
We are able to extract
from $\C$ a path (i) with one endpoint equal to $d$ and the other either $a$ or $b$ such that $w(\P) \geq w(\C)/2$ and $\P$ does not go through any even endpoint of $Q$ or (ii) with two even endpoints of $\Q$ and positive weight. If $\P$ is as in (ii), we can apply Lemma \ref{helpbas}.
We construct $\Q_1= Z \cup \P$. Note that $d$ is even. $d$ is therefore fine in $\Q_1$. Also, $c$ is fine. If $\Q_1$ does not go through any endpoint of $\Q$, it forms a canonical path with the endpoints $a$ and $c$. Otherwise we split it and obtain a canonical path with two odd endpoints or apply Lemma \ref{lematZ}.
\noindent {\bf Case: $c \notin \Q$, both $c$ and $d$ are fine.}
Note that $d \in \Q$. Otherwise $Z$ would form a canonical path with positive weight. If $Z$ goes through some even endpoint(s) of $\Q$, we split it and apply Lemma \ref{lematZ} or Lemma \ref{help}.
If $d$ lies on a cycle $\C \subset \Q$ incident to an odd endpoint $a$ of $\Q$, we extract from $\C$ a path $\P$ of positive weight and then $\P \cup Z$ forms a canonical path of positive weight.
If $d$ lies on a path $P$ between $a$ and $b$, we take either one part of $\Q$ and $Z$ or the other and $Z$ and obtain a canonical path of weight greater than $w(\Q)$.
\noindent {\bf Case: $a$ and $b$ are odd, a cycle $\C \subset \R$ incident to $c$ goes through $a$ and $b$.} It means that $\R$ does not contain any cycle incident to $d$ that goes through $a$ or $b$ and thus that $d$ is fine in $\Q \cup Z$. Also we may assume that $d \in \Q$ - the other case is already covered above. We split $\C$ into three meta-paths $\P_1=\P(a,b), \P_2=\P(a,c), \P_3=\P(b,c)$.
We observe that every cycle contained in $\R$ has non-positive weight. Therefore $w(R_{max})>0$ because $w(\R)>0$.
We will show that $\C \cup \R_{max}$ contains two paths $\S_1=\P(a,d)$ and $\S_2=\P(b,d)$, each of which has positive weight. We know that $w(\C \cup \R_{max})>0$. If $w(\P_3) <0$, then $\S_1= \P_1 \cup \P_2 \cup \\R_{max}$ has positive weight.
Otherwise $\S_1 = \P_3 \cup \R_{max}$ has positive weight. Similarly, if $w(\P_2) <0$, then $\S_2= \P_1 \cup \P_2 \cup \R_{max}$ has positive weight
and otherwise $\S_2= \P_2 \cup \R_{max}$ has positive weight. Using Lemma \ref{helpbas}, we know that any path $\P \subset \Q$ between $a$ and $b$ has positive weight, which means that the whole $\Q$ has positive weight. It is so because every cycle contained in $\Q$ is incident to exactly one odd endpoint of $\Q$ and has positive weight.
Let us observe that $d$ cannot coincide with either $a$ or $b$ as $\R$ contains only one path between $c$ and $d$. If $c$ coincides with either $a$ or $b$ the arguments above hold.
\noindent {\bf Case: $a$ and $b$ are odd, a cycle $\C_1 \subset \R$ incident to $c$ goes through $a$ and a cycle $\C_2 \subset \R$ incident to $c$ goes through $b$.}
Again, we may assume that $d \in \Q$. We again observe that every cycle contained in $\R$ has non-positive weight. Therefore $w(R_{max})>0$ because $w(\R)>0$.
We show that $w(\Q)>0$. To this end it suffices to show that the path $\P(a,b) \subseteq \Q$ has positive weight.
We extract from $\C_1$ and $\C_2$ paths $\P_1$ and $\P_2$, correspondingly between $a$ and $c$ and $b$ and $d$ such that $w(\P_1) \geq w(\C_1)/2$ and
$w(\P_2) \geq w(\C_2)/2$. It means that $w(\R_{max} \cup \P_1)>0$ and $w(\R_{max} \cup \P_2)>0$. This in turn means that $w(\P(a,d) \subseteq \Q)>0$
and $w(\P(b,d) \subseteq \Q)>0$. Hence $w(\P(a,b) \subseteq \Q)>0$.
Similarly as in the case above $d$ cannot coincide with either $a$ or $b$ and if $c$ coincides with either $a$ or $b$ the arguments above hold.
\noindent {\bf Case: $d \in \Q$ is fine, $c$ is not and $c \notin \Q$.}
It means that there exists $C \subset R$ incident to $c$ that goes through exactly one odd endpoint of $\Q$ - $a$ - other cases are dealt with above.
We proceed as follows. We extract from $\C$ a path $\P_1$ with the endpoints $a$ and $c$ such that $w(\P_1) \geq w(\C)/2$. We extend the set $Z$ so that it contains every cycle $C \subset R$ that goes through $a$ and does not contain $\P_1$.
$\Q_2$ consists of $Z$ and a path $\P \subseteq \Q$ between $a$ and $d$. $w(\P)>0$ because $w(\P_1 \cup \R_{max}) >0$. Therefore $\Q_2$ has positive weight and is a canonical path with the endpoints $a$ and $c$.
We are left with the following case.
\noindent {\bf Case: (i) both $c\in \Q$ and $d \in \Q$ and (ii) $d$ is fine and (iii) (a) $c$ is fine or (b) $a$ is an odd endpoint of $\Q$ and every cycle $C \subset R$ incident to $c$ that goes through an endpoint of $\Q$, goes through $a$ and through no other endpoint of $\Q$.}
We extend the set $Z$ so that it contains every cycle $\C \subset \R$ that goes through $a$. Now, both $c$ and $d$ are fine.
We may assume that neither $c$ nor $d$ lies on a cycle $\C' \subset \Q$ of non-positive weight that goes through an even endpoint of $\Q$.
We may also assume that either every path connecting $c$ and $d$ contained in $\Q$ goes through some end-point of $Q$ or that $R_{max}$ goes through
an odd end-point of $\Q$ - otherwise we can apply Lemma \ref{helpbas}.
\begin{claim} \label{Cdod}
If $b$ is even and $\Q$ contains a path $\T$ connecting $b$ and $d$ with non-positive weight and such that $\T$ does not go through $c$ or any even end-point of $\Q$, then $\Q'=\Q \cup Z \setminus \T$ is a canonical path w.r.t $M$ having weight greater than $w(\Q)$.
\end{claim}
We are left with the following cases:
\begin{enumerate}
\item $c$ and $d$ both lie on the path connecting $a$ and $b$ in $\Q$ and $\R_{max}$ goes through $a$.
\item $c$ lies on a cycle of $\Q$ incident to $a$ and $d$ on a path between $a$ and $b$ in $\Q$.
\item two different cycles of $\Q$ incident to $a$.
\item $c$ and $d$ lie on two different paths connecting $a$ and $b$.
\end{enumerate}
In the other cases we may use Lemma \ref{help}. In each of the above cases we remove from $\Q$: $\T$ - a path contained in $\Q$ connecting $d$ and $b$ and also all cycles going through $b$ but not going through $c$ or $d$. We obtain an edge-set $Q'$ which is a canonical path w.r.t. $M$ with the endpoints $a$ and $c$. We show that $w(Q')>0$.
In the first case it is enough to show that a path $\P$ connecting $a$ and $d$ that belongs to $\Q \cap \Q'$ has non-negative weight.
We split $\P$ and $\R_{max}$ into two paths: correspondingly $\P_1=\P(a,c)$ and $\P_2=\P(c,d)$ and $\S_1=\P(c,a)$ and $\S_2=\P(a,d)$. The weight of
$\P_2$ is positive because $w(\R_{max})>0$ and by Lemma \ref{helpbas}. It holds that $w(\S_1) >0$ or $w(\S_2) >0$. If $w(\S_2)>0$, then $w(\P)>0$ and we are done.
In the other case, $w(\P_1)>0$ (because $w(\S_1)>0$ and by Lemma \ref{helpbas}). We also already know that $w(\P_2)>0$, which means that $w(\P)>0$.
In the second case let us note that any path $\T' \subset \Q$ connecting $c$ and $d$ has positive weight by Lemma \ref{helpbas} and the fact that $w(\R_{max})>0$. Let us notice that the part of $\Q'$ that is contained in $\Q$ consists of one such path $\T'$ and some number of cycles incident to $a$, all of which have non-negative weight. Since $\Q' =(\Q' \cap \Q) \cup Z)$, we are done.
In the third case the cycle $\C$ contained in $Q$ going through $a$ and $d$ has positive weight and if we split it into two paths connecting $a$ and $d$, while building $\Q'$ we can remove that path, whose weight is not bigger. Therefore $\Q' \cap \Q$ consists of one such path contained in $\C$ and some number of cycles contained in $\Q$ and going through $a$.
In the fourth case, we use the fact that every path contained in $\Q$ connecting $b$ and $d$ has positive weight, because we may assume that we cannot use Claim \ref{Cdod}. \koniec
\begin{lemma}\label{lematR}
If $\R$ contains two edge-disjoint paths between $c$ and $d$, then there exists a canonical path of weight greater than $\Q$.
\end{lemma}
\dowod
Exactly one of the end-points of $\R$ is odd w.r.t. $\Q$, assume it is $c$. Let us note that if $c \in \Q$, then $c$ is also even.
Suppose first that $R$ contains a cycle $\C$ of positive weight. If $\C$ does not contain any odd vertices, $\C$ constitutes a canonical path and we are done. By Lemma \ref{cykl}, if $\C$ contains any of the vertices $\{a,b\}$, we are also done.
Let us notice that $\R$ always contains some cycle $\C$ of positive weight. Any cycle $\C' \subset \R$ going through $c$ and not $d$ is of positive weight. Such cycle $\C'$ for sure does not go through $a$ or $b$ (by Lemmas \ref{helpbas} and \ref{help}). Also, if $\C'$ exists, it means that $c \notin \Q$.
If such $\C'$ does not exist, then $\R$ contains two edge-disjoint paths $\R_1, \R_2$ between $c$ and $d$ such that $w(\R_1 \cup \R_2)>0$. Such edge-set must contain some cycle $\C$ of positive weight.
The only possibility that $\C \subset \R$ of positive weight does not imply the existence of a canonical path with positive weight is when $\C$ goes through $c$, $c$ does not belong to $\Q$ ($c$ is odd) and goes through neither $a$ nor $b$. For the rest of the proof suppose that this is the case.
Suppose now that some path $\S \subset \R$ between $c$ and $d$ contains some endpoint of $\Q$. We consider the set $Z \subseteq \R$ that consists of edge-disjoint paths $\S_1, \ldots, \S_k, \S$, each with the endpoints $c$ and $d$ and such that either (i) $\R$ does not contain $\C'$ as above and then no path $\S_i$ contains any end-point of $\Q$ and $k \geq 2$ or (ii) $\R$ contains some $\C'$ as above and then $Z$ contains additionally every such cycle and $k=1$; also $w(Z) >0$. Let us note that such $Z$ always exists.
Suppose that $\S$ contains exactly one endpoint of $\Q$ - $a$ which is odd or exactly one even endpoint - $b$ and that $Z$ is as in case (i). If $k$ is odd, we consider $\S' \subset \S$ - a path between $c$ and the distinguished endpoint. If $w(\S') \leq 0$, $Z \setminus \S'$ is either a canonical path
with the endpoints $a$ and $c$ with positive weight or we can apply Lemma \ref{lematZ} to it. If $w(\S') >0$, again $\S'$ is either a canonical path
with the endpoints $a$ and $c$ with positive weight (if the distinguished endpoint is an odd endpoint $a$) or we can apply Lemma \ref{lematZ} to it. If $k$ is even, we proceed in the same way but considering $\S'' \subset \S$ - a path between $d$ and the distinguished endpoint.
If $\S$ contains two odd endpoints of $\Q$ or two even endpoints, we act similarly but split $\S$ into three paths with the endpoints $a$ and $b$, $a$ and $c$, and $b$ and $d$.
Suppose now that no path $\S \subset \R$ between $c$ and $d$ contains any endpoint of $\Q$.
It means that there exists a cycle $\C \subset \R$ that goes through some endpoint $a$ of $\Q$. It also goes through $d$ and not $c$ and also has non-positive weight.
We split $C$ either into three paths $P_1=P(a,b), P_2=P(a,d), P_3=P(b,d)$ - if $\C$ goes also through $b$,
or two paths with the endpoints $a$ and $d$.
If $w(P_1) >0$ and both $a$ and $b$ are even or both $a$ and $b$ are odd, we are done - by Lemmas \ref{helpbas} and \ref{help}. Otherwise we are able to extract
from $\C$ a path with one endpoint equal to $d$ and the other either $a$ or $b$ such that $w(P) \geq w(C)/2$ and $P$ does not go through any even endpoint of $Q$.
We construct $\Q_1$. It consists of every path $\S \subset \R$ with the endpoints $c$ and $d$.
$P$ and each cycle contained in $R$ incident to $c$ but not $d$. Clearly $w(\Q_1) >0$ as in order to obtain $\Q_1$, we have removed from $\R$ at most $w(\C)/2$ which has non-positive weight and possibly some cycles of $\R$ incident to $d$ but not $c$, each one also with non-positive weight.
Note that $d$ is even in $\Q_1$. It is also fine in $\Q_1$ as $deg_{\Q_1}(d) < deg_{\Q}(d)$. Also $c$ is fine in $\Q_1$ as well as $d$ is fine in $\Q\cup \Q_1$ - the degrees of $c$ are the same in $\Q_1$ and $\Q \cup \Q_1$.
If $P$ ends at an odd end-point, say $a$, of $Q$ - $Q_1$ forms a canonical path with the endpoints $a$ and $c$.
Otherwise we can treat $\Q_1$ as $Z$ from Lemma \ref{lematZ}.
\koniec
\begin{lemma}
\label{lem:shortcut_one_endpoint}
Suppose that $\Q$ has one endpoint $a$ and there is a positive meta-path $S$ between $a$ and $c$. Suppose also that $c$ is incident to $\Q$ and is fine in $\Q \cup S$ and if $S$ contains $d$ then $d$ is even. Then there exists a canonical path of weight greater than $\Q$.
\end{lemma}
\dowod
If $\Q$ contains a non-positive meta-cycle incident to $c$ we split it into two paths and replace lighter of them with $S$. Otherwise $c$ with all cycles of $\Q$ incident to $c$ is a positive canonical path, because not all cycles of $\Q$ are positive and so $a$ is fine.
\koniec
\begin{lemma}
\label{lem:one_endpoint_Z}
Suppose that $\Q$ has one endpoint $a$ and $Z$ contains a meta-path $S$ between $a$ and $c$ and possibly some positive cycles incident to $c$, but not containing $d$. Suppose also that $Z$ is positive, $c$ is fine in $\Q \cup Z$ and if $\S$ goes through $d$ then $d$ is even. Then there exists a canonical path of weight greater than $\Q$.
\end{lemma}
\dowod
If any cycle of $Z$ is incident to $a$ then we use Lemma \ref{L1}. If $c\notin \Q$ then $Z$ is a positive canonical path. If $c\in \Q$ and $Z$ contains some cycle we use Lemma \ref{cykl}. Finally if $c\in \Q$ and $Z$ does not contain any cycle we use Lemma \ref{lem:shortcut_one_endpoint}.
\koniec
\begin{lemma}
\label{lem:Q_one_endpoint}
Suppose that $\Q$ has one endpoint $a$ and $\R$ has two endpoints $c$ and $d$. Then there exists a canonical path of weight greater than $\Q$.
\end{lemma}
\dowod
If $\Q$ is a single meta-cycle and $1\notin B(a)$ then $\R\cup \Q$ is a canonical path.
Let $R_{max}$ denote meta-path of $\R$ between $c$ and $d$ of maximum weight.
\noindent {\bf Case: $a=c$ and $d\notin \Q$.}
If $\R$ contains a positive cycle $C$ incident to $c$, but not to $d$, we replace any non-positive cycle of $\Q$ with $C$. If $d$ is odd, then let $\C$ denote cycles of $\R$ incident to $d$, but not to $c$. Each of them has positive weight, so $\C \cup R_{max}$ also has positive weight and is a canonical path. If $d$ is even and there are at least two meta-paths of $\R$ between $c$ and $d$ then we choose from them two heaviest paths and they form a cycle incident to $c$ and $d$ and we replace one of cycles of $\Q$ with it. Finally if $d$ is even and there is exactly one meta-path of $\R$ between $c$ and $d$ then $R_{max}$ with cycles of $\R$ incident to $d$ form a canonical path of positive weight.
\noindent {\bf Case: $a=c$ and $d\in \Q$.}
If there is a positive meta-cycle of $\R$ incident only to one of its endpoints we use Lemma \ref{L1}. Otherwise set of all meta-paths of $\R$ between $c$ and $d$ have positive weight. If $d$ is odd w.r.t. $\Q$ then we use Lemma \ref{lem:shortcut_one_endpoint} by setting $S=R_{max}$. Otherwise we choose two heaviest meta-paths of $\R$ between $c$ and $d$ and they form a cycle $C$ of positive weight. Then we replace any non-positive cycle of $\Q$ with $C$.
\noindent {\bf Case: $a\notin \R$.}
If there is a cycle of $\Q$ incident to both endpoints of $\R$ then these endpoints are even or not incident to any meta-cycle of $\R$, as otherwise we would use Lemma \ref{L1}. Therefore set of meta-paths between endpoints of $\R$ has positive weight, so also $R_{max}$ has positive weight. Then we use Lemma \ref{help} with $S=R_{max}$.
Otherwise, if it exists, let $C$ be the meta-cycle of $\Q$ of non-positive weight incident only to one endpoint of $\R$, say $c$. Once again $c$ is even or not incident to any meta-cycle. Let $\R'$ be $\R$ without meta-cycles incident to $c$, but not to $d$ (its weight is greater that $\R$). Let us split $C$ into two paths between $a$ and $c$ and let $P$ be lighter of them. Then $\R' \cup \Q \setminus P$ is a canonical path of weight greater than $\Q$.
Finally suppose that all cycles of $\Q$ incident to any endpoint of $\R$ are positive. Let $\C$ denote these cycles, let $C$ be one of them incident to $c$ and let $P$ be lighter sub-path of $C$ between $a$ and $c$. Then $\R' \cup \C \setminus P$ is a canonical path of positive weight.
\noindent {\bf Case: $a\in \R$}
If $a$ lies on positive cycle of $\R$ incident to only one of its endpoints then we use Lemma \ref{cykl}. Let us consider the case when $a$ lies on some non-positive cycle $C$ of $\R$ incident to only one of its endpoints. Let $c$ be endpoint of $\R$ incident to $C$, which by Corollary \ref{cor:summary} is even, let $P$ be the heavier subpath of $C$ between $a$ and $c$ and let $\C$ be set of meta-cycles of $\R$ incident to $c$. Let $D$ be any cycle of $\Q$ of non-positive weight. If $D$ does not contain $d$ then $\Q \setminus D \cup \R \setminus \C \cup P$ is a canonical path of weight greater than $\Q$. If $D$ contains $d$, then $d$ is even or not incident to any cycle of $\R$ (otherwise we use Lemma \ref{L1}). Let $P'$ be lighter subpath of $D$ between $a$ and $d$. Then $\Q \setminus P' \cup \R \setminus \C \cup P$ is a canonical path of weight greater than $\Q$.
Now we assume that $a$ does not lie on any meta-cycle of $\R$, but it lies on some meta-path of $\R$. If $\R$ contains only one meta-path, then $a$ splits $\R$ into two paths and let $R'$ be heavier of them. If $R'$ is a canonical path we are done. Otherwise let $c$ be the endpoint of $\R'$. If all cycles of $\Q$ incident to $c$ are positive we add them to $\R'$, thus creating positive canonical path. If $c$ is incident to positive cycle, we use Lemma \ref{L1}. Otherwise path of $\R'$ between $c$ and $a$ is positive, which we will denote as $S$. Let $C$ be non-positive cycle of $\Q$ incident to $c$. We split $C$ into two paths between $a$ and $c$ and let $P$ be lighter of them. Then $\Q \setminus P \cup S$ is a canonical path of weight greater than $\Q$.
Finally let us assume that $\R$ has many meta-paths and therefore $c$ is odd and $d$ is even. Firstly let us assume that in $\R$ there is a positive meta-path $S$ between one of its endpoints and $a$, that does not go through the other endpoint. If the endpoint of $S$ is $c$ we use Lemma \ref{lem:one_endpoint_Z}. If the endpoint of $S$ is $d$ let $R_1$ be the path of $\R$ containing $S$ and let $R_{max}$ be the heaviest path of $\R$, unless $R_1$ is heaviest and then let $R_max$ be the second heaviest path. Let $\C$ denote cycles of $\R$ incident to $c$, but not to $d$. We may assume that $R_1\setminus S$ is non-positive, as we would have used previous case, so $S \cup R_{max} \cup \C$ is positive. If $R_{max}$ is not incident to $a$ we use Lemma \ref{lem:one_endpoint_Z} with $Z = c\notin \Q$ $S \cup R_{max}\cup \C$. If $R_{max}$ is incident to $a$, then we split it into $P_1$ between $a$ and $c$ and $P_2$ between $a$ and $d$. If $P_1 \cup \C$ is positive we use Lemma \ref{lem:shortcut_one_endpoint} with $Z=P_1\cup \C$. Otherwise $P_2 \cup S$ is positive so we choose any non-positive cycle of $\Q$ and replace it with $P_2 \cup S$.
In case when all meta-paths of $\R$ incident to $a$ are non-positive, let $\S$ denote those paths that are not incident to $a$ (it might be empty) and let $\C$ be cycles incident to $c$. We can assume that all paths of $\R$ between $c$ and $a$ are non-positive, because otherwise we use Lemma \ref{lem:one_endpoint_Z}. Let us consider $\S \cup \C$, which has positive weight as we only removed non-positive paths and cycles incident to even endpoint. If it is a canonical path we are done. If $c$ is fine and $d$ is not we choose any path between $a$ and $d$ and add it to $\R\setminus\S$ (we still remove only non-positive paths). If $d$ is fine and $c$ is not then let $P$ be a path between $a$ and $c$ and let $R_1$ be path between $c$ and $d$ containing $P$. If $R_1\setminus P$ is positive then $R_1\setminus P$ with maximum meta-path of $\S$ and $\C$ is positive (because either maximum meta-path is positive or we remove only non-positive paths) so we use Lemma \ref{lem:one_endpoint_Z}. Otherwise $\S\cup\C\cup P$ is a positive canonical path. Both $c$ and $d$ are not finee in $\S \cup \C$ only if $\S$ contains odd number of paths and $c\in \Q$. Then we choose a cycle of $\R$ incident to $c$ (either one of $\C$, or if it is empty we form cycle from two heaviest paths between $c$ and $d$) and use Lemma \ref{L1}.
\koniec
\begin{lemma}
\label{lem:R_one_endpoint}
Suppose that $\R$ has one endpoint, denoted by $c$. Let $a$ and $b$ be the endpoints of $\Q$ (if $\Q$ has only one endpoint it will be denoted as $a$). Then there exists a canonical path of weight greater than $\Q$.
\end{lemma}
\dowod
If $\R$ is a single meta-cycle and $1\notin B(c)$ then $\Q\cup \R$ is a canonical path.
We know that $\R$ has positive weight, so there is at least one cycle in $\R$ of positive weight. In such case, if $c$ lies on $\Q$ then by Lemma \ref{L1} we are done.
If both endpoints of $\Q$ are fine in $\R$ then $\R$ is a canonical path with respect to $M$.
Now we assume that one endpoint of $\Q$, say $a$, is not fine in $\R$. $a$ is incident to some number of cycles of $\R$. If any of them, let us call it $C$, has non-positive weight, then we split it into two meta-paths between $a$ and $c$. We remove the lighter of these paths and obtain that way a canonical path, as degree of $c$ is odd and we decreased degree of $a$ by $1$.
If all cycles of $\R$ incident to $a$ have positive weight, we consider any of them and let us call it $C$. We split $C$ into two paths, the same way as before. Let us consider heavier of these paths and call it $P$. Let $\C$ be set of cycles of $\R$ incident to $a$ except the one with $P$. Then $\C \cup P$ is a positive canonical path.
In case when both endpoints of $\Q$ are not fine in $\R$ and incident to $\R$ we proceed similarly. If there is no cycle incident to both $a$ and $b$ and all cycles incident to $a$ are non-positive then we remove all those cycles and proceed with $b$ as before. If all cycles incident to $a$ are positive we form a canonical path from all them except one path between $a$ and $c$ (similarly as above). If there is cycle $C$ incident to both $a$ and $b$ we consider sub-path of $C$ between $a$ and $b$ that does not contain $c$. If it is non-positive we remove it and obtain a positive canonical path. Otherwise we use Lemma \ref{help}.
\koniec
\koniec
\bibliographystyle{plain}
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2008.09370
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\section{Introduction}
Modeling imaging sensor noise is an important task for many image processing and computer vision applications.
Besides low-level applications such as image denoising~\cite{Unprocessing,CBDNet,DnCNN,zhang2018ffdnet}, many high-level applications, such as detection or recognition~\cite{he2017mask,liu2016ssd,redmon2016you,ren2015faster}, can benefit from a better noise model.
Many existing works assume statistical noise models in their applications.
The most common and simplest one is signal-independent additive white Gaussian noise (AWGN)~\cite{DnCNN}.
A combination of Poisson and Gaussian noise, containing both signal-dependent and signal-independent noise, is shown to be a better fit for most camera sensors~\cite{foi2008pg,CBDNet}.
However, the behavior of real-world noise is very complicated.
Different noise can be induced at different stages of an imaging pipeline.
Real-world noise includes but is not limited to photon noise, read noise, fixed-pattern noise, dark current noise, row/column noise, and quantization noise.
Thus simple statistical noise models can not well describe the behavior of real-world noise.
Recently, several learning-based noise models are proposed to better represent the complexity of real-world noise in a data-driven manner~\cite{NoiseFlow,GCBD,GRDN}.
In this paper, we propose a learning-based generative model for signal-dependent synthetic noise.
The synthetic noise generated by our model is perceptually more realistic than existing statistical models and other learning-based methods.
When used to train a denoising network, better denoising quality can also be achieved.
Moreover, the proposed method is camera-aware.
Different noise characteristics of different camera sensors can be learned simultaneously by a single generative noise model.
Then this learned noise model can generate different synthetic noise for different camera sensors respectively.
Our main contributions are summarized as follows:
\begin{itemize}
\item propose a learning-based generative model for camera sensor noise
\item achieve camera awareness by leveraging camera-specific Poisson-Gaussian noise and a camera characteristics encoding network
\item design a novel feature matching loss for signal-dependent patterns, which leads to significant improvement of visual quality
\item outperform state-of-the-art noise modeling methods and improve image denoising performance
\end{itemize}
\section{Related Work} \label{sec:related}
Image denoising is one of the most important applications and benchmarks in noise modeling.
Similar to the recent success of deep learning in many vision tasks, deep neural networks also dominate recent advances of image denoising.
DnCNN~\cite{DnCNN} shows that a residual neural network can perform blind denoising well and obtains better results than previous methods on additive white Gaussian noise (AWGN).
However, a recent denoising benchmark DND~\cite{DND}, consisting of real photographs, found that the classic BM3D method~\cite{BM3D} outperforms DnCNN on real-world noise instead.
The main reason is that real-world noise is more complicated than AWGN, and DnCNN failed to generalize to real-world noise because it was trained only with AWGN.
Instead of AWGN, CBDNet~\cite{CBDNet} and Brooks \emph{et al}\onedot~\cite{Unprocessing} adopt Poisson-Gaussian noise and demonstrate significant improvement on the DND benchmark.
Actually, they adopt an approximated version of Poisson-Gaussian noise by a heteroscedastic Gaussian distribution:
\begin{align} \label{eq:pg}
n \sim \mathcal{N}(0, \delta_{\mathrm{shot}} I + \delta_{\mathrm{read}}) \,,
\end{align}
where $n$ is the noise sampling, $I$ is the intensity of a noise-free image, and $\delta_{\mathrm{shot}}$ and $\delta_{\mathrm{read}}$ denote the Poisson and Gaussian components, respectively.
Moreover, $\delta_{\mathrm{shot}}$ and $\delta_{\mathrm{read}}$ for a specific camera sensor can be obtained via a calibration process~\cite{NLF}.
The physical meaning of these two components corresponds to the signal-dependent and signal-independent noise of a specific camera sensor.
Recently, several learning-based noise modeling approaches have been proposed~\cite{NoiseFlow,GCBD,GRDN}.
GCBD~\cite{GCBD} is the first GAN-based noise modeling method.
Its generative noise model, however, takes only a random vector as input but does not take the intensity of the clean image into account.
That means the generated noise is not signal-dependent.
Different characteristics between different camera sensors are not considered either.
The synthetic noise is learned and imposed on sRGB images, rather than the raw images.
These are the reasons why GCBD didn't deliver promising denoising performance on the DND benchmark~\cite{DND}.
GRDN~\cite{GRDN} is another GAN-based noise modeling method.
Their model was trained with paired data of clean images and real noisy images of smartphone cameras, provided by NTIRE 2019 Real Image Denoising Challenge~\cite{NTIRE2019}, which is a subset of the SIDD benchmark~\cite{SIDD}.
In addition to a random seed, the input of the generative noise model also contained many conditioning signals: the noise-free image, an identifier indicating the camera sensor, ISO level, and shutter speed.
Although GRDN can generate signal-dependent and camera-aware noise, the denoising network trained with this generative noise model only improved slightly.
The potential reasons are two-fold: synthetic noise was learned and imposed on sRGB images, not raw images; a plain camera identifier is too simple to represent noise characteristics of different camera sensors.
Noise Flow~\cite{NoiseFlow} applied a flow-based generative model that maximizes the likelihood of real noise on raw images, and then exactly evaluated the noise modeling performance qualitatively and quantitatively.
To do so, the authors proposed using Kullback-Leibler divergence and negative log-likelihood as the evaluation metrics.
Both training and evaluation were conducted on SIDD~\cite{SIDD}.
To our knowledge, Noise Flow is the first deep learning-based method that demonstrates significant improvement in both noise modeling and image denoising capabilities.
However, they also fed only a camera identifier into a gain layer to represent complex noise characteristics of different camera sensors.
\begin{figure}[ht!]
\footnotesize
\centering
\includegraphics[width=\linewidth]{fig/net.pdf}
\caption{\textbf{An overview of our noise-modeling framework.} The proposed architecture comprises two sub-networks: the Noise-Generating Network and the Camera-Encoding Network. First, a clean image \cleanImg{s}{i} and the initial synthetic noise $\tilde{\mathbf{n}}_\mathrm{init}$ sampled from Poisson-Gaussian noise model are fed into the generator $G$. In addition, a latent vector $\emph{\textbf{v}}$ provided by the camera encoder $E$, which represents the camera characteristics, is concatenated with the features of the middle layers of $G$. Eventually, the final synthetic noise ${\tilde{\mathbf{n}}}$ is generated by $G$. To jointly train $G$ and $E$, a discriminator $D$ is introduced for the adversarial loss $L_\mathrm{Adv}$ and the feature matching loss $L_\mathrm{FM}$. Moreover, a triplet loss $L_\mathrm{Triplet}$ is proposed to let the latent space of $\emph{\textbf{v}}$ be more reliable}
\label{fig:net}
\end{figure}
\section{Proposed Method} \label{s:method}
Different from most existing works, the proposed learning-based approach aims to model noise characteristics for each camera sensor.
Fig.~\ref{fig:net} shows an overview of our framework, which comprises two parts: the Noise-Generating Network and the Camera-Encoding Network.
The Noise-Generating Network, introduced in Sec.~\ref{subs:noise-generating-network}, learns to generate synthetic noise according to the content of a clean input image and the characteristics of a target camera.
The target camera characteristics are extracted via the Camera-Encoding Network from noisy images captured by that target camera, which is illustrated in Sec.~\ref{subs:camera-encoding-network}.
Finally, Sec.~\ref{subs:loss} shows how to train these two networks in an end-to-end scheme.
\subsection{Noise-Generating Network} \label{subs:noise-generating-network}
As depicted in the upper part of Fig.~\ref{fig:net}, a clean image \cleanImg{s}{i} from the $s^\mathrm{th}$ camera and the initial synthetic noise $\tilde{\mathbf{n}}_\mathrm{init}$ are fed into a noise generator $G$ and then transformed into various feature representations through convolutional layers. At the last layer, the network produces a residual image $R(\tilde{\mathbf{n}}_\mathrm{init} | \cleanImg{s}{i})$ that approximates the difference between real noise $\mathbf{n} \sim \mathbb{P}_{r}$ and $\tilde{\mathbf{n}}_\mathrm{init}$, where $\mathbb{P}_{r}$ indicates the real noise distribution. Ideally, we can generate realistic synthetic noise $\tilde{\mathbf{n}} \approx \mathbf{n}$ from the estimated residual image as
\begin{align} \label{eq:g_output}
\tilde{\mathbf{n}} = G(\tilde{\mathbf{n}}_\mathrm{init} | \cleanImg{s}{i}) = \tilde{\mathbf{n}}_\mathrm{init} + R(\tilde{\mathbf{n}}_\mathrm{init} | \cleanImg{s}{i})\,.
\end{align}
To achieve this objective, we adopt adversarial learning for making the generated noise distribution $\mathbb{P}_{g}$ fit $\mathbb{P}_{r}$ as closely as possible. A discriminator $D$ is used to measure the distance between distributions by distinguishing real samples from fake ones, such that $G$ can minimize the distance through an adversarial loss $L_\mathrm{Adv}$. Therefore, we need to collect pairs of clean images and real noise $(\cleanImg{s}{i}, \realNoise{s}{i})$ as the real samples.
A real noise sample $\realNoise{s}{i}$ can be acquired by subtracting $\cleanImg{s}{i}$ from the corresponding noisy image $\noisyImg{s}{i}$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\realNoise{s}{i} = \noisyImg{s}{i} - \cleanImg{s}{i}$. Note that a clean image could have many corresponding noisy images because noisy images can be captured at different ISOs to cover a wide range of noise levels. For simplicity, we let $i$ denote not only the scene but also the shooting settings of a noisy image.
In addition to measuring the distance in adversarial learning, the discriminator $D$ also plays another role in our framework. It is observed that some signal-dependent patterns like spots or stripes are common in real noise; hence we propose a feature matching loss $L_\mathrm{FM}$ and treat $D$ as a feature extractor. The feature matching loss forces the generated noise $\tilde{\mathbf{n}}$ and the clean image $\cleanImg{s}{i}$ to share similar high-level features because we assume these signal-dependent patterns should be the most salient traits in clean images.
It is worthwhile to mention that a noise model should be capable of generating a variety of reasonable noise samples for the same input image and noise level. GANs usually take a random vector sampled from Gaussian distribution as the input of the generator to ensure this stochastic property. In most cases, this random vector is not directly relevant to the main task. However, our goal is exactly to generate random noise, which implies that this random vector could be treated as the initial synthetic noise. Moreover, Gaussian distribution can be replaced with a more representative statistical noise model. For this reason, we apply Poisson-Gaussian noise model to the initial synthetic noise $\tilde{\mathbf{n}}_\mathrm{init}$ as in (\ref{eq:pg}):
\begin{align} \label{eq:npg}
\tilde{\mathbf{n}}_{\text{init}} \sim \mathcal{N}(0, {\delta_\mathrm{shot}}_i^s \cleanImg{s}{i} + {{\delta_\mathrm{read}}_i^s}) \,,
\end{align}
where ${\delta_\mathrm{shot}}_i^s$ and ${\delta_\mathrm{read}}_i^s$ are the Poisson and the Gaussian component for $\noisyImg{s}{i}$, respectively. Note that these two parameters not only describe the preliminary noise model for the $s^\mathrm{th}$ camera but also control the noise level of $\tilde{\mathbf{n}}_\mathrm{init}$ and $\tilde{\mathbf{n}}$.
\subsection{Camera-Encoding Network} \label {subs:camera-encoding-network}
FUNIT~\cite{funit} has shown that encoding the class information is helpful to specify the class domain for an input image. Inspired by their work, we would like to encode the camera characteristics in an effective representation. Since ${\delta_\mathrm{shot}}_i^s$ and ${\delta_\mathrm{read}}_i^s$ are related to the $s^\mathrm{th}$ camera in (\ref{eq:npg}), the generator $G$ is actually aware of the camera characteristics from $\tilde{\mathbf{n}}_\mathrm{init}$. However, this awareness is limited to the assumption of the Poisson-Gaussian noise model. We, therefore, propose a novel Camera-Encoding Network to overcome this problem.
As depicted in the lower part of Fig.~\ref{fig:net}, a noisy image $\noisyImg{s}{j}$ is fed into a camera encoder $E$ and then transformed into a latent vector $\emph{\textbf{v}} = E(\noisyImg{s}{j})$. After that, the latent vector $\emph{\textbf{v}}$ is concatenated with the middle layers of $G$. Thus, the final synthetic noise is rewritten as
\begin{align} \label{eq:g_output2}
\tilde{\mathbf{n}} = G(\tilde{\mathbf{n}}_\mathrm{init} | \cleanImg{s}{i}, \emph{\textbf{v}}) = \tilde{\mathbf{n}}_\mathrm{init} + R(\tilde{\mathbf{n}}_\mathrm{init} | \cleanImg{s}{i}, \emph{\textbf{v}}) \,.
\end{align}
We consider $\emph{\textbf{v}}$ as a representation for the characteristics of the $s^\mathrm{th}$ camera and expect $G$ can generate more realistic noise with this latent vector.
Aiming at this goal, the camera encoder $E$ must have the ability to extract the core information for each camera, regardless of the content of input images. Therefore, a subtle but important detail here is that we feed the $j^\mathrm{th}$ noisy image rather than the $i^\mathrm{th}$ noisy image into $E$, whereas $G$ takes the $i^\mathrm{th}$ clean image as its input. Specifically, the $j^\mathrm{th}$ noisy image is randomly selected from the data of the $s^\mathrm{th}$ camera. Consequently, $E$ has to provide latent vectors beneficial to the generated noise but ignoring the content of input images.
Additionally, some regularization should be imposed on $\emph{\textbf{v}}$ to make the latent space more reliable. FUNIT calculates the mean over a set of class images to provide a representative class code. Nevertheless, this approach assumes that the latent space consists of hypersphere manifolds. Apart from FUNIT, we use a triplet loss $L_\mathrm{Triplet}$ as the regularization. The triplet loss is used to minimize the intra-camera distances while maximizing the inter-camera distances, which allows the latent space to be more robust to image content. The detailed formulation will be shown in the next section.
One more thing worth clarifying is why the latent vector $\emph{\textbf{v}}$ is extracted from the noisy image $\noisyImg{s}{j}$ rather than the real noise sample $\realNoise{s}{j}$. The reason is out of consideration for making data preparation easier in the inference phase, which is shown as the violet block in Fig.~\ref{fig:net}. Collecting paired data $(\cleanImg{s}{j}, \noisyImg{s}{j})$ to acquire $\realNoise{s}{j}$ is cumbersome and time-consuming in real world. With directly using noisy images to extract latent vectors, there is no need to prepare a large number of paired data during the inference phase.
\subsection{Learning} \label{subs:loss}
To jointly train the aforementioned networks, we have briefly introduced three loss functions: 1) the adversarial loss $L_\mathrm{Adv}$, 2) the feature matching loss $L_\mathrm{FM}$, and 3) the triplet loss $L_\mathrm{Triplet}$. In this section, we describe the formulations for these loss functions in detail.
\subsubsection{Adversarial Loss.}
GANs are well-known for reducing the divergence between the generated data distribution and real data distribution in the high-dimensional image space. However, there are several GAN frameworks for achieving this goal. Among these frameworks, we choose WGAN-GP~\cite{WGAN-GP} to calculate the adversarial loss $L_\mathrm{Adv}$, which minimizes Wasserstein distance for stabilizing the training. The $L_\mathrm{Adv}$ is thus defined as
\begin{align} \label{eq:advG}
L_\mathrm{Adv} = - \underset{\tilde{\mathbf{n}} \sim \mathbb{P}_{g}} {\mathbb{E}}[D(\tilde{\mathbf{n}} | \mathbf{I}_C)]\,,
\end{align}
where $D$ scores the realness of the generated noise. In more depth, scores are given at the scale of patches rather than whole images because we apply a PatchGAN~\cite{PatchGAN} architecture to $D$. The advantage of using this architecture is that it prefers to capture high-frequency information, which is associated with the characteristics of noise.
On the other hand, the discriminator $D$ is trained by
\begin{align} \label{eq:advD}
L_{D} = \underset{\tilde{\mathbf{n}} \sim \mathbb{P}_{g}} {\mathbb{E}}[D(\tilde{\mathbf{n}} | \mathbf{I}_C)] - \underset{\mathbf{n} \sim \mathbb{P}_{r}}{\mathbb{E}}[D(\mathbf{n} | \mathbf{I}_C)] + \lambda_\mathrm{gp} \underset{\hat{\mathbf{n}} \sim \mathbb{P}_{\hat{\mathbf{n}}}} {\mathbb{E}}[(\|\nabla_{\hat{\mathbf{n}}}D(\hat{\mathbf{n}} | \mathbf{I}_C) \|_{2}-1)^{2}]\,,
\end{align}
where $\lambda_\mathrm{gp}$ is the weight of gradient penalty, and $\mathbb{P}_{\hat{\mathbf{n}}}$ is the distribution sampling uniformly along straight lines between paired points sampled from $\mathbb{P}_{g}$ and $\mathbb{P}_{r}$.
\subsubsection{Feature Matching Loss.}
In order to regularize the training for GANs, some works \cite{funit,FMLoss} apply the feature matching loss and extract features through the discriminator networks. Following these works, we propose a feature matching loss $L_\mathrm{FM}$ to encourage $G$ to generate signal-dependent patterns in synthetic noise. The $L_\mathrm{FM}$ is then calculated as
\begin{align} \label{eq:fm}
L_\mathrm{FM} = \underset{\tilde{\mathbf{n}} \sim \mathbb{P}_{g}} {\mathbb{E}}[\| D_{f}(\tilde{\mathbf{n}} | \mathbf{I}_C) - D_{f}(\mathbf{I}_C | \mathbf{I}_C) \|_{1} ]\,,
\end{align}
where $D_{f}$ denotes the feature extractor constructed by removing the last layer from $D$. Note that $D_f$ is not optimized by $L_\mathrm{FM}$.
\subsubsection{Triplet Loss.}
The triplet loss was first proposed to illustrate the triplet relation in embedding space by~\cite{TripletLoss}. We use the triplet loss to let the latent vector $\emph{\textbf{v}} = E(\noisyImg{s}{j})$ be more robust to the content of noisy image. Here we define the positive term $\emph{\textbf{v}}^{+}$ as the latent vector also extracted from the $s^\mathrm{th}$ camera, and the negative term $\emph{\textbf{v}}^{-}$ is from a different camera on the contrary. In particular, $\emph{\textbf{v}}^{+}$ and $\emph{\textbf{v}}^{-}$ are obtained by encoding the randomly selected noisy images $\noisyImg{s}{k}$ and $\noisyImg{t}{l}$, respectively. Note that $\noisyImg{s}{k}$ is not restricted to any shooting setting, which means the images captured with different shooting settings of the same camera are treated as positive samples. The objective is to minimize the intra-camera distances while maximizing the inter-camera distances. The triplet loss $L_{\text{Triplet}}$ is thus given by
\begin{align} \label{eq:camera}
L_\mathrm{Triplet} = \underset{\emph{\textbf{v}}, \emph{\textbf{v}}^{+}, \emph{\textbf{v}}^{-} \sim \mathbb{P}_{e}}
{\mathbb{E}} \left[\max(0, \left\lVert \emph{\textbf{v}} - \emph{\textbf{v}}^{+} \right\rVert_2 - \left\lVert \emph{\textbf{v}} - \emph{\textbf{v}}^{-} \right\rVert_2 + \alpha) \right] \,,
\end{align}
where $\mathbb{P}_e$ is the latent space distribution and $\alpha$ is the margin between positive and negative pairs.
\subsubsection{Full Loss.}
The full objective of the generator $G$ is combined as
\begin{align} \label{eq:fullG}
L_\mathrm{G} = L_\mathrm{Adv} + \lambda_\mathrm{FM} L_\mathrm{FM} + \lambda_\mathrm{Triplet} L_\mathrm{Triplet} \,,
\end{align}
where $\lambda_\mathrm{FM}$ and $\lambda_\mathrm{Triplet}$ control the relative importance for each loss term.
\section{Experimental Results} \label{s:results}
In this section, we first describe our experiment settings and the implementation details.
Then, Sec.~\ref{subs:results:nm} shows the quantitative and qualitative results.
Sec.~\ref{subs:results:ablation} presents extensive ablation studies to justify our design choices. The effectiveness and robustness of the Camera-Encoding Network are evaluated in Sec.~\ref{subs:results:camera:encoding}.
\subsubsection{Dataset.}
We train and evaluate our method on Smartphone Image Denoising Dataset (SIDD)~\cite{SIDD},
which consists of approximately 24{,}000 pairs of real noisy-clean images.
The images are captured by five different smartphone cameras: Google Pixel, iPhone 7, Samsung Galaxy S6 Edge, Motorola Nexus 6, and LG G4.
These images are taken in ten different scenes and under a variety of lighting conditions and ISOs.
SIDD is currently the most abundant dataset available for real noisy and clean image pairs.
\subsubsection{Implementation Details.}
We apply Bayer preserving augmentation~\cite{BayerAug} to all SIDD images, including random cropping and horizontal flipping.
At both training and testing phases, the images are cropped into $64\times64$ patches.
Totally 650{,}000 pairs of noisy-clean patches are generated.
Then we randomly select 500{,}000 pairs as the training set and 150{,}000 pairs as the test set.
The scenes in the training set and test set are mutually exclusive to prevent overfitting. Specifically, the scene indices of the test set are 001, 002 and 008, and the remaining indices are used for the training set.
To synthesize the initial synthetic noise $\tilde{\mathbf{n}}_\mathrm{init}$, we set the Poisson component ${\delta_{\mathrm{shot}}}_i^s$ and Gaussian component ${\delta_{\mathrm{read}}}_i^s$ in (\ref{eq:npg}) to the values provided by SIDD, which are estimated using the method proposed by~\cite{NLF}.
The weight of gradient penalty of $L_D$ in (\ref{eq:advD}) is set to $\lambda_\mathrm{gp} = 10$, and the margin of $L_{\mathrm{Triplet}}$ in (\ref{eq:camera}) is set to $\alpha = 0.2$.
The loss weights of $L_G$ in (\ref{eq:fullG}) are set to $\lambda_\mathrm{FM} = 1$ and $\lambda_\mathrm{Triplet} = 0.5$.
We use the Adam optimizer~\cite{kingma2014adam} in all of our experiments, with an initial learning rate of 0.0002, $\beta_{1} = 0.5$, and $\beta_{2} = 0.999$.
Each training batch contains 64 pairs of noisy-clean patches. The generator $G$, discriminator $D$, and camera encoder $E$ are jointly trained to convergence with 300 epochs. It takes about 3 days on a single GeForce GTX 1080 Ti GPU.
All of our experiments are conducted on linear raw images.
Previous works have shown that many image processing methods perform better in Bayer RAW domain than in sRGB domain~\cite{DND}.
For noise modeling or image denoising, avoiding non-linear transforms (such as gamma correction) or spatial operations (such as demosaicking) is beneficial because we can prevent noise characteristics from being dramatically changed by these operations.
\subsubsection{Methods in Comparison.}
We compare our method with two mostly-used statistical models: Gaussian noise model and Poisson-Gaussian noise model, and one state-of-the-art learning-based method: Noise Flow~\cite{NoiseFlow}.
\subsection{Quantitative and Qualitative Results} \label{subs:results:nm}
To perform the quantitative comparison, we adopt the Kullback-Leibler divergence ($D_\mathrm{KL}$) as suggested in~\cite{NoiseFlow}.
Table~\ref{table:kld} shows the average $D_\mathrm{KL}$ between real noise and synthetic noise generated by different noise models.
Our method achieves the smallest average Kullback-Leibler divergence, which means that our method can synthesize more realistic noise than existing methods.
\setlength{\tabcolsep}{4pt}
\begin{table}[tb]
\footnotesize
\begin{center}
\caption{\textbf{Quantitative evaluation of different noise models.} Our proposed noise model yields the best Kullback-Leibler divergence ($D_\mathrm{KL}$). Relative improvements of our method over other baselines are shown in parentheses}
\label{table:kld}
\begin{tabular}{c|cccc}
\toprule
& Gaussian & Poisson-Gaussian & Noise Flow & Ours \\
\midrule
$D_\mathrm{KL}$ & 0.54707 (99.5\%) & 0.01006 (74.7\%) & 0.00912 (72.0\%) & \pmb{0.00159} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
Fig.~\ref{fig:solvepg} shows the synthetic noise generated in linear RAW domain by all noise models and then processed by the camera pipeline toolbox provided by SIDD~\cite{SIDD}.
Each two consecutive rows represent an image sample for different ISOs (indicated by a number) and different lighting conditions (L and N denote low and normal lighting conditions respectively).
Our method can indeed generate synthetic noise that is more realistic and perceptually closer to the real noise.
\def21{02_0100_693}
\def3{08_0100_106}
\def1240{180_0100_394}
\def52_1600_701{52_1600_701}
\def14_3200_631{14_3200_631}
\setlength{\tabcolsep}{1.4pt}
\begin{figure}[ht!]
\tiny
\centering
\begin{tabular}{ccccccc}
Clean & & Gaussian & \makebox[2em][c]{Poisson-Gauss.} & Noise Flow & Ours & Real Noise \\
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_0.jpg} &
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noisy}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_6.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_7.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_10.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_8.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_9.jpg}
\\
&
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noise}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_1.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_2.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_3.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_4.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/21_5.jpg}
\\
\multicolumn{7}{c}{(a) 100-L} \\
%
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_0.jpg} &
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noisy}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_6.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_7.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_10.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_8.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_9.jpg}
\\
&
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noise}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_1.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_2.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_3.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_4.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/3_5.jpg}
\\
\multicolumn{7}{c}{(b) 100-N}
\\
%
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_0.jpg} &
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noisy}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_6.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_7.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_10.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_8.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_9.jpg}
\\
&
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noise}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_1.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_2.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_3.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_4.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/1240_5.jpg}
\\
\multicolumn{7}{c}{(c) 100-N}
\\
%
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_0.jpg} &
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noisy}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_6.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_7.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_10.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_8.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_9.jpg}
\\
&
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noise}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_1.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_2.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_3.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_4.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/52_1600_701_5.jpg}
\\
\multicolumn{7}{c}{(d) 1600-L}
\\
%
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_0.jpg} &
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noisy}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_6.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_7.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_10.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_8.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_9.jpg}
\\
&
\raisebox{3.0\normalbaselineskip}[0pt][0pt]{\rotatebox[origin=c]{90}{Noise}} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_1.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_2.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_3.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_4.jpg} &
\includegraphics[width=\figwidth\columnwidth]{fig/nm/14_3200_631_5.jpg}
\\
\multicolumn{7}{c}{(e) 3200-N}
\\
\end{tabular}
\caption{\textbf{Visualization of noise models.}
The synthetic noise samples of several noise modeling methods on different clean images with different ISO/lighting conditions are illustrated
%
Quantitatively, the proposed method outperforms others in terms of $D_\mathrm{KL}$ measurement with real noise distribution.
Furthermore, ours have many clear structures that fit the texture of clean images and real noise.
%
Note that the noise value is scaled up for better visualization purpose
}
\label{fig:solvepg}
\end{figure}
\subsection{Ablation Studies} \label{subs:results:ablation}
In this section, we perform ablation studies to investigate how each component contributes to our method, including the feature matching loss $L_\mathrm{FM}$, the Camera-Encoding Network $E$, the triplet loss $L_\mathrm{Triplet}$, and the initial synthetic noise $\tilde{\mathbf{n}}_\mathrm{init}$.
The results are shown in Table~\ref{table:ablation} and~\ref{table:ablation_n_init}.
\setlength{\tabcolsep}{4pt}
\begin{table}[tb]
\footnotesize
\begin{center}
\caption{\textbf{Ablation study of our model.} $L_\mathrm{adv}$: the adversarial loss, $L_\mathrm{FM}$: the feature matching loss, $E$: the Camera-Encoding Network, $L_\mathrm{Triplet}$: the triplet loss.
The Kullback-Leibler divergence $D_\mathrm{KL}$ is measured in different settings}
\label{table:ablation}
\begin{tabular}{l|cccc}
\toprule
$L_\mathrm{Adv}$ & $\surd$ & $\surd$ & $\surd$ & $\surd$ \\
$L_\mathrm{FM}$ & & $\surd$ & $\surd$ & $\surd$ \\
$E$ & & & $\surd$ & $\surd$ \\
$L_\mathrm{Triplet}$ & & & & $\surd$ \\
\midrule
$D_\mathrm{KL}$ & 0.01445 & 0.01374 & 0.01412 & \pmb{0.00159} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsubsection{Feature Matching Loss $L_\mathrm{FM}$.}
Fig.~\ref{fig:FM} shows that the feature matching loss is effective in synthesizing signal-dependent noise patterns and achieving better visual quality.
With the feature matching loss $L_\mathrm{FM}$, the network is more capable of capturing low-frequency signal-dependent patterns.
As shown in Table~\ref{table:ablation}, the Kullback-Leibler divergence can also be improved from 0.01445 to 0.01374.
\begin{figure}[ht!]
\footnotesize
\centering
\footnotesize
\renewcommand{\tabcolsep}{1pt}
\renewcommand{\arraystretch}{1}
\begin{tabular}{cccc}
w/o $L_\mathrm{FM}$ & with $L_\mathrm{FM}$ & Real Noise & Clean \\
\includegraphics[width=\figwidthFM\columnwidth]{fig/fm/2_1.jpg} &
\includegraphics[width=\figwidthFM\columnwidth]{fig/fm/2_2.jpg} &
\includegraphics[width=\figwidthFM\columnwidth]{fig/fm/2_3.jpg} &
\includegraphics[width=\figwidthFM\columnwidth]{fig/fm/2_4.jpg} \\
\end{tabular}
\caption{\textbf{Visualization of synthetic noise with and without feature matching loss.}
With the feature matching loss $L_\mathrm{FM}$, the generated noise is highly correlated to the image content.
%
Hence, they have more distinct structures resembling the texture of clean images.
%
Besides, the $D_\mathrm{KL}$ measurements are slightly improved
}
\label{fig:FM}
\end{figure}
\subsubsection{Camera-Encoding Network and Triplet Loss.}
The Camera-Encoding Network is designed to represent the noise characteristics of different camera sensors.
However, simply adding a Camera-Encoding Network alone provides no advantage (0.01374 $\rightarrow$ 0.01412), as shown in Table~\ref{table:ablation}.
The triplet loss is essential to learn effective camera-specific latent vectors, and the KL divergence can be significantly reduced from 0.01412 to 0.00159.
The camera-specific latent vectors can also be visualized in the $t$-SNE space~\cite{maaten2008visualizing}.
As shown in Fig.~\ref{fig:tSNE}, the Camera-Encoding Network can effectively extract camera-specific latent vectors from a single noisy image with the triplet loss.
\begin{figure}[ht!]
\footnotesize
\centering
\footnotesize
\renewcommand{\tabcolsep}{1pt}
\renewcommand{\arraystretch}{1}
\begin{tabular}{cc}
w/o $L_\mathrm{Triplet}$ & with $L_\mathrm{Triplet}$\\
\includegraphics[width=\figwidthtSNE\columnwidth]{fig/notrip.pdf} &
\includegraphics[width=\figwidthtSNE\columnwidth]{fig/trip.pdf} \\
\end{tabular}
\caption{\textbf{Ablation study on the distributions of latent vectors from a camera encoder trained with or without $L_\mathrm{Triplet}$.} We project the encoded latent vectors $\emph{\textbf{v}}$ of noisy images from five different cameras with $t$-SNE.
%
The Camera-Encoding Network trained with $L_\mathrm{Triplet}$ can effectively group the characteristics of different cameras
}
\label{fig:tSNE}
\end{figure}
\subsubsection{Initial Synthetic Noise $\tilde{\mathbf{n}}_{\text{init}}$.}
Table~\ref{table:ablation_n_init} shows the average KL divergence when using Gaussian or Poisson-Gaussian noise as the initial noise $\tilde{\mathbf{n}}_{\text{init}}$.
The KL divergence severely degrades from 0.00159 to 0.06265 if we use Gaussian noise instead of Poisson-Gaussian noise.
This result shows that using a better synthetic noise as initial and predicting a residual to refine it can yield better-synthesized noise.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\footnotesize
\begin{center}
\caption{\textbf{Ablation study of the initial synthetic noise $\tilde{\mathbf{n}}_{\text{init}}$} Using Poisson-Gaussian as initial synthetic noise model performs better than using Gaussian}
\label{table:ablation_n_init}
\begin{tabular}{l|cc}
\toprule
$\tilde{\mathbf{n}}_{\text{init}}$ & Gaussian & Poisson-Gaussian \\
\midrule
$D_\mathrm{KL}$ & 0.06265 & \pmb{0.00159} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsection{Robustness Analysis of the Camera-Encoding Network}\label{subs:results:camera:encoding}
To further verify the behavior and justify the robustness of the Camera-Encoding Network, we design several experiments with different input conditions.
\subsubsection{Comparing Noise for Different Imaging Conditions or Different Cameras.}
Given a clean image $\cleanImg{s}{i}$ and a noisy image $\noisyImg{s}{j}$ also from the $s^{\text{th}}$ camera,
our noise model should generate noise $\tilde{\mathbf{n}}_{A} = G(\tilde{\mathbf{n}}_{\text{init}} | \cleanImg{s}{i}, E(\noisyImg{s}{j}))$. The Kullback-Leibler divergence $D_{\text{KL}}(\tilde{\mathbf{n}}_{A}\|\mathbf{n}_i^s)$ between the generated noise and the corresponding real noise should be very small (0.00159 in Table~\ref{table:ablation_ce}). On the other hand, $D_{\text{KL}}(\tilde{\mathbf{n}}_{A}\|\mathbf{n}_j^s)$ between the generated noise and a non-corresponding real noise should be quite large (0.17921 in Table~\ref{table:ablation_ce}), owing to the different imaging conditions, even though the real noise $\mathbf{n}_j^s$ is from the same $s^{\text{th}}$ camera.
If the latent vector is extracted by a noisy image of the $t^{\text{th}}$ camera instead of the $s^{\text{th}}$ camera, the generated noise becomes $\tilde{\mathbf{n}}_{B} = G(\tilde{\mathbf{n}}_{\text{init}} | \cleanImg{s}{i}, E(\noisyImg{t}{k}))$. Because the latent vector is from a different camera, we expect that $D_{\text{KL}}(\tilde{\mathbf{n}}_{A}\|\mathbf{n}_i^s) < D_{\text{KL}}(\tilde{\mathbf{n}}_{B}\|\mathbf{n}_i^s)$. Table~\ref{table:ablation_ce} also verifies these results.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\footnotesize
\begin{center}
\caption{\textbf{Analysis of noisy images from different cameras.} The comparison of Kullback-Leibler divergence for different cameras of the noisy image, where $\tilde{\mathbf{n}}_{A} = G(\tilde{\mathbf{n}}_{\text{init}} | \cleanImg{s}{i}, E(\noisyImg{s}{j}))$ and $\tilde{\mathbf{n}}_{B} = G(\tilde{\mathbf{n}}_{\text{init}} | \cleanImg{s}{i}, E(\noisyImg{t}{k}))$
}
\label{table:ablation_ce}
\begin{tabular}{c|ccc}
\toprule
& $(\tilde{\mathbf{n}}_{A}\|\mathbf{n}_i^s)$ & $(\tilde{\mathbf{n}}_{A}\|\mathbf{n}_j^s)$ & $(\tilde{\mathbf{n}}_{B}\|\mathbf{n}_i^s)$ \\
\midrule
$D_{\text{KL}}$ & 0.00159 & 0.17921 & 0.01324 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsubsection{Analysis of Different Noisy Images from the Same Camera.}
Another important property of the Camera-Encoding Network is that it must capture camera-specific characteristics from a noisy image, and the extracted latent vector should be irrelevant to the image content of the input noisy image.
To verify this, we randomly select five different noisy images from the same camera.
These different noisy images are fed into the Camera-Encoding Network, while other inputs for the Noise Generating Network are kept fixed.
Because these noisy images are from the same camera, the generated noise should be robust and consistent.
Table~\ref{table:noisydiff} shows that the $D_\mathrm{KL}$ between the generated noise and real noise remains low for different noisy images.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\footnotesize
\begin{center}
\caption{\textbf{Analysis of different noisy images from the same camera.} The Kullback-Leibler divergence results from five randomly selected noisy images but fixed inputs for the generator}
\label{table:noisydiff}
\begin{tabular}{c|ccccc}
\toprule
Noisy image sets & $1^\mathrm{st}$ & $2^\mathrm{nd}$ & $3^\mathrm{rd}$ & $4^\mathrm{th}$ & $5^\mathrm{th}$ \\
\midrule
$D_{\text{KL}}$ & 0.00159 & 0.00180 & 0.00183 & 0.00163 & 0.00176 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\section{Application to Real Image Denoising} \label{subs:results:nr}
\subsection{Real-world Image Denoising}
We conduct real-world denoising experiments to further compare different noise models.
For all noise models, we follow Noise Flow~\cite{NoiseFlow} to use the same 9-layer DnCNN network~\cite{DnCNN} as the baseline denoiser.
Learning-based noise models (Noise Flow and ours) are trained with SIDD dataset. We then train a denoiser network with synthetic training pairs generated by each noise model separately.
Table~\ref{table:nr} shows the average PSNR and SSIM~\cite{wang2004image} on the test set.
The denoisers trained with statistical noise models (Gaussian and Poisson-Gaussian) are worse than those trained with learning-based noise models (Noise Flow and Ours), which also outperform the denoiser trained with real data only (the last row of Table~\ref{table:nr}).
This is because the amount of synthetic data generated by noise models is unlimited, while the amount of real data is fixed.
Our noise model outperforms Noise Flow in terms of both PSNR and SSIM while using more training data for training noise models leads to better denoising performance.
Table~\ref{table:nr} also shows that using both real data and our noise model results in further improved PSNR and SSIM.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\footnotesize
\begin{center}
\caption{\textbf{Real-World image denoising.}
The denoising networks using our noise model outperform those using existing statistical noise models and learning-based models.
\textcolor{red}{\pmb{Red}} indicates the best and \textcolor{blue}{\underline{blue}} indicates the second best performance
(While training using both synthetic and real data, Ours + Real, synthetic and real data are sampled by a ratio of $5:1$ in each mini-batch) } \label{table:nr}
\begin{tabular}{l|cc|cc}
\toprule
& \# of training data &\# of \emph{real} training && \\
Noise model & for noise model & data for denoiser & PSNR & SSIM\\
\midrule
Gaussian & - & - & 43.63 & 0.968 \\
\midrule
Poisson-Gaussian & - & - & 44.99 & 0.982 \\
\midrule
\multirow{2}{*}{Noise Flow~\cite{NoiseFlow}} & 100k & - & 47.49 & 0.991 \\
& 500k & - & 48.52 & 0.992 \\
\midrule
\multirow{2}{*}{Ours} & 100k & - & 47.97 & 0.992 \\
& 500k & - & \textcolor{blue}{\underline{48.71}} & \textcolor{blue}{\underline{0.993}} \\
\midrule
\multirow{2}{*}{Ours + Real} & 100k & 100k & 47.93 & \textcolor{red}{\pmb{0.994}} \\
& 500k & 500k & \textcolor{red}{\pmb{48.72}} & \textcolor{red}{\pmb{0.994}} \\
\midrule
\multirow{2}{*}{Real only} & - & 100k & 47.08 & 0.989 \\
& - & 500k & 48.30 & \textcolor{red}{\pmb{0.994}} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\subsection{Camera-Specific Denoising Networks}
To verify the camera-aware ability of our method, we train denoiser networks with our generative noise models, which are trained with and without the Camera-Encoding Network (and with and without the triplet loss) respectively.
For our noise model without the Camera-Encoding Network, we train a single generic denoiser network for all cameras.
For our noise models with the Camera-Encoding Network, we train camera-specific denoiser networks with and without the triplet loss for each camera.
The denoising performance is shown in Table~\ref{table:cameradiff}.
The results show that the Camera-Encoding Network with the triplet loss can successfully capture camera-specific noise characteristics and thus enhance the performance of camera-specific denoiser networks.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\footnotesize
\begin{center}
\caption{\textbf{Real-world image denoising using camera-aware noise model.} Grouping the proposed Camera-Encoding Network and triplet loss $L_{\text{Triplet}}$ can extract camera-specific latent vectors and thus improve camera-specific denoiser networks}
\label{table:cameradiff}
\begin{tabular}{l|ccccc}
\toprule
& \multicolumn{5}{c}{PSNR on test cameras} \\
Model & IP & GP & S6 & N6 & G4 \\
\midrule
w/o $(E+L_{\text{Triplet}})$ & 57.4672 & 44.5180 & 40.0183 & 44.7954 & 51.8048 \\
with $E$, w/o $L_{\text{Triplet}}$ & 49.8788 & 45.7755 & 40.4976 & 41.8447 & 51.8139 \\
with $(E+L_{\text{Triplet}})$ & \pmb{58.6073} & \pmb{45.9624} & \pmb{41.8881} & \pmb{46.4726} & \pmb{53.2610} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\section{Conclusions} \label{s:conclusion}
We have presented a novel learning-based generative method for real-world noise.
The proposed noise model outperforms existing statistical models and learning-based methods quantitatively and qualitatively.
Moreover, the proposed method can capture different characteristics of different camera sensors in a single noise model.
We have also demonstrated that the real-world image denoising task can benefit from our noise model.
As for future work, modeling real-world noise with few-shot or one-shot learning could be a possible direction. This could reduce the burden of collecting real data for training a learning-based noise model.
\def21{1014}
\def3{1073}
\def1240{1240}
\setlength{\tabcolsep}{1.4pt}
\begin{figure}[ht]
\centering
\tiny
\begin{tabular}{ccccccc}
Real Noisy & Poisson-Gauss. & Real Only & Noise Flow & Ours & Ours + Real & Ground Truth \\
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/21_2.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/21_4.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/21_5.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/21_6.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/21_7.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/21_8.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/21_1.jpg} \\
%
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/3_2.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/3_4.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/3_5.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/3_6.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/3_7.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/3_8.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/3_1.jpg} \\
%
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/1240_2.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/1240_4.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/1240_5.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/1240_6.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/1240_7.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/1240_8.jpg} &
\includegraphics[width=\figwidthnr\columnwidth]{fig/nr/1240_1.jpg} \\
\end{tabular}
\caption{\textbf{Results of denoisers trained on different noise models.} We compare the denoised results trained on different settings, 1) only real pairs, 2) synthetic pairs with Noise Flow or the proposed method, and 3) the mixture of synthetic pairs from ours and real pairs
}
\label{fig:nr}
\end{figure}
\clearpage
\bibliographystyle{splncs04}
|
2008.09252
|
\section{Introduction}
\label{sec:introduction}
\noindent Quantum computers are at the edge of pushing our computational capabilities beyond the boundaries set by classical machines \cite{arute2019quantum,boixo2018characterizing,lund2017quantum}. One of the fields where quantum computers are particularly promising is the simulation of gauge theories, which describe the interactions of elementary particles. While current numerical simulations led to several breakthroughs \cite{aoki2020flag, aoki2017review, PhysRevD.100.114501}, they are ultimately restricted in their predictive
capabilities. On one side, limitations originate from the inherent difficulty faced by classical computers
in simulating quantum properties. On the other, the sign problem \cite{Troyer:2004ge, Gattringer:2016kco} affects simulations of both equilibrium (e.g., phase diagrams), and non-equilibrium (e.g., real-time dynamics) physics. Therefore, classical simulations based on tensor networks \cite{Banuls:2018jag, Banuls:2019rao, Dalmonte_2016,Magnifico_2021} or Markov Chain Monte Carlo face hard numerical problems \cite{robaina2020simulating, Banuls:20198n}. Quantum simulations are a promising solution, with proof-of-concept demonstrations in 1D \footnote{Here and in the following, the considered dimensions are spatial only. As such, `1D' indicates a lattice in one spatial dimension, while `2D' a lattice in two spatial dimensions. Indeed, since we use a Hamiltonian formulation of LGTs, time is not discretized.} gauge theories \cite{Mil:2019pbt, martinez2016real, muschik2017u, klco2018quantum, klco20192, bauls2019simulating, kokail2019self, schweizer2019floquet, gorg2019realization, yang2020observation, lu2019simulations} already being achieved. Extending quantum simulations of fundamental particle interactions to higher spatial dimensions represents an enormous scientific opportunity to address open questions which lie beyond the capabilities of classical computations. However, the step from one to two (or higher) spatial dimensions is extremely challenging due to an inherent increase of complexity of the models. Moreover, current limitations of so-called Noisy Intermediate-Scale Quantum (NISQ) devices \cite{preskill2018quantum} make this leap even more difficult. In this work, we develop a protocol that ease the requirements on the quantum hardware and as a consequence allows for quantum simulations of gauge theories in higher spatial dimensions.\\
\begin{figure*}
\includegraphics[width=1.75\columnwidth]{Fig1.pdf}
\caption{
Blueprint for quantum simulations of lattice gauge theories. As described in Sec.~\ref{sec:main_results}, we use a VQE to simulate QED. The ingredients to perform the VQE are quantum circuits (Secs.~\ref{subsec:trappedIonResults_openBoundaryConditions} and \ref{subsec:trappedIonResults_periodicBoundaryConditions}), a classical optimizer (Sec.~\ref{sec:main_results}), and a measurement scheme to estimate the energy of a quantum state (App.~\ref{app:optimalPartitioning}). As described in Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding}, the VQE requires a Hamiltonian $\hat{H}_{\rm qubit}$ cast in terms of qubit operators. $\hat{H}_{\rm qubit}$ is optimized for ion trap quantum computers (see picture), but can be used with any platform. In turn, $\hat{H}_{\rm qubit}$ is obtained from an effective Hamiltonian $\hat{H}_{\rm eff}$, which is derived in Sec.~\ref{sec:encodedHamiltonian} from the results of Ref.~\cite{paper1}. As schematically indicated in the ``Hamiltonian formulation'' box, Ref.~\cite{paper1} uses the Kogut-Susskind \cite{kogut1975hamiltonian} formulation of 2D LGTs whose continuum limits corresponds to QED. The outcomes of simulated VQEs are presented in Secs.~\ref{subsec:trappedIonResults_openBoundaryConditions} and \ref{subsec:trappedIonResults_periodicBoundaryConditions}, while the corresponding analytical results in Sec.~\ref{sec:quantumSimulationOf2DEffects}. Images of the lattice and ion trap quantum computer are from Refs.~\cite{lattice_img} and \cite{kokail2019self}, respectively.
}
\label{fig:paperStructure}
\end{figure*}
\\Current quantum simulation protocols include analog, digital, and variational schemes \cite{bauls2019simulating, wiese2013ultracold,Ferguson2021}. Analog protocols \cite{Zohar_2013, kasper2016schwinger, zohar2012simulating, rico2018so, marcos2014two, hauke2013quantum, celi2020emerging, tagliacozzo2013simulation, davoudi2020towards,PhysRevD.95.094507, Kuno_2015, PhysRevLett.111.115303, PhysRevA.94.063641} aim at implementing the Hamiltonian of the simulated theory directly on quantum hardware. While this approach is challenged by the current practical difficulty of implementing gauge-invariance and the required many-body terms in the lab, first promising proof-of-concept demonstrations have been realized \cite{Mil:2019pbt, yang2020observation, schweizer2019floquet, gorg2019realization}. Noteworthy are approaches based on ultracold atoms, which allow for large system sizes and have the advantage that fermions can be used to simulate matter fields. Digital protocols \cite{bender2018digital, martinez2016real, muschik2017u, klco20192, Shaw_2020,brower2020lattice} face similar practical challenges, but have the advantage to be universally programmable and allow for the simulation of both real-time dynamics and equilibrium physics. Lastly, hybrid quantum-classical variational approaches \cite{kokail2019self, klco2018quantum, dumitrescu2018cloud, shehab2019toward} are in an early development stage and can be used to address equilibrium phenomena. These schemes do not require the simulated Hamiltonian to be realized on the quantum device. Along with their inherent robustness to imperfections \cite{mcclean2016theory}, this feature makes variational schemes suitable for NISQ technology.\\
\\As shown in Fig.~\ref{fig:paperStructure}, we use a variational approach to simulate ground state properties of lattice gauge theories (LGTs). In contrast to previous schemes, our protocols provide the novel opportunity to use existing quantum hardware \cite{kokail2019self, nam2020ground} to simulate 2D effects in LGTs, including dynamical matter and non-minimal gauge field truncations, with a perspective to go to the continuum limit. We consider quantum electrodynamics (QED), the gauge theory describing charged particles interacting through electromagnetic fields. In contrast to 1D QED, where the gauge field can be eliminated \cite{martinez2016real,muschik2017u,hamer1982massive, hamer1997series}, in higher dimensions both appear non-trivial magnetic field effects and the Fermi statistics of the matter fields become important. As a consequence, many-body terms appear in the Hamiltonian and implementations on currently available quantum hardware become challenging. In this work, we outline novel approaches to overcome these difficulties and to render near-term demonstrations possible.\\
\\Specifically, we provide effective simulation techniques for simulating quasi-2D and 2D lattice-QED systems
with open and periodic boundary conditions. To address the problem of efficiently finding the ground state of these models on NISQ
hardware, we develop a variational quantum eigensolver (VQE) algorithm \cite{mcclean2016theory} for current qubit-based quantum computers.
In the quest of simulating LGT employing this VQE algorithm, we address the crucial points of
\begin{enumerate}
\item developing a formulation of the problem within the resources of NISQ devices,
\item implementing the model efficiently on the quantum hardware,
\item having a clear procedure to scale up to larger, more complex systems, and
\item verifying the results in known parameter regimes.
\end{enumerate}
For the first step, we cast the model into an effective Hamiltonian description, as done in Ref.~\cite{paper1}. The total Hilbert space is then reduced to a smaller gauge-invariant subspace by eliminating redundant degrees of freedom (at the cost of introducing non-local interactions). In order to measure this effective Hamiltonian on the quantum hardware (second step), we find an encoding for translating fermionic and gauge operators into qubit operators. The quantum circuits for the VQE are subsequently determined, respecting the symmetries of both the encoding and the Hamiltonian. From one side, this allows for an optimal exploration of the subsector of the Hilbert space in which gauge-invariant states lie. From the other, our circuits are Hamiltonian inspired, and can be scaled up to bigger systems and to less harsh truncations (third step). The fourth and last step in the list above is taken care by comparing the outcomes of the VQE algorithm with analytical results, in parameter regimes that are accessible to both. In the following, we resort to perturbation theory and exact diagonalization, but more advanced techniques \cite{PhysRevD.90.074501,PhysRevD.86.094504,PhysRevD.81.094505} can be in principle used.\\
\\Our work allows to run quantum simulations of 2D lattice gauge theories on lattices with arbitrary values of the lattice spacing $a$. With the results in Ref.~\cite{paper1}, it is now possible to reach a well-controlled continuum limit $a \rightarrow 0$ while avoiding the problem of autocorrelations inherent to Markov Chain Monte Carlo (MCMC) methods. This offers the exciting long-term perspective to compute physical (i.e., continuum) quantities, such as bound state spectra, non-perturbative matrix elements, and form factors that can be related to collider and low energy experiments. Studying these physical observables requires large lattices whose simulation is inaccessible to present-day quantum devices. However, various local quantities describe fundamental properties of a theory, and can be simulated on small lattices that are accessible today. An example is the plaquette expectation value, as used in the pioneering work of Creutz \cite{creutz1983monte}. Despite its simplicity, this observable can be related to the renormalized running coupling that we consider below. We emphasize that the methods presented here can be used as a launch pad to further developments in the realm of quantum simulations of LGTs, that are aiming at higher dimensions, larger lattice sizes and/or non-abelian theories. Furthermore, the LGT models can be extended to include topological terms, or finite fermionic chemical potentials, that are currently hard or even inaccessible to MCMC due to the sign problem.\\
\\We treat the case of open boundary conditions within a ladder system [see Fig.~\ref{fig:openBoundaryConventions}(a)]. Although this
system does not encompass the physics of the full 2D plane, it allows one to explore magnetic field properties
on currently available quantum hardware. For that reason, we provide a simulation protocol for the
basic building block of 2D LGTs, a single plaquette including matter, that demonstrates dynamically generated gauge fields. This is shown by observing the effect of particle-antiparticle pair creations on
the magnetic field energy. In particular, both positive and negative fermion masses are considered. In the
latter case, MCMC methods cannot be applied due to the zero mode problem \cite{Troyer:2004ge, Gattringer:2016kco}.\\
\\Additionally, we consider a single plaquette with periodic boundary conditions (see Fig.~\ref{fig:periodicBoundaryConventions}) to demonstrate
that our approach provides an important first step towards calculating the so-called ``running coupling" in gauge theories \cite{aoki2020flag, aoki2017review}. The running of the coupling, i.e., the dependence of the charge on the energy scale on which it is probed, is fundamental to gauge theories and is absent in 1D QED. For example, its precise determination in quantum chromodynamics is crucial for analyzing particle collider experiments. Here, we propose a first proof-of-concept quantum simulation of the running coupling for pure gauge QED.\\
\\
The paper is structured as follows (see Fig.~\ref{fig:paperStructure}). In Sec.~\ref{sec:encodedHamiltonian}, we introduce lattice QED and present the effective Hamiltonian description on which the proposal is based. The VQE algorithm is then outlined in Sec.~\ref{sec:main_results}, along with a short description of the available quantum hardware on which it may be implemented, along with possible classical optimization routines. In Sec.~\ref{sec:main_result}, we demonstrate our VQE algorithm for an ion-based quantum computer. The all-to-all connectivity available on this platform is an excellent match for our approach in the NISQ era, since the elimination of redundant degrees of freedom results in non-local interactions. We present a detailed experimental proposal, along with a classical simulation of the proposed experiments to demonstrate that an implementation with present-day quantum computers is feasible. Finally, we describe the physical 2D phenomena that we aim to study in Sec.~\ref{sec:quantumSimulationOf2DEffects}. Conclusions and outlooks are presented in Sec.~\ref{sec:conclusionsAndOutlook}.
\section{Simulated models}
\label{sec:encodedHamiltonian}
\noindent In this section, we present the models to be simulated in our proposal. In Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions}, we review the Hamiltonian formulation of lattice QED in $2$ dimensions along with the truncation applied to gauge degrees of freedom. The specific systems considered in the rest of this paper are then described in Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForOpenBoundaryConditions} [open boundary conditions (OBC)] and Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions} [periodic boundary conditions (PBC)].
\subsection{Lattice QED in 2 dimensions}
\label{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Fig2.pdf}
\caption{Conventions for lattice QED in 2D. \textbf{(a)} Ladder system with open boundary conditions. Using Gauss' law, the number of gauge degrees of freedom can be reduced to one per plaquette (blue ellipses). \textbf{(b)} A single plaquette with matter sites at the vertices and gauge fields on the links. Circles (squares) represent odd (even) sites, and black (grey) corresponds to unoccupied (occupied) fermionic fields. Positive field direction is to the right and up. \textbf{(c)} Table showing the mapping between fermionic sites, particles ($e$) and antiparticles ($p$), and spins. \textbf{(d)} Conventions for Gauss' law. \textbf{(e)} The two gauge-field configurations that minimize the electric field energy for a single plaquette with two particle-antiparticle pairs. As explained in the main text, these configurations are relevant for the 2D effects discussed in Sec.~\ref{subsec:quantumSimulation_openBoundaryConditions} (see App.~\ref{app:newAppendix} for more details).}
\label{fig:openBoundaryConventions}
\end{figure}
\noindent In this work, we consider two-dimensional lattices, with matter and gauge fields defined on the vertices and on the links, respectively. Using staggered fermions \cite{kogut1975hamiltonian}, electrons and positrons are represented by single component fermionic field operators $\hat{\phi}_i$ for each site $i$. As shown in Figs.~\ref{fig:openBoundaryConventions}(b) and \ref{fig:openBoundaryConventions}(c), odd (even)-numbered lattice sites hold particles (antiparticles), that carry a $+1$ $(-1)$ charge $q_i$.
Gauge fields on the links between sites $i$ and $j$ are described by the operators $\hat{E}_{ij}$ (electric fields) and $\hat{U}_{ij}$. Electric field operators take integer eigenvalues $e_{ij} = 0,~\pm 1,~\pm 2, \dots$ with $\hat{E}_{ij}\ket{e_{ij}} = e_{ij}\ket{e_{ij}}$, while $\hat{U}_{ij}$ acts as a lowering operator on electric field eigenstates, i.e., $\hat{U}_{ij} \ket{e_{ij}} = \ket{e_{ij} -1}$, with $[\hat{U}_{ij}, \hat{U}^{\dag}_{kl}] = 0$ and $[\hat{E}_{ij}, \hat{U}_{kl}] = -\delta_{i,k} \delta_{j,l}\hat{U}_{ij}$.\\
\\Using the Kogut-Susskind formulation \cite{kogut1975hamiltonian}, the Hamiltonian consists of an electric, a magnetic, a mass, and a kinetic term; $\hat{H} = \hat{H}_{\textrm{E}} + \hat{H}_{\textrm{B}} + \hat{H}_{\textrm{m}} + \hat{H}_{\textrm{kin}}$, where
\begin{subequations}\label{eq:HtotOpenBoundary}
\begin{align}
\hat{H}_{\textrm{E}} =& \frac{g^2}{2} \sum_{ \substack{i, \\ i \stackrel{+}\longrightarrow j}} \hat{E}^2_{ij}, \label{eq:HEOpenBoundary}\\
\hat{H}_{\textrm{B}} =& -\frac{1}{2g^2a^2} \sum_{n = 1}^{N}\Big(\hat{P}_n + \hat{P}^{\dag}_n\Big), \label{eq:HBOpenBoundary} \\
\hat{H}_{\textrm{m}} =& m \sum_{i \in \textrm{sites}} (-1)^{i+1}\hat{\phi}_i^{\dag} \hat{\phi}_i, \label{eq:HmOpenBoundary} \\
\hat{H}_{\textrm{kin}} =& \Omega\Big( \sum_{\substack{i~\textrm{odd},\\i \stackrel{+}\longrightarrow j}} \hat{\phi}_i^{\dag} \hat{U}_{ij} \hat{\phi}_j \nonumber \\&+ \sum_{\substack{i~\textrm{even}, \\i \stackrel{+}\longrightarrow j}} \hat{\phi}_i \hat{U}^{\dag}_{ij} \hat{\phi}_j^{\dag} \Big) + \textrm{H.c.}. \label{eq:HkinOpenBoundary}
\end{align}
\end{subequations}
In the summations, we use $i \stackrel{+} \longrightarrow j$ to denote the link between lattice sites $i$ and $j$ with positive orientation [see Fig.~\ref{fig:openBoundaryConventions}(a) and (b)]. We denote the bare coupling by $g$, $m$ is the fermion mass, $a$ the lattice spacing and $\Omega $ the kinetic strength.
We use natural units $\hbar = c = 1$ and all operators in Eqs.~\eqref{eq:HtotOpenBoundary} are dimensionless \cite{paper1}.\\
\\In the Hamiltonian above, we introduced the operator $\hat{P}_n = \hat{U}_{ij}^{\dag} \hat{U}_{jk}^{\dag} \hat{U}_{il} \hat{U}_{lk}$, where sites $(i,j,k,l)$ form a closed loop clockwise around the plaquette $n$ as in Fig.~\ref{fig:openBoundaryConventions}(b) \cite{wilson1974confinement}. This allows us to define the plaquette operator
\begin{align} \label{eq:PlaquetteOperator}
\Box &= \frac{1}{2N} \sum_{n = 1}^N\Big(\hat{P}_n + \hat{P}_n^{\dag}\Big),
\end{align}
with $N$ being the number of plaquettes.\\
\\At each vertex $i$, gauge invariance is imposed by the symmetry generators \cite{kogut1975hamiltonian, Zohar_2013} $\hat{G}_i = \hat{E}_{li} - \hat{E}_{ij} + \hat{E}_{ki} - \hat{E}_{im} - \hat{q}_i$ [see Fig.~\ref{fig:openBoundaryConventions}(d) for the definition of $l$, $j$, $k$, $m$], where $\hat{q}_i = \hat{\phi}_{i}^{\dag} \hat{\phi}_{i} - \frac{1}{2}\left[1 + (-1)^{i}\right]$ is the charge operator. Relevant, i.e., gauge-invariant quantum states are defined by Gauss' law $\hat{G}_{i}\ket{\Psi_{\textrm{phys}}} = \epsilon_{i}\ket{\Psi_{\textrm{phys}}}$ for each vertex of the lattice, where the eigenvalue $\epsilon_{i}$ corresponds to the static charge at site $i$. We consider the case $\epsilon_i = 0~\forall i$, but background charges can be easily included in the derivations below. In the continuum limit $a \rightarrow 0$ and three spatial dimensions, Gauss' law takes the familiar form $\nabla \hat{E} = \hat{\rho}$, where $\hat{\rho}$ is the charge density.\\
\\Gauss' law can be used to lower the requirements for a quantum simulation.
More specifically, within the Hamiltonian formulation of a gauge theory, only an exponentially small part of the whole Hilbert space consists of gauge-invariant states. In QED, this subspace is selected by applying Gauss' law with a specific choice of the static charges.
As a result, it is possible to eliminate some of the gauge fields and obtain an effective Hamiltonian which is constrained to the chosen subspace (see also Ref.~\cite{bender2020gauge}).
Practically, this reduces the number of qubits that are required to simulate the gauge fields -- which is particularly important in the NISQ era.
As can be seen in the following sections, the trade-off of this resource-efficient encoding is the appearance of long-range many-body interactions in the effective Hamiltonian.
Nevertheless, universal quantum computers offer the possibility to perform gates between arbitrary pairs of qubits (long-range interactions). Moreover, the VQE approach is promising when dealing with interactions that cannot be directly implemented in the quantum hardware. Therefore,
resorting to the effective Hamiltonians of the models considered for our quantum simulation in Sec.~\ref{sec:main_result} is advantageous in terms of experimental feasibility of our protocols.
\subsection{Effective Hamiltonian for open boundary conditions}
\label{subsec:encodedHamiltonian_effectiveHamiltonianForOpenBoundaryConditions}
\noindent In this section, we consider a one-dimensional ladder of plaquettes with OBC [see Fig.~\ref{fig:openBoundaryConventions}(a)]. Although this system does not encapsulate the full 2D physics of QED, it allows us to study important aspects of gauge theories that are not present in one spatial dimension, such as magnetic phenomena.\\
\\The OBC for a ladder of $N$ plaquettes appear as dashed lines in Fig.~\ref{fig:openBoundaryConventions}(a), indicating null background field. While there are $3N+1$ gauge fields in the ladder, applying Gauss' law to all vertices reduces the independent gauge degrees of freedom to $N$. Given the freedom in choosing the independent gauge fields, we select the $(2n, 2n+1)$ links for each plaquette $n$, as shown by the blue ellipses in Fig.~\ref{fig:openBoundaryConventions}(a). On each link that does not hold an independent gauge field, the corresponding unitary operator $\hat{U}$ is set to the identity.
As a result, the plaquette operator becomes $\hat{P}_n = \hat{U}_{2n, 2n+1}$ and both the kinetic and magnetic terms are consequently simplified. The effective Hamiltonian for the ladder is derived in App.~\ref{app:ElectricEnergyContribution}.\\
\\We now focus on the basic building block of the 2D ladder system, the plaquette [see Fig.~\ref{fig:openBoundaryConventions}(b)]. Setting $N=1$ in Eqs.~\eqref{eq:OpenBoundarySpinHamiltonian}, the effective Hamiltonian for a single plaquette with OBC becomes
\begin{subequations} \label{eq:singlePlaquetteHamiltonian}
\begin{align}
\hat{H}_{\textrm{E}} = &\frac{g^2}{2}\Big[\Big(\hat{E}_{23}\Big)^2 + \Big(\hat{E}_{23} + \hat{q}_2\Big)^2 \nonumber \\
&+ \Big(\hat{E}_{23} - \hat{q}_{3}\Big)^2 + \Big(\hat{E}_{23} + \hat{q}_{1} + \hat{q}_{2}\Big)^2 \Big], \label{eq:singlePlaquetteHamiltonianHE}\\
\hat{H}_{\textrm{B}} = &-\frac{1}{2g^2}\Big(\hat{U}_{23} + \hat{U}^\dagger_{23}\Big), \\
\hat{H}_{\textrm{m}} =& m \sum_{i =1}^4 (-1)^{i+1}\hat{\phi}_i^{\dag} \hat{\phi}_i, \label{eq:singlePlaquetteHamiltonianHm}\\
\hat{H}_{\textrm{kin}} = &\Omega \Big(\hat{\phi}_1^{\dag} \hat{\phi}_2 + \hat{\phi}_1^{\dag} \hat{\phi}_4 + \hat{\phi}_2 \hat{U}^\dagger_{23} \hat{\phi}_3^{\dag} + \hat{\phi}_4 \hat{\phi}_3^{\dag}\Big) + \textrm{H.c.}. \label{eq:singlePlaquetteHamiltonianHkin}
\end{align}
\end{subequations}
The notation for describing the state of the plaquette with OBC is a tensor product of two kets, with the first ket representing the matter sites $1-4$ ($v$, $e$, and $p$ are the vacuum, particle, and antiparticle, respectively) and the second ket the state of the gauge field on the $(2,3)$ link [see Fig.~\ref{fig:openBoundaryConventions}(e) for examples].\\
\\We remark that in 1D QED, all gauge fields can be eliminated for systems with OBC. As a result, only two-body terms remain in the Hamiltonian \cite{hamer1982massive, hamer1997series, martinez2016real, muschik2017u}. For the plaquette, many-body terms are unavoidable since gauge degrees of freedom survive. This represents one of the main challenges in simulating LGTs in more than one spatial dimensions.
\subsection{Effective Hamiltonian for periodic boundary conditions}
\label{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{Fig3.pdf}
\caption{Single periodic boundary plaquette for a pure gauge theory. \textbf{(a)} Representation in terms of the eight gauge fields (blue arrows), that are associated with the links of the lattice. \textbf{(b)} Representation in terms of independent gauge-invariant (i.e., physical) operators called ``rotators''. As explained in Ref.~\cite{paper1}, the plaquette can be seen as an infinite lattice of plaquettes. Moreover, to explore ground state properties, only three independent rotators $\hat{R}_1$, $\hat{R}_2$, and $\hat{R}_3$ (shown in solid blue) are sufficient for describing the system.}
\label{fig:periodicBoundaryConventions}
\end{figure}
\noindent For the second model, we consider a square lattice with PBC and without fermionic matter. This system has been considered in Ref.~\cite{paper1}, but for the benefit of the reader we summarize the main results that are required for the VQE simulation. As explained in Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions}, Gauss' law can be used to eliminate redundant gauge fields, resulting in an unconstrained effective Hamiltonian.
The basic building block, a single plaquette with PBC, includes eight gauge fields [see Fig.~\ref{fig:periodicBoundaryConventions}(a)], and is equivalent to an infinite 2D lattice of four distinct plaquettes [see Fig.~\ref{fig:periodicBoundaryConventions}(b)].
Due to the absence of matter, we consider the pure gauge Hamiltonian $\hat{H}_{\textrm{gauge}} = \hat{H}_{\textrm{E}} + \hat{H}_{\textrm{B}}$, given by Eqs.~\eqref{eq:HEOpenBoundary} and \eqref{eq:HBOpenBoundary}.\\
\\Since we are interested in studying ground state properties, it is sufficient to consider three independent gauge degrees of freedom (see Ref.~\cite{paper1}), called ``rotators" $\hat{R}_{i}$, $i \in \{1,2,3\}$ shown in Fig.~\ref{fig:periodicBoundaryConventions}(b). Rotators are a convenient representation for the gauge fields, as they correspond to the circulating electric field in a specific plaquette. As such, the operators $\hat{P}_{n}$ ($\hat{P}^{\dag}_{n}$) in Eqs.~\eqref{eq:HBOpenBoundary} and \eqref{eq:PlaquetteOperator} are directly their descending (ascending) operators. Indeed, the commutation relations between a rotator $\hat{R}$ and its associated operator $\hat{P}$ are the same as the ones of an electric field $\hat{E}$ and its lowering operator $\hat{U}$, presented in Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions}. The notation for describing the state of the plaquette with PBC is thus a tensor product of three kets, each corresponding to one of the three rotators represented by full lines in Fig.~\ref{fig:periodicBoundaryConventions}(b).\\
\begin{comment}
For large values of the bare coupling $g$, the electric field term $\hat{H}_{\textrm{E}}$ in $\hat{H}_{\textrm{gauge}}$ is dominant -- accordingly, the Hamiltonian representation with diagonal electric term described in Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions} is efficient. For small $g$, however, the magnetic term $\hat{H}_{\textrm{B}}$ is dominant, and the eigenstates of $\hat{H}_{\textrm{gauge}}$ are superpositions of all electric field basis states. As we will see below in {\color{blue} Sec X}, any truncation scheme necessarily leads to discrepancies with the full model. To mitigate the truncation error, we apply the novel technique introduced in Ref.~\cite{paper1} and resort to two different formulations of the Hamiltonian. The electric formulation described in Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions} is applied in the region $g^{-2} < 1$, while for $g^{-2} >1$, a so-called magnetic basis is used, for which $\hat{H}_{\textrm{B}}$ is diagonal \cite{stryker2018gauss}. Since the vector potential $\theta$ is a continuous degree of freedom, the magnetic formulation needs to optimize the angle $\Delta \theta$ covered by the truncated theory, and the resolution of the discretized vector potential $\delta \theta$. This is necessary to allow for an efficient description of the ground state over sufficiently large values of $g$ using only a limited number of basis states. We achieve this by employing the cyclic group $\mathbb{Z}_{2L+1}$ ($L \in \mathbb{N}$) to approximate the continuous group U(1). The magnetic representation is obtained by first taking an approximation in the form of a series expansion that is exact for $L \rightarrow \infty$. We then apply the Fourier transform, and subsequently truncate the group $\mathbb{Z}_{2L+1}$ by only considering $l < L$ states in order to achieve equivalence to a truncated U(1) theory.
As such, the basis becomes $\ket{-l},...,\ket{0},...,\ket{l}$. The optimal choice of the parameter $L$, in addition to the technical details of approximating the vector potential $\theta$ by means of a practical discretized truncation, is studied in Ref.~\cite{paper1}.
\end{comment}
\\Usually, LGT Hamiltonians are formulated in the basis of the electric field, where $\hat{H}_{\textrm{E}}$ in $\hat{H}_{\textrm{gauge}}$ is diagonal. For large values of the bare coupling $g$, the term $\hat{H}_{\textrm{E}}$ is dominant, and this representation is efficient.
For small $g$, however, the magnetic term $\hat{H}_{\textrm{B}}$ is dominant, and the eigenstates of $\hat{H}_{\textrm{gauge}}$ are superpositions of all electric field basis states.
Since the Hilbert space of these operators is of infinite dimension, this necessarily leads to truncation errors when the Hamiltonian is mapped to any quantum device (as in Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding}).
To mitigate the truncation error, we apply the novel technique introduced in Ref.~\cite{paper1}, and resort to two different formulations of the Hamiltonian. The electric formulation in Eqs.~\eqref{eq:HtotOpenBoundary} is applied in the region $g^{-2} \lesssim 1$, while for $g^{-2} \gtrsim 1$ a so-called magnetic basis is used, for which $\hat{H}_{\textrm{B}}$ is diagonal (see also Ref.~\cite{stryker2018gauss}).
This magnetic representation is obtained using the cyclic group $\mathbb{Z}_{2L+1}$ ($L \in \mathbb{N}$) to approximate the continuous group U(1) associated to each rotator. We first take an approximation in the form of a series expansion that is exact for $L \rightarrow \infty$. Then, we apply the Fourier transform and truncate the group $\mathbb{Z}_{2L+1}$ by only considering $2l+1 < 2L+1$ states in order to achieve equivalence to a truncated U(1) theory. As such, the rotator's basis becomes $\ket{-l},...,\ket{0},...,\ket{l}$. This leaves one with $L$ as a free parameters, whose optimal choice, along with the technical details of approximating the U(1) theory with a truncated $\mathbb{Z}_{2L+1}$ model, is studied in Ref.~\cite{paper1}.\\
\\We thus arrive at the Hamiltonian $\hat{H}_{\textrm{gauge}}^{(\gamma)} = \hat{H}_{\textrm{E}}^{(\gamma)} + \hat{H}_{\textrm{B}}^{(\gamma)}$, where $\gamma = e$ indicates the electric representation,
\begin{subequations} \label{eq:periodicHamiltonianElectric}
%
\begin{align}
\hat{H}_{\textrm{E}}^{(e)} = &~2g^2 \Big[(\hat{R}_1)^2 + (\hat{R}_2)^2 + (\hat{R}_3)^2 \nonumber\\&- \hat{R}_2 (\hat{R}_1 + \hat{R}_3)\Big], \label{eq:periodicHamiltonianElectricE}\\
\hat{H}_{\textrm{B}}^{(e)} = &-\frac{1}{2g^2}\Big(\hat{P}_1 + \hat{P}_2 + \hat{P}_3 \nonumber\\&+ \hat{P}_1 \hat{P}_2 \hat{P}_3 + \textrm{H.c.}\Big) \label{eq:periodicHamiltonianElectricB},
\end{align}
%
\end{subequations}
and we denote the magnetic representation with $\gamma = b$,
\begin{subequations} \label{eq:periodicHamiltonianMagnetic}
%
\begin{align}
\hat{H}_{\textrm{E}}^{(b)} = &~g^2 \sum_{\nu = 1}^{2L} \Big\{f_{\nu}^c \sum_{i=1}^{3} (\hat{P}_{i})^{\nu} + \frac{f_{\nu}^{s}}{2}\big[(\hat{P}_{2})^{\nu} - (\hat{P}_{2}^\dagger)^{\nu}\big] \nonumber\\
&~\times \sum_{\mu = 1}^{2L} f_{\mu}^{s} \big[(\hat{P}_{1})^{\mu} + (\hat{P}_{3})^{\mu}\big]\Big\} + \textrm{H.c.},\label{eq:periodicHamiltonianMagneticE}\\
\hat{H}_{\textrm{B}}^{(b)} = &-\frac{1}{g^2} \Big[ \sum_{i = 1}^{3} \cos \Big(\frac{2 \pi \hat{R}_{i}}{2L+1}\Big) \nonumber \\
& + \cos \Big(\frac{2 \pi (\hat{R}_{1} + \hat{R}_{2} + \hat{R}_{3})}{2L+1}\Big)\Big]. \label{eq:periodicHamiltonianMagneticB}
\end{align}
%
\end{subequations}
The coefficients $f_k^c$ and $f_k^s$ come from the Fourier transform, and their analytic form is
\begin{align}
f_{\nu}^{s} = &\frac{(-1)^{\nu+1}}{2 \pi} \Big[\psi_{0} \Big(\frac{2L+1+\nu}{2(2L+1)}\Big) \nonumber \\
& -\psi_{0} \Big(\frac{\nu}{2(2L+1)}\Big)\Big], \\
f_{\nu}^{c} = &\frac{(-1)^{\nu}}{4 \pi^{2}} \Big[ \psi_{1} \Big(\frac{\nu}{2(2L+1)}\Big) \nonumber \\
&- \psi_{1} \Big(\frac{2L+1+\nu}{2(2L+1)}\Big)\Big],
\end{align}
where $\psi_{k}(\cdot)$ is the $k$-th polygamma function \cite{paper1}. We note that making use of electric and magnetic representations in the case of the plaquette with OBC (see Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForOpenBoundaryConditions}) is straightforward. For clarity, however, we use this formulation only when considering PBC.
\section{Foundations for variational quantum simulations}
\label{sec:main_results}
\noindent As detailed in Sec.~\ref{sec:encodedHamiltonian}, the Hamiltonians associated with the considered models contain exotic long-range interactions and many-body terms that are beyond the capabilities of current analogue quantum simulations.
This is particularly true once the infinite dimensional operators $\hat{E}$ and $\hat{U}$, corresponding to the gauge field and its connection \cite{kogut1975hamiltonian}, respectively, are encoded in terms of qubit operators (see Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding}).
Resorting to digital approaches, trotterized adiabatic state preparation is possible in theory \cite{jordan2012quantum}. However, the high complexity of LGTs beyond one dimension renders this approach infeasible for the technology available today.
Due to the absence of large-scale and universal fault-tolerant quantum computers we thus employ a hybrid quantum-classical strategy, using a VQE protocol that renders the observation of the effects described in Sec.~\ref{sec:quantumSimulationOf2DEffects} within reach of present-day quantum devices.
In fact, those protocols have already proven their validity in quantum chemistry \cite{mcclean2016theory, o2016scalable, hempel2018quantum, peruzzo2014variational}, nuclear physics \cite{dumitrescu2018cloud, shehab2019toward} and even classical applications \cite{farhi2014quantum,borle2020quantum,bravo2020variational}.\\
\\In the following, we first introduce the VQE protocol (Sec.~\ref{subsec:variationalQuantumSimulation_variationalQuantumSimulation}) and discuss feasibility on currently available quantum platforms (Sec.~\ref{subsec:trappedIonResults_trappedIons}). Then, we describe our minimization algorithm, which is run on the classical device to find the desired state (Sec.~\ref{subsec:optimizationRoutine}).
\subsection{Hybrid quantum-classical simulations}
\label{subsec:variationalQuantumSimulation_variationalQuantumSimulation}
\noindent The VQE is a closed feedback loop between a quantum device and an optimization algorithm performed by a classical computer \cite{peruzzo2014variational, mcclean2016theory}.
The quantum device is queried for a cost function $\mathcal{C}(\boldsymbol{\theta})$, which is expressed in terms of a parameterized variational state $\ket{\Psi(\boldsymbol{\theta})} = U(\boldsymbol{\theta})\ket{\Psi_{\textrm{in}}}$.
Here, $\ket{\Psi_{\textrm{in}}}$ is an initial state that can be easily prepared, while $U(\boldsymbol{\theta})$ entails the application of a quantum circuit involving the variational parameters $\boldsymbol{\theta}$. The classical optimization algorithm minimizes $\mathcal{C}(\boldsymbol{\theta})$, and provides an updated set of variational parameters $\boldsymbol{\theta}$ after each iteration of the feedback loop.
Since our quantum simulations aim at preparing the ground state (see Sec.~\ref{sec:quantumSimulationOf2DEffects}), our VQE cost function is defined as the expectation value of the system's Hamiltonian $\hat{H}$ with respect to the variational state, i.e. $\mathcal{C}(\boldsymbol{\theta}) = \bra{\Psi(\boldsymbol{\theta})} \hat{H} \ket{\Psi(\boldsymbol{\theta})}$.\\
\\Importantly, the Hamiltonian is never physically realized on the quantum hardware. Instead, its expectation value is estimated by measuring $\hat{H}$ with respect to the variational state $\ket{\Psi(\boldsymbol{\theta})}$ \cite{peruzzo2014variational, mcclean2016theory} (see App.~\ref{app:optimalPartitioning}). Thus, the VQE is advantageous whenever the studied model contains complicated long-range, many-body interactions (as in this case), that cannot be realized in current devices. Moreover, this approach is insensitive to several systematic errors, such as offsets in the variational parameters $\boldsymbol{\theta}$, since these are compensated by the classical optimization routine.
Both these properties relax the quantum resource requirements and facilitate future experimental implementations of the VQE routines presented in the remainder of this paper.\\
\subsection{Quantum hardware considerations}
\label{subsec:trappedIonResults_trappedIons}
\noindent
In this work, we consider qubit-based platforms for the realization of a proof-of-principle experiment with present-day technology. Promising quantum computing platforms include configurable Rydberg arrays \cite{bernien2017probing, labuhn2016tunable}, superconducting architectures \cite{arute2019quantum, corcoles2015demonstration}, and trapped ions \cite{lanyon2011universal, debnath2016demonstration}. Comparing these approaches, Rydberg-based systems offer the advantage that $2$D and $3$D arrays can be realized using optical tweezers, which translates to a large number of available qubits. The ability to implement large qubit registers will make future generations of Rydberg arrays a very promising candidate for our schemes, once higher levels of controllability and gate fidelity become available. Superconducting and ion-based architectures offer both high controllability and gate fidelities already today. For superconducting qubits, entangling gates are inherently of nearest-nearest neighbour type, which implies that entangling operations between non-neighbouring qubits have to be realized through a number of swap gates. While this is entirely possible, especially for next-generation devices, our models entail long-range interactions that result from the elimination of redundant gauge fields (see Secs.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForOpenBoundaryConditions} and \ref{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions}) and thus suggest the necessity of creating highly entangled states which would require significant gate overhead on superconducting platforms.
In contrast, ion-based quantum computers \cite{benhelm2008towards, brown2011single} have all-to-all connectivity, allowing for addressed entangling gates between arbitrary qubits. This aligns well with our target models, motivating their use for this proposal. Despite their comparatively slow readout limits the measurement budget, we solve this issue with both an efficient measurement strategy (see App.~\ref{app:optimalPartitioning}) and an optimized classical algorithm (see Sec.~\ref{subsec:optimizationRoutine}). A realistic budget for currently available ion-based quantum computers is given by about $10^7$ measurement shots. This includes the time for the quantum computer to initialize, modify and measure the states, and the time for the classical computer to minimize the cost function $\mathcal{C}(\boldsymbol{\theta})$.\\
\\Here, we describe the properties of ion-based computers in more detail. We consider a string of ions confined in a macroscopic linear Paul trap \cite{lanyon2011universal, debnath2016demonstration}. The qubit states $\ket{0}$ and $\ket{1}$ are encoded in the electronic states of a single ion and can be manipulated using laser light in the visible spectrum. A universal set of quantum operations is available, which can be combined to implement arbitrary unitary operations. More specifically, the available gate-set consists of local rotations $\hat{U}_{j}(\phi) = \exp (-i \frac{\phi}{2} \hat{\sigma}_{j}^{\alpha})$ and addressed M\o lmer-S\o rensen (MS) gates \cite{sorensen1999quantum} between arbitrary pairs of qubits. These gates are implemented with fidelities exceeding $98\%$ for both single- and multi-qubit gates \cite{benhelm2008towards, brown2011single, zhang2017observation}.
While currently available ion-based hardware provides a sufficient number of qubits to carry our proposed protocols, future LGT quantum simulations addressing larger lattices will require large-scale quantum devices. Ion based quantum computers can be scaled up using segmented 2D traps \cite{kielpinski2002architecture} and networking approaches that connect several traps together \cite{monroe2014large, Ragg_2019} and therefore offer a pathway for developing quantum simulations of increasing size and complexity.
\subsection{Classical Optimization Routine}
\label{subsec:optimizationRoutine}
\noindent The classical optimization routine employed for the VQE needs to be chosen depending on the requirements from both the hardware and the cost function $\mathcal{C}(\boldsymbol{\theta})$. Indeed, the stochastic nature of the latter has to be taken into account, and the experimental repetition rate poses limitations on the number of data points that the classical machine can use for minimization. In the following, we consider different optimization routines, discuss their strengths and weaknesses, and motivate our specific choice.\\
\\Algorithms based on the scheme of gradient descent are promising candidates, particularly when the number of variational parameters grows large. Recently, a variety of these algorithms have been modified to take the stochastic nature of the cost function into account \cite{Kingma2014aa, Ruder2017An-overview}, and there are versions which are tailored to the quantum regime \cite{Stokes2020Quantum}.
Methods to measure the gradient of an operator \cite{Li2017} have been proposed, but they are costly when applied to multi qubit gates and require additional ancilla qubits \cite{Schuld:2019aa}. In addition, finite step size approximations of the gradient require a large number of experimental evaluations to obtain reliable values for the gradient.\\
\\A limitation of the available quantum hardware is the small number of experimental shots, particularly when resorting to ion-based platforms.
This drawback is especially limiting for gradient based optimization. Indeed, to prevent the VQE to settle in a local minimum, one has to perform multiple runs from different initial conditions. Furthermore, already acquired data cannot easily be recycled in scenarios where the algorithm is applied to different parameters regimes (e.g. variations in $g$ in Sec.~\ref{sec:main_result}). A family of algorithms which can recycle previous data is the mesh-based. These evaluate the cost function on a grid in parameter space, with refinements where the global minimum is expected to be \cite{Audet2006Mesh}. Mesh and values of the cost function can be stored, and reused for a different set of parameters, to accelerate the optimization.
The feature of building a data repository is shared with Bayesian optimizers, which build a regression model of the energy landscape based on Gaussian processes \cite{Rasmussen2003}. These algorithms explore the variational parameters' space by performing cost function evaluations at the location of potential minimizing points, suggested by their regression model. Hence, they are able to reduce the required number of measurements substantially. However their effectiveness is limited to around $20$ variational parameters \cite{Frazier2018aa}.\\
\begin{comment}
However, the latter is generally costly to be computed. Indeed, if one estimates the gradient from several values of the cost function $\mathcal{C}(\boldsymbol{\theta})$, finite step size approximations result in a large number of experimental evaluations. The same is true if the gradient is directly computed on the quantum device, particularly when (some of) the variational parameters are associated to entangling gates \cite{Li2017}. Methods have been proposed to reduce this overhead, but resort to ancilla qubits \cite{Schuld:2019aa} and are out of reach for current quantum hardware. Moreover, gradient-based optimization may settle in a local minimum and hence requires multiple runs from different initial conditions.\\
\\Another class of algorithm for the cost function minimization is mesh-based. In this case, $\mathcal{C}(\boldsymbol{\theta})$ is evaluated on a grid, with refinements where the global minimum is expected. A Bayesian oracle {\color{green} is ok?} builds a regression model of the parameters' landscape that suggests the location of points with more potential. Moreover, since the cost function is evaluated on a grid, data acquired for a previous set of physical parameters ($g$, $m$ and $\Omega$) can be recycled when these parameter are changed. Along with the bayesian oracle, this results in a drastic reduction of the required resources, that makes mesh-based algorithms particularly promising. However, they are currently limited to around $20$ variational parameters \cite{Frazier2018aa}.\\
\\The choice of building the classical VQE routine on the gradient-based or the mesh-based algorithm has to be taken gauging advantages and disadvantages of these algorithms, once applied to a specific quantum hardware. Two important factors are the possibility of evaluating the gradient of the cost function, and the number of variational parameters in the optimization. Here, we employ the optimization algorithm previously used in Ref.~\cite{kokail2019self}, which is a modified version of the Dividing Rectangles algorithm (DIRECT) \cite{JonesOriginalDirect93, DirectConvergence2004, NicholasDirect2014, LiuDirect2015}. This algorithm is mesh-based and divides the parameters' space into so-called hypercells, each of them contains a single sample point representative of the cost function. Hypercells might be divided into smaller cells, depending on their size and value of $\mathcal{C}(\boldsymbol{\theta})$. The bayesian oracle is used for extracting properties on the landscape and thus for selecting the hypercells to be divided \cite{Frazier2018aa}. Our choice is motivated from the small number of measurement shots that can be performed on an ion-based quantum computer (see Sec.~\ref{subsec:variationalQuantumSimulation_variationalQuantumSimulation}), and from the limited number of variational parameters $\boldsymbol{\theta}$ required in our models (see Sec.~\ref{sec:main_result}).
\end{comment}
\\Here, we employ an optimization algorithm similar to the one used in Ref.~\cite{kokail2019self}, which is a modified version of the Dividing Rectangles algorithm (DIRECT) \cite{hooke1961direct,JonesOriginalDirect93, DirectConvergence2004, NicholasDirect2014, LiuDirect2015}. This algorithm divides the search space into so-called hypercells. Each hypercell contains a single sample point representative of the cost function value in that cell. The algorithm selects promising hypercells, to be divided into smaller cells, based on the cost function value as well as the cell size. Larger cells contain more unexplored territory and are hence statistically more likely to harbour the global minimum. During the optimization, the algorithm maintains a regression model as used in Bayesian optimization. This metamodel is used for an accurate function value estimation that aids in selecting the hypercells to be divided. Furthermore, at regular intervals, one or more direct Bayesian optimization steps are carried out \cite{Frazier2018aa}.
Our choice is motivated from the small number of measurement shots that can be performed on an ion-based quantum computer (see Sec.~\ref{subsec:variationalQuantumSimulation_variationalQuantumSimulation}), and from the limited number of variational parameters $\boldsymbol{\theta}$ required in our models (see Sec.~\ref{sec:main_result}).
\section{Quantum simulation of 2D LGTs}\label{sec:main_result}
\noindent In this section, we present the VQE protocol for simulating the two models introduced in Sec.~\ref{sec:encodedHamiltonian}. We give numerical results from a classical simulation, which includes the projection noise error.
Both proposed experiments prepare the ground state of the theory and measure the ground state expectation value of the plaquette operator $\expval{\Box} \sim \expval{\hat{H}_\mathrm{B}}$, as defined in Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions}.
For OBC, this allows to study the dynamical generation of gauge fields by pair creation processes, while in the case of PBC it is related to the renormalization of the coupling at different energy scales (see Sec.~\ref{sec:quantumSimulationOf2DEffects}).
\subsection{Encoding for quantum Hardware}
\label{subsec:variationalQuantumSimulation_qubitEncoding}
\noindent To run a VQE, we first require an encoding of the models outlined in Sec.~\ref{sec:encodedHamiltonian} for qubit-based quantum hardware.
While we follow the Jordan-Wigner transformation \cite{jordan1928pauli} for mapping the fermionic operators, there is not a unique procedure to represent the gauge field operators (alternatives are given in Refs. \cite{lewis2019qubit, Shaw_2020}). Here, the encoding is chosen to reduce the complexity of the quantum circuits and to respect symmetries that both suit the simulated models and the quantum platform.\\
\\Gauge field operators are defined on infinite dimensional Hilbert spaces and the gauge field takes the values $0,\,\pm1,\,\pm2, \dots$ (see Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions}).
To simulate them using finite-dimensional quantum systems, a truncation scheme is required.
Let us take $2l+1$ basis states into account, i.e. the gauge field can take at most the values $\pm l$. Consequently, we substitute the electric field operator $\hat{E}$ with the $z$-th component of a spin $\hat{\vec{S}} = (\hat{S}^x,\hat{S}^y,\hat{S}^z)$ of length $|l| = \sqrt{l(l+1)}$,
\begin{equation}\label{eq:SpinTruncation}
\hat{E} \longmapsto \hat{S}^z = \sum_{i=-l}^{l} i \lvert i \rangle \langle i \rvert.
\end{equation}
This opens two different paths to implement the truncation for $\hat{U}$.
The first employs the spin lowering $\hat{S}^{-} = \hat{S}^{\textrm{x}} - i\hat{S}^{y}$ and raising $\hat{S}^{+} = \hat{S}^x + i\hat{S}^y$ operators, such that $\hat{U} \mapsto \hat{S}^{-}/|l|$.
The second prescribes
\begin{gather}\label{eq:S-operator}
\hat{U} \longmapsto
\begin{bmatrix}
0 & \dots & \dots & 0 \\
1 & \dots & \dots & 0 \\
0 & \ddots & \vdots & 0 \\
0 & \dots & 1 & 0\\
\end{bmatrix}.
\end{gather}
For $|l| \rightarrow \infty$, both mappings for $\hat{U}$ ensure $\hat{U}^{\dagger}\hat{U} = 1$ and the correct commutation relations between $\hat{E}$ and $\hat{U}$. The errors introduced by finite $|l|$ have been studied in Refs.~\cite{verstraete2008matrix,paper1}, where it is proven that are negligible in most scenarios \footnote{We remark that mappings for the operator $\hat{U}$ that preserve unitarity are possible, as in Refs.~\cite{PhysRevLett.111.115303,Kuno_2015,PhysRevD.95.094507,PhysRevA.94.063641}}. The specific choice of the truncation scheme depends on the quantum hardware that is employed. As an example, for quantum systems that allow for the implementation of interacting spin chains with large spins, such as $l = 1$ \cite{senko2015realization}, the definition based on the spin lowering $\hat{S}^{-}$ and raising $\hat{S}^+$ is ideal.
For qubit-based quantum hardware, it is convenient to employ Eq.~\eqref{eq:S-operator} for the representation of $\hat{U}$, as done in the following.\\
\\For a given $l$, each gauge degree of freedom is then described by the $2l+1$ states $\ket{e},~e = -l$, $-l+1, \dots, ~0, \dots, ~l-1, ~l$. We map this vector space onto $2l+1$ qubits using
\begin{align} \label{eq:encodedstates}
\ket{-l + j} &= \ket{ \overbrace{0 \hdots 0}^{j} 1 \overbrace{0 \hdots 0}^{2l-j}},
\end{align}
where $0 \leq j \leq 2l$. With this encoding, the gauge field operators might be replaced by the simple forms
\begin{subequations}\label{eq:encodedoperators}
%
\begin{align}
\hat{E}&\mapsto \hat{S}^z = \frac{1}{2} \sum_{i = 1}^{2l} \prod_{j=1}^{i} \hat{\sigma}_{j}^{z}, \\
\hat{U}&\mapsto \sum_{i = 1}^{2l} \hat{\sigma}_{i}^{-} \hat{\sigma}_{i+1}^{+}, \label{eq:encodedoperatorsU}
\end{align}
%
\end{subequations}
where $\hat{\sigma}_i^{\pm} = \frac{1}{2}(\hat{\sigma}_i^x \pm i\hat{\sigma}_i^{y})$, and $\hat{\sigma}_i^{\textrm{x}}$, $\hat{\sigma}_i^{y}$, $\hat{\sigma}_i^{z}$ are the Pauli operators associated with the $i^{\textrm{th}}$ qubit.
From these mapping we directly recover the relations $\hat{E}\ket{e} = e\ket{e}$ and $\hat{U}\ket{e} = (1 - \delta_{e, -l})\ket{e-1}$ for all $-l \leq e \leq l$. As an example, for $l=1$ the states in the gauge field basis become
\begin{subequations} \label{eq:truncatedgaugestates}
\begin{align}
\ket{1} = &~\ket{001}, \\
\ket{0} = &~\ket{010}, \\
\ket{-1} = &~\ket{100}.
\end{align}
\end{subequations}
\\For our protocol, the required resources scale linearly in terms of both the parameter $l$ and the number of gauge and matter fields.
The encoding presented in Eq.~\eqref{eq:encodedstates} requires $2l+1$ qubits for storing $2l+1$ states. In principle, the same information can be stored in $\log (2l+1)$ qubits \cite{lewis2019qubit, Shaw_2020}.
The reasons for which we choose the qubit encoding in Eq.~\eqref{eq:encodedstates} are the following.
First, as demonstrated in Ref.~\cite{paper1}, small values of $l$ allow for a good description of the untruncated model. Second, the terms assembling the operator $\hat{U}$ in Eq.~\eqref{eq:encodedoperatorsU} are easily implementable on different hardware platforms and are even native in trapped ion systems in the form of MS-gates. Finally, the states in Eq.~\eqref{eq:encodedstates} form a subspace of fixed magnetization, which is decoherence-free under the action of correlated noise (e.g. a globally fluctuating magnetic field) and allows for detection of single qubit bit-flip errors. Importantly, the latter is separately true for each individual gauge field and for the fermionic state. Since all the utilized quantum states lay in the single excitation subspace, a measurement resulting in states outside of this subspace can only be due to errors occurring during the quantum evolution, as long as the applied gates conserve the excitations. Hence, erroneous outcomes can be discriminated from faithful ones.\\
\\In the following, we will use the term ``physical states" to refer to the computationally relevant qubit states.
For the gauge fields, this means that the qubit states lie in the computational space spanned by the states in Eqs.~\eqref{eq:truncatedgaugestates}.
For the matter fields whose computational states are obtained via the Jordan-Wigner transformation, we operate in the zero-charge subsector (see Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions}), which translates into matter states of zero magnetization. As a final remark, we highlight that our encoding for the electric field operators $\hat{E}$ and $\hat{U}$ applies equally well to the rotator $\hat{R}$ and plaquette $\hat{P}$ operator used for the plaquette with PBC (see Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions}).
\subsection{Open boundary conditions: dynamically generated magnetic fields}
\label{subsec:trappedIonResults_openBoundaryConditions}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig4.pdf}
\caption{VQE circuit for preparing the ground state of a plaquette with open boundary conditions. \textbf{(a)} Quantum circuit with variational parameters $\theta_l$ shown for each gate. By identifying symmetries and redundancies via classical simulation, gates shaded in grey can be eliminated, and we set $\theta_{16} = \theta_{19} = \theta_{21} = \pi/2$, leaving eleven variational parameters. \textbf{(b)} Definitions of the gates used.}
\label{fig:openCircuit}
\end{figure}
\noindent In the case of a plaquette with OBC, the simulation involves four qubits for the matter fields and $2l+1$ qubits for the gauge field. Here, we consider $l=1$ and thus the system consists of seven qubits. According to Fig.~\ref{fig:openCircuit}(c), we number the matter qubits as $1$, $2$, $3$, and $4$, and the gauge qubits $5$, $6$, and $7$. Plugging the encoding presented in Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding} into the Hamiltonian in Eq.~\eqref{eq:singlePlaquetteHamiltonian}, we get
\begin{subequations} \label{eq:singlePlaquetteHamiltonianEncoded}
\begin{align}
\hat{H}_{\textrm{E}} = &~\frac{g^2}{4}\Big\lbrace \hat{\sigma}_{5}^z\left[ \hat{\sigma}_{1}^z-\hat{\sigma}_{3}^z + \hat{\sigma}_{6}^z(\hat{\sigma}_{1}^z - \hat{\sigma}_{3}^z - 2) - 1 \right] \nonumber \\
& + \hat{\sigma}_{2}^z \left[ \hat{\sigma}_{1}^z + 2 \hat{\sigma}_{5}^z \left( \hat{\sigma}_{6}^z + 1 \right) - 1 \right] + 4\hat{\sigma}_{6}^z \Big\rbrace, \\
\hat{H}_{\textrm{B}} = &-\frac{1}{2g^2}\left[ \hat{\sigma}_{6}^+ \left( \hat{\sigma}_{5}^- + \hat{\sigma}_{7}^- \right) + \hat{\sigma}_{6}^- \left( \hat{\sigma}_{5}^+ + \hat{\sigma}_{7}^+ \right) \right], \\
\hat{H}_{\textrm{m}} = &~\frac{m}{2} \Big(\hat{\sigma}_1^z - \hat{\sigma}_2^z + \hat{\sigma}_3^z - \hat{\sigma}_4^z\Big), \\
\hat{H}_{\textrm{kin}} = &~-i\Omega \Big[\hat{\sigma}_{1}^+ \hat{\sigma}_{2}^- + \hat{\sigma}_{1}^+ \hat{\sigma}_{4}^- + \hat{\sigma}_{4}^-\hat{\sigma}_{3}^+ \nonumber\\
& - \hat{\sigma}_{2}^- \left( \hat{\sigma}_{5}^+ \hat{\sigma}_{6}^- + \hat{\sigma}_{6}^+ \hat{\sigma}_{7}^- \right) \hat{\sigma}_{3}^+ \Big] + \textrm{H.c.}. \label{eq:singlePlaquetteHamiltonianHkinEncoded}
\end{align}
\end{subequations}
The matter qubit states are also given in Fig.~\ref{fig:openBoundaryConventions}(c).
Recall that we chose the encoding such that physical states have total magnetization $\expval{\hat{S}_{\textrm{tot}}^{z}} = 1$. \\%To take advantage of the chosen encoding and its described merits, we design the variational circuit focusing on these points.\\
\\The VQE quantum circuit shown in Fig.~\ref{fig:openCircuit}(a) preserves not only the total magnetization of the system, but also the magnetization of each of the gauge and matter subsystems. Hence, as mentioned in Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding}, our magnetization-preserving quantum circuit used in combination with physical input states confines the VQE to the space of physical states.\\
\\The VQE circuit in its unreduced form [i.e., including the grey-shaded gates in Fig.~\ref{fig:openCircuit}(a)] is motivated by the form of the Hamiltonian in Eqs.~\eqref{eq:singlePlaquetteHamiltonian}.
All qubits are initialized in the input state $\ket{0}$, and NOT gates prepare the bare vacuum $\ket{vvvv}\ket{0}$ [see Fig.~\ref{fig:openBoundaryConventions}(c)] as the initial state for the VQE.
For the gauge field subsystem, the application of the parameterized $i$SWAP gates allows for accessing all three gauge field states $\ket{1}$, $\ket{0}$, and $\ket{-1}$, ensuring that the free ground state for $\Omega=0$ could be produced.
Similarly, the parameterized $i$SWAP gates on the qubits $1$ to $4$ are used to allow for all physical basis states within the matter subsystem, and resemble the hopping terms of the kinetic Hamiltonian in Eq.~\eqref{eq:singlePlaquetteHamiltonianHkinEncoded}. These gates correspond to particle-antiparticle pair creation/annihilation in the model, and as a consequence, all matter basis states in the zero-charge subsector are made available by this part of the circuit.
The kinetic Hamiltonian is likewise responsible for the entanglement between the subsystems, as the pair creation/annihilation processes are combined with a correction of the gauge field.
The layer of parameterized controlled-$i$SWAP gates hence takes the role of the annihilation operator $\hat{U}$ and entangles the matter and gauge subsystems.
In the effective Hamiltonian of Eqs.~\eqref{eq:singlePlaquetteHamiltonian}, the gauge degree of freedom lies on the $(2,3)$ link and is directly coupled to matter sites $2$ and $3$. Accordingly, the circuit couples the gauge field with only these two fermions, which act as controls in the layer of controlled-$i$SWAP gates.
Finally, parameterized $i$SWAP gates are applied on the matter qubits again to adjust the state after entangling the two subsystems, and a layer of single-qubit $z$-rotations is utilized to correct for relative phases. Other single-qubit operations are avoided as they are generally not magnetization preserving. We highlight that this circuit (as well as the ones in Fig.~\ref{fig:periodicCircuit}), being Hamiltonian inspired, from one side ensures the capability of exploring the subsector of physical states in the Hilbert space. From the other, it avoids redundant gates and limits the circuit depth, ensuring that barren plateaus \cite{mcclean2018barren,Uvarov_2020} in the energy landscape are prevented \cite{cerezo2021cost} when increasing the truncation $l$ and/or the number of plaquettes.\\
\begin{figure}
\includegraphics[width=\columnwidth]{Fig5.pdf}
\caption{Classical simulation of the proposed experiment for observing the dynamical generation of magnetic fields where $\Omega = 5$ and $m = 0.1$ (see Sec.~\ref{subsec:quantumSimulation_openBoundaryConditions}) using the circuits given in Fig.~\ref{fig:openCircuit}. The blue and black dots represent data points obtained by variational minimization with a total finite measurement budget of $10^7$ measurements for the entire plot. Half of this budget is spent for the black dot alone. The black solid lines are determined via exact diagonalization of the Hamiltonians in Eqs.~\eqref{eq:singlePlaquetteHamiltonianEncoded}. \textbf{(a)} Energy of the variational ground state. \textbf{(b)} Plaquette expectation value $\expval{\Box}$ as a function of $g^{-2}$. All dots are calculated using the exact state corresponding to the optimal variational parameters found by the VQE.}
\label{fig:openResults}
\end{figure}
\\The quantum circuit described above involves a total of $21$ variational parameters. While not strictly necessary, it is beneficial for currently available quantum hardware to reduce the number of variational parameters. This can be done by identifying the intrinsic symmetries of the ground state. By classically simulating the circuit in Fig.~\ref{fig:openCircuit}, we find that the solution space is still accessible by fixing $\theta_{16} = \theta_{19} = \theta_{21} = \pi/2$, and removing the gates shaded in grey in Fig.~\ref{fig:openCircuit}(a), leaving a total of eleven parameters. Furthermore, $\theta_{11}$, $\theta_{12}$ and $\theta_{13}$ can be set to zero outside the transition region $2 \lesssim g^{-2} \lesssim 5$, within which an almost vanishing energy gap between the ground and first excited state requires more precision for correctly estimating the ground state.\\
\\Since the circuit design is based on the structure of the Hamiltonian, the same design principles can be applied to larger scale systems. When adding more plaquettes, additional parametric $i$SWAP gates are used to populate all basis states within the matter and gauge subsystems. Then, the gauge degrees of freedom are coupled to their respective neighbouring matter sites using additional controlled-$i$SWAP gates [in correspondence with Eq.~\eqref{eq:OpenBoundarySpinHamiltoniankin}]. Finally, $z$-rotations adjust the relative phases of all qubits. When increasing the truncation $l$, additional $i$SWAP gates are inserted for populating the newly introduced gauge field states, and controlled-$i$SWAP gates are added for entangling them with the respective matter sites. In both cases -- adding more plaquettes and increasing the truncation cut-off -- a linear increase in the number of qubits and $i$SWAP gates and a quadratic increase in the number of controlled-$i$SWAP operations is expected.\\
\\A classical simulation of the proposed experiment, including statistical noise on the cost function $\mathcal{C}(\boldsymbol{\theta})$ (representative of the probabilistic nature of quantum state measurements -- see App.~\ref{app:optimalPartitioning}), is shown in Fig.~\ref{fig:openResults}. The data points (blue and black dots) correspond to the lowest energies found by the VQE. The energies and the plaquette expectation values are calculated using the exact state corresponding to the optimal variational parameters found for each value of $g^{-2}$. We verified that the VQE resorts to statistical errors affecting $\mathcal{C}(\boldsymbol{\theta})$ that are always lower than the ground and first excited states' energy gap. The black solid lines are obtained via exact diagonalization of the Hamiltonian in Eqs.~\eqref{eq:singlePlaquetteHamiltonianEncoded}. Using the measurement procedure described in App.~\ref{app:optimalPartitioning} and taking statistical error into account, the entire plot corresponds to approximately $10^7$ measurements to be performed on the quantum device. Half of this budget is used for the point at $g^{-2} \simeq 2.18$, where the energy difference between the ground and first excited states is much smaller if compared to other values of $g^{-2}$ (see above).\\
\\The ground state energy found by the VQE approximates well the energy of the exact ground state, as shown in Fig.~\ref{fig:openResults}(a). The plaquette expectation value is sensitive to small changes in the variational state, which leads to relatively large deviations with respect to the results obtained via exact diagonalization in Fig.~\ref{fig:openResults}(b), even for states whose energy is very close to the exact energy. Yet, the variational optimization is able to accurately resolve transitions in the order parameter. The fidelity of the variational ground state with respect to the exact ground state is particularly high in the extremal regions, exceeding $98\%$. All points achieve a fidelity greater than $90\%$.\\
\subsection{Periodic boundary conditions: running coupling}
\label{subsec:trappedIonResults_periodicBoundaryConditions}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig6.pdf}
\caption{VQE circuits for preparing the ground state of a plaquette with periodic boundary conditions. Gate definitions are given in Fig.~\ref{fig:openCircuit}. Gates of the same color in each circuit share a variational parameter after eliminating redundant parameters. The half-shaded gates include an offset of $+\pi/2$ added to the shared parameter, while the grey-shaded gates can be eliminated entirely. \textbf{(a)} Circuit for the electric representation of the Hamiltonian, and \textbf{(b)} for the magnetic representation.}
\label{fig:periodicCircuit}
\end{figure}
\noindent In this section, we provide a VQE protocol for simulating the running coupling in LGTs. As explained in Sec.~\ref{subsec:quantumSimulation_periodicBoundaryConditions}, the running coupling is a genuine 2D effect that can be studied experimentally in a proof-of-concept demonstration by first preparing the ground state of a plaquette with PBC and subsequently measuring the expectation value $\langle \Box \rangle$.\\
\\Differently from the plaquette with OPC, the electric and magnetic representations of the Hamiltonian [see Eqs.~\eqref{eq:periodicHamiltonianElectricEncoded} and \eqref{eq:periodicHamiltonianMagneticEncoded}] are used for different regions of the bare coupling $g^{-2}$. This is done to obtain better convergence to the untruncated result, as the two representations are well suited for the strong and weak coupling regimes, respectively (see Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions} and Ref.~\cite{paper1}). Noting that the definitions in Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding} presented for the electric gauge field and their lowering operators are trivially extended to rotators and plaquette operators, we encode the plaquette with PBC into nine qubits. Rotator $1$ is represented by qubits $1$ through $3$, rotator $2$ by qubits $4$ through $6$, and rotator $3$ by qubits $7$ through $9$ (see Fig.~\ref{fig:periodicCircuit}). Thus, the Hamiltonian becomes
\begin{subequations} \label{eq:periodicHamiltonianElectricEncoded}
\begin{align}
\hat{H}_{\textrm{E}}^{(e)} = & -\frac{g^2}{2} \Big\lbrace \hat{\sigma}_4^z\left( \hat{\sigma}_5^z + 1 \right)\left( \hat{\sigma}_1^z + \hat{\sigma}_7^z + \hat{\sigma}_7^z \hat{\sigma}_8^z \right) \nonumber\\& + \hat{\sigma}_2^z \left[ \hat{\sigma}_1^z \hat{\sigma}_4^z \left( \hat{\sigma}_5^z + 1 \right) -2 \right] -2 \left( \hat{\sigma}_5^z + \hat{\sigma}_8^z + 3 \right) \Big\rbrace, \\
\hat{H}_{\textrm{B}}^{(e)} = &-\frac{1}{2g^2}\Big[ \hat{\sigma}_1^- \hat{\sigma}_2^+ +\hat{\sigma}_2^- \hat{\sigma}_3^+ + \hat{\sigma}_4^- \hat{\sigma}_5^+ + \hat{\sigma}_5^- \hat{\sigma}_6^+ \nonumber\\& + \hat{\sigma}_7^- \hat{\sigma}_8^+ + \hat{\sigma}_8^- \hat{\sigma}_9^+ + \left( \hat{\sigma}_1^- \hat{\sigma}_2^+ + \hat{\sigma}_2^- \hat{\sigma}_3^+ \right) \nonumber\\& \times \left( \hat{\sigma}_4^- \hat{\sigma}_5^+ + \hat{\sigma}_5^- \hat{\sigma}_6^+ \right) \left( \hat{\sigma}_7^- \hat{\sigma}_8^+ + \hat{\sigma}_8^- \hat{\sigma}_9^+ \right) \Big] + \textrm{H.c.}
\end{align}
\end{subequations}
in case $g^{-2} \lesssim 1$, and
\begin{subequations} \label{eq:periodicHamiltonianMagneticEncoded}
%
\begin{align}
\hat{H}_{\textrm{E}}^{(b)} = &~g^2 \sum_{\nu = 1}^{2L} \Bigg\{f_{\nu}^c \sum_{i=1}^{3} \left( \hat{\sigma}_{3i-2}^- \hat{\sigma}_{3i-1}^+ + \hat{\sigma}_{3i-1}^- \hat{\sigma}_{3i}^+ \right)^{\nu} \nonumber\\
& + \frac{f_{\nu}^{s}}{2}\left[\left( \hat{\sigma}_{4}^- \hat{\sigma}_{5}^+ + \hat{\sigma}_{5}^- \hat{\sigma}_{6}^+ \right)^{\nu} - \left( \hat{\sigma}_{4}^+ \hat{\sigma}_{5}^- + \hat{\sigma}_{5}^+ \hat{\sigma}_{6}^- \right)^{\nu}\right] \nonumber\\
& \times \sum_{\mu = 1}^{2L} f_{\mu}^{s} \Big[\left(\hat{\sigma}_{1}^- \hat{\sigma}_{2}^+ + \hat{\sigma}_{2}^- \hat{\sigma}_{3}^+ \right)^{\mu} \nonumber\\
& + \left(\hat{\sigma}_{7}^- \hat{\sigma}_{8}^+ + \hat{\sigma}_{8}^- \hat{\sigma}_{9}^+ \right)^{\mu}\Big]\Bigg\} + \textrm{H.c.}, \\
\hat{H}_{\textrm{B}}^{(b)} = &-\frac{1}{g^2} \Bigg[ \sum_{i = 1}^{3} \cos \left(\frac{\pi \left( \hat{\sigma}_{3i-2}^z + \hat{\sigma}_{3i-2}^z \hat{\sigma}_{3i-1}^z \right)}{2L+1}\right) \nonumber \\
& + \cos \left(\frac{\pi \sum_{i=1}^3 \left(\hat{\sigma}_{3i-2}^z + \hat{\sigma}_{3i-2}^z \hat{\sigma}_{3i-1}^z \right)}{2L+1}\right)\Bigg]
\end{align}
%
\end{subequations}
whenever $g^{-2} \gtrsim 1$. \\
\\As for the case of OBC, the circuit design for a plaquette with PBC is motivated by the structure of the Hamiltonian and employs the same gate set. Due to the differences between the electric and magnetic representations, we use two different VQE circuits which are shown in Fig.~\ref{fig:periodicCircuit}.
Contrary to the plaquette with OBC, the controlled $i$SWAP gate are not used to entangle the matter with the gauge subsystems. Instead, they are motivated by the coupling of the rotators and plaquette operators in the last terms of Eqs.~\eqref{eq:periodicHamiltonianElectricE} and \eqref{eq:periodicHamiltonianElectricB}, respectively, and the corresponding ones in the magnetic representation.\\
\\Both circuits have $19$ variational parameters and allow us to thoroughly explore the associated Hilbert spaces. However, since we are solely interested in the system ground state, it is convenient for NISQ technology to reduce the number of variational parameters by exploiting the symmetries between rotators $1$ and $3$ [which are apparent from the Hamiltonians in Eqs.~\eqref{eq:periodicHamiltonianElectric} and \eqref{eq:periodicHamiltonianMagnetic}] and by identifying additional redundant parameters through classical simulation of the VQE. As a result, in Fig.~\ref{fig:periodicCircuit}(a) we use a single parameter for each of the following sets: $\{\theta_{1},\theta_{2}, \theta_{11}, \dots, \theta_{19}\}$, $\{\theta_{3} ,\dots \theta_{6}\}$, and $\{\theta_{7},\dots, \theta_{10}\}$, as indicated by the color coding. Parameters $\theta_{11}$, $\theta_{13}$, $\theta_{14}$, $\theta_{16}$, $\theta_{17}$, and $\theta_{19}$ include an offset of $+\pi/2$ added to the shared variational parameter, which is indicated by the half-shaded gates.
For the circuit in Fig.~\ref{fig:periodicCircuit}(b), we make the groupings $\{\theta_{1}, \theta_{2}, \theta_{3},\theta_{4}\}$, $\{\theta_{5} ,\dots, \theta_{8}\}$, and $\{\theta_{9}, \theta_{10}\}$, while $\theta_{12}$, $\theta_{15}$, and $\theta_{18}$ can be eliminated, and $\theta_{11}$, $\theta_{13}$, $\theta_{14}$, $\theta_{16}$, $\theta_{17}$, and $\theta_{19}$ are fixed at $\pi/2$. This leaves just three variational parameters for each circuit.\\
\\The circuit for the electric representation of Eqs.~\eqref{eq:periodicHamiltonianElectricEncoded} is shown in Fig.~\ref{fig:periodicCircuit}(a). All qubits are initialized in the input state $\ket{0}$, and NOT gates prepare the vacuum state $\ket{0} = \ket{010}$ for each of the three rotators as the intial state for the VQE. The layer of controlled-$i$SWAP gates reflects the coupling between the rotators in the electric Hamiltonian $\hat{H}_{\textrm{E}}^{(e)}$ [see Eq.~\eqref{eq:periodicHamiltonianElectricE}], which takes the form $-\hat{S}^z_2 (\hat{S}^z_1 + \hat{S}^z_3)$. This term results from the elimination of rotator $4$ as a redundant degree of freedom, and introduces an asymmetry between rotator $2$ and rotators $1$ and $3$. When increasing $g^{-2}$, the ground state spreads from $\ket{0} \ket{0} \ket{0}$ (in the rotator basis) to all other electric levels, and states in which all three rotators have the same sign ($\ket{1}\ket{1}\ket{1}$ and $\ket{-1} \ket{-1} \ket{-1}$) receive the strongest negative contribution. To encourage the VQE to prepare the correct superposition of states, the parameterized controlled-$i$SWAP gates are connected to control the spread of population within rotators $1$ and $3$ based on the population of rotator $2$.\\
\\The circuit for the magnetic representation of Eqs.~\eqref{eq:periodicHamiltonianMagneticEncoded} is shown in Fig.~\ref{fig:periodicCircuit}(b). Its construction is similar to the circuit for the electric representation described above, but rather encourages the flip-flop interactions between rotators described by $\hat{H}_{\textrm{E}}^{(b)}$ [see Eq.~\eqref{eq:periodicHamiltonianMagneticE}]. Both the electric and magnetic circuits maintain constant magnetization of each gauge field, which prevents access to unphysical states.\\
\\Designing the VQE circuit based on the form of the Hamiltonian allows for a scalable architecture. For systems with additional plaquettes, the coupling between rotators remains pairwise, which translates into the addition of controlled-$i$SWAP gates between all pairs of coupled gauge fields, as was described above for OBC. When considering larger truncations $l$, additional $i$SWAP gates are introduced to allow for all gauge field basis states. Additional controlled-$i$SWAP gates are then added to share entanglement in a similar fashion as for the case $l=1$ considered here. In both cases, the scaling is the same as for the OBC circuit, i.e., the number of qubits and the number of $i$SWAPs scale linearly, while the number of controlled-$i$SWAP gates scales quadratically in the worst-case scenario.\\
\\We simulate the proposed experiment classically, including statistical noise on the cost function $\mathcal{C}(\boldsymbol{\theta})$. Our results are shown in Fig.~\ref{fig:periodicResults}. Points obtained with the electric and the magnetic representation of the Hamiltonian are shown in blue and red, respectively, while the black solid lines come from exact diagonalization of the Hamiltonians. As for the OBC, the energy and the plaquette expectation value $\langle \Box \rangle$ are calculated using the exact state obtained with the optimal variational parameters found by the VQE. Using the measurement procedure described in App.~\ref{app:optimalPartitioning} and taking statistical errors into account, the entire plot corresponds to $6\times10^5$ measurements to be performed on the quantum device.\\
\\The VQE protocol reaches the correct ground state energy [see Fig.~\ref{fig:periodicResults}(a)], and the expectation value of the plaquette operator is accurate if compared to the exact truncated results [Fig.~\ref{fig:periodicResults}(b)]. The fidelity of the variational ground state with respect to the exact ground state exceeds $96\%$ for all points, and for the majority of points it exceeds $99\%$.\\
\begin{figure}
\includegraphics[width=\columnwidth]{Fig7.pdf}
\caption{Classical simulation of the proposed experiment for observing the running of the coupling (see Sec.~\ref{subsec:quantumSimulation_periodicBoundaryConditions}) using the circuits given in Fig.~\ref{fig:periodicCircuit}. The red and blue data points are obtained from variational minimization with a total finite measurement budget of $6 \times 10^5$ measurements for the entire plot. The electric (magnetic) representation is shown in blue (red), and the black solid lines are determined via exact diagonalization of the Hamiltonians in Eqs.~\eqref{eq:periodicHamiltonianElectric} and \eqref{eq:periodicHamiltonianMagnetic}. \textbf{(a)} Energy of the variational ground state using the electric representation in the region $g^{-2} < 1$ and using the magnetic representation in the region $g^{-2} > 1$. \textbf{(b)} Plaquette expectation value $\expval{\Box}$ as a function of $g^{-2}$. All dots are calculated using the exact state corresponding to the optimal variational parameters found by the VQE.}
\label{fig:periodicResults}
\end{figure}
\subsection{Errors in ion platforms}
\label{subsec:trappedIonResults_errors}
\noindent In this section, we discuss the effects of experimental imperfections, as well as statistical noise, on our protocols. Since the results in Secs.~\ref{subsec:trappedIonResults_openBoundaryConditions} and \ref{subsec:trappedIonResults_periodicBoundaryConditions} are derived assuming an ion-based quantum computer \cite{zhang2017observation,shehab2019noise,RevModPhys.93.025001,kim2010quantum,rajabi2019dynamical,chertkov2021holographic}, we consider here the main sources of errors in such platforms. Our considerations are based on the experimental apparatus in Refs.~\cite{kokail2019self,marciniak2021optimal}, that has been previously used for LGT simulations in 1D. If the reader is interested in using a superconducting device, we refer to Refs.~\cite{arute2019quantum, corcoles2015demonstration,PhysRevLett.122.080504}.\\
\\In the following, we distinguish between extrinsic and intrinsic errors. Extrinsic errors are determined by the characteristics of the experimental apparatus and include imperfect gate operations, dephasing, and systematic errors such as offsets in the variational parameters $\boldsymbol{\theta}$. By contrast, intrinsic errors are inherent to the VQE protocol and as such unavoidable. These are statistical errors, and follow from the stochastic nature of quantum measurements (see App.~\ref{app:optimalPartitioning}).\\
\\As discussed in Refs.~\cite{kokail2019self,marciniak2021optimal}, for low circuit depths intrinsic errors are the dominant source of noise. Since the cost function $\mathcal{C}(\boldsymbol{\theta})$ is estimated from independent contributions (see App.~\ref{app:optimalPartitioning}),
we add up the uncertainties due to each term for separately. This substantially increases the uncertainty of $\mathcal{C}(\boldsymbol{\theta})$, even for eigenstates of the considered Hamiltonian. As an example, for each value of the cost function, obtaining a statistical error smaller than the energy gap between the ground and first excited state requires averaging several thousands of experimental shots. This is expected to be the main, and largest overhead from noise sources in our proposal.\\
\\
Extrinsic errors have limited impact for the small system sizes considered here. Indeed, VQE schemes are robust against systematic over- or under-rotation in one- and two-qubits gates \cite{mcclean2016theory}. This follows from the fact that the classical optimizer determines the variational parameters $\boldsymbol{\theta}$ by minimizing the cost function. As such, since the shifts in $\boldsymbol{\theta}$ are slowly varying if compared to the time required for the cost function minimization, these errors are compensated by the classical subroutine. Furthermore, imperfect gates and finite coherence time limit the number of gates and the circuit depth that can be used, respectively. This results in a limitation in the number of plaquettes and truncation $l$ that can be studied. Such limitation can however be relaxed by technological improvements.\\
\section{Analytical interpretation of the results}
\label{sec:quantumSimulationOf2DEffects}
\noindent In this section, we use the effective Hamiltonian description provided in Sec.~\ref{sec:encodedHamiltonian} to study lattice QED with OBC (Sec.~\ref{subsec:quantumSimulation_openBoundaryConditions}) and PBC (Sec.~\ref{subsec:quantumSimulation_periodicBoundaryConditions}). While next generation quantum computers will widen the scope of our approach, we focus on phenomena that can be studied with small system sizes and for which quantum simulations can be carried out on current quantum hardware \cite{martinez2016real, nam2020ground, arute2019quantum, corcoles2015demonstration, bernien2017probing, labuhn2016tunable}. Here, we motivate the relevance of the results of the quantum simulations in Sec.~\ref{sec:main_result}.
\subsection{Dynamical matter and magnetic fields}
\label{subsec:quantumSimulation_openBoundaryConditions}
\noindent We describe a minimal example that allows for the study of 2D effects in lattice QED with OBC. More specifically, we examine the appearance of dynamically generated magnetic fields due to particle-antiparticle creation processes within a single plaquette. This effect manifests itself in an abrupt change of the ground state as a result of the competition between the kinetic and magnetic terms in the QED Hamiltonian of Eqs.~\eqref{eq:singlePlaquetteHamiltonian}. There are two parameter regimes in which this phenomenon occurs. In one, the parameter $\Omega$ [see Eq.~\eqref{eq:singlePlaquetteHamiltonianHkin}] is dominant if compared to the mass and the bare coupling $g$, allowing the kinetic and the magnetic terms to be the leading contributions to the energy. Alternatively, the ground state's shift appears when the mass is smaller than zero (or with positive mass and non-zero background field). This last scenario is hard to simulate using MCMC methods, in which the use of the inverse of the lattice Dirac operator in combination with a negative mass induces zero modes \cite{Gattringer:2010zz}, leading to unstable simulations. We note that in 1D QED, considering a negative fermion mass is equivalent to a theory with positive mass in the presence of a topological term $\theta = \pi$ (see, e.g., \cite{funcke2020topological}) \footnote{In principle, the fermion mass can be written with a phase factor $m e^{i\theta}$. In 1D QED, the anomaly equation (see the appendix of Ref.~\cite{funcke2020topological}) can be used to eliminate this phase by a rotation, and it appears as the topological term in the gauge field. This means that by working with a negative fermion mass, the model is equivalent to a theory with a $\theta$ term at $\theta = \pi$.}. It will be interesting to investigate the relation of the negative fermion mass to a theory in the presence of a topological term in $2$ dimensions.\\
\\The proposed experiment consists of preparing the ground states of Eqs.~\eqref{eq:singlePlaquetteHamiltonian} for different values of the coupling $g$, and subsequently measuring the magnetic field energy, which is proportional to $\expval{\Box}$ [see Eq.~\eqref{eq:PlaquetteOperator}]. In the strong coupling regime $g^{-2} \ll 1$, the magnetic field energy vanishes $\expval{\Box} = 0$, since the ground state approaches the bare vacuum $\ket{vvvv}\ket{0}$. For weak couplings, $g^{-2} \gg 1$, the magnetic term dominates and the ground state is a superposition of all electric field basis elements. As such, $\expval{\Box}$ converges to 1 when $g^{-2} \rightarrow \infty$. However, the truncation described in Sec.~\ref{subsec:encodedHamiltonian_LatticeQEDIn2+1Dimensions} bounds $\expval{\Box}$ to a smaller value \cite{paper1}.\\
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Fig8.pdf}
\caption{\textbf{(a)} Plaquette expectation value as a function of $g^{-2}$ for a single plaquette with mass $m = -50$ and kinetic strength $\Omega = 5$. Regions $1$, $2$, and $3$ correspond to the numbered regions in part (b) below. Exact diagonalization of the truncated model is represented by the black dashed line, and perturbation theory was used for the blue, green, and orange lines. The insets are magnifications of parts of regions 1 and 2 (blue and green, respectively), plotting the difference between exact diagonalization and perturbation theory of order reported in the legend to confirm convergence to the full U(1) theory. The difference is not completely vanishing due to the small contribution of the magnetic term which is ignored in perturbative calculations. \textbf{(b)} Schematic representation of the ground and first excited state's energy.}
\label{fig:plaquettePerturbationPlot}
\end{figure}
\\To explain the underlying physics in the intermediate region between the strong and weak coupling regimes, we first examine the case in which the kinetic term of the Hamiltonian can be treated as a perturbation (Fig.~\ref{fig:plaquettePerturbationPlot}). Perturbative calculations are valid for $1 \ll \Omega \ll |4m|$, and we show results for $\Omega = 5$ and $m = -50$. We then proceed to the non-perturbative case (Fig.~\ref{fig:plaquetteNonPerturbative}) with parameters $\Omega = 5$ and $m = 0.1$. Our analytical results within the perturbative regime are determined using our iterative algorithm described in App.~\ref{app:PerturbationTheory}, and are unaffected by truncation effects in the parameter $l$. Therefore, they also serve as a test of validity for exact results.\\
\\For $\Omega = 5$ and $m = -50$, the kinetic term can be treated as a perturbation, and we take the sum of the electric and mass terms in Eqs.~\eqref{eq:singlePlaquetteHamiltonianHE} and \eqref{eq:singlePlaquetteHamiltonianHm} as the bare Hamiltonian. We can thus well describe the strong coupling regime $g^{-2} \ll 1$, in which a 2D effect occurs that is characterized by a jump of the expectation value of the plaquette operator. A plot of $\expval{\Box}$ as a function of $g^{-2}$ is shown in Fig.~\ref{fig:plaquettePerturbationPlot}(a), where the black dashed line is obtained using exact diagonalization of the Hamiltonian in Eqs.~\eqref{eq:singlePlaquetteHamiltonian} for a truncation $|l| = 1$, and the coloured lines correspond to perturbative calculations. We begin by analyzing the system for $g^{-2} \ll 1$ (marked as region $1$ in Fig.~\ref{fig:plaquettePerturbationPlot}), and examine increasing values of $g^{-2}$ as we move through regions $2$
and $3$.\\
\\In region $1$, the ground state of the Hamiltonian is essentially the bare vacuum $\ket{vvvv}\ket{0}$, which is characterized by null energy and $\expval{\Box} = 0$. The dominating electric term prevents the creation of electric field, despite the mass term incentivizing particle-antiparticle pair creation ($m <0$). The magnetic term is negligible in the strong coupling regime. With increasing $g^{-2}$, the cost of creating electric fields is reduced, until it becomes favourable to create particle-antiparticle pairs. In particular, for $g^{-2} = -(4m)^{-1}$, the energy relief $2m$ of creating a pair compensates the cost $g^2/2$ of creating an electric field on the link connecting the pair.\\
\\The kinetic term $\hat{H}_{\textrm{kin}}$ is responsible for the energy anti-crossing between regions $1$ and $2$ as it allows the creation of particle-antiparticle pairs and the corresponding electric fields. It couples the vacuum to the plaquette states which contains the maximum number of particle-antiparticle pairs (we refer to this as a fully-filled plaquette). However, Gauss' law allows two different gauge field configurations for the fully-filled plaquette, as shown in Fig.~\ref{fig:openBoundaryConventions}(e). These states are given by $\ket{f_{\pm}^{(0)}} = \frac{1}{\sqrt{2}}\ket{epep}(\ket{0} \pm \ket{1})$, and while they are degenerate with respect to the bare Hamiltonian $\hat{H}_{\textrm{E}} + \hat{H}_{\textrm{m}}$, $\expval{\hat{H}_{\textrm{kin}}}$ is minimized for $\ket{f_{-}}$ and $\expval{\hat{H}_{\textrm{B}}}$ for $\ket{f_{+}}$. The existence of these two configurations in the presence of the perturbation $\hat{H}_{\textrm{kin}}$ and the magnetic term $\hat{H}_{\textrm{B}}$ creates competition between two quasi-degenerate vacua in regions $2$ and $3$ of Fig.~\ref{fig:plaquettePerturbationPlot}(b).
The ground states in these regions are described by the corresponding corrected states in perturbation theory $\ket{f_{\pm}} = \sum_{n} \ket{f_{\pm}^{(n)}}$ (see App.~\ref{app:PerturbationTheory}), as shown in Fig.~\ref{fig:plaquettePerturbationPlot}(b). Here, we see that the kinetic term facilitates the creation of particle-antiparticle pairs and drives the ground state from the vacuum in region $1$ towards $\ket{f_{-}}$ in region $2$.\\
\\The remarkable 2D feature of the theory is the jump of $\expval{\Box}$ between regions $2$ and $3$, which is shown by the black dotted line in Fig.~\ref{fig:plaquettePerturbationPlot}(a). This jump corresponds to the sharp energy anti-crossing in Fig.~\ref{fig:plaquettePerturbationPlot}(b), and follows from the competition between the quasi-degenerate vacua in the presence of the kinetic and magnetic terms. As the relative weights of $\hat{H}_{\textrm{kin}}$ and $\hat{H}_{\textrm{B}}$ change with $g^{-2}$, an anti-crossing occurs and the ground state changes from $\ket{f_{-}}$ to $\ket{f_{+}}$. More specifically, there is a value $g_{\textrm{c}}$ such that, for any $g^{-2} > g_{\textrm{c}}^{-2}$, $E_{\textrm{B}}^{(+)} + E_{\textrm{kin}}^{(+)} < E_{\textrm{B}}^{(-)} + E_{\textrm{kin}}^{(-)}$, where $E_{\alpha}^{(\pm)} = \bra{f_{\pm}} \hat{H}_{\alpha} \ket{f_{\pm}}$. Despite the fact that the magnetic Hamiltonian is not included in our perturbative analysis, we can analytically calculate this value $g_{\textrm{c}}$ (see App.~\ref{app:PerturbationTheory}).
By requiring the magnetic and kinetic contributions from the energy to be equal, we find $g_{\textrm{c}} = 0.012 + o(\Omega^8)$, which is in excellent agreement with the results obtained from exact diagonalization [see Fig.~\ref{fig:plaquettePerturbationPlot}(a)]. Following the jump, in region $3$, the ground state is thus $\ket{f_+}$, until we reach the weak coupling regime and the magnetic term becomes dominant compared to all other contributions.\\
\\The analysis above shows how pair creation processes can lead to dynamically generated magnetic fluxes that result in negative values of the magnetic field energy. For negative mass, the considered effect of competing vacua occurs over a wide range of parameters. The strength of the kinetic term $\Omega$ broadens the dip and shifts the position of the jump $g_{\textrm{c}}$.\\
\begin{figure}
\includegraphics[width=\columnwidth]{Fig9.pdf}
\caption{Ground state properties in the non-perturbative regime. \textbf{(a)} Plaquette operator expectation value $\expval{\Box}$ and entanglement entropy $\mathcal{S}(\rho_g)$ as a function of $g^{-2}$. \textbf{(b)-(d)} The probability by component of the ground state for the indicated $g^{-2}$. At $g^{-2} = 10^{-3}$, the lowest energy state is the vacuum $\ket{vvvv}\ket{0}$. For $g^{-2} = 1$, the ground state approximates that of the kinetic term. For large $g^{-2}$, the magnetic Hamiltonian in the presence of the kinetic Hamiltonian yields the tensor product between a superposition of matter states and the gauge ground state of the magnetic term. Through (a) - (d), we used $\Omega = 5$ and $m = 0.1$, and a gauge field truncation $l =1$.}
\label{fig:plaquetteNonPerturbative}
\end{figure}
\\In the following, we consider the regime in which the kinetic term cannot be treated as a perturbation. For $\Omega = 5$ and $m = 0.1$, exact diagonalization results are shown in Fig.~\ref{fig:plaquetteNonPerturbative}(a). In the intermediate region where $g \simeq 1$, the kinetic term has significant weight, which leads to entanglement between matter and gauge degrees of freedom in the ground state. In particular, in the limit $\Omega \gg |m|,g^{2},g^{-2}$ and for $l \rightarrow \infty$, it can be shown for $N$ plaquettes \cite{lieb1994flux} that the energy is minimized for a magnetic flux of $\pi$, which is generated by pair creation processes and corresponds to $\expval{\Box} = -1$ (which is not reached in Fig.~\ref{fig:plaquetteNonPerturbative}(a) due to the effects of truncation). Figures~\ref{fig:plaquetteNonPerturbative}(b)-(d) show the probabilities of the components of the ground state at different values of $g^{-2}$.
In the strong and weak coupling regimes, the ground states are the vacuum $\ket{vvvv}\ket{\textrm{GS}^{(e)}}$ and $\frac{1}{2}(\ket{vvvv}-\ket{epep} + i\ket{epvv} + i\ket{vvep}) \ket{\textrm{GS}^{(b)}}$, respectively, where $\ket{\textrm{GS}^{(e)}} = \ket{0}$ is the gauge component of the ground state of the electric term and $\ket{\textrm{GS}^{(b)}}$ is that of the magnetic term (see App.~\ref{app:newAppendix}). For $g^{-2}=1$, however, the ground state approximates the ground state of the truncated kinetic term for $l = 1$ with $92\%$ fidelity (which can be increased by incrementing $\Omega$). To quantify the ground state entanglement between matter and gauge degrees of freedom, we calculate the entanglement entropy $\mathcal{S}(\rho_g) = -\textrm{Tr}[\rho_g \log \rho_g]$, where $\rho_g$ is the density matrix of the gauge degrees of freedom that remain after eliminating those that are redundant. The entanglement entropy is plotted as the red dashed line in Fig.~\ref{fig:plaquetteNonPerturbative}(a).\\
\subsection{Running coupling}
\label{subsec:quantumSimulation_periodicBoundaryConditions}
\noindent In this section, we consider 2D effects in QED on a lattice subject to PBC. The method described in Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions} and in Ref.~\cite{paper1} is based on the Hamiltonian formalism of LGTs, which allows for simulations that are unaffected by the problem of autocorrelations inherent to MCMC methods. Thus, we can compute physical observables at arbitrary values of the lattice spacing, ultimately allowing for reaching a well-controlled continuum limit. As such, future quantum hardware could be used to efficiently calculate, for instance, the bound state mass spectrum of the theory, properties of the inner structure of such bound states, or form factors which are important for experiments \cite{Gattringer:2010zz, aoki2020flag}.\\
\\We are therefore bound to consider local observables which only need a small number of lattice points.
We consider the plaquette operator $\Box$ as a simple example, as was done in the pioneering work by Creutz \cite{creutz1983monte} in the beginning of MCMC simulations
of LGTs. Despite its local nature, the operator $\Box$ can be related to a fundamental parameter of the theory, namely, the
renormalized coupling $g_{\rm ren}$ \cite{Booth:2001qp}. Importantly, the 1D Schwinger model is
super-renormalizable, meaning that the coupling does not get renormalized, while, as we will show, renormalization is necessary for our 2D model.\\
\\Renormalization appears in quantum field theories through quantum fluctuations, i.e., the spontaneous generation of particle-antiparticle pairs from the vacuum. This phenomenon leads to a charge shielding (or anti-shielding for non-Abelian gauge theories) \cite{Peskin:1995ev}, which in turn changes the strength of the charge depending on the distance (or the energy scale) at which it is probed. As such, the charge becomes {\em scale dependent} and in this work we choose the inverse lattice spacing $1/a$ as the scale. The scale dependence, which is a renormalization effect, is referred to as the {\em running} of the coupling and its knowledge is fundamental in understanding the interactions between elementary particles.
In particular, the running coupling serves as an input to interpret results from
collider experiments, such as the Large Hadron Collider. Hence, the ability to compute the running
coupling from the plaquette operator can have
a direct impact on such experiments and our knowledge of elementary particle interactions.\\
\\As explained in Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions}, we study the case of two-dimensional QED without matter. Despite the absence of matter, and in contrast to the previously studied 1D Schwinger model, renormalization is needed, and hence the calculation of the running coupling. A definition of the renormalized coupling $g_{\textrm{ren}}$ can be given through the ground state expectation value of the plaquette $\expval{\Box}$ \cite{Booth:2001qp}
\begin{equation}
g_{\rm ren}^2 = \frac{g^2}{\langle \Box \rangle^{1/4}},
\label{eq:plaquettecoupling}
\end{equation}
where $g$ is the bare coupling from the Hamiltonian [see Eqs.~\eqref{eq:periodicHamiltonianElectric} and \eqref{eq:periodicHamiltonianMagnetic}].
By looking at Eq.~\eqref{eq:plaquettecoupling}, it follows that to cover the scale dependence of the coupling, we need to evaluate $\expval{\Box}$ over a broad range of the lattice spacing $a$. Equivalently (we set $a=1$ above), this means that we need to perform simulations at many values of $g^{-2}$ , covering the whole spectrum between the extremal regions of strong and weak couplings, $g^{-2} \ll 1$ and $g^{-2} \gg 1$, respectively. In particular, the interesting region where perturbation theory is no longer applicable and bound states can be computed on not too large lattices is where $g^{-2} \simeq 1$. Covering such a broad range of scales is the major problem in standard lattice gauge simulations. In fact, while approaching the weak coupling regime, autocorrelation effects prevent calculations from reaching very small values
of the lattice spacing, making continuum extrapolations and hence convergence to meaningful results difficult. We remark that in large scale lattice simulations, much more sophisticated definitions of the renormalized coupling $g_{\rm ren}$ are generally used \cite{Bruno:2017gxd, aoki2020flag}. These alternative definitions allow, besides other things, to disentangle cut-off effects, inherent to the plaquette coupling, from the true physical running. However, the alternative forms of $g_{\textrm{ren}}$ described in \cite{Bruno:2017gxd, aoki2020flag} are meaningful for large lattices only, and are hence beyond the capabilities of current quantum simulators. In this work, we allow for a proof-of-concept demonstration which paves the way to future improvements.\\
\\With our method (see Sec.~\ref{subsec:encodedHamiltonian_effectiveHamiltonianForPeriodicBoundaryConditions} and Ref.~\cite{paper1}), we can simulate the system both in the strong and in the weak coupling regimes, with very modest truncations. The most difficult region to simulate is characterized by $g^{-2} \simeq 1$, where convergence is studied in Ref.~\cite{paper1}. We demonstrate this in Fig.~\ref{fig:plaquettePeriodicPlot}, where the ground state expectation value of the plaquette operator $\expval{\Box}$ is plotted against $g^{-2}$. For PBC, the plaquette operator is given by $\Box = -\frac{g^2}{4} \hat{H}_{\textrm{B}}^{(\gamma)}$ where $\gamma = e$ ($\gamma = b$) refers to the electric (magnetic) representation [see Eqs.~\eqref{eq:periodicHamiltonianElectricB} and
\eqref{eq:periodicHamiltonianMagneticB}].
Accordingly, the blue and red lines are determined with these two representations, while the grey, dotted lines are obtained using perturbation theory, and describe well the exact results in the extremal regions. The left graph is for a truncation $l=1$ and the right for $l=2$.\\%{\color{green} delete the following? The following is a repetition from \cite{paper1}...}\\
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Fig10.pdf}
\caption{Ground state expectation value of the plaquette operator $\Box$ for a pure gauge periodic boundary plaquette (see Fig.~\ref{fig:periodicBoundaryConventions}). Results obtained using the electric and magnetic representation are shown as blue and red lines, respectively. Perturbative calculations are shown as grey dashed lines. The plots correspond to two different truncations $l=1$ and $l=2$, as indicated above. The blue (red) shaded region indicates the values of $g^{-2}$ for which the the electric (magnetic) representation achieves the best results.}
\label{fig:plaquettePeriodicPlot}
\end{figure}
\\In Sec.~\ref{sec:main_results}, we provide a protocol for quantum simulating
the system with a truncation $l = 1$ of the gauge field using 9 qubits.
As shown in Fig.~\ref{fig:plaquettePeriodicPlot}, using a truncation $l = 2$ around the region of the plot where $g^{-2} \simeq 1$ brings us much closer to the converged result. Using the same protocol as for $l = 1$ (see Sec.~\ref{subsec:trappedIonResults_periodicBoundaryConditions}),
the simulation for $l = 2$ requires $15$ qubits.
\begin{comment}
\section{Details on the formulation for quantum hardware}
\label{sec:variationalQuantumSimulationWithQubits}
\noindent In the previous sections, we presented an effective Hamiltonian description along with $2$D phenomena to be studied. In the following, we explain the software protocols for simulating our models on quantum hardware. Assuming a universal qubit-based quantum computer, in Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding} we provide an efficient encoding of the gauge fields in terms of qubits. Since currently available quantum computers have limited system size and number of gates, it is advantageous to employ a VQE-based approach, as explained in Sec.~\ref{subsec:variationalQuantumSimulation_variationalQuantumSimulation}. To facilitate a VQE of the complicated models described in Secs.~\ref{subsec:quantumSimulation_openBoundaryConditions} and \ref{subsec:quantumSimulation_periodicBoundaryConditions}, in Sec.~\ref{subsec:variationalQuantumSimulation_measurementDecomposition} we present an efficient measurement scheme suitable for any qubit-based quantum device. This scheme reduces the experimental requirements for implementation by efficiently grouping commuting observables. Concrete VQE circuits will be discussed in Sec.~\ref{sec:main_results}. For NISQ hardware, the question of optimal circuit design becomes platform-specific, and we will provide a detailed experimental proposal for trapped ions.
\end{comment}
\section{Conclusions \& outlook}
\label{sec:conclusionsAndOutlook}
\noindent In this work, we proposed a protocol to observe 2D effects in LGTs on currently available quantum computers. By using the methods in Ref.~\cite{paper1}, we provided a practical VQE-based framework to simulate two toy models using NISQ devices. Importantly, we include the numerics for observing 2D phenomena in a basic building block of 2D QED with present-day quantum resources. \\
\\The effective models studied here include both dynamical matter and a non-minimal gauge field truncation, providing the novel opportunity to study several 2D effects in LGTs. More specifically, we showed how to observe dynamical generation of magnetic fields as a result of particle-antiparticle pair creation, and paved the way for an important first step towards simulating short distance quantities such as the running coupling of QED. While the protocols presented in Sec.~\ref{sec:main_results} are designed for trapped ion systems, our approach can be easily adapted to suit different types of quantum hardware.\\
\\One of the most appealing characteristics of our approach is that it can be generalized to more complex systems. Immediate extensions of our results include simulations of a 2D plane of multiple plaquettes, and of QED in three spatial dimensions. Moreover, including fermionic matter in the case of PBC can be done by following the same procedure as for OBC \cite{paper1}. It will also be interesting to explore implementations on non-qubit based hardware that is capable of representing gauge degrees of freedom with spin-$l$ systems ($l>1/2$) \cite{senko2015realization}. Another extension is the development of Trotter-type protocols to simulate real-time evolution
using the encoding given in Sec.~\ref{subsec:variationalQuantumSimulation_qubitEncoding}. As part of the quest to move towards quantum simulations of QCD, another possible extension is to progress from a U(1) gauge theory (QED) to a non-Abelian gauge theory, such as SU(2), and eventually to SU(3). Importantly, simulations of LGTs with larger lattice sizes will become feasible with future advancements in quantum computing, allowing for exciting possibilities, such as the ability to relate the running coupling of a gauge theory to a physical parameter and to make connections between quantum simulations and experiments in high energy physics. Ultimately, future quantum computers may offer the potential to also simulate models with a topological term, non-zero chemical potential, or real-time phenomena, effects which are very hard or even impossible to access with MCMC techniques.\\
\\The field of quantum simulations of LGTs is in its early exploratory stages and is rapidly developing. The type of problems considered here will act as test-bed for quantum computer implementations. By providing practical and experimentally feasible solutions for simulating gauge theories beyond 1D, our work opens up new pathways towards accessing regimes that are classically out of reach.
\section*{Acknowledgements}
\noindent We thank Rainer Blatt, Philipp Schindler, Thomas Monz, Raymond Laflamme, and Michele Mosca for fruitful and enlightening discussions, and Luca Masera for computational consulting.
This work has been supported by Transformative Quantum Technologies Program (CFREF), NSERC and the New Frontiers in Research Fund.
JFH acknowledges the Alexander von Humboldt Foundation in the form of a Feodor Lynen Fellowship.
CM acknowledges the Alfred P. Sloan foundation for a Sloan Research Fellowship.
AC acknowledges support from the Universitat Aut\'{o}noma de Barcelona Talent Research program,
from the Ministerio de Ciencia, Inovaci\'{o}n y Universidades (Contract No. FIS2017-86530-P),
from the the European Regional Development Fund (ERDF) within the ERDF Operational Program of Catalunya (project QUASICAT/QuantumCat),
and from the the European Union's Horizon 2020 research and innovation programme under the Grant Agreement No. 731473 (FWF QuantERA via QTFLAG I03769). Research in Innsbruck is supported by the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 817482 (PASQuanS).\\
|
2111.02945
|
\section{Introduction}
\label{sec:intro}
We have long known that the
halo of our own Milky Way galaxy extends over many kiloparsecs and that it is principally populated
with metal-poor stars \citep{ELS62, searle+zinn78, freeman1987}. The high level of inhomogeneity, the clear presence of stellar streams
and substructure, and the possible dual nature of the halos of Milky Way and M31
are among the more recent results that have changed our understanding of galaxy halo assembly
\citep[e.g.,][]{ferguson+02,belokurov+06,carollo+07,bell+08,
sesar+11,drake+13,ibata+14}.
A major difference for studies of stellar halos around nearby galaxies beyond the Local Group was made by the high resolution and high sensitivity of the Hubble Space Telescope (HST) cameras, which can resolve and measure individual halo stars routinely in galaxies at distances out to about 10~Mpc \citep{harris+07a,rejkuba+11,peacock+15,monachesi+16,cohen+20}. With special efforts to accumulate longer exposure times, galaxies out to the distance of the Virgo system are within reach \citep{williams+07,bird+10}.
Direct starcounts of RGB stars make a highly
effective route to tracing the halo population outward to levels of
equivalent surface brightness that are difficult to achieve by other means
\citep[e.g.,][in addition to the papers cited above]{pritchet_vandenbergh1994,harris+07b,gilbert+2012,harmsen+2017}.
Integrated-light studies can be quite effective at identifying the spatial distribution and structure of the halo \citep{mihos+13, duc+15, merritt+16, iodice+16, iodice+19, beilek+20}, in spite of challenging photometry at extremely low surface brightness levels (see \citealt{mihos19}
for a recent review of advances and challenges related to deep imaging of diffuse light around galaxies). However, integrated-light studies do not allow an unambiguous description of the properties of the stellar populations because of the age-metallicity degeneracy \citep[see, e.g.,][]{mihos+13}, and furthermore, they do not give information on the \textit{\textup{distributions}} of age and metallicity.
These underlying distributions
can be addressed by analyzing the color-magnitude diagram (CMD) of resolved stars.
Even with the HST, only a few large early-type galaxies (ETGs) are especially amenable to resolved stellar population studies. Of these,
NGC 5128 (often referred to by its well-known radio source designation
Centaurus A or Cen A) has a special place: At a distance of 3.8~Mpc \citep{rejkuba04,
harris+10}, it is by far the nearest easily observable giant ETG.
Because of its proximity, NGC 5128 was the first giant ETG
to have its stellar halo
resolved into individual red giant branch (RGB) stars
\citep{soria+96}, and it has since been observed with the HST within several programs. It has also been imaged under very good seeing conditions from the ground.
Several earlier studies concentrated on the visible star formation in the inner halo
\citep{fassett+graham00,mould+00,rejkuba+01,rejkuba+02,graham+fassett02,peng+02,crockett+12},
which is associated with the so-called inner and outer filaments \citep{blanco+75, santoro+15}
and is possibly triggered by the radio jet \citep{oosterloo+morganti05}. Although visibly quite prominent, these features contain only a small fraction of the halo mass
\citep{rejkuba+04,oosterloo+morganti05}, have a low star formation efficiency \citep{salome+16},
and are confined to a region extending out to$\simeq 35$ kpc along the northeastern major axis \citep{neff+14}. Within a radius of $\sim 20$~kpc lie also well-known shells around the galaxy \citep{malin+83,peng+02} that were likely formed during a past
accretion of a companion galaxy that deposited the gas and dust that has long been noted as a characteristic of NGC~5128 \citep{charmandaris+00}. \citet{struve+10} pointed out that the H\i\, fraction in Centaurus~A ($M_{\mathrm{HI}}/L_B = 0.01$) is rather low for an ETG, suggesting that the most recent accretion involved a relatively small (Small Magellanic Cloud like) galaxy about $1.6 - 3.2 \times 10^8$~yr ago. An alternative major-merger origin was proposed by numerical simulations that reproduced some selected properties of NGC~5128 \citep{bekki+peng06,wang+20}. Setting this in a broader context:
\citet{tal+09} found that 73\%
of nearby luminous elliptical galaxies show tidal disturbance signatures in their stellar
bodies; similar results are evident from the MATLAS survey \citep{duc+15}. Thus
NGC~5128, far from being ``peculiar'', is a quite typical giant elliptical \citep[see also the review of][]{harris10}.
A systematic survey of the stellar halo properties in NGC~5128 was performed with HST imaging
that reached at least 1.5 mag below the RGB tip (TRGB), deep enough to sample its full metallicity distribution. This survey started with the
works of G.~Harris and collaborators over 20 years ago. Inner- and mid-halo fields were
observed with the WFPC2 camera at 8, 21, and 31 kpc distance from the center
\citep{harris+99,harris+harris00,harris+harris02}, establishing for the first time a (then) surprisingly broad
metallicity distribution of the halo RGB stars, with a peak metallicity
close to solar in the innermost field and only mildly subsolar in the other two
fields. Low-metallicity stars reaching [Fe/H] $\simeq -2$~dex are present in all these
fields, but are very much in the minority.
After the HST servicing mission 3B and the installation of the Advanced Camera
for Surveys (ACS), we carried out a much deeper photometric probe in
a field 38 kpc SSE of the center, with sufficiently faint limits to detect the core helium burning stars
located in the red clump \citep{rejkuba+05}. This deep view of the halo not
only further confirmed the metal-rich nature of the halo stars, but also permitted a
quantitative estimate of the halo age distribution: 70-80\% of the stars formed $12\pm 1$~Gyr ago, and the
remaining 20-30\% population is best fit with 2-4 Gyr old
models \citep[][henceforward R11]{rejkuba+11}.
Remarkably, these deep data indicate that the full metallicity range of the
models ($Z=0.0001-0.04$) combined with old ages needs to be used to reproduce the
colors of the reddest RGB stars in this field.
The last Hubble servicing mission 4 enabled imaging with a wider field of
view by combining the newly installed Wide Field Camera 3 (WFC3) in parallel with ACS.
We designed a program to map five pairs of new locations in
the outer halo of NGC~5128, two along the minor and three along the northeastern major
axis, with the intention of exploring the extent of the halo and the properties of the stars located
in its most extreme regions. The first results from this program were summarized briefly in
\citet[][Paper I]{rejkuba+14}.
We found a transition from a metal-rich inner galaxy
to a lower-metallicity outer halo, with a shallow metallicity gradient
and hints
of possible substructures in the outer halo based on metallicity and number density
variations in neighboring parallel fields. The shape of
the halo was found to be elongated (roughly consistent with the inner halo),
with an excess of stars along the major axis
above the $r^{1/4}$ law fit to the star counts from earlier HST studies. Most
remarkably, halo RGB stars are still present in fields as far out as 140~kpc (25 effective radii
$R_e$) along the major axis, and 90~kpc (16 $R_e$)
along the minor axis. In short, no clear `end' to this galaxy halo has been found.
Complementing the narrow pencil-beam studies conducted with the HST,
the extended halo of NGC 5128 was also surveyed from the
ground with the VIMOS\footnote{VIsible MultiObject Spectrograph (VIMOS) was mounted on the Unit Telescope 3 (UT3) of the Very Large Telescope (VLT) at the European Southern Observatory (ESO) Paranal Observatory and it included imaging in addition to its primary spectroscopic modes.} optical imager on the 8m ESO VLT
\citep{crnojevic+13, bird+15}. The Magellan Megacam imager at Las Campanas Observatory was used to observe NGC 5128
\citep{crnojevic+16} as part of the Panoramic Imaging Survey
of Centaurus and Sculptor \citep[PISCeS;][]{crnojevic+16iaus}.
These wide surveys uncovered a vast amount of substructure, including several dwarfs in the process of being accreted. This process creates overdense regions in the halo. However, the intermediate Galactic latitude of NGC 5128 unfortunately means that large numbers of foreground stars are present, which adds to unresolved background galaxies.
When the field contamination issue is combined with uneven completeness due to observations taken under a range of observing conditions,
studying its halo properties based on RGB star counts is challenging from the ground.
We believe that the best way forward is
to combine the strengths of the wide-area surveys that provide a global view of the halo and substructure with the higher resolution and deeper observations from space that enable detailed investigation of
its stellar composition.
Since our last HST-based study (Paper I), new ACS and WFC3 imaging was secured that primarily focused on confirming the newly discovered dwarf galaxies and substructures \citep{crnojevic+19}, but also added further parallel pointings in the halo of NGC~5128. Armed with the information from the recent wide-area PISCeS survey of the Cen A halo,
we assembled all HST observations taken so far to present a homogeneous analysis of the radial structure of the halo, its metallicity distribution, and metallicity gradients across a vast area of the halo of NGC 5128.
\begin{figure*}
\centering
\resizebox{\hsize}{!}{
\includegraphics{Fig1_HSTfields_CenAdwarfs_insert.png}}
\caption{Distribution of the fields imaged with the HST in F606W and F814W filters
as listed in Table~\ref{tab:HSTobslog} relative to the
center of NGC~5128 projected on the plane of the sky. Gray squares are fields F1-F9, orange squares are for WFC3 parallels W1-W16 from GO13856, and the blue inverted triangle is for the ACS parallel from GO15426.
North is at the top and east at the left. The halo isophotal major
axis is oriented 35\degree \ east of north \citep[counterclockwise;][]{dufour+79}. The elliptical contours
have an axis ratio of 0.77 as determined from the inner halo \citep{dufour+79} and are plotted for
1.5, 4, 5.5, 7, 10.5, 15.5, and 25 $R_e$ distance (as indicated). The green stars
are dwarf galaxies in the Cen A group as listed in Table 4 in \citet{mueller+19}. Most of them have recently confirmed Cen A group membership based on the primary pointings from GO13856 \citep{crnojevic+19}.
The large black square in the center indicates the relative size of the $35'\times 35'$ ($38.6 \times 38.6$~kpc$^2$)
unsharp-masked image of the shells in the central parts of the galaxy shown in
the right panel \citep[][image credit: Eric Peng (JHU), Holland Ford (JHU/STScI), Ken Freeman
(ANU), Rick White (STScI), and NOAO/AURA/NSF]{peng+02}.}
\label{fig:fieldsrel}
\end{figure*}
In this paper we describe in more detail the data and photometric
measurements done for the Cycle 20 observations,
for which initial results were presented in Paper I.
We add 15 more halo fields that were obtained as Parallel pointings within the more recent
Cycle 22 and Cycle 25 HST programs (PI: Crnojevi\'c), essentially doubling the database.
We also carry out a homogeneous analysis of all these halo locations. These parallel fields were only used as control fields for the dwarf galaxy CMD analysis by \citet{crnojevic+19} and were not analyzed in terms of their contribution to the overall stellar halo properties before.
Furthermore, we discuss the foreground and background field contamination using several methods: a control field, Milky Way models, and a combination of optical and near-IR photometry. We finally adopt a statistical decontamination of our CMDs based on the control field.
From the newly calibrated and decontaminated CMDs, we derive a new self-consistent surface-density profile extending outward to the currently detectable limits of this galaxy's giant halo and measure the stellar halo ellipticity.
We discuss the possible presence of intermediate-age asymptotic giant branch (AGB) stars in the halo by
analyzing the numbers of stars populating the CMDs just above the RGB tip.
Throughout the paper, we adopt the distance to NGC 5128 of 3.82~Mpc (distance modulus $(m-M)_0 = 27.91$ mag), which is based on average distances from four well-established distance determination methods: the Cepheid period-luminosity relation, TRGB, the planetary nebulae luminosity function, and the Mira variables period-luminosity relation (see \citealt{harris+10} for detailed discussion of the distance determinations). Based on the intrinsic brightness of the RGB tip $M_I(TRGB)=-4.05$ \citep{rizzi+07,freedman+20}, the TRGB in NGC~5128 fields is found at $I_0 = 23.86$ mag.
\begin{table*}
\caption{HST observations of the NGC 5128 halo fields.
}
\label{tab:HSTobslog}
\centering
\begin{tabular}{lclrrlllrrrr}
\hline\hline
\multicolumn{1}{c}{Field} &
\multicolumn{1}{c}{Date} &
\multicolumn{1}{c}{Prg.} &
\multicolumn{1}{c}{Exptime} &
\multicolumn{1}{c}{Exptime} &
\multicolumn{1}{c}{HST} &
\multicolumn{1}{c}{RA} &
\multicolumn{1}{c}{DEC} &
\multicolumn{1}{c}{E(B$-$V)} &
\multicolumn{1}{c}{$R$} &
\multicolumn{1}{c}{$\theta_a$} &
\multicolumn{1}{c}{$a/R_e$} \\
\multicolumn{1}{c}{ID$^1$} &
\multicolumn{1}{c}{yyyy-mm} &
\multicolumn{1}{c}{ID} &
\multicolumn{1}{c}{F814W} &
\multicolumn{1}{c}{F606W} &
\multicolumn{1}{c}{Camera} &
\multicolumn{1}{c}{J2000} &
\multicolumn{1}{c}{J2000} &
\multicolumn{1}{c}{(mag)} &
\multicolumn{1}{c}{(kpc)} &
\multicolumn{1}{c}{(deg)} \\
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{(sec)} &
\multicolumn{1}{c}{(sec)} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{hh:mm:ss} &
\multicolumn{1}{c}{dd:mm:ss} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{} \\
\hline
F1 & 1999-06 & 8195 &12100 & 17500 & WFPC2/PC & 13:24:51 & -43:04:33 & 0.121 & 8.3 & 152.0 & 1.8 \\
F2 & 1997-08 & 5905 &12800 & 12800 & WFPC2/WF & 13:25:29 & -43:19:09 & 0.115 & 20.0 & -144.2 & 4.8 \\
F3 & 1999-07 & 8195 &17500 & 17800 & WFPC2/WF+PC & 13:25:16 & -43:28:01 & 0.123 & 29.9 & -149.5 & 6.8 \\
F4 & 2002-07 & 9373 &30880 & 30880 & ACS/WFC & 13:25:15 & -43:34:30 & 0.123 & 37.1 & -149.0 & 8.4 \\
F5A & 2013-05 & 12964 &2137 & 2270 & ACS/WFC & 13:27:32 & -42:12:24 & 0.115 & 59.8 & 10.0 & 11.0 \\
F5W & 2013-05 & 12964 &2376 & 2496 & WFC3/UVIS & 13:28:01 & -42:14:37 & 0.113 & 60.5 & 3.9 & 10.8 \\
F6A & 2013-01 & 12964 &2137 & 2270 & ACS/WFC & 13:29:19 & -41:56:51 & 0.141 & 85.7 & 1.7 & 15.2 \\
F6W & 2013-01 & 12964 &2376 & 2496 & WFC3/UVIS & 13:29:10 & -41:51:21 & 0.139 & 90.0 & 4.8 & 16.0 \\
F7A & 2013-06 & 12964 &2137 & 2270 & ACS/WFC & 13:31:12 & -41:11:57 & 0.091 & 140.5 & 5.0 & 25.0 \\
F7W & 2013-06 & 12964 &2376 & 2496 & WFC3/UVIS & 13:31:29 & -41:16:45 & 0.093 & 137.8 & 2.7 & 24.4 \\
F8A & 2013-05 & 12964 &2137 & 2270 & ACS/WFC & 13:22:52 & -42:44:49 & 0.139 & 36.5 & 95.1 & 11.9 \\
F8W & 2013-05 & 12964 &2376 & 2496 & WFC3/UVIS & 13:23:20 & -42:47:10 & 0.143 & 30.3 & 94.1 & 9.9 \\
F9A & 2013-01 & 12964 &2137 & 2270 & ACS/WFC & 13:20:42 & -42:35:19 & 0.158 & 64.9 & 98.7 & 21.1 \\
F9W & 2013-01 & 12964 &2376 & 2496 & WFC3/UVIS & 13:20:30 & -42:30:00 & 0.144 & 69.9 & 95.2 & 22.8 \\
W1 & 2015-08 & 13856 &1278 & 1210 & WFC3/UVIS & 13:30:21 & -41:58:43 & 0.110 & 91.8 & -5.7 & 16.4 \\
W2 & 2016-01 & 13856 &1278 & 1210 & WFC3/UVIS &13:29:51 & -41:47:41 & 0.112 & 97.9 & 1.8 & 17.3 \\
W3 & 2015-06 & 13856 &1278 & 1210 & WFC3/UVIS &13:23:18 & -41:51:02 & 0.110 & 82.3 & 53.7 & 23.4 \\
W4 & 2015-06 & 13856 &1278 & 1210 & WFC3/UVIS &13:20:08 & -42:03:27 & 0.107 & 91.6 & 80.4 & 29.6 \\
W5 & 2015-06 & 13856 &1278 & 1210 & WFC3/UVIS &13:26:12 & -41:09:35 & 0.094 & 124.3 & 30.84 & 28.2 \\
W6 & 2016-01 & 13856 &1278 & 1210 & WFC3/UVIS &13:26:29 & -42:38:31 & 0.096 & 28.1 & 8.6 & 5.1 \\
W7 & 2015-06 & 13856 &1278 & 1210 & WFC3/UVIS &13:27:07 & -43:03:44 & 0.089 & 20.2 & -63.1 & 6.2 \\
W8 & 2015-04 & 13856 &1278 & 1210 & WFC3/UVIS &13:26:51 & -43:30:40 & 0.099 & 36.9 & -117.7 & 11.1 \\
W9 & 2015-04 & 13856 &1278 & 1210 & WFC3/UVIS & 13:24:33 & -43:25:25 & 0.114 & 29.1 & -167.4 & 5.5 \\
W10 & 2015-04 & 13856 &1278 & 1210 & WFC3/UVIS & 13:24:29 & -43:23:30 & 0.116 & 27.5 & -170.6 & 5.0 \\
W11 & 2015-06 & 13856 &1278 & 1210 & WFC3/UVIS & 13:24:39 & -43:19:50 & 0.115 & 23.0 & -170.4 &4.2 \\
W12 & 2015-04 & 13856 &1278 & 1210 & WFC3/UVIS & 13:24:40 & -43:19:37 & 0.115 & 22.7 & -170.2 & 4.2 \\
W13 & 2015-06 & 13856 &1278 & 1210 & WFC3/UVIS & 13:21:57 & -43:08:40 & 0.130 & 43.5 & 136.1 & 11.4 \\
W14 & 2016-01 & 13856 &1278 & 1210 & WFC3/UVIS & 13:24:06 & -42:03:37 & 0.106 & 66.1 & 49.5 & 18.1 \\
W16 & 2015-06 & 13856 &1278 & 1210 & WFC3/UVIS & 13:30:06 & -43:35:15 & 0.117 & 67.8 & -88.8 & 22.3 \\
\\
F10A$^2$ & 2018-04 & 15426 & 2228 & 2379 & ACS/WFC & 13:17:47 & -44:12:24 & 0.096 & 121.8 & 165.2 & 23.4 \\
\hline
\end{tabular}
\tablefoot{
Columns RA and DEC report the center of the field coordinates, E(B$-$V) is the reddening value toward the field center based on \citet{schlegel+98} reddening maps, and $R$ is the projected radial distance in kpc from the center of the galaxy. In the last two columns, $\theta_a$ is the azimuthal angle of the field location measured clockwise from the isophotal major axis (see Fig.~\ref{fig:fieldsrel}), while $a$ is the semimajor axis of the elliptical isophote that goes through the field (assuming e=0.54 ellipticity) in units of $R_e=305" = 5.6$~kpc. \\
\tablefoottext{1}{Our numbering of the Wxx fields in the first column (Field ID) follows the MAST Archive listing, in which there
is no ``W15''.\\}
\tablefoottext{2}{The field F10A listed at the bottom is used as a background control field (see Sect.~\ref{sec:decontstat}).}
}
\end{table*}
\section{Data}
\subsection{HST imaging observations}
\label{sec:HSTimaging}
In Table~\ref{tab:HSTobslog} we list the basic information about all HST fields used
in this study.\footnote{Additional images for other
fields in NGC~5128 exist in the
HST archive, but they are unsuitable for a homogeneous analysis because of one of the following reasons: (i) located at the center of known overdensities \citep{crnojevic+19}, (ii) are not deep enough, or (iii) were obtained with different filters, or only in one filter.} Their distribution on the sky, in a coordinate system centered on NGC 5128, is shown in Fig.~\ref{fig:fieldsrel}.
In the right panel we also show the unsharp-masked image from \citet{peng+02}. The comparison emphasizes the wide span
of the halo probed by our target fields, most of which lie far beyond the inner regions (i.e., the inner square in Fig.~\ref{fig:fieldsrel})
containing arcs and shells, dust, and star-forming regions that are remnants of the recent merger episode(s).
Concerns about contamination of our target fields from these factors are thus mitigated.
By contrast, we need to be aware of possible contamination from the presence of outer-halo dwarf
satellites and substructure \citep{crnojevic+16}. We show therefore in Fig.~\ref{fig:fieldsrel} the location of known dwarf satellites of Cen~A within the area covered by our HST fields. Having effective radii between $\sim 150-600$~pc, these dwarfs typically cover up to $ \sim 1'$ on sky\footnote{1' at the distance of Cen A corresponds to $\sim 1.1$~kpc.} and thus do not contribute stars to adjacent parallel pointings. One exception is the CenA-MM-Dw3 stream \citep{crnojevic+19} that crosses one of our fields. This is further discussed below.
The observations of the inner WFPC2 fields F1-F3 taken within HST Cycles 5 and 8 were described
in detail by \citet{harris+99} and \citet{harris+harris00, harris+harris02}, while field F4 is our deepest probe into the halo population,
where the ACS photometry reached the core-helium burning stars \citep{rejkuba+05,rejkuba+11}. This deep field received 12 orbit-long exposures for each of the two filters. The fields
F5-F9 are from our Cycle 20 data (Paper I) and have two entries each, one for the WFC3 camera (e.g., F5W), and the other for the ACS (e.g., F5A) images taken in parallel mode. The Cycle 20 observations were organized such that two orbits for each target were placed within a spacecraft visit to obtain the images in two filters at the same field orientation. Within each orbit, a set of three dithered images using the WFC3-UVIS-DITHER-LINE-3PT pattern were taken for each filter.
Fields W1 -- W16 are the WFC3
observations from the Cycle 22 program GO13856 (PI: Crnojevi\'c). In this program, ACS fields were placed on faint dwarf satellites \citep{crnojevic+19}, while the WFC3 data were the parallels located on blank halo fields
adjacent to them. The two cameras are separated by more than $5'$ in the focal plane of the HST, corresponding
to a linear separation of $\sim 5.5$ kpc center to center at the distance of NGC 5128. This distance is much larger than
the physical dimensions of any of the dwarf satellites, therefore the W1 -- W16 pointings are expected to sample pure field halo
populations. Fields W11 and W12 heavily overlap in location due to the closeness of the primary ACS targets, and there is no W15 field (see Table \ref{tab:HSTobslog}). The Wxx set therefore really just covers 14 independent pointings.
Finally, we added one more field (F10A, listed in the last row of Table \ref{tab:HSTobslog}).
This is the parallel image from program GO15426 (PI Crnojevi\'c) taken with the ACS,
located on a remote halo field $\sim 120$ kpc to the southwest. As we describe below,
it provides us with what is likely to be the best available control field for gauging the field
contamination.
For all the targets listed in Table \ref{tab:HSTobslog}, the filters were F606W and F814W, which transform well
into $(V, I)$ and which provide a color index $(V-I)$ that is reasonably sensitive to metallicity
for old RGB stars over the entire range from [Fe/H] $\simeq -2$~dex to above-solar abundance.
The "Exptime" columns in Table~\ref{tab:HSTobslog} report the total exposure time per filter
(for F5-F9, these are essentially full one-orbit exposures, while for W1-W16, they are half-orbit
exposures, leading to slightly shallower CMDs). The camera(s) that were used determine the total field of view (FOV) per field:
for F1 (WFPC2/PC1), this is as small as 0.3403 arcmin$^2$, F2 covers 5.33 arcmin$^2$, and F3, which combines all four WFPC2 detectors, reaches 5.674 arcmin$^2$. For the rest,
the ACS camera FOV is 11.33 arcmin$^2$ , and WFC3 covers 7.29 arcmin$^2$.
In total the target fields span almost 4 degrees across the sky. We therefore expect
differences in Galactic (Milky Way) foreground extinction from one field to the next that are large enough to call for individual
correction. The adopted Galactic extinctions toward each field are listed in Table \ref{tab:HSTobslog} and are computed from the \citet{schlegel+98} reddening value E(B$-$V) adopting the \citet{schlafly+finkbeiner11} recalibration\footnote{source: http://irsa.ipac.caltech.edu/applications/DUST/} and $A_V=2.79*E(B-V), A_I=1.55*E(B-V)$.
Table~\ref{tab:HSTobslog} also gives the on-sky projected radial distance $R$ in kiloparsecs from the center of NGC 5128\footnote{RA$_0$=13:25:27.6, DEC$_0$=-43:01:09 from NED} assuming an intrinsic distance modulus
$(m-M)_0 = 27.91$ \citep{harris+10}. The position angle $\theta_a$ is the azimuthal angle (measured north of east, or clockwise in the figure)
from the major axis $a$
(where the ellipticity is $e = (b/a) = 0.54$, see Sect.~\ref{sec:AGB} below).
Here $a$ is listed in units
of $R_e$, where we adopt $R_e = 305\arcsec = 5.6$ kpc from \citet{dufour+79} as determined from the integrated-light
profile of the inner spheroid.
\subsection{Photometry}
All observations used the same filters ($F606W, F814W$) and the photometric data reduction procedures were the same. We started from the pipeline processed \emph{*.drc} images downloaded from the HST archive that are the multidrizzled combinations of the individual exposures including CTE corrections.
While the stars in the target halo fields are quite uncrowded in any absolute sense and aperture photometry would be possible for them, we preferred to run the
\emph{daophot} and \emph{allstar} suite of photometric codes in IRAF \citep{stetson87} to perform the photometry via PSF fitting.
In addition to a homogeneous procedure applied to all fields, this
gives us the advantage of having stellar PSF-fitting parameters that can be used to
distinguish objectively between slightly resolved background galaxies and point sources.
In each field, a master image consisting of all exposures in both filters was constructed to provide
the deepest possible source for object detection.
SourceExtractor \citep{bertin_arnouts1996} was used on the master image
to detect candidate objects and perform a preliminary culling out of
nonstellar objects (half-light radii $r_{1/2} < 1.0$ or $> 1.6$ px were rejected). From there, \emph{daophot}
was used to carry out small-aperture (r = 2 px) photometry, construction of a PSF for each filter, and
then final photometry with \emph{allstar}. The independently determined PSFs on each field
were built from typically 30 to 70 individual bright uncrowded stars.
Although these
proved to be quite consistent with each other, in the end, to improve internal
consistency, exactly the same set of $(F606W,F814W)$ PSFs picked from the highest S/N cases for a given camera and filter were used
on all the fields. Further culling of the starlists was done by rejecting any objects with
\emph{allstar} parameters $\chi > 2$ or $err > 0.2$ mag in either filter. Because crowding is
not a factor, any such rejected objects are almost always nonstellar.
Empirically derived aperture corrections to r = 10 px were added to
the \emph{allstar} measured magnitudes, and the large-aperture data were then converted into
filter magnitudes $(F606W, F814W)$ with filter zeropoints from the ACS or WFC3 webpages.
These were finally converted into standard $(V,I)$ with the linear color transformations noted below.
\subsection{Calibrations for the ACS and WFC3 photometry}
Our photometric calibration for the ACS fields follows the prescription in \citet{sirianni+05}
and is on the VEGAMAG system. To bring
the instrumental magnitude measurements to the HST system, we used the zeropoint calculator
published on the HST web pages.
We applied the following calibration equations:
\begin{equation}
F606W = -2.5\times \log{\frac{f606w_{inst}}{\mathrm{exptime}}} + zpt_{F606W} + apcor_{F606W}
\end{equation}
\begin{equation}
F814W = -2.5\times \log{\frac{f814w_{inst}}{\mathrm{exptime}}} + zpt_{F814W} + apcor_{F814W}
,\end{equation}
where $zpt_{\mathrm{filter}}$ is the VEGAMAG zeropoint for the given filter, and $apcor_{\mathrm{filter}}$ is the corresponding aperture correction from our PSF measurement to a 10 px aperture, and then from 10 px
to infinite radius \citep{bohlin12}. For the ACS data,
$zpt_{F606W}=26.407$ and $zpt_{F814W}=25.523$.
For the WFC3 camera, the aperture correction to 10 px = $0\farcs4$ radius was made, after which filter zeropoints can be applied that already include the step to infinite radius. We adopted $zpt_{F606W}=25.8843$, $zpt_{F814W}=24.5730$.
In Paper I we used the WFC3 zeropoints from the HST webpages to convert the $(F606W, F814W)$ magnitudes into $(V,I)$ , but out of temporary necessity, we applied the color terms ($c_1$) for the same filters from the ACS/WFC detector. Since then, Sahu et al. (2014) provided additional data for WFC3/UVIS, listing the zeropoint differences $(I-F814W), (V-F606W)$ for stars over a range of blackbody
temperatures and spectral types. By plotting these differences versus $(V-I)$,
we reconstructed the color terms in the Vegamag system for the transformations. As with ACS,
the slope of the color term $c_1$ is very small for $I$, but for $V$ , it appears to be slightly shallower than for the ACS.
The transformations for WFC3 that we adopt here are
\begin{equation}
(V-I) = 1.1396 \times (F606W-F814W)
\end{equation}
\begin{equation}
V = F606W + 0.1545 \times (V-I)
\end{equation}
\begin{equation}
I = F814W + 0.032 \times (V-I)
.\end{equation}
Empirical transformations of the standard WFC3 filters into $BVI$ have also
been derived by \citet{harris2018} from combined ACS and WFC3 photometry in a
47 Tucanae standard field; encouragingly, for F606W and F814W, these are quite similar to those given above. The effect of adopting the revised transformations above was to make the $(V-I)$ colors from WFC3 bluer than in Paper I by typically $\Delta(V-I) \simeq 0.1$ mag. This
shift brings the CMDs for ACS and WFC3 into closer agreement.
\subsection{Completeness and photometric error analysis}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig2_completeness_3panel.pdf}
}
\caption{Samples of \emph{daophot/addstar} experiments for three different
fields in our study. The completeness fraction $f$ is plotted vs.
V or I magnitude in each panel. The measured $f-$values per 0.25-magnitude bin
are shown as the black (I) or red (V) points with error bars. The solid lines
(black for I, red for V) show the smooth fitted curves of the form
$f = 1 / (1 + e^{\alpha (m-m_0)})$
described in the text.
}
\label{fig:completeness}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig3_mag_error.pdf}}
\caption{Sample of photometric measurement precision for Field F5A (top) and W12 (bottom panels) for the I band. The magnitude difference $\Delta(m)$ (measured -- input) for 10000 input artificial
stars is plotted against input magnitude for F5A (upper left) and W12 (lower left panel). The 50\% detection completeness
levels are marked with the vertical dotted red lines. The dotted green line is the magnitude of the TRGB. The blue lines indicate $\pm 0.25$mag around the TRGB. In the right panels we plot the distributions of magnitude differences (measured -- input) for stars within $\pm 0.25$~mag of the TRGB, binned to 0.01 mag for F5A (upper right) and W12 (lower right panel). The Gaussian curves plotted over the magnitude difference histograms have the mean and sigma as indicated in the panels.
}
\label{fig:delmag}
\end{figure}
\begin{table}
\caption{Photometric uncertainty and completeness parameters.}
\label{tab:f}
\centering
\begin{tabular}{llll}
\hline \hline
Parameter & 12964(ACS) & 12964(WFC3) & 13856(WFC3) \\
\hline
$\alpha_V$ & 1.8 & 1.8 & 1.8 \\
$m_{0,V}$ & 28.1 & 28.25 & 27.70 \\
$\alpha_I$ & 1.6 & 1.8 & 2.0 \\
$m_{0,I}$ & 26.8 & 26.8 & 26.42 \\
\\
$\beta_{0,V}$ & 0.02 & 0.02 & 0.035 \\
$\beta_{1,V}$ & 0.053 & 0.056 & 0.063 \\
$\beta_{2,V}$ & 27.0 & 27.0 & 27.0 \\
$\beta_{0,I}$ & 0.02 & 0.02 & 0.025 \\
$\beta_{1,I}$ & 0.050 & 0.042 & 0.050 \\
$\beta_{2,I}$ & 25.5 & 25.5 & 25.0 \\
\hline
\end{tabular}
\end{table}
\begin{figure*}
\resizebox{\hsize}{!}{
\includegraphics{Fig4_CMDs_scatter_H20calib_dereddened_trgb.png}
}
\caption{Measured CMDs for fields F1-F9 as listed in
Table \ref{tab:HSTobslog}, including the correction for Galactic extinction. The older data for locations at 8, 20, and 30 kpc (F1, F2, and F3) were taken with
the WFPC2 camera. For the fields F5 to F9, data from ACS pointings are plotted as black dots
and and from WFC3 as red dots. The solid lines show the 50\% detection completeness
levels (ACS and WFPC2 in blue, and WFC3 in orange). The dotted green line indicates the expected location of the TRGB. For F4, the photometric
limits are much deeper than for the other fields. }
\label{fig:cmdf1_9}
\end{figure*}
\begin{figure*}
\resizebox{\hsize}{!}{
\includegraphics{Fig5_CMDs_scatter_H22calib_dereddened_trgb.png}
}
\caption{CMDs for fields W1-W16 corrected for Galactic extinction. These
fields have half the exposure time of F5-F9 and thus have brighter limiting magnitudes
and noticeably brighter completeness cutoff levels in the bluer filter. The solid (orange) lines show the 50\% detection completeness levels. The dotted green line indicates the expected location of TRGB.}
\label{fig:cmdW1_16}
\end{figure*}
The \emph{addstar} task in \emph{daophot} was used to measure the detection completeness of
our photometry and the internal measurement uncertainties. Artificial stars (scaled PSFs) were
added, 1000 at a time, into the images, and we then remeasured with exactly the same procedures as
described above. The completeness $f$ is the recovery fraction: the number of artificial stars detected in a given magnitude bin divided by the number of artificial stars inserted. A sample of the fraction of recovered stars versus magnitude is shown in Fig.~\ref{fig:completeness},
for the three different combinations of filter and exposure time that we have (the ACS fields from
GO12964, WFC3 from GO12964, and WFC3 from GO13856). Completeness experiments for fields F1--F4 are described in detail in the papers that reported their initial analysis \citep{harris+99, harris+harris00, harris+harris02, rejkuba+05}.
As is typical for very uncrowded images such as we have here, the detection fraction declines smoothly
from near-100\% down to near-zero over a two-magnitude run. Analytic curves of the form
$f = 1/(1+ e^{\alpha (m-m_0)})$ \citep{harris+16} are superimposed on the data in Fig.~\ref{fig:completeness}.
In this relation, $m_0$ represents the magnitude
where $f = 0.5$, and $\alpha$ represents the steepness of decline.
Table \ref{tab:f} lists the pairs of parameters for the three sets of
images. The most important feature of the adopted completeness curves
is that they accurately describe the
shape of $f(m)$ for $f > 0.5$; we did not use any data below this
50\% point.
The exposure times and therefore also the completeness are comparable for F10A
(GO15426) and for the Cycle 20 data from GO12964.
As expected, with respect to these, $m_0$ is about one magnitude brighter than for the
fields in GO13856, which have half the exposure time of the other two sets.
As we show below, however, this difference does not seriously
hamper our goals to derive the halo RGB population
density.
The artificial star tests were also used to determine the measurement
uncertainties of the data. Sample artificial star runs showing the recovered magnitudes compared with
their input values are shown in Fig.~\ref{fig:delmag} for a range of magnitudes
between $\sim2$~mag brighter than the TRGB down to well below the completeness limit. These tests showed that to a close approximation,
the measurement uncertainties are about 1.4 times larger than the numbers
returned by \emph{daophot/allstar}. A useful interpolation equation for the
dependence of measurement uncertainty on magnitude is
\begin{equation}
e_m = \beta_0 + \beta_1 e^{(m-\beta_2)}
,\end{equation}
where the $\beta-$values appropriate for the different sets of exposures are listed in Table \ref{tab:f}.
The artificial star tests can also be used to check for the possible presence of blends. These are stars that are measured brighter due to underlying (unresolved) background and thus shift to brighter bins in the luminosity function. In the right panels of Fig.~\ref{fig:delmag} we show the difference of magnitude (measured$-$input) for stars within $\pm 0.25$~mag of the TRGB. The tail of the distribution extending to negative (brighter measured) magnitudes is negligibly small even in our most densely populated field (W12), and it is completely absent in a typical halo field (F5A). The mean magnitude of measured stars is 0.01 mag brighter for W12 field, and it shows no systematic shift for less populated fields.
\section{Color-magnitude diagrams}
The CMDs for our complete set of fields are shown in Figs.~\ref{fig:cmdf1_9} and \ref{fig:cmdW1_16}. In Fig.~\ref{fig:cmdf1_9}, a small but important check on the internal consistency of our
photometry is that for each pair of ACS/WFC3 fields, the CMDs overlie each other closely,
within the scatter of points along the RGB in each field.
Here, they match better than we found in Paper I, a direct result of the improved WFC3 color
transformations in the present work.
A broad RGB component is present in all fields, as is the clear TRGB located at $I_0 \simeq 23.86$ (indicated with the dotted green line) and
slanting downward at redder colors due to increasing bolometric correction.
A population of foreground stars is also clearly present in all fields over a wide range
of colors, and it extends well above the RGB population. In the outermost fields
(e.g., W3, W4, W5, and W16), the contamination appears to be comparable to the number of NGC 5128 stars.
Therefore the first necessary task for the analysis is to remove this contamination as well as
possible so that unbiased measures of the radial density distribution can be made.
The field contamination is due to a
combination of foreground stars and very small, faint background galaxies that
managed to pass the selection criteria described above.
Although the well-populated
inner fields are completely dominated by the NGC 5128 halo RGB stars and thus contamination
is of relatively little concern, for our outermost fields, the objective removal of contaminants is critical.
It is possible to model the foreground population of Milky Way stars as a function of the position on the sky, but faint background galaxies provide more of a challenge. In the following, we first discuss the bright foreground stellar component in comparison with the Milky Way stellar population models. This is used to evaluate the ability of the models to reproduce the foreground Milky Way component and also to assess the possible presence of a bright AGB component in the halo of NGC~5128. A contribution from bright AGB giants is expected in some inner fields given the detection of a population of bright long period variable (LPV) stars \citep{rejkuba+03_LPVnir} and intermediate-age globular clusters (GCs) \citep{woodley+10a}, but it is so far unknown how far it extends into the halo \citep[see also the discussion in][]{crnojevic+13}. The population of contaminants in the CMD can also be evaluated empirically. In Sect.~\ref{sec:decontstat} we describe this alternative method, which we eventually preferred and adopted
for the remaining analysis.
\subsection{Modeling the foreground component}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig6_TrileCMD.pdf}
}
\caption{CMD of the TRILEGAL simulation for a field size of 0.026 sq.deg equal to five pairs of ACS+WFC3 pointings, which best represents the foreground population (see text). The polygon appearing in Fig. \ref{fig:HST_5fieldsall}, where it encloses the majority of stars belonging to the halo of NGC~5128, is overplotted.
}
\label{fig:cmd_fore}
\end{figure}
Models of the Milky Way stellar population can be used to simulate the number of stars and their magnitude-color distribution along the line of sight of NGC 5128. In Paper I we adopted the TRILEGAL \citep{girardi+05} simulation because it had a higher density of faint stars than the Besan\c{c}on \citep{robin+03} model. In this paper we explore in more detail the ability of the TRILEGAL model to reproduce the foreground Milky Way population in comparison with our data (see also Appendix A). In order to avoid issues with different completeness due to different exposure times, we restrict the comparison of the models to the fields observed during Cycle 20 (F5-F9).
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig7_H20combinedCMD.pdf}
}
\caption{Combined CMD for fields F5+F6+F7+F8+F9. Stars detected in WFC3 pointings are shown in red and in ACS pointings with black dots. The blue polygon encloses most of the stars belonging to the halo of NGC~5128.
}
\label{fig:HST_5fieldsall}
\end{figure}
The TRILEGAL simulator prompts for many different inputs. Keeping the parameters of the galactic components to their default values, we simulated a field of 0.026 square degrees, equal to the total surveyed area of 5 ACS + 5 WFC3 pointings observed during Cycle 20, in the direction of the NGC 5128 coordinates. Most of the simulated objects are main-sequence stars, but with a few white dwarfs populating the bluer colors of the distribution.
The TRILEGAL model with a Chabrier IMF and a binary fraction of 0.3 (Fig.~\ref{fig:cmd_fore}) has 1128 stars in the magnitude range $19<I<22$. In the same magnitude range, the Besan\c{c}on model has 1097 stars. The models obtained with other prescriptions have a slightly different number of stars in this range, although these differences may reflect the normalization of the TRILEGAL simulator. To compare the simulations among themselves, we normalized their counts in the range $20 < I < 22$. The simulations adopting different options (e.g., binaries on or off) yield similar results, except for the
case of a Salpeter IMF, in which a large component of faint ($I \gtrsim 25$) and red ($V-I \gtrsim 3$) dwarfs appears that has no counterpart on the observed CMD (Fig.~\ref{fig:HST_5fieldsall}). However, our data are largely incomplete in this faint red magnitude range in all fields except in F4, where we still do not observe large numbers of faint red dwarfs. We thus consider the Chabrier log-normal IMF and 30\% binaries as the most relevant TRILEGAL rendition of the foreground.
To maximize the statistical significance of the foreground population for comparison with the model, we combined the CMDs for fields F5-F9, as shown in Fig.~\ref{fig:HST_5fieldsall}. In this diagram, the blue polygon encompasses the area in which the contribution of RGB stars from NGC 5128 halo dominates.
The TRGB stands out very clearly around $I \sim 24$ mag, slanting toward fainter magnitudes at redder colors. The plume of stars brighter than the TRGB is mainly due to Milky Way contaminants having 1095 stars between $19 < I < 22$ mag. Its luminosity function (LF) is well reproduced by TRILEGAL and matches the Besan\c{c}on model very closely. In this magnitude range, the color distribution is also better reproduced by the Besan\c{c}on model. In Appendix A we examine in more detail the color distributions of TRILEGAL and Besan\c{c}on models. We find further differences with respect to observations in the bright magnitude range.
\begin{figure}
\centering
\resizebox{\hsize}{!}{
\includegraphics[angle=0,clip]{Fig8_H20LFs_bckgTRILE.pdf}
}
\caption{Bright portion of the luminosity functions for the five Cycle 20 ACS+WFC3 field pairs, shown as the colored histograms.
The prediction from the TRILEGAL model, normalized to the same area, is shown as the dashed black line. The dotted light green histogram shows the LF of the field F10A that best represents the empirical foreground population, appropriately scaled to the WFC3+ACS area.
}
\label{fig:lf_fields}
\end{figure}
We next investigated the LF of the stars above the TRGB.
In Fig.~\ref{fig:lf_fields} we show the observed LF data for F5 - F9, broken out for the five (ACS+WFC3) pointings.
Each of the colored lines shows the
sum of the observed ACS+WFC3 counts, while the dashed black line shows the TRILEGAL model, scaled to the ACS+WFC3 area.
At magnitudes $I \leq 22,$ the LF is very well reproduced by the simulator. For $I > 24,$ the observed LF increases dramatically, and we are into the main RGB population of NGC 5128. In the range $22 < I < 24,$ the data show a noticeable excess of stars with respect to the simulation, which may signal a contribution from bright intermediate-age stars in NGC 5128 or excess field contamination that is not accounted for in the model.
On closer inspection, there is a difference between the excess counts in the three innermost
fields (F5, F6, and F8) and in the two outer fields (F7 and F9) that sit lower. In Fig.~\ref{fig:lf_fields} we also show as the shaded dotted light green histogram the LF of the observed field F10A (see Fig.~\ref{fig:fieldsrel} for its location), also scaled to WFC3+ACS area, which we use below for empirical removal of contamination (Sec.~\ref{sec:decontstat}).
At fainter magnitudes (I>24.5), that is, below the TRGB and not shown in this diagram, both F7 and F9 show a mild excess of stars with respect to field F10A.
In total, in these five field pairs there are 1581 objects with $22<I<24$,
while the simulator predicts 1055 foreground stars for the same area. The excess is thus
($ 526 \pm 23$) stars (where the quoted uncertainty is a lower limit due to Poisson statistics only),
or $\sim (33 \pm 2)\%$. However, given that field F10A, which does not show any appreciable sign of RGB stars belonging to NGC 5128 in its CMD, also has an excess of star counts with respect to the TRILEGAL model, we argue that most of this excess in the outer halo fields ($R_{gc} \gtrsim 30$~kpc) is due to field contamination that is underestimated in the model rather than to a genuine AGB component. This is further explored using the match between near-IR and optical data described in Appendix B.
The TRILEGAL simulation also predicts a component of stars fainter than the magnitude of the NGC~5128 TRGB, with colors to the red of the black polygon and fainter than the RGB. These stars, which consist of dwarfs with mass $\sim$ 0.15 M$_\odot$, consist of $\sim$ 490 objects, while on the observed CMD, there are virtually none. In this
part of the CMD, the data are rather incomplete, which likely accounts for the discrepancy with respect to the model. Overall, the total numbers of foreground objects from the TRILEGAL simulation at magnitudes below the TRGB and within the NGC 5128 RGB polygon are much smaller than our
sampled NGC 5128 population.
It is important to note that in addition to the small mismatch between the observed bright star counts and the model predictions and the possible small contribution of the AGB component, the additional faint background galaxy contamination remains unaccounted for. Therefore we also explore the possibility of a strictly empirical method to subtract the field contamination in the next subsection.
subsection{Statistical decontamination of the CMDs}
\label{sec:decontstat}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig9_CMDcleaning.png}
}
\caption{Illustration of the point-by-point cleaning process described in the text.
The lower left panel shows the CMD for field F10A. This field has little to no trace
of a clear RGB population, and we adopt it to be the control field for the remaining target fields. In the upper panels, we show the CMD for field F5W before (left)
and after (right) cleaning. All CMDs are dereddened. Most of the
stars above the TRGB have been removed, as have a number of other stars along and around the RGB.}
\label{fig:cleaning}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig10_CountsBright.pdf}
}
\caption{Number of stars per arcmin$^2$ in all observed fields as a function of
distance in extinction-corrected I-band magnitude bins as labeled in each panel. The bottom x-axis is the distance corresponding to the semimajor axis of the elliptical isophote that goes through the field (assuming ellipticity of the outer halo e=0.54 for all fields) in log and top x-axis in linear scale. Red (blue) dots highlight fields along, or close to, the major (minor) axis, and fields plotted with black dots are in between. The hashed horizontal band in each panel indicates the number of stars (with 1$\sigma$ Poissonian error) in the background field F10A in the same magnitude bin.}
\label{fig:counts_bright}
\end{figure}
Ideally, the preferred and purely empirical
way to determine the field contamination independently of any modeling
would be to use HST pointings just outside the NGC 5128 halo, observed with the same filters and similar exposure times to those used in our program.\footnote{In our original observing program
design, we anticipated that our most remote targets, such as F7, would give us
suitable control fields for this purpose. However, the halo proved to be much more
extended than expected (at least along the NE axis).}
An extensive search of the MAST Archive for fields matching these criteria was unsuccessful,
\emph{} except for field F10A (see Table \ref{tab:HSTobslog} and Fig.~\ref{fig:fieldsrel}). At a projected distance of $R \sim 120$ kpc from the
center of NGC 5128, it is on the opposite
side of the galaxy (to the southwest) from our other outer fields. The CMD for F10A is shown
in the lower left panel of Fig.~\ref{fig:cleaning}. It reveals a swath of stars crossing the
diagram in the upper right corner. These stars are mostly foreground Milky Way stars, which appear in every one of our other CMDs, and they are also clearly visible in the TRILEGAL model of the Milky Way (Fig.~\ref{fig:cmd_fore}). The F10A CMD also contains some very faint objects in the lower left corner, but little else.
There is little or no clear trace of an RGB population that would be found in the
range $I = 24-26$ and $(V-I) \sim 1 - 2$, as is the case in the CMDs for
all other fields. Of all the HST fields in the region that are available to us,
it has the lowest population of stars in the RGB region that is our target. The fact that it is
not quite as distant from the galaxy center as our fields F7 and W5 and yet has a smaller
population of stars suggests to us simply that
the halo of NGC 5128 is not ideally smooth and symmetric at these outermost distances.
More or less of necessity, we adopted F10A as our control field.
Because our target fields are spread over a large area spanning
almost 4 degrees in diameter (Fig.~\ref{fig:fieldsrel}), differences in the
number density of contaminants larger than simple Poisson statistics may be present.
Neither the foreground stars nor the faint background galaxies are
uniformly spread across an area this wide.
That is, the assumption of a single mean contamination level is
an approximation. This field-to-field variance sets an ultimate limit on our
ability to decontaminate the CMDs using a single background field.
Figure~\ref{fig:counts_bright} shows the number density of bright stars in all the fields with respect to our nominal background field F10A (see below). Stars are selected according to the specified magnitude bins without further selection by color, but the correction for incompleteness is applied according to parameters given in Table~\ref{tab:f}. Between $20<I_0<23$, that is, more than one magnitude brighter than the TRGB, the field-to-field variations in the number of Milky Way foreground contaminants are fairly consistent within $\sim 1-2 \sigma$. This gives us confidence that statistical decontamination with a single F10A field works. There is also a notable excess in surface density in the bottom, and more marginally, also in the middle panels in Fig.~\ref{fig:counts_bright} in the inner halo fields. We return to this in Sect.~\ref{sec:AGB}.
\begin{figure*}
\resizebox{\hsize}{!}{
\includegraphics{Fig11_CMDs_scatter_H20clean.pdf}
}
\caption{CMDs for fields F1-F9 after the point-by-point fore- and background decontamination described in the text. Data from ACS pointings are plotted as black dots,
and those for WFC3 as red dots. The solid lines show the 50\% detection completeness
levels (ACS and WFPC2 in blue, WFC3 in orange).}
\label{fig:cmdf1_9_clean}
\end{figure*}
\begin{figure*}
\resizebox{\hsize}{!}{
\includegraphics{Fig12_CMDs_scatter_H22clean.pdf}
}
\caption{CMDs for fields W1-W16 after cleaning. The solid (orange) lines show the 50\% detection completeness levels.}
\label{fig:cmdW1_16_clean}
\end{figure*}
With the control field F10A in hand, star-by-star subtraction of its starlist
was performed in the conventional way by searching for nearest
neighbors in the two CMDs and matching them. Specifically, we assumed that the target field
(the one needing cleaning) consists of a list of stars with magnitudes and colors ($m_i, c_i$). Similarly,
the background field consists of a list ($m_j, c_j$). Each star
in the target field list is surrounded by an ellipse of axial
length $x_c$ in $(V-I)$ and $x_m$ in $I$. Its nearest
neighbor in the control field list is then found, where the distance $d$ between
them is defined as
\begin{equation}
d^2 = \frac{\Delta(V-I)^2}{x_c^2} + \frac{\Delta I^2}{x_m^2}.
\end{equation}
\noindent If that neighbor
falls within the ellipse ($d_{min} < 1$), then both stars are removed from their lists.
After experimentation with different ellipse axes, we adopted
$x_c = x_m = 0.2$ mag. A key consistency test of the method is that
the stars in the CMD brighter than the TRGB ($I < 24$, i.e., the
region of the CMD that is almost entirely contaminants) should be
successfully removed to within Poisson $\sqrt{n}$ statistics, where $n$ is the
number of control-field stars in that part of the CMD.
F10A is an ACS field, so its entire starlist was used for cleaning the other
ACS fields F4-F9. For the WFC3 fields, which have 0.64 times the area of ACS,
64\% of the stars in F10A were selected at random.
Before the CMD matching was made, the control field and target field starlists
were both individually dereddened according to $E_{V-I} = 1.24 E_{B-V}$ and
$A_I = 1.55 E_{B-V}$ and with the different
foreground reddenings of the fields, as given in Table \ref{tab:HSTobslog}.
The method is illustrated in Fig.~\ref{fig:cleaning}.
The results are shown for field F5W, which has a clear RGB component, but
also a noticeable contamination fraction.
The CMDs for all the program fields after this point-by-point cleaning
are shown in Figs. \ref{fig:cmdf1_9_clean} and \ref{fig:cmdW1_16_clean}.
As suggested above, the most noticeable changes
in the CMDs are in the bright regions ($I < 24$), but the RGB populations
in the sparsest fields (such as F7, F9, W3, W4, W5, W16) now also stand
out more clearly.
It is also clear that some residual contamination is still present;
in relative terms, this is most noticeable in the sparsest fields.
For these sparse outermost fields, a very small amount of ``noise'' (the contaminants) must be subtracted from a similarly small
``signal'' (the RGB stars), so the results are strongly subject to small-number
statistical fluctuations. In addition, as noted above, both components show physical density
differences across the entire halo that do not strictly
follows Poisson $\sqrt{n}$ statistics. In short,
the cleaning process for this problem was limited by the intrinsic field-to-field statistical
variance of the halo stellar populations and the background and not by the details of our cleaning method.
To gain an alternative look at a mean pure-background population of stars, we also experimented with
an iterative approach using the target fields themselves. This approach takes
the starting point that the number density $\phi$ of stars in each field has
two components: the RGB stars belonging to NGC 5128, and a contaminant
population,
\begin{equation}
\phi = \phi_{RGB} + \phi_{bkgd} .
\end{equation}
\noindent We then assumed (necessarily, although not quite correctly) that $\phi_{bkgd}$ is the same
for all the fields, and also that $\phi_{RGB}$ follows a power-law
decline with radius $\phi_{RGB} \simeq R^{-\alpha}$ , as found in Paper I.
Assuming that the power-law slope $\alpha$ remains known and constant at all $R$
then gives enough leverage to solve for $\phi_{bkgd}$. In practice, we took
one of the sparsest outer fields (such as F7 or F10A) and subtracted its CMD from one
of the inner fields that was more dominated by the RGB component (such as F5). This left
a CMD that contained only RGB stars to within the field-to-field statistical scatter. This
pure-RGB CMD was then scaled by $(R, \alpha$) to subtract the RGB
component from any other field, leaving only $\phi_{bkgd}$. Iteration of this approach yields much
cleaner and contamination-free CMDs. However, the assumption of
a single power-law form for $\phi_{RGB}$ imposes a rather strong condition
on the approach, and it did not yield results that were different from
or better than the purely empirical approach with the single control field
described above.
For the purposes of the present paper, we adopted the cleaned CMDs as determined
from the use of only F10A as a control field. Our conclusion is that we can trace
the RGB population outward to the point where $\phi_{RGB} \sim \phi_{bkgd}$,
as can be seen by comparing, for example, the cleaned versions of F7, F9, W3, or W5
in Figs.~11-12 with their original CMDs in Figs.~4-5.
Tracing the halo to still larger radii requires star counts that reach deeper magnitude levels than we have at present, and that cover a wider area.
\begin{figure*}
\centering
\resizebox{\hsize}{!}{
\includegraphics[angle=0,clip]{Fig13a_cmds_clean2.png}
\includegraphics[angle=0,clip]{Fig13b_isochs.png}
}
\caption{Bright portion of CMDs of the five fields that show the highest excess of stars above the TRGB with stars brighter than the TRGB (dashed horizontal line at $I_0=23.86$) plotted as crosses (left). These CMDs have been statistically cleaned from foreground contamination using observations in F10A shown in the lower right CMD. Padova stellar evolutionary isochrones for different metallicities listed at the top, and ages 2, 4, 6 and 10 Gyr (right). The isochrones have been shifted to the distance of NGC~5128 and are color-coded according to stellar evolutionary stage: RGB (red circles), early AGB (blue circles), and TP-AGB (green circles). The boxes indicate regions for which the mass-specific production factor $P_j$ has been computed from the isochrones (see text).}
\label{fig:cmd_isochs}
\end{figure*}
\section{Intermediate-age AGB component in the inner halo}
\label{sec:AGB}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig14_plratio_10Gyr.pdf}
}
\caption{Number ratio of the bright AGB vs. RGB stars selected within the boxes described in Fig.~\ref{fig:cmd_isochs} for a theoretical stellar population including two bursts (see legend) as a function of mass of the younger population. The shaded band encompasses the ratio of stars in our inner halo fields (W6, W9, W10, W11, and W12) counted in the boxes plotted with solid lines in Fig.~\ref{fig:cmd_isochs}. The horizontal line is the observed ratio when brighter stars are also included within the dashed box ($22.25<I_0<23.85$) with respect to counts in the RGB box ($23.85<I_0<25.45$).}
\label{fig:plratio}
\end{figure}
In the bottom panel of Fig.~\ref{fig:counts_bright}, within the magnitude bin $23\leq I_0 \leq 23.86$ that is just above the tip of the RGB, the increase in the observed surface density of stars in the fields located within $\sim 30$~kpc from the center is strong. The excess is strongest in fields W11 and W12, which are nearly collocated at about 23 kpc galactocentric distance along the southwestern major axis. The nearby fields W9 and W10, located at $R_{gc}\sim$29 and $\sim$28~kpc along the southwestern major axis, respectively, and field W6, which is at a similar distance as W10 ($\sim$28 kpc), but on the other side of the galaxy along the northeastern major axis (see~Fig.~\ref{fig:fieldsrel}), show a slightly lower, but still highly significant excess of stars above the TRGB. Counts in the same magnitude bin in W8 and F4, which are at $\sim 37$~kpc in between major and minor axis toward the southeast and south, have stellar densities at the upper limit of the foreground field F10A, but are consistent with its stellar density within $1\sigma$. The innermost field, W7 at $R_{gc}\sim 20$~kpc close to the minor axis, also shows an excess of stars brighter than the TRGB, although only at the $\sim 2\sigma$ level. Considering projected distances on sky alone (assuming spherical halo), it appeared that the bright star excess is lowest in W7, the field that has lowest projected galactocentric distance. However, after accounting for a higher ellipticity of the halo of e=0.54 (see Sec.~5) and arranging fields according to their distances that correspond to the semimajor axis of the elliptical isophote that goes through the field, the minor axis field W7 is at 34.7~kpc (see Fig.~\ref{fig:counts_bright}). This shows that when the ellipticity in the halo is accounted for, the bright star excess increases monotonically inward. Because of older cameras used for observations in F1-F3 and their different completeness and crowding corrections,
we decided to exclude these pointings from this analysis.\footnote{F3 located at distance corresponding to $\sim 38$~kpc, when accounting for ellipticity, shows bright star excess intermediate between W7 and F4, thus continuing the observed trend. F2, located near the same isophote as W6 and W10, has significantly fewer stars than these two fields. F1, the innermost field at 1.8~Re, has a very high excess of bright stars, but part of this might be due to blends. }
This excess of stars above the TRGB might be due to bright AGB giants in NGC~5128 or to a contribution from an additional foreground component. It might also be due to scatter of stars from the top of the RGB due to photometric errors. We examine each option in turn.
\subsection{Blends and photometric errors?}
Larger photometric errors and scattering of stars into a brighter magnitude bin due to blends of two or more RGB stars close to the TRGB would result in an excess of stars just above the TRGB, as observed in the bottom panel of Fig.~\ref{fig:counts_bright}. We note, however, that fields showing this excess also have a marginal excess in the brighter magnitude bin ($22<I_0<23$, middle panel), which cannot be populated in the same way because too few stars lie around $I_0 \sim 23$ and the photometric errors are smaller. Furthermore, our fields are not crowded, and our extensive completeness tests exclude the possibility of such large errors and blends (see Fig.~\ref{fig:delmag}).
Therefore this hypothesis is rejected.
\subsection{Additional foreground component?}
An additional foreground component may seem a more plausible explanation for the excess of contaminants in W9+W10+W11+W12, which are very close on the sky. We combined the $I_0$ LF of these four nearby fields and used the Sobel filter to measure the discontinuity in the LF, which could correspond to the TRGB. While the TRGB belonging to the NGC~5128 is clearly detected at $I_0 = 23.85 \pm 0.02$~mag, the brighter stars show a quite sharp increase in the LF at $I_0= 23.3 \pm 0.1$~mag. If this sharp feature in the LF were due to a TRGB, this would place the foreground object at a distance of 2.95~Mpc, compatible with a dwarf satellite in the Centaurus A group \citep{mueller+19}. Indeed, NGC~5128 is known to host a rich system of dwarf satellites \citep{karachentsev+07, mueller+17,crnojevic+19}. However, at the distance of 2.95 Mpc W11-W9 are separated by $4.88$~kpc, which is much too large for a dwarf galaxy. The similar excess observed in W6 is even more problematic. W6 is even farther away on the other side of the galaxy. The physical distance between these fields makes the hypothesis of a foreground object implausible.
We also checked the dependence of the excess of stars on the MW latitude and found no correlation. Based on these arguments, we deem it unlikely that foreground contamination primarily causes the observed bright stars in the inner halo fields.
\subsection{Intermediate-age AGB component}
Intermediate-age AGB stars could explain the excess counts above the TRGB. We note that in addition to the excess in stellar density in the bin immediately above the TRGB, as seen in the bottom panel in Fig.~\ref{fig:counts_bright}, the middle panel also indicates a mild excess in the next brighter bin ($22 \leq I_0 \leq 23$) in the same fields.
Figure~\ref{fig:cmd_isochs} compares the bright portion of the statistically decontaminated CMDs in the fields that show excess counts (left panels) with the theoretical isochrones, which are shifted to the distance of NGC~5128. The isochrones were constructed using the CMD tool\footnote{\tt{http://stev.oapd.inaf.it/cgi-bin/cmd}} maintained by L. Girardi and are based on the Padova set of stellar evolutionary models that match PARSEC isochrones \citep{bressan+12} ending at the first thermal pulse with COLIBRI models \citep{marigo+13} that describe the thermally pulsing AGB (TP AGB) phase. The latter are calibrated on the characteristics of the population of AGB stars in the Magellanic Clouds
\citep{pastorelli+19, pastorelli+20}. The theoretical isochrones are plotted for the models ranging from the most metal-poor ([M/H]$=-2$~dex; bluest model) to super-solar metallicity ([M/H]$=+0.2$~dex; reddest model). The red dots show RGB, blue show early AGB, and green presents the TP AGB evolutionary phase. The
density of points
along the isochrones is proportional to stellar evolutionary lifetime: Each isochrone is a simulation of a simple stellar population with a given age (2, 4, 6, or 10 Gyr) and metallicity, as indicated in the panels. The boxes drawn over the isochrones are the same as those overplotted on the observed CMDs on the left. The magenta box (hereafter RGB box) spans the magnitude range of the upper RGB up to the TRGB ($23.85<I_0<25.45$), the solid purple box (hereafter AGB box) samples 0.6 mag above the TRGB ($23.25<I_0<23.85$), and the dashed purple box spans a brighter range $22.25<I_0<23.85$. The isochrones show that for old ages, only the most metal-poor populations may contribute stars brighter than the TRGB. There are in any case too few of these and they are too faint to account for the excess observed in the inner fields. Clearly, a younger population of 4 Gyr or younger is needed to account for the observed excess of stars.
We can estimate the contribution of this young component using the observed star counts ($N_j$) in the RGB and AGB boxes and the theoretical mass specific production in the same boxes ($P_j = N_j/M_{SSP}$; \citealt{greggio+renzini2011}). The $P_j$ factors, derived from the simulated simple stellar populations, multiplied by the number of observed stars in the diagnostic boxes, yield the initial mass of their parent SSP \citep{greggio02}.\footnote{The model populations were computed assuming a Salpeter IMF, but the IMF is irrelevant for the computation of the $N_{AGB}/N_{RGB}$ as long as it is the same as for both components.}
We find that younger stellar populations are more efficient producers of stars in the RGB box, that is, they provide more objects per unit mass in this location on the CMD. The dependence of $P_{RGB}$ on the metallicity is instead weak. Conversely, $P_{AGB}$ shows a strong dependence on metallicity at ages younger than $\sim$ 4 Gyr, with a peak at $[M/H] \simeq -0.5$.
Adopting a median metallicity for the stars in the inner halo of NGC~5128 of [M/H]$=-0.5$~dex (see Paper~I), we computed the expected ratio of star counts in the AGB and RGB boxes as function of the contribution of the young component to the total mass of the stellar population sampled in the field for different combinations of the ages of the old and the young component. Figure~\ref{fig:plratio} shows the results for a 10 Gyr old stellar population coupled with a young component of various ages. The shaded area represents the range of this ratio as measured in the inner fields under discussion. Dashed lines refer to the case in which the AGB box extends up to $I_0 = 22.25$.
The observed $N_{AGB}/N_{RGB}$ ratio is compatible with a ~10\% fraction of $\sim 2$~Gyr component along with an older population. Alternatively, an even larger ($\sim 30-40$\%) fraction of 3 Gyr old stars is required. It is impossible to reproduce the observed counts above the TRGB without a population of at least 3 Gyr or younger.
Based on the above, we conclude that in addition to the bulk of the old stellar population, there is the excess of stars above the TRGB within ($\la 35$~kpc) due to an intermediate-age AGB component with 2-3 Gyr old stars contributing between 10-30\% in mass. In the fields within $\la 25$~kpc, which show excess also in the $22 < I_0 < 23$~mag bin, the younger population may contribute even up to 40\% of mass if its age is 3~Gyr.
This result is a fourth piece of evidence that the inner halo of NGC~5128 had an extended star formation or multiple star formation bursts. The variability period analysis of the luminous
LPV stars discovered in two fields located $\sim 18$~kpc to the northeast and $\sim 10$~kpc to the south \citep{rejkuba+03_LPVcat} revealed a fraction of $\sim 10$\% of LPVs with periods longer than 500~days, which are due to more massive and hence younger progenitors with ages ranging between 1.5-5 Gyr, depending on the metallicity \citep{rejkuba+03_LPVnir}. The spectroscopic study of
GCs using Lick indices \citep{woodley+10a} showed that 68\% of the NGC 5128 GCs have old ages ($>8$ Gyr), 14\% have intermediate ages (5--8 Gyr), and 18\% are younger than $5$~Gyr. Finally, \citet{rejkuba+11} compared the deep CMD of F4, which reaches the core helium burning red clump stars, with simulated CMDs. They found that two-burst models with 70--80\% of the stars formed $12\pm 1$ Gyr ago and 20--30\% contribution of $2 - 4$ Gyr old stars provide the best agreement with the data. Almost all GCs are located within $\sim 30$~kpc, simply because this was the area of the galaxy explored at the time. More recent surveys \citep{taylor+17_SCABS, hughes+21} have found many more GC candidates extending out to more than 100~kpc, but not much is known about their age and metallicity distributions.
We note that no high excess of bright AGB stars is present in F4; see the bottom panel of Fig.~\ref{fig:counts_bright}, which shows that this field has a stellar density just above the upper range of our foreground field F10A. The two results agree if the majority of the intermediate-age stars in this field are between 3-4 Gyr old because we showed that stellar populations older than 3 Gyr do not contribute much to the bright AGB counts (Fig.~\ref{fig:cmd_isochs}), except for the very metal-poor ones, which are not strongly present in the field F4 \citep{rejkuba+05}.
Given the active assembly history of NGC~5128, it is not surprising to find such an intermediate-age population fraction even out to $\sim 30$~kpc in the halo. In a recent paper, \citet{wang+20} proposed that NGC~5128 is a result of a major merger, based on simulations that started 6 Gyr ago with two nearly equal-size progenitors and ended with a fusion 2 Gyr ago. The simulations reproduced several observables fairly well (e.g., planetary nebula velocity field, morphology of the halo and that of shells and streams observed in the PISCeS survey, metallicity distribution, halo metallicity gradient, and star formation in the central disk of NGC~5128), and made predictions that do not have sufficient observables to constrain the model. One of these is the stellar age distribution in the halo, which shows an increase of intermediate-age stars and younger mean age in the inner halo, in qualitative agreement with our finding (for more details, see Fig.~6 in \citealt{wang+20}).
Other studies using deep near-IR data combined with optical observations could help to disentangle the Milky Way foreground dwarfs from giants in NGC~5128, and thus to better quantify the fraction of the intermediate-age population. An example is shown in Appendix B for the major-axis field F5, where we find very few luminous AGB stars.
When the NGC~5128 distance modulus of 27.91, the average reddening $E(B-V)=0.1$, and metallicity [Fe/H]$=-0.4$~dex are taken into account, the MS turnoff magnitude for a 2 Gyr population would have I=19.26, J=29, H=28.9, and K=28.86 mag. A 4 Gyr population MS turnoff is fainter (I=30.76, J=30.45, H=30.2, and K=30.15). The bright MS turnoff for the intermediate-age population younger than $\sim 4$~Gyr will be within reach of the MICADO camera at the Extremely Large Telescope (ELT) throughout the halo of NGC~5128 beyond 1 $R_e$ \citep{schreiber+14, davies+21}.
\section{Stellar density profile and ellipticity of the halo}
In Paper I, we suggested based on a smaller dataset (only F1-F9, and with a less
thorough CMD cleaning procedure) that the outer halo of NGC 5128 might be more elongated than the inner halo, consistent with results from other large galaxies.
The present, more extended set of data can be used to carry out a more detailed
measurement of the structure of the diffuse halo. For this purpose, we used the star counts for the cleaned fields (Fig.~\ref{fig:cmdf1_9_clean} and \ref{fig:cmdW1_16_clean}).
RGB stars over all metallicities are included in this simple indicator,
but no corrections are included for any intrinsic age differences of the stellar population as a function
of galactocentric distance.
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig15_density_e077.pdf}
}
\caption{\emph{} Number density of halo stars in NGC 5128 as a function of galactocentric
distance $a$, normalized to the effective radius $R_e$ (upper panel). This calculation assumes a halo elongation
$e = 0.77$ from the inner-halo photometry of \citet{dufour+79}. Black symbols are the ACS and WFPC2 fields from Table \ref{tab:HSTobslog},
and open red symbols are the WFC3 fields. The equation for the power-law fit line is given in the text. Error bars are calculated from Poisson statistics, including
the raw number counts in each field, the number of subtracted stars from the decontamination procedure, and the completeness correction.\emph{} Residuals from the fitted line in the upper panel, plotted vs. azimuthal angle
from the isophotal major axis (lower panel). The residuals clearly show a systematic trend with position angle $\theta_a$,
indicating that an incorrect ellipticity $e$ has been assumed.}
\label{fig:density1}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig16_density_e054.pdf}
}
\caption{Same data as in the previous figure, now recalculated with the
assumption that the halo eccentricity is $e = 0.54$ (upper panel).
The new fitted power-law line is given in the text.
The \emph{dashed line} gives the best-fitting de Vaucouleurs $r^{1/4}$
profile to the same data. Residuals from the power-law solution in the upper panel, plotted vs. azimuthal distance
from the isophotal major axis (lower panel). Significant scatter remains, but there is no remaining trend with $\theta_a$.}
\label{fig:density2}
\end{figure}
To calculate the number density of RGB stars $\phi_{RGB}$
versus galactocentric distance, we used the
decontaminated (cleaned) CMDs and took all stars within the selection box $24 < I_0 < 26, 1.0 < (V-I)_0 < 1.4$. The total numbers of stars within this region were explicitly corrected for
photometric incompleteness: that is, each individual star was multiplied by $(1/f),$
where $f$ is the completeness factor at its given magnitude and color.
For the ACS and WFC3 fields F5-F9, the mean completeness correction
factor is just 1.1, while for the shorter-exposure fields W1-W16, the correction factor
averages about 1.34. For F1-F3, which were taken with the less sensitive WFPC2 camera but
with relatively long exposures, the correction factor is about 1.06. Finally, for the
very deep F4, the correction is 1.00 over our magnitude range of interest.
Dividing by the area of the camera then gives $\phi_{RGB}$, now fully
corrected for contamination and incompleteness.
The location $(\alpha, \delta)$ of each field on the sky
was converted into coordinates $(x, y)$ projected onto the major and minor axes of
the galaxy, assuming that the major axis is oriented 55\degree\ N of E (clockwise; see Fig.~1).
From the basic equation of the ellipse and its eccentricity,
\begin{equation}
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \,\, {\rm and} \, \, b = ea
,\end{equation}
we obtain the necessary relation to convert any location $(x, y)$ into its equivalent
semimajor axis $a$ value,
\begin{equation}
a^2 = x^2 + \frac{y^2}{e^2}.
\end{equation}
For a first iteration, we assumed\emph{} that (a) the eccentricity $e$ is the same as the
value $e = 0.77$ for the inner halo isophotes (as used in Paper I); and (b) the orientation
angle of the major axis (55\degree N of E, i.e., clockwise) is a constant and valid at all radii.
However, because we have target fields over a wide area that reaches far beyond the inner halo
where $e$ was determined, we can also investigate if $e$ might change at large radii.
After calculating $a$(equiv) and $\phi_{RGB}$ for every target field, we plotted a first
estimate of the density profile.
The result is shown in Fig.~\ref{fig:density1}. Here $a$ is normalized to the isophotal
effective radius adopted as $R_e = 305''$ (Paper I).
A least-squares fit with equal weights to all points in Fig.~\ref{fig:density1} gives
\begin{equation}
{\rm log} ~ \phi_{RGB} \, = \, (5.117 \pm 0.201) - (3.088 \pm 0.193)~{\rm log}~(a/R_e)
,\end{equation}
where the units of $\phi$ are number per arcmin$^2$. The residuals around this fit
have an rms scatter $\sigma({\rm log} \phi) = 0.290$ dex. However, the residuals (shown in the lower panel, plotted versus azimuthal angle $\theta_a$) clearly
depend on position angle.
That is, the fields closer to the \emph{\textup{minor}} axis ($\theta_a \rightarrow 90\degree$)
have lower densities than expected under our assumptions. An obvious interpretation is
that the equivalent $a-$values for these same fields should actually be higher than
we first assumed: $e$ is even lower than we thought, and the halo is in fact more elongated at larger radii than it is in the inner halo.
We can therefore iterate the calculations of $a$ assuming different values of $e$, until a
result for the radial profile is achieved that leaves no systematic residuals in $\phi$ versus position angle $\theta_a$. This same solution will also minimize the
field-to-field scatter around the best-fit curve.
We find that the solution converges at $e = 0.54$ with an estimated uncertainty $\pm 0.02$.
This solution is shown in Fig.~\ref{fig:density2}.
The new equation for the density profile in the same power-law form is
\begin{equation}
{\rm log} ~ \phi_{RGB} \, = \, (5.328 \pm 0.153) - (3.077 \pm 0.138)~{\rm log}~(a/R_e) \, .
\end{equation}
The scatter around the best-fit curve is now minimized, at $\sigma({\rm log} \phi) = 0.212$ dex.
An alternative well-known form for halo light is the de Vaucouleurs-Sersic profile
log $\phi = A + B (a/R_e)^{1/n}$ with $n=4$. For the same assumption that $e = 0.54$
for the outer halo, the best-fit $r^{1/4}$ curve is also shown in Fig.~\ref{fig:density2}.
The result is
\begin{equation}
{\rm log} ~ \phi_{RGB} \, = \, (7.685 \pm 0.271) - (3.008 \pm 0.142) (a/R_e)^{1/4} \, .
\end{equation}
It is nearly as accurate as the simpler power-law form (the residual rms scatter is $\pm 0.224$, insignificantly different from the power law).
The residual scatter of the data points in Fig.~\ref{fig:density2} is clearly
larger than expected from Poisson count statistics alone. The scatter is
also noticeably the largest for the outermost fields, where the numbers of
counted RGB stars are particularly low. This may be evidence that the halo becomes more patchy due to substructure at several $R_e$ and beyond, but
deeper photometry and larger area coverage are needed to pursue this interpretation further.
\begin{figure}
\resizebox{\hsize}{!}{
\includegraphics{Fig17_profile_e054.pdf}
}
\caption{Surface brightness profile along the isophotal major axis of NGC 5128 in the
form of the $a^{1/4}$ law, as described in the text. Our star count data are shown as the
green, black, and red points from the three different HST cameras, along with the best-fitting
$a^{1/4}$ profile given in Fig.~\ref{fig:density2}, and they assume $e=0.54$ for the outer halo,
as described above. The \emph{dashed line} shows the inner-halo surface brightness profile
from \cite{dufour+79}, where $\mu_V$ is in mag arsec$^{-2}$. The magnitude scale at the right
has been shifted vertically to match the inner surface brightness profile with the outer star counts at
$(a/R_e) = 1.158$.}
\label{fig:profile}
\end{figure}
Last, for this section, we show in Fig.~\ref{fig:profile} the starcount data obtained here
combined with the surface brightness profile for the inner part of the galaxy as obtained
by \cite{dufour+79}, which is
$\mu_V = 8.32 [(a/R_e)^{1/4} - 1] + 22.00$ mag arcsec$^{-2}$. The purpose of this plot is simply to give a comprehensive radial profile
of the galaxy along its major axis, extending from $a \simeq 0.23 R_e$ out to almost $30 R_e$.
Our very innermost HST field F1 just barely overlaps the \emph{\textup{outermost}} limit of the direct
surface brightness photometry \citep{dufour+79}, so splicing the two sets of data together
is intrinsically uncertain. Nevertheless, they can be successfully matched for a conversion offset where log $\phi = 0.0$ (1 RGB star per arcmin$^{-2}$ between $I = 24 - 26$) is equivalent to a surface brightness $\mu_V \simeq 33.82$ mag arcsec$^{-2}$ \footnote{For a stellar population model based estimate of stellar density within 2 mag of the TRGB vs. surface brightness see {\tt https://eso.org/sci/meetings/2015/StellarHalos2015/ \\ talks\_presentation/greggio\_stellarhalo.pdf}}.
As Fig.~\ref{fig:profile} shows, our RGB star-count data in the most remote
parts of the halo reach $\mu_V \sim 32$ mag arcsec$^{-2}$ (roughly
equivalent to $\mu_B \simeq 33$; see the figure in Appendix C). These results exemplify the ability of
resolved-star photometry to penetrate to surface brightness levels that can scarcely
be obtained by other means.
Early evidence that the NGC 5128 halo has higher
ellipticity at larger galactocentric distance
was provided by \citet{hesser+84} on the
basis of the spatial distribution of the halo
GCs
and on the isophotal contours
of the outer halo from
deep Schmidt-telescope images that were available then \citep{cannon1981}.
More recently, \citet{crnojevic+13} used VLT/VIMOS ground-based imaging of selected fields along the
major and minor axes. Compared with our HST fields, their data have the advantage of covering a larger total area,
but the disadvantage of much higher field contamination.
Their study traces the NGC 5128 halo out to 85 kpc (slightly more than half as
far as our current study) and finds that the data match an outward extension of the
de Vaucouleurs profile established in the
inner regions \citep{dufour+79} moderately well. Interestingly, they also suggest that
for $R_{gc} \gtrsim 55$ kpc, the halo shows evidence of becoming more elliptical, in the range
$e \sim 0.5-0.6$. Their estimate is very much in line with our finding of
$e = 0.54 \pm 0.02$ for $R_{gc} \gtrsim 30$ kpc.
A systematic increase of stellar halo ellipticity
with radius is also a commonly observed feature of large ETGs \citep[e.g.,][]{tal+vandokkum11,huang+2018}. The \citet{cooper+13} simulations reproduced the observed stellar density profiles well. These simulations combined the semianalytic galaxy formation model of \citet{guo+11}, based on halo merger trees from Millennium II N-body simulation \citep{boylan-kolchin+09}, with the particle-tagging technique of \citet{cooper+10}
to resolve
low surface brightness outskirts of galaxies on scales from Milky Way-like galaxies to central galaxies of groups and poor clusters. The 2D images of simulated galaxies, scaled appropriately to display also the low surface brightness features, have quite elongated stellar halos that reach $\mu_V \ga 32$~mag/arcsec$^2$ at $150-200$~kpc projected galactocentric distance for galaxies in the mass range of NGC~5128 ($M_\star \sim 2 \times 10^{11}$~M$_\sun$; \citealt{fall+romanowski18}). The halo surface brightness and stellar mass density profiles of simulated galaxies are (by construction) best fit by the sum of two Sersic profiles. A single Sersic profile is only a good fit for the circularly averaged total stellar surface density for galaxy halos in which the accreted component dominates at all radii \citep{cooper+13}. The accretion-dominated galaxies have more extended profiles and a higher Sersic index than in situ dominated galaxies, but even for these galaxies, the in situ stars are an important component within the inner $\la 10$~kpc. In a recent study, \citet{pulsoni+21} examined the impact of galaxy merger history on the intrinsic shape profiles of early-type galaxies in the TNG100 Illustris simulation \citep{nelson+19}. They found that galaxies with higher stellar mass and accreted fractions are less flattened and more triaxial, and that mergers contribute to more spherical-triaxial stellar halo shapes.
A ground-based imaging study with the Magellan Megacam
covers a still larger and contiguous area out to $R_{gc} \simeq 150$ kpc \citep[][part of the PISCeS survey]{crnojevic+16}. Their study
reported two low-surface-brightness streams in the outer halo,
as well as faint satellites. The HST fields in our current study typically avoid
these satellites or tidal features and thus are closer to measuring the ``smooth halo''
of the galaxy (as far as such a smooth underlying distribution exists in these
outer regions). However, we note that several of our pointings that are close on the sky show larger scatter, and the most notable exception to the smooth-halo field is pointing F6A, which is traversed by the disrupting dw3 dwarf stream \citep{crnojevic+19}.
\section{Summary and conclusions}
We have analyzed HST imaging of the halo of NGC 5128 to study the
radial density profile of its halo, covering the distance range $R_{gc} = 8$ to 140 kpc
(equivalent to 1.5 to 25 $R_e$). Data from the WFPC2, ACS, and WFC3 cameras
for a total of 29 distinct pointings across the smooth halo were used,
and one of these pointings acted as a control field for background
subtraction. These
target fields span a diameter of 4\degree \ on the sky.
The projected number density of red giant halo stars in the magnitude range
$I_0 = 24 - 26$ over all metallicities was used as the halo tracer after suitable
subtraction of background field contamination and correction for photometric
incompleteness.
\begin{itemize}
\item For magnitudes $I_0 < 24$, we find that the dominant source of field
contamination is from Milky Way foreground stars that are brighter
than the NGC 5128 RGB population, as determined either from a control field
or from TRILEGAL models.
\item In the most remote fields along either the major or minor axes
of the halo, we were able to successfully trace the RGB population
down to
an equivalent surface brightness $\mu_V \simeq 32$ mag arcsec$^{-2}$. With wider area coverage and/or deeper data,
still fainter equivalent surface brightness levels could be achieved. The Nancy Grace Roman Space Telescope will be an ideal instrument for mapping extended nearby galaxy halos \citep{williams+19}.
\item Over the full radial range covered by our study, the profile is well
matched by a classic $r^{1/4}$ curve or (within the scatter of the data)
a simple power-law form $\phi \sim r^{-3.1}$. Despite the fairly steep
decline of the profile, we have not yet discovered any final
cutoff or ``end'' of the halo of this giant galaxy, even with the best
available data. We detect some substructures shown by the scatter in stellar surface density around the best-fit profile, which agrees with the contiguous panoramic imaging by the PISCeS survey \citep{crnojevic+16}. However, most of our fields avoid obvious substructures and measure the ``smooth halo'', as much as such a smooth component exists in the outskirts of a large galaxy.
\item Classic isophotal measurements showed that NGC 5128 has an inner-halo
ellipticity $e= (b/a) = 0.77$. However, our data show that for $R_{gc} \gtrsim 30$ kpc, the halo flattens to $e = 0.54 \pm 0.02$.
\item The analysis of the luminosity function shows that
over $22 < I_0 < 23.86$, that is, above the TRGB, a measurable excess of stars above the background level predicted by the Milky Way population models (both TRILEGAL and Besan\c{c}on) is
present. Beyond 30 kpc, this excess is constant and consistent with an additional Milky Way dwarf-star population that is missing from the models, as is evident also from the combination of optical and near-IR photometry (see Appendix B). At galactocentric distances smaller than 30 kpc, a 2-3 Gyr old AGB component contributes between 10-40\% of the stellar mass. This confirms earlier results from LPVs, GCs, and red clump stars \citep{rejkuba+03_LPVnir, woodley+10a, rejkuba+11}. Further studies combining optical and near-IR data can provide powerful diagnostics distinguishing the Milky Way dwarfs from AGB giants in NGC 5128, and the ELT will be able to resolve the turnoff for this intermediate-age population.
\end{itemize}
The observed stellar density profile, shape, and substructures in the stellar halo of NGC~5128 testify to its growth through hierarchical mergers and show the dominant contribution from accreted stars in its outskirts. It does not answer the question whether the galaxy formed through a relatively recent ($<5-6$~Gyr) major merger of two spiral galaxies \citep{bekki+peng06, wang+20} or had a two-phase formation \citep{oser+10} with an early assembly stage that formed the elliptical, which subsequently grew through minor-merger episodes and accretion of smaller satellites \citep{naab+09}. In its inner parts, NGC~5128 shows clear evidence of a past merger history, and our study uncovered a quite extended distribution of $2-3$~Gyr old intermediate-age stellar population. The accretion of a small gas-rich spiral may have provided the fuel for the recent star formation in the central parts as suggested by e.g., \citet{quillen+93}. However, the short time since the merger in this model ($< 2 \times 10^8$ yr) would imply that the bright intermediate-age component we observe within the inner 30~kpc was deposited directly from the accreted galaxy or originated from a different merger event. In a following paper, we will use this dataset to derive
the metallicity distribution function of the RGB stars along with their
progressive change with radius.
\begin{acknowledgements}
We thank the referee for the thorough review of the paper.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research is based on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
The HST observations are associated with programs 5905, 8195, 9373, 12964, 13856, and 15426.
This paper also used data collected at the European Southern Observatory under ESO programme 290.B-5040(A) and 290.B-5040(B).
Research by DC is supported by NSF grant AST-1814208, and by NASA through grants number HST-GO-15426.007-A and HST-GO-15332.004-A from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.
\end{acknowledgements}
\bibliographystyle{aa}
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2105.00079
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\section{Introduction}
\label{sec:introduction}
\begin{figure}[t]
\centering
\includegraphics[clip, width=0.8\columnwidth]{./example_3.pdf}
\caption{A more informative response (Response B in the figure) can provide information that helps to infer the query content given the dialogue context.}
\vspace{-0.5 cm}
\label{fig:back-reasoning}
\end{figure}
Recently developed end-to-end dialogue systems are trained using large volumes of human-human dialogues to capture underlying interaction patterns~\citep{li2015diversity,li2017adversarial,xing2017topic,khatri2018advancing,vinyals2015neural,zhang2019dialogpt,bao2019plato}. A commonly used approach to designing data-driven dialogue systems is to use an encoder-decoder framework: feed the dialogue context to the encoder, and let the decoder output an appropriate response. Building on this foundation, different directions have been explored to design dialogue systems that tend to interact with humans in a coherent and engaging manner~\citep{li2016deep,li2019dialogue,wiseman2016sequence,baheti2018generating,xing2016topic,zhang2018personalizing}. However, despite significant advances, there is still room for improvement in the quality of machine-generated responses.
An important problem with encoder-decoder dialogue models is their tendency to generate generic and dull responses, such as \emph{``I don't know''} or \emph{``I'm not sure''}~\citep{li2015diversity,baheti2018generating,li2016deep,jiang-why-2018}.
There are two types of methods for dealing with this problem.
The first introduces updating signals during training, such as modeling future rewards (e.g., ease of answering) by applying reinforcement learning~\citep{li2016deep,li2019dialogue}, or bringing variants or adding constraints to the decoding step~\citep{wiseman2016sequence,li2015diversity,baheti2018generating}.
The second type holds that, by itself, the dialogue history is not enough for generating high-quality responses, and side information should be taken into account, such as topic information~\citep{xing2016topic,xing2017topic} or personal user profiles~\citep{zhang2018personalizing}. Solutions relying on large pre-trained language models, such as DialoGPT~\citep{zhang2019dialogpt}, can be classified into the second family as well.
In this paper, we propose to train dialogue generation models \emph{bidirectionally} by adding a backward reasoning step to the vanilla encoder-decoder training process.
We assume that the information flow in a conversation should be coherent and topic-relevant. Given the dialogue history, neighboring turns are supposed to have a tight topical connection to infer the partial content of one turn given the previous turn \emph{and vice versa}.
Inferring the next turn given the (previous) conversation history and the current turn is the traditional take on the dialogue generation task.
We extend it by adding one more step: given the dialogue history and the next turn, we aim to infer the content of the current turn. We call the latter step \emph{backward reasoning}. We hypothesize that this can push the generated response to be more informative and coherent: it is unlikely to infer the dialogue topic given a generic and dull response in the backward direction.
An example is shown in Figure~\ref{fig:back-reasoning}. Given the dialogue context and \emph{query},\footnote{We use \emph{query} to distinguish the current dialogue turn from the context and the response; \emph{query} is not necessarily a real query or question as considered in search or question-answering tasks.} we can predict the reply following a traditional encoder-decoder dialogue generation setup. In contrast, we can infer the content of \emph{query} given the context and reply as long as the reply is informative.
Inspired by~\citet{zheng2019mirror}, we introduce a latent space as a bridge to simultaneously train the encoder-decoder model from two directions.
Our experiments demonstrate that the resulting dialogue generation model, called \emph{Mirror}, benefits from this bidirectional training process.
Overall, our work provides the following contributions:
\begin{enumerate}[leftmargin=*,label=\textbf{C\arabic*},nosep]
\item We introduce a dialogue generation model, \emph{Mirror}, for generating high quality responses in open-domain dialogue systems;
\item We define a new way to train dialogue generation models bidirectionally by introducing a latent variable; and
\item We obtain improvements in terms of dialogue generation performance with respect to human evaluation on two datasets.
\end{enumerate}
\section{Related Work}
Conversational scenarios being considered today are increasingly complex, going beyond the ability of rule-based dialogue systems~\citep{weizenbaum1966eliza}. \citet{ritter2011data} propose a data-driven approach to generate responses, building on phrase-based statistical machine translation. Neural network-based models have been studied to generate more informative and interesting responses~\citep{sordoni2015neural,vinyals2015neural,serban2016hred}.
\citet{serban2017hierarchical} introduce latent stochastic variables that span a variable number of time steps to facilitate the generation of long outputs.
Deep reinforcement learning methods have also been applied to generate coherent and interesting responses by modeling the future influence of generated responses \citep{li2016deep,li2019dialogue}. Retrieval-based methods are also popular in building dialogue systems by learning a matching model between the context and pre-defined response candidates for response selection~\citep{qiu2020if,tao2019multi,wu2016sequential,gu2020speaker}. Our work focuses on response \emph{generation} rather than \emph{selection}.
Since encoder-decoder models tend to generate generic and dull responses, \citet{li2015diversity} propose using maximum mutual information as the objective function in neural models to generate more diverse responses.
\citet{xing2017topic} consider incorporating topic information into the encoder-decoder framework to generate informative and interesting responses.
To address the dull-response problem, \citet{baheti2018generating} propose incorporating side information in the form of distributional constraints over the generated responses. \citet{su2020diversifying} propose a new perspective to diversify dialogue generation by leveraging non-conversational text. Recently, pre-trained language models, such as GPT-2~\citep{radford2019language}, Bert~\citep{devlin2018bert}, XL-Net~\citep{yang2019xlnet}, have been proved effective for a wide range of natural language processing tasks. Several authors make use of pre-trained transformers to attain performance close to humans both in terms of automatic and human evaluation~\citep{zhang2019dialogpt,wolf2019transfertransfo,golovanov2019large}. Though pre-trained language models can perform well for general dialogue generation, they may become less effective without enough data or resources to support these models' pre-training. In this work, we show the value of developing dialogue generation models with limited data and resources.
The key distinction compared to previous efforts~\citep{li2015diversity,baheti2018generating} is our work is the first to use the original training dataset through a differentiable backward reasoning step, without external information.
\section{Method: \emph{Mirror}}
\label{sec:method}
\subsection{Problem setting}
In many conversational scenarios, the dialogue context is relatively long and contains a lot of information, while the reply (\emph{Response}) is short (and from a different speaker). This makes it difficult to predict the information in the context by only relying on the response in the backward direction.
Therefore, we decompose the dialogue context into two different segments: the context $c$ and query $x$ (Figure~\ref{fig:back-reasoning}).
Assuming that we are predicting the response at turn $t$ in a dialogue, the context $c$ will consist of the dialogue turns from $t-m$ to $t-2$ and the query $x$ corresponds to turn $t-1$.
Here, we use the term \emph{query} to distinguish the dialogue turn at time step $t-1$ from the context $c$ and response $y$; as explained before, the term \emph{query} should not be confused with a query or question as in search or question-answering tasks.
The value $m$ indicates how many dialogue turns we keep in the context $c$.
We use $c_{all}$ to represent the concatenation of $c$ and $x$, which is also the original context before being decomposed.
Our final goal is to predict the response $y$ given dialogue context $c$ and query $x$.
\subsection{Mirror-generative dialogue generation}
\label{section:mirror-method}
\citet{shen2017conditional} propose to maximize the conditional log likelihood of generating response $y$ given context $c_{all}$, $\log p(y \mid c_{all})$, and they introduce a latent variable $z$ to group different valid responses according to the context $c_{all}$.
The lower bound of $\log p(y \mid c_{all})$ is given as:
\begin{equation}
\begin{split}
\log p(y \mid c_{all}) \geq{} & \mathbb{E}_{z \sim q_\phi(z \mid c_{all}, y)} \log p_\theta (y \mid c_{all}, z) - {}\\
& D_\mathit{KL}(q_\phi(z \mid c_{all}, y) \| p_\theta(z \mid c_{all})).
\end{split}
\label{eq:loss-vhred}
\end{equation}
In Eq.~\ref{eq:loss-vhred}, $q_\phi(z \mid c_{all}, y)$ is the posterior network while $p_\theta(z \mid c_{all})$ is the prior one.
Instead of maximizing the conditional log likelihood $\log p(y \mid c_{all})$, we propose to maximize $\log p(x,y \mid c)$, representing the conditional likelihood that $\langle x,y\rangle$ appears together given dialogue context $c$.
The main assumption underlying this change is that in a conversation, the information flow between neighboring turns should be coherent and relevant, and this connection should be bidirectional.
For example, it is not possible to infer what the query is about when a generic and non-informative reply ``I don't know'' is given as shown in Figure~\ref{fig:back-reasoning}.
By taking into account the information flow from two different directions, we hypothesize that we can build a closer connection between the response and the dialogue history and generate more coherent and informative responses. Therefore, we propose to optimize $\log p(x,y \mid c)$ instead of $\log p(y\mid c_{all})$.
Following \cite{kingma2013auto,shen2017conditional}, we choose to maximize the variational lower bound of $\log p(x,y \mid c)$, which is given as:
\begin{equation}
\begin{split}
\log p(x,y \mid c) \geq {} & \mathbb{E}_{z \sim q_\phi(z \mid c, x, y)} \log p_\theta (x, y \mid c, z) - {}\\
& D_\mathit{KL}(q_\phi(z \mid c, x, y) \| p_\theta(z \mid c)),
\end{split}
\label{eq:loss-lb}
\end{equation}
where $z$ is a shared latent variable between context $c$, query $x$ and response $y$.
Next, we explain how we optimize a dialogue system by maximizing the lower bound shown in Eq.~\ref{eq:loss-lb} from two directions.
\begin{figure}[t]
\centering
\includegraphics[clip, width=0.75\linewidth]{./framework_half.pdf}
\caption{The main architecture of our model, \emph{Mirror}. It consists of three steps: information encoding, latent variable generation, and target decoding. }
\vspace*{-0.5\baselineskip}
\label{fig:framework}
\end{figure}
\subsubsection{Forward generation in dialogue generation}
With respect to the forward dialogue generation, we interpret the conditional likelihood $\log p_\theta (x, y \mid c, z)$ in the forward direction:
\begin{equation}
\begin{split}
\mbox{}\hspace*{-3mm}
\log p_\theta (x, y \mid c, z)
= \log p_\theta (y \mid c, z, x) + \log p_\theta (x \mid c, z).
\end{split}
\hspace*{-2mm}\mbox{}
\label{eq:cond-backward}
\end{equation}
Therefore, we can rewrite Eq.~\ref{eq:loss-lb} in the forward direction as:
\begin{equation}
\begin{split}
\log\, & p(x,y \mid c)\\
\geq{} & \mathbb{E}_{z \sim q_\phi(z \mid c, x, y)} [\log p_\theta (y \mid c, x, z) + \log p_\theta (x \mid c, z)] {}\\& - D_\mathit{KL}(q_\phi(z \mid c, x, y) \| p_\theta(z \mid c)).
\end{split}
\label{eq:forward-loss}
\end{equation}
We introduce $q_\phi(z \mid c, x, y)$ as the posterior network, also referred to as the recognition net, and $p_\theta(z \mid c)$ as the prior network.
\subsubsection{Backward reasoning in dialogue generation}
As in the forward direction, if we decompose the conditional likelihood $\log p_\theta (x, y \mid c, z)$ in the backward direction, we can rewrite Eq.~\ref{eq:loss-lb} as:
\begin{equation}
\begin{split}
\log\,& p(x,y \mid c) \\
\geq {} & \mathbb{E}_{z \sim q_\phi(z \mid c, x, y)} [\log p_\theta (x \mid c, y, z) + \log p_\theta (y \mid c, z)] {}\\
&- D_\mathit{KL}(q_\phi(z \mid c, x, y) \| p_\theta(z \mid c)).
\end{split}
\label{eq:backward-loss}
\end{equation}
\subsubsection{Optimizing dialogue systems bidirectionally} Since the variable $z$ is sampled from the shared latent space between forward generation and backward reasoning steps, we can regard $z$ as a bridge to connect the training in two different direction and this opens the possibility to train dialogue models effectively. By merging Eq.~\ref{eq:forward-loss} and Eq~\ref{eq:backward-loss}, we can rewrite the lower bound Eq.~\ref{eq:loss-lb} as:
\begin{equation}
\begin{split}
\mbox{}\hspace*{-2mm}
\log {} &p(x,y \mid c) \geq \mathbb{E}_{z \sim q_\phi(z \mid c, x, y)} \left[
\frac{1}{2} \log p_\theta (x \mid c, z, y) \right. \\
& \phantom{XX}+ \frac{1}{2} \log p_\theta (y \mid c, z) + \frac{1}{2}\log p_\theta (y \mid c, z, x) \\
& \phantom{XX}+ \left.\frac{1}{2} \log p_\theta (x \mid c, z) -D_\mathit{KL}(q_\phi(z \mid c, x, y) \| p_\theta(z \mid c))\vphantom{\frac{1}{2}}\right]\\
={}&L(c, x, y; \theta, \phi),
\end{split}
\label{eq:loss}
\end{equation}
which is the final loss function for our dialogue generation model.
\subsubsection{Model architecture} The complete architecture of the proposed joint training process is shown in Figure~\ref{fig:framework}.
It consists of three steps: (1) information encoding, (2) latent variable generation, and (3) target decoding.
With respect to the information encoding step, we utilize a context encoder $Enc_{ctx}$ to compress the dialogue context $c$ while an utterance encoder $Enc_{utt}$ is used to compress the query $x$ and response $y$, respectively.
To model the latent variable $z$, we assume $z$ follows the multivariate normal distribution, the posterior network $q_\phi(z\mid c, x, y) \sim N(\mu, \sigma^2 I)$ and the prior network $p_\theta(z \mid c) \sim N(\mu^\prime, \sigma^{\prime2} I)$. Then, by applying the reparameterization trick \citep{kingma2013auto}, we can sample a latent variable $z$ from the estimated posterior distribution $N(\mu, \sigma^2 \bm{I})$. During testing, we use the prior distribution $N(\mu^\prime, \sigma^{\prime2} \bm{I})$ to generate the variable $z$. The KL-divergence distance is applied to encourage the approximated posterior $N(\mu, \sigma^2 \bm{I})$ to be close to the prior $N(\mu^\prime, \sigma^{\prime2} \bm{I})$.
According to Eq.~\ref{eq:loss}, the decoding step in the right side of Figure~\ref{fig:framework} consists of four independent decoders, $Dec_1$, $Dec_2$, $Dec_3$, and $Dec_4$, corresponding to $\log p(y\mid c,z,x)$, $\log p(x\mid c,z)$, $\log p(x\mid c,z,y)$ and $\log p(y\mid c,z)$, respectively. Decoder $Dec_1$ is used to generate the final response during the testing stage. To make full use of the variable $z$, we attach it to the input of each decoding step. Since we have the shared latent vector $z$ as a bridge, training for the two directions is not independent, and updating one direction will definitely improve the other direction as well. In the end, both directions will contribute to the final dialogue generation process.
\section{Experimental Setup}
\label{sec:experiments}
\subsection{Datasets}
We use two datasets.
First, the MovieTriples dataset~\citep{serban2016hred} has been developed by expanding and preprocessing the Movie-Dic corpus~\citep{banchs2012movie} of film transcripts and each dialogue consists of 3 turns between two speakers. We regard the first turn as the dialogue context while the second and third one as the query and response, respectively. In the final dataset, there are around 166k dialogues in the training set, 21k in the validation set and 20k in the test set. In terms of the vocabulary table size, we set it to the top 20k most frequent words in the dataset.
Second, the DailyDialog dataset~\citep{li2017dailydialog} is a high-quality multi-turn dialogue dataset. We split the dialogues in the original dataset into shorter dialogues by every three turns as a new dialogue. The last turn is used as the target response and the first as the context and the third one as the query. After preprocessing, we have 65k, 6k, and 6k dialogs in the training, testing and validation sets, respectively. We limit the vocabulary table size to the top 20k most frequent words for the DailyDialog dataset.
\subsection{Baselines}
\begin{description}[leftmargin=\parindent,nosep]
\item[\textbf{Seq2SeqAtt}] This is a LSTM-based~\citep{hochreiter1997long} dialogue generation model with attention mechanism~\citep{bahdanau2014neural}.
\item[\textbf{HRED}] This method~\citep{serban2016hred} uses a hierarchical recurrent encoder-decoder to sequentially generate the tokens in the replies.
\item[\textbf{VHRED}] This extension of HRED incorporates a stochastic latent variable to explicitly model generative processes that possess multiple levels of variability~\citep{serban2017hierarchical}. This is also the model trained with Eq.~\ref{eq:loss-vhred}.
\item[\textbf{MMI}] This method first generates response candidates on a Seq2Seq model trained in the direction of context-to-target, $P(y\mid c,x)$, then re-ranks them using a separately trained Seq2Seq model in the direction of target-to-context, $P(x\mid y)$, to maximize the mutual information~\citep{li2015diversity}.
\item[\textbf{DC}] This method incorporates side information in the form of distributional constraints, including topic constraints and semantic constraints~\citep{baheti2018generating}.
\item[\textbf{DC-MMI}] This method is a combination of \textbf{MMI} and \textbf{DC}, where the decoding step takes into account mutual information together with the proposed distribution constraints in the method \textbf{DC}.
\end{description}
\subsection{Training details}
We implement our model, \emph{Mirror}\footnote{Codebase: \url{https://github.com/cszmli/mirror-sigir}}, with PyTorch in the OpenNMT framework~\citep{opennmt}. The utterance encoder is a two-layer LSTM~\citep{hochreiter1997long} and the dimension is 1,000. The context encoder has the same architecture as the utterance encoder but the parameters are not shared. The four decoders have the same design but independent parameters, and each one is a two-layer LSTM with 1,000 dimensions.
In terms of the dimension of the hidden vector $z$, we set it to $160$ for the DailyDialog dataset while $100$ for MovieTriples. The word embedding size is $200$ for both datasets. We use Adam~\citep{kingma2014adam} as the optimizer. The initial learning rate is $0.001$ and learning rate decay is applied to stabilize the training process.
\subsection{Evaluation}
We conduct a human evaluation
on Amazon MTurk guided by~\citep{li2019acute}.
For each two-way comparison of dialogue responses (against Mirror), we ask annotators to judge which of two responses is more appropriate given the context.
For each method pair (Mirror, Baseline) and each dataset, we randomly sample $200$ dialogues from the test datasets; each pair of responses is annotated by $3$ annotators.
\section{Results and Analysis}
\label{sec:results}
In Table~\ref{Table:results}, we show performance comparisons between Mirror and other baselines on two different datasets. According to Table~\ref{Table:results}(top), it is somewhat unexpected to see that HRED can achieve such close performance compared to Mirror on DailyDialog, given its main architecture is a hierarchical encoder-decoder model. We randomly sample some dialogue pairs for which HRED outperforms Mirror to see why annotators prefer HRED over Mirror. For many of these cases, Mirror fails to generate appropriate responses, while HRED returns generic but still acceptable responses given the context. When we have the back reasoning step in Mirror, we expect that it will lead to more informative generations. Still, it also increases the risk of generating responses with incorrect syntax or relevant but inappropriate responses. A possible reason for the latter is that the backward reasoning step has dominated the joint training process, which can degenerate the forward generation performance.
The performance gap between Mirror and all approaches (including HRED) is large on the DailyDialog dataset (see Table~\ref{Table:results}(bottom)).
\begin{table}[t]
\caption{Human evaluation using the MovieTriple and DailyDialog datasets.}
\label{Table:results}
\centering
\begin{tabular}{l l*{4}{c}}
\toprule
& \textbf{Method pair} & Wins & Losses &Ties \\
\midrule
\parbox[t]{4mm}{\multirow{6}{*}{\rotatebox[origin=c]{90}{\em (a) MovieTriple~}}}
& Mirror vs. Seq2SeqAttn &0.53 &0.37 &0.10 \\
& Mirror vs. HRED &0.41 &0.40 &0.19 \\
& Mirror vs. VHRED &0.45 &0.38 &0.17 \\
& Mirror vs. MMI &0.48 &0.42 &0.10 \\
& Mirror vs. DC &0.50 &0.33 &0.17 \\
& Mirror vs. DC-MMI &0.39 &0.35 &0.26 \\
\midrule
\parbox[t]{4mm}{\multirow{6}{*}{\rotatebox[origin=c]{90}{\em (b) DailyDialog~}}}
& Mirror vs. Seq2SeqAttn &0.50 &0.26 &0.24 \\
& Mirror vs. HRED &0.49 &0.32 &0.19 \\
& Mirror vs. VHRED &0.48 &0.37 &0.15 \\
& Mirror vs. MMI &0.40 &0.34 &0.26 \\
& Mirror vs. DC &0.45 &0.38 &0.17 \\
& Mirror vs. DC-MMI &0.47 &0.35 &0.18 \\
\bottomrule
\end{tabular}
\vspace*{-0.5\baselineskip}
\end{table}
Due to space limitations, we only present one dialogue example in Table~\ref{Table:case_table}. The example is a typical case of why the response generated by DC has high embedding scores, but the human evaluation result is not promising. In this example, the response from DC has high semantic similarity with the context because of words like ``ask you'', ``apartment'', and ``questions''. However, it cannot be regarded as an appropriate and meaningful response in the given context. Comparing Mirror with methods that have use MMI (MMI, DC-MMI), the performance gap is relatively small. This is evidence showing the effectiveness of maximizing mutual information in improving the response quality. The Mirror method can be treated as a way to maximize mutual information implicitly. The advantage is that we can train dialogue models in two directions simultaneously.
\begin{table}[!ht]
\centering
\caption{Example generated responses by different models when the dialogue context is given.}
\label{Table:case_table}
\resizebox{0.8\linewidth}{!}{
\begin{tabular}{c}
\toprule
\textbf{Context}\\
\midrule
\makecell[lt]{\textbf{Speaker A:} here ' s my license . \\
\textbf{Speaker B:} i ' m afraid i ' m going to have to ask you to\\ remain in the apartment . the narcotics squad will be arriving \\any moment now . they want to ask you a few questions . } \\
\midrule
\textbf{Response} \\
\midrule
\makecell[lt]{\textbf{Reference:} squad ? what do they want with me ?\\ i don ' t even use aspirin !\\
\textbf{Seq2Seq:} no .\\
\textbf{HRED:} i don ' t think so .\\
\textbf{VHRED:} oh , i ' m sorry . \\
\textbf{MMI:} i ' m sorry . i ' m sorry . i don ' t know what you ' re \\talking about . i don ' t know what i ' m afraid of . \\
\textbf{DC:} i ' m not going to ask you . but he will be in the apartment for \\a moment -- and we can have some questions with that one of them !\\
\textbf{DC-MMI:} i ' m going to ask you . \\
\textbf{Mirror:} well , i ' m sure they ' ll have to wait . } \\
\bottomrule
\end{tabular}
}
\vspace*{-0.5\baselineskip}
\end{table}
\section{Conclusion and Future Work}
\label{sec:conclusion}
We have presented a novel approach to generating informative and coherent responses in open-domain dialogue systems, called \emph{Mirror}. First, we reformulate the original response generation task from two sides: context and response, to three sides: context, query, and response. Given the dialogue context and query, predicting the response is exactly like the traditional dialogue generation setup. Thus, \emph{Mirror} has one more step: inferring the query given the dialogue context and response.
By incorporating the backward reasoning step, we implicitly push the model to generate responses that have closer connections with the dialogue history.
By conducting experiments on two datasets, we have demonstrated that Mirror improves the response quality compared to several competitive baselines without incorporating additional sources of information, which comes with additional computational costs and complexity.
For future work, Mirror's bidirectional training approach can be generalized to other domains, such as task-oriented dialogue systems and question-answering tasks.
\clearpage
\bibliographystyle{ACM-Reference-Format}
|
2109.01388
|
\section{introduction}
The residual fragments cross sections in spallation reactions are key infrastructure data for nuclear applications in many aspects, such as
nuclear physics, radiation damage to electronics and radio-protection of astronauts \cite{r.protect}, extraterrestrial bodies history via the radioisotopes produced inside \cite{iso.history}, tracing the transport history of cosmic rays \cite{ray.history} and abundance of Li, Be, and B elements \cite{taleofnuclei}, neutron sources or as radioactive isotope beams like the China Spallation Neutron Source (CSNS) facility \cite{CSNS}, the Beijing Rare Ion beam Facility (BRIF) facility \cite{BRIF}, the Accelerator-Driven System (ADS) \cite{ADS1999,ADS2000}, and even in situ proton therapy tomography \cite{PT1,PT2}. Traditional methods to predict fragment productions in spallation reactions include the transport models like quantum molecular dynamics (QMD) \cite{QMD2009,QMD2020,QMD2019}, statistical muti-fragmentation model (SMM) \cite{SMM1995,SMM2001,SMM2005} and the Li\`{e}ge intranuclear cascade (INC) \cite{INCL2013,INCL2014,INCL2015} model, etc. A de-excitation (principally evaporation or fission) simulation is always performed after the QMD, SMM, INC models for better predictions. Semi-empirical formula like EPAX \cite{EPAX1,EPAX2,EPAX3} and SPACS \cite{SPACS} have also been frequently adopted to predict the fragment cross sections in spallation reactions for incident energy higher than 100 MeV/u. Most of the models mentioned above have participated in the international benchmark done under the OECN/NEA \cite{NEA1993,NEA1994,NEA1995,NEA1997} in the mid-nineties and the auspices of the International Atomic Energy Agency (IAEA) in 2010. However, difficulties still exist for the reasons that a wide range of incident energy and a vast of fragments are involved in the spallation reactions. The inadequate precise of present models prevent their applications in many key problems.
Machine learning is efficient to form new model based on the big-data learning, which has been involved in various industries and basic science researches, such as data mining, medical diagnosis, handwriting recognition, biological field, engineering application, automatic driving, stock analysis, and so on. Machine learning techniques have made various novel applications in physics, e.g., nuclear mass \cite{Nmass1,Nmass2,Nmass3,Nmass4}, nuclear charge radii \cite{Nradii}, nuclear $\beta$-decay half-life \cite{Nhalf-life}, neutron-induced reactions \cite{N-induced}, fission yields \cite{Nfission,QiaoChY}, emitting nuclei \cite{N-emitting}, quantum many-body problem \cite{QMB2017,QMB2020}, strong gravitational lenses \cite{SGL}, phases of matter \cite{PM1,PM2,PM3,PM4}, temperature determination in heavy-ion collision \cite{YDSong}, single crystal growth \cite{SCG}, experimental control \cite{EXPC} and nuclear liquid-gas phase transition \cite{RWang2020}. As one of the machine learning techniques, the Bayesian neural networks (BNN) have many advantages, such as automatic complexity control, possibility to use prior information and hierarchical models for the hyperparameters, predictions for outputs and giving uncertainty qualification \cite{Neal1,Neal2}.
Effort has also been paid for the construction of new approaches to describe spallation reactions using the BNN technologies. The direct learning by BNN and the physical guided BNN + SPACS approaches have been performed in previous works \cite{Nspallation1,Nspallation2}. In them, the adopted database has about 4,000 data, which have been measured at GSI, Darmstadt \cite{GSI-Fe,GSI-Xe200,GSI-Xe500,GSI-Xe1000}, and the Lawrence Berkeley Laboratory (LBL) \cite{LBL-Ar,LBL-Ca}. Because the deficient information about incident energies and light fragments in the sample data, unphysical predictions have been found in BNN method. With the guidance of SPACS empirical formula, good predictions have been found in the BNN + sEPAX method for reactions within the incident energies ranging from 300 MeV/u to 1 GeV/u, but poor extrapolation results arise for light fragments and incident energies lower than 300 MeV/u.
In this work, to improve the model accuracy and generalization ability, more than 10,000 cross sections measured at RIKEN \cite{RIKEN-Nb113,RIKEN-Zr105,RIKEN-Pd118+196,RIKEN-Cs+Sr} and excitation functions for proton induced reactions \cite{Michel2014} have been incorporated. To provided reasonable physical guides, a simplified EPAX formula (named as sEPAX) will be proposed. A physical guided BNN + sEPAX model will be constructed to quantify the patterns of systematic deviations between theory and experiment
The paper is organized as follows. In Section \ref{method}, the Bayesian theory and sEPAX formula will be briefly introduced. In Section \ref{result}, the accuracy and generalization of BNN and BNN + sEPAX models are demonstrated. Finally, conclusion are presented in Section \ref{summary}.
\section{model descriptions}
\label{method}
The main concepts of the Bayesian method, the simplified EPAX formula and the model structure for BNN and BNN + sEPAX method will be briefly introduced in this section.
\subsection{Bayesian method\label{bnn}}
The BNN method is one of the typical multilayer perceptron (MLP) networks. It is a ``back-propagation'' or ``feedforward'' network. In the general ideas of MLP networks, some numbers of layers of hidden unities will be constructed, via which the output values $f_{k}(x)$ can be exported from the input sets $x^{(i)}$. In a typical network of one hidden layer, the outputs are calculated as follows,
\begin{equation}\label{MLP}
f_{k}(x;\theta)= a_{k}+\sum_{j=1}^{H}b_{jk} \tanh(c_j+\sum_{i=1}^{I} d_{ji} x_i),
\end{equation}
where $H$ denotes the number of hidden unites, and $I$ is the number of input variables $x=\{x^{(i)}\}$. $d_{ij}$($b_{jk}$) are the weights on the connection from input unit $i$ (hidden unite $j$) to hidden unit $j$ (output unites $k$). The $c_{j}$ and $a_{k}$ are the biases of the hidden and output unites. The weights and bias are the parameters of the network, i.e., $\theta =\{a, b_{j}, c_j, d_{ji}\}$. Each output $f_{k}(x)$ is a weighted sum of hidden unit values plus a bias. Each hidden unite computes a similar weight sum of input values, and then passes it through a nonlinear activation function. The activation function has chosen to be the hyperbolic tangent ($\tanh$) in this work.
For a regression task involving the prediction of a noisy vector $t$ of target variables given the input vector $x$, the likelihood function $p(D/\theta)$ might be defined to be Gaussian-type, with $t$ having a mean of $f_{k}(x)$ (k=1) and a standard deviation of $\alpha$,
\begin{equation}\label{likelihood}
\begin{split}
p(D|\theta)=\exp(-\chi^{2}/2), \\
\chi^{2}=\sum\limits_{i}^{N} [t^{(i)}-f(x^{(i)};\theta)]^2 / {\alpha_{i}^2},
\end{split}
\end{equation}
where $D=(x^{(i)},t^{(i)})$ ($i =$ 1, 2, ... , $N$), and $x^{(i)}$ ( $t^{(i)}$) are the inputs (outputs) of the network structure.
\subsection{Simplified EPAX formula (sEPAX)}
\label{sEPAX}
Following the EPAX parametrizations in Ref. \cite{EPAX1}, the cross section ($\sigma$) for a fragment with mass and charge ($A, Z$) produced from a projectile nucleus ($A_{p}$, $Z_{p}$) impinging on a target nucleus (with $A_{t}=Z_{t}=1$ for proton) is written as,
\begin{equation}
\sigma(A,Z) = Y_{A}\sigma_{Z}(Z_{prob}-Z) = Y_{A}n \exp(-R\mid Z_{prob}-Z\mid^{U}), \label{SigAZ}
\end{equation}
in which $Y_{A}$ is the mass yield. $\sigma_{Z}$ is the ``charge dispersion'' referring to the elemental distribution of given mass number $A$ around the maximum of charge dispersion $Z_{prob}$. The shape of the charge dispersion is governed by the width parameter $R$ and the exponent $U$. The normalization factor $n=\sqrt{R/\pi}$ assures the unity of the integral charge dispersion.
The mass yield, $Y_{A}$, is assumed to exponentially depend on the mass difference between projectile and fragment ($A_{p}-A$),
\begin{equation}
Y_A = S\cdot P\cdot \exp[-P\cdot(A_{p}-A)].
\end{equation}
The slope $P$ depends on the mass of the projectile,
\begin{equation}
P=\exp(-1.731-0.01339\cdot A_p).
\end{equation}
An overall scaling factor $S$ accounts for the peripheral reaction, which depends both on the mass of projectile and target nuclei,
\begin{equation}
S=0.27[(A_p)^{1/3} + (A_t)^{1/3}-1.8].
\end{equation}
The parameters $R$, $Z_{prob}$, and $U$ are strongly correlated to each other, which are difficult to be uniquely obtained using the least-squares fitting technique. $U$ is assorted to $U_{p}$ and $U_{n}$ for the neutron-deficient and neutron-rich sides of the valley of $\beta$-stability, which have different values in the three EPAX versions \cite{EPAX1,EPAX2,EPAX3}. In this work, $U=U_{p}=U_{n}=2.0$ are taken for simplification. $Z_{prob}$ is parameterized according to the $\beta$-stability line,
\begin{equation}
Z_{prob} \simeq Z_\beta = A/(1.98+0.0155\cdot A^{2/3}).
\end{equation}
Similar to $Z_{prob}$, the width parameter $R$ is taken to depend on the mass of fragment,
\begin{equation}
R \simeq \exp(-0.015\cdot A+3.2\times 10^{5}\cdot A^2).
\end{equation}
\subsection{Model construction}
\label{construction}
Two types of database for spallation reactions have been adopted in this work. One is the residual production cross sections from various spallation reactions measured by bombarding one projectile at hundreds of MeV/u on a liquid-hydrogen target using the reverse kinematics technique at GSI, LBL and RIKEN (as listed in Table \ref{measdata}), which are named as D1. The other one is the data of excitation functions for fragments produced in proton induced reactions aiming at describing productions of cosmogenic nuclides in extraterrestrial matter by solar and galactic cosmic ray protons, which are named as D2. The D2 cover productions of nuclides from $^{nat}$C, $^{nat}$N, $^{nat}$O $^{nat}$F, $^{nat}$Mg,$^{27}$Al, $^{nat}$Si, $^{nat}$Ca, $^{nat}$Ti, $^{nat}$V, $^{55}$Mn, $^{nat}$Fe, $^{59}$Co, $^{nat}$Ni $^{nat}$Cu, $^{nat}$Sr, $^{89}$Y, $^{nat}$Zr, $^{93}$Nb, $^{nat}$B, and $^{197}$Au \cite{Michel2014}. In D1 the number of data is about 10,000, while in D2 it is about 3,000. It should be noted that only the reactions of incident energy higher than 30 MeV/u and the measuring uncertainty less than 30 percent are adopted in the learning set. Besides, the data for pick-up fragments are excluded from the learning set.
\begin{table}[thbp]
\caption{(Color online) A list of the adopted data for the measured residue cross sections in the $X$ + p spallation reactions at GSI, LBL and RIKEN.}
\label{measdata}
\centering
\begin{tabular}{p{60pt}<{\centering}|p{60pt}<{\centering}|p{60pt}<{\centering}|p{60pt}<{\centering}}
\hline
\hline
$^{A}X$ + $p$ &E(MeV/u) & Charge Range &Reference\\
\hline
&300; 500; 750; & \\
$^{56}$Fe + $p$ &1000; 1500 &8-27 &\cite{GSI-Fe} \\
\hline
$^{36}$Ar + $p$ &361; 546; 765 &9-17 &\cite{LBL-Ar} \\
\hline
$^{40}$Ar + $p$ &352 &9-17 &\cite{LBL-Ar} \\
\hline
$^{40}$Ca + $p$ &356; 565; 763 &10-20 &\cite{LBL-Ca} \\
\hline
&200 &48-55 &\cite{GSI-Xe200} \\
$^{136}$Xe + $p$ &500 &41-56 &\cite{GSI-Xe500} \\
&1000 &3-56 &\cite{GSI-Xe1000} \\
\hline
$^{113}$Nb + $p$ &113 &37-42 &\cite{RIKEN-Nb113} \\
\hline
$^{93}$Zr + $p$ &105 &36-41 &\cite{RIKEN-Zr105} \\
\hline
$^{107}$Pd + $p$ &118; 196 &42-47 &\cite{RIKEN-Pd118+196}\\
\hline
$^{137}$Cs + $p$ &185 &51-56 &\cite{RIKEN-Cs+Sr} \\
$^{90}$Sr + $p$ &185 &34-39\\
\hline
\hline
\end{tabular}\\
\end{table}
In constructing the models, the minimum numbers of parameters in the input set is chosen, which are $\{x^{(i)}\} = \{A_{p}^{(i)},Z_{p}^{(i)},E^{(i)},Z_{f}^{(i)},A_{f}^{(i)}$\}, with $Z_{p(f)}^{(i)}$($A_{p(f)}^{(i)}$) being the mass (charge) number of the projectile (fragment), and $E^{(i)}$ the incident energies. The output set is $t^{(i)}$ = lg(${\sigma}_{exp}^{(i)}$) for BNN method, and $t^{(i)}$ = lg(${\sigma}_{exp}^{(i)}$)- lg($\sigma_{th}^{(i)}$) for BNN + sEPAX method, with $\sigma^{(i)}_{exp}$ being the measured data and $\sigma^{(i)}_{th}$ the theoretical calculations by sEPAX formula. A 5-32-1 structure is adopted both for BNN and BNN + sEPAX models, i.e., 5 inputs $\{x^{(i)}\} = \{A_{p}^{(i)},Z_{p}^{(i)},E^{(i)},Z_{f}^{(i)},A_{f}^{(i)}$\}, one single layer with 32 hidden unites and one output set $t^{(i)}$ = lg(${\sigma}_{exp}^{(i)}$).
It is noted that the incident energy is the main variable in the D2 dataset, which accounts for the main weight in the network, and the weight of fragments is very small. In the D1 dataset, the weights are contrary to that of D2. Since the data in D1 and D2 have significant difference, in constructing the BNN and BNN + sEPAX models, the D2 dataset ($N_{l}$ = 11,807) is firstly adopted as the learning set and D1 as the testing set to verify whether the have the same fragment production mechanisms.
\section{Results and discussion}
\label{result}
Based on the constructed BNN and BNN + sEPAX models, the discussion will be concentrated on the isotopic distributions, mass distributions, and fragment excitation functions. And the extrapolation ability of the two models will also be tested.
\subsection{ISOTOPIC DISTRIBUTIONS}
\label{isotopic}
The isotopic distributions for the 356 MeV/u $^{40}$Ca + $p$ and 1 GeV/u $^{136}$Xe + $p$ reactions (see Table \ref{measdata}) are employed to show the performance of BNN and BNN + sEPAX models with $N_{l}$ = 11807. The predicted and measured results are compared in FIG. \ref{40Ca11807} for 356 MeV/u $^{40}$Ca + $p$ reaction and in FIG. \ref{136Xe11807} for 1 GeV/u $^{136}$Xe + $p$ reaction, respectively. For the relatively spallation small system of $^{40}$Ca + $p$ reaction, the predicted isotopic cross sections by BNN and BNN + sEPAX models are within an order of magnitude difference of the experimental values and their trends of isotopic distributions are consistent. For the $^{136}$Xe + $p$ reaction, which is a relatively large spallation system, the BNN model are poorly reproduce the existing experimental isotopic distributions, because of inadequate fragments in the learning set. While with the physical guidance of sEPAX formula, the predictions by BNN + sEPAX model are in good agreement with experimental data both for the heavy fragments and the quasiprojectile fragments. Poorly predictions for the light neutron-deficient fragments are observed, indicating that further improvement is needed.
\begin{figure}
\centering
\includegraphics[width=8.6cm]{40Ca356-11807.eps}
\caption{(Color online) Test isotopic cross section distributions by the BNN and BNN + sEPAX models with D2 ($N_{l}$=11,807) as the learning set for the 356 MeV/u $^{40}$Ca + $p$ reaction. The experimental and models' error bars are too small to be shown.}
\label{40Ca11807}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.6cm]{136Xe1000-11807.eps}
\caption{(Color online) Similar to FIG. \ref{40Ca11807} but for the 1 GeV/u $^{136}$Xe + $p$ reaction.}
\label{136Xe11807}
\end{figure}
According to the above comparison, it can be concluded that the fragment production mechanisms of dataset D1 and D2 are the same, and the BNN and BNN + sEPAX method can be applied to extrapolate the spallation cross sections. In the following, D1 and D2 are merged (the total number of data is $N_{l}$=13,786) as the new learning set to construct the new BNN and BNN + sEPAX models. In this manner, the weights of for fragments and incident energies in the learning set are improved. The extrapolated isotopic cross sections of the 356 MeV/u $^{40}$Ca + $p$ and 1 GeV/u $^{136}$Xe + $p$ reactions by BNN and BNN + sEPAX models ($N_{l}$=13,786) are shown in FIGs. \ref{40Ca13786} and \ref{136Xe13786}, respectively.
For the $^{40}$Ca + $p$ reaction, the extrapolations of two models are consistent for the $Z$ = 10 and $Z$ = 13 isotopes, while the BNN model predict much larger cross sections for $Z$ = 16 and $Z$ = 19 neutron-rich isotopes. For the $^{136}$Xe + $p$ reaction, the extrapolations of BNN and BNN + sEPAX models for the neutron-deficient fragments are consistent for the isotopes from $Z=6$ to $Z=54$, while the BNN model predict much larger cross sections than the BNN + sEPAX model for neutron-rich isotopes from the light to medium ones. The extrapolations for the quasiprojectiles are consistent to the measured results. The extrapolation ability of these two models will be further tested later.
\begin{figure}
\centering
\includegraphics[width=8.6cm]{40Ca356-13786.eps}
\caption{(Color online) The extrapolation results of BNN (red circles) and BNN + sEPAX (blue triangles) models with $N_{l}$ = 13,786 learning set for 356 MeV/u $^{40}$Ca + $p$ reaction. The experimental and models' error bars are plotted.}
\label{40Ca13786}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.6cm]{136Xe1000-13786.eps}
\caption{(Color online) Similar to Fig. \ref{40Ca11807} but for the 1 GeV/u $^{136}$Xe + $p$ reaction.}
\label{136Xe13786}
\end{figure}
\subsection{MASS DISTRIBUTIONS}
\label{massyield}
The mass yield $Y(A)$ is a key factor for fragment predictions in Eq. (\ref{SigAZ}). It is interesting to see how well the BNN and BNN + sEPAX models can reproduce the mass yield. In FIG. \ref{mass}, the predicted mass distributions for the 356 MeV/u $^{40}$Ca + $p$ and 1 GeV/u $^{136}$Xe + $p$ reactions are shown in panel (a) and (b), respectively. Compared to those models with the $N_{l}$ = 11,807 learning set, the BNN and BNN + sEPAX models using the $N_{l} =$ 13,786 learning set can better reproduce the $\sigma(A)$ distributions for both the two reactions. For the $^{136}$Xe + $p$ reaction, obviously distorted distributions are found in the $A >$ 120 fragments predicted by the BNN and BNN + sEPAX models, for which are very close to the mass number of the projectile nucleus. Since these two reactions are included in the learning set, the result only shows the mass distribution trends of the small and large spallation systems, and cannot demonstrate the generalization ability of the models.
\begin{figure*}[htbp]
\centering
\subfigure{\includegraphics[width=7cm]{40Ca356-mass.eps}}
\subfigure{\includegraphics[width=7cm]{136Xe1000-mass.eps}}
\caption{(Color Online) The mass cross sections ($\sigma_{A}$) predicted by BNN and BNN + sEPAX models with different number of learning set ($N_{l} =$ 11,807 and $N_{l} =$ 13,786) for the 356 MeV/u $^{40}$Ca + $p$ [in panel (a)] and 1 GeV/u $^{136}$Xe + $p$ [in panel (b)] reactions. The black squares denote the experimental data. The red triangles (origin triangles) and blue circles (green circles) denote the BNN + sEPAX (BNN) methods with D2 ($N_{l} =$ 11,807) and D1 + D2 ($N_{l} =$ 13,786) as the learning set, respectively.}
\label{mass}
\end{figure*}
\subsection{Fragment Excitation Functions}
\label{excitaiton}
It should be noted that, inheriting the main formulas of EPAX, the sEPAX formulas do not incorporate the incident energy term. The incident energy dependence of fragments in the BNN and BNN + sEPAX models are born from the learning of massive data in D2, which makes it possible to yield the excitation functions of fragments in reactions. The excitation functions for the residual fragments have been predicted for the $^{22}$Na, $^{24}$Na, $^{28}$Mg, $^{26}$Al in the $^{40}$Ca + $p$ reactions, $^{24}$Na, $^{36}$Cl, $^{47}$Sc, and $^{52}$Mn in the $^{56}$Fe + $p$ reactions, $^{54}$Mn, $^{75}$Se, $^{105}$Ag, $^{138}$Ba in the $^{138}$Ba + $p$ reactions, and $^{85}$Sr, $^{95}$Nb, $^{160}$Er, $^{173}$Hf in the $^{197}$Ag + $p$ reactions (the data of the measured excitation functions are taken from Ref. \cite{Michel2014}). The range of the incident energy has been selected to be from 30 MeV/u to 3 GeV/u. The selected incident energies range from 30 MeV/u to 3 GeV/u. The results are shown in FIG. \ref{excurve}.
It can be seen that predicted excitation functions well reproduce the experimental data in the four reactions. Considering the mass number of projectile nucleus, for the relatively small system (for example, $^{40}$Ca and $^{56}$Fe), the extrapolation results of BNN and BNN + sEPAX model are consistent, while for the heavy system, relatively large differences emerge in the incident energy below 300 MeV/u (for example, the $^{197}$Ag). Considering the mass number of fragments, the extrapolation of heavy projectile fragments and quasiprojectiles fragments by the BNN and BNN + sEPAX models are consistent with the measured results, but large differences arise for the light and medium fragments (for example, $^{24}$Na in the $^{56}$Fe reaction, $^{46}$Sc and $^{75}$Se in the $^{138}$Ba reactions). In the $^{197}$Au + $p$ reactions, the BNN and BNN + sEPAX predict different trends for the $^{22}$Na and $^{46}$Sc fragments when the incident energy is below 400 MeV/u. This could be caused by the reason that no data from system larger than $^{136}$Xe is included in the learning set except $^{197}$Au.
\begin{figure*}
\centering
\includegraphics[width=15cm]{excurve.eps}
\caption{(Color online) The fragment excitation functions predicted by BNN and BNN + sEPAX models with $N_{l}$ = 13786 learning set for the $^{40}$Ca + $p$, $^{56}$Fe + $p$, $^{138}$Ba + $p$ and $^{197}$Au + $p$ reactions. The range of incident energy is from 30 MeV/u to 3 GeV/u. The experimental data are taken from Ref. \cite{Michel2014}.}
\label{excurve}
\end{figure*}
\subsection{Model validation}
Due to the large difference between the extrapolations for the BNN and BNN + sEPAX models, an empirical relationship between the cross section and the average binding energy per nucleon of fragment $B'$ in spallation reaction is adopted to test the extrapolation abilities.
\begin{equation}
\sigma = Cexp[(B'-8)/\tau],\\
\end{equation}
where $C$ and $\tau$ are free parameters. $B' = (B - \varepsilon_{p})/A$, in which $\varepsilon_{p}$ is the pairing energy,
\begin{equation}
\varepsilon_{p} = 0.5[(-1)^N + (-1)^Z]\varepsilon_{0} \cdot A^{-3/4}.
\end{equation}
Based on the canonical ensemble theory, this empirical formula has been shown to be reasonable for both neutron-deficient and neutron-rich fragments in multi-fragmentation reaction \cite{Tsang2007,SongYD2018,MaCW2018,MaCW2019,SongYD2019}. The $\sigma\sim B'$ correlation for fragments by BNN and BNN + sEPAX models of $Z =$ 10 and 16 isotopes in 356 MeV/u $^{40}$Ca + $p$ reaction and of $Z =$ 21 and 40 isotopes in 1 GeV/u $^{136}$Xe + $p$ reaction are plotted in Fig. \ref{Binding}. The binding energy of isotopes are taken from AME2020 \cite{AME2020} and $\varepsilon_{0} =$ 30 MeV is adopted \cite{Tsang2007}. Due to the limited experimental data, only the fitting lines for the $Z =$ 16 neutron-deficient isotopes (see panel (b)) and $Z =$ 40 neutron-rich isotopes (see panel (d)) are plotted.
In Fig. \ref{40Ca13786}, it is shown that the extrapolations of BNN and BNN + sEPAX models for the $Z$ = 10 isotopes produced in $^{40}$Ca + $p$ reaction are consistent, and the extrapolations of BNN model are higher than BNN + sEPAX model for $Z =$ 16 neutron-deficient isotopes. In FIG. \ref{Binding}(b), the BNN + sEPAX model predictions agree the fitting function better than the BNN model. In FIG. \ref{136Xe13786}, it is seen that the extrapolations of BNN model are higher than BNN + sEPAX model for $Z <$ 33 neutron-rich isotopes, while both of them are consistent for $Z >$ 33 isotopes. Due to the deficient experimental, the linear fitting results in panel (c) cannot be given, but it is obvious that the BNN predictions for $I >$ 10 fragments (red circles) are excessively upturned. While in panel (d), it is shown that the extrapolations of BNN and BNN + sEPAX models for the $Z =$ 40 neutron-rich isotopes should be both reasonable.
\begin{figure*}
\centering
\includegraphics[width=15cm]{Binding.eps}
\caption{(Color online) The correlation between $\sigma$ and B' for isotopes with $Z$ = 10 (in pale (a)), 16 (in pale (b)) in 356 MeV/u$^{40}$ca + $p$ and for $Z$ = 21 (in pale (c)), 40 (in pale (d)) in 1 GeV/u $^{136}$Xe + $p$. The black squares, red circles and blue triangles denote the experimental data, BNN and BNN + sEPAX predictions, respectively. The lines denote the linear fit of the partial experimental points}
\label{Binding}
\end{figure*}
Based on the above discussions, it is suggested that the BNN + sEPAX model with $N_{l}$ = 13,786 learning set can provide precise predictions to fragment cross sections for proton-induced spallation systems smaller than $^{136}$Xe within the incident energy range from tens of MeV/u to above a few GeV/u. Considering the related nuclear applications, it can be used to the residual fragments in proton therapy (above tens of MeV/u to 1 GeV/u), the ADS system and nuclear waste disposal (below 1 GeV/u), solar cosmic ray physics (sub-GeV region and even higher \cite{solarcosmicrayppnp}), shielding at accelerator, etc.
\section{summary}
\label{summary}
The BNN and BNN + sEPAX methods are applied to construct new predictive models for fragment cross sections in proton-induced spallation reactions. Two types of data have been adopted to constructed the predictive models. One type is the isotopic cross sections with incident energy from around 100 MeV/u to 1 GeV/u (D1), and the other type is the fragment excitation functions in reactions of from 30 MeV/u to 2.6 GeV/u (D2). The BNN + sEPAX method
can achieve reasonable extrapolations with less information compared with BNN method. It is suggested that the BNN + sEPAX model based on D1 + D2 learning dataset can be applied to predict the isotopic cross sections and fragment excitation functions for the proton induced reactions within an incident energy range from tens of MeV/u to a few GeV/u for system smaller than $^{136}$Xe, which can provide precision predictions to both nuclear physics and the related disciplines in nuclear astrophysics, proton therapy, nuclear energy and radioactive ion beam (RNB) facilities.
Furthermore, with plenty of data available, the Bayesian approaches are efficient to provide predictions in various areas. A similar prediction requirement has been emerged in the modern radioactive nuclear beam experiments which search the rare isotopes near/beyond the drip lines using the projectile fragmentation reaction \cite{PF2021PPNP}. For the weak predictive ability in rare isotopes of existing methods, some new semi-empirical methods have been proposed \cite{FRACS,FRACSc,SongYD2019,SCICh192,OESt21MeiPRC}. It is indicative that the empirical formula provided by these works will be very helpful to establish new interesting BNN approaches for projectile fragmentation reactions.
\section*{acknowledgments}
This work is supported by the National Natural Science Foundation of China (Grant No. 11975091, 1210050535 and U1732135), the Program for Innovative Research Team (in Science and Technology) in University of Henan Province (Grant No. 21IRTSTHN011), China.
|
1610.07307
|
\section{Introduction}
Throughout this paper, groups are assumed to be finite, and graphs are assumed to be finite, connected, simple and undirected.
For a graph $\Gamma$, we denote by $V(\Gamma)$ the set of all vertices of $\Gamma$, by $E(\Gamma)$ the set of all edges of $\Gamma$, by $A(\Gamma)$ the set of all arcs (ordered paries of adjacent vertices) of $\Gamma$, and by $\hbox{\rm Aut\,}(\Gamma)$ the full automorphism group of $\Gamma$. For $u, v\in V(\Gamma)$, denote by $\{u, v\}$ the edge incident to $u$ and $v$ in $\Gamma$.
For the group-theoretic and the graph-theoretic terminology not defined here we refer the reader to \cite{Bondy,Wielandt}.
Let $\Gamma$ be a graph. If $\hbox{\rm Aut\,}(\Gamma)$ is transitive on $V(\Gamma)$, $E(\Gamma)$ or $A(\Gamma)$, then $\Gamma$ is said to be {\em vertex-transitive}, {\em edge-transitive} or {\em arc-transitive}, respectively.
An arc-transitive graph is also called a symmetric graph.
A graph $\Gamma$ is said to be {\em semisymmetric} if $\Gamma$ has regular valency and is edge- but not vertex-transitive.
Let $G$ be a permutation group on a set $\Omega$ and $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in \Omega$.
Denote by $G_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}$ the stabilizer of $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ in $G$, that is the subgroup of $G$ fixing the point $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$.
We say that $G$ is {\em semiregualr} on $\Omega $ if $G_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=1$ for every $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in \Omega$ and {\em regular} if $G$ is transitive and semiregular.
A graph is said to be a {\em bi-Cayley graph} over a group $H$ if it admits $H$ as a semiregular automorphism group with two orbits (Bi-Cayley graph is sometimes called {\em semi-Cayley graph}).
Note that every bi-Cayley graph admits the following concrete realization. Given a group $H$, let $R$, $L$ and $S$ be subsets of $H$ such that $R^{-1}=R$, $L^{-1}=L$ and $R\cup L$ does not contain the identity element of $H$. The {\em bi-Cayley graph} over $H$ relative to the triple $(R, L, S)$, denoted by BiCay($H, R, L, S$), is the graph having vertex set the union of the right part $H_{0}=\{h_{0}~|~h\in H\}$ and the left part $H_{1}=\{h_{1}~|~h\in H\}$, and edge set the union of the right edges $\{\{h_{0},~g_{0}\}~|~gh^{-1}\in R\}$, the left edges $\{\{h_{1},~g_{1}\}~|~gh^{-1}\in L\}$ and the spokes $\{\{h_{0},~g_{1}\}~|~gh^{-1}\in S\}$.
Let $\Gamma={\rm BiCay}(H, R, L, S)$. For $g\in H$, define a permutation $R(g)$ on the vertices of $\Gamma$ by the rule
$$h_i^{R(g)}=(hg)_i, \forall i\in{\mathbb Z}_2, h\in H.$$
Then $R(H)=\{R(g)\ |\ g\in H\}$ is a semiregular subgroup of $\hbox{\rm Aut\,}(\Gamma)$ which is isomorphic to $H$ and has $H_0$ and $H_1$ as its two orbits.
When $R(H)$ is normal in $\hbox{\rm Aut\,}(\Gamma)$, the bi-Cayley graph $\Gamma={\rm BiCay}(H,R,L,S)$ will be called a {\em normal bi-Cayley graph} over $H$ (see \cite{Zhouaut}).
A natural problem in the study of bi-Cayley graphs is: for a given finite group $H$, to classify bi-Cayley graphs with specific symmetry properties over $H$. Some partial answers for this problem have been obtained. For example, in \cite{Boben} Boben et al. studied some properties of cubic $2$-type bi-Cayley graphs over cyclic groups and the configurations arising from these graphs, in \cite{Pisanski} Pisanski classified cubic bi-Cayley graphs over cyclic groups, in \cite{BiCayley.2} Kov\'{a}cs et al. gave a classification of arc-transitive one-matching abelian bi-Cayley graphs,
and more recently, Zhou et al. \cite{Zhoucubic} gave a classification of cubic vertex-transitive abelian bi-Cayley graphs.
In this paper, we shall investigate cubic edge-transitive bi-Cayley graphs over metacyclic $p$-groups where $p$ is an odd prime. Following up \cite{NET}, we call a bi-Cayley graph over a metacyclic $p$-group a {\em bi-$p$-metacirculant}.
Another motivation for us to consider bi-Cayley graphs over metacyclic $p$-groups is the observation that the Gray graph \cite{Bouwer1}, the smallest trivalent semmisymmetric graph, is a bi-Cayley graph over a non-abelian metacyclic group of order $27$. In \cite{NET}, the cubic edge-transitive bi-Cayley graphs over abelian groups have been classified. So, we shall restrict our attention to bi-Cayley graphs over non-abelian metacyclic $p$-groups.
Our first result characterizes the automorphism groups of cubic edge-transitive bi-$p$-metacirculants.
\begin{theorem}\label{5no}
Let $\Gamma$ be a connected cubic edge-transitive bi-Cayley graph over a non-abelian metacyclic $p$-group $H$ with $p$ an odd prime.
Then $p=3$, and either $\Gamma$ is isomorphic to the Gray graph or $\Gamma$ is a normal bi-Cayley graph over $H$.
\end{theorem}
Applying the above theorem, our second result gives a classification of connected cubic edge-transitive bi-Cayley graphs over a inner-abelian metacyclic $p$-group. A non-abelian group is called an {\em inner-abelian group} if all of its proper subgroups are abelian.
\begin{theorem}\label{result}
Let $\Gamma$ be a connected cubic edge-transitive bi-Cayley graph over an inner-abelian metacyclic $3$-group $H$. Then $\Gamma$ is isomorphic to either $\Gamma_{t}$ or $\Sigma_{t}$ (see Section~\ref{sec-5} for the construction of these two families of graphs).
\end{theorem}
Theorem~\ref{5no} also enables us to give a short proof of the main result in \cite{2p3}.
\begin{cor}{\rm~\cite[Theorem 1.1]{2p3}}\label{cor2p3}
Let $p$ be a prime. Then, with the exception of the Gray graph on $54$ vertices, every cubic edge-transitive graph of order $2p^3$ is vertex-transitive.
\end{cor}
\section{Preliminaries}
In this section, we first introduce the notation used in this paper. For a positive integer $n$, denote by $\mathbb{Z}_n$ the cyclic group of order $n$ and
by $\mathbb{Z}_n^*$ the multiplicative group of $\mathbb{Z}_n$ consisting of numbers coprime to $n$.
For a finite group $G$, the full automorphism group, the center, the derived subgroup and the Frattini subgroup of $G$ will be denoted by $\hbox{\rm Aut\,}(G)$, $Z(G)$, $G'$ and $\Phi(G)$, respectively. For $x, y\in G$, denote by $[x, y]$ the commutator $x^{-1}y^{-1}xy$. For a subgroup $H$ of $G$,
denote by $C_G(H)$ the centralizer of $H$ in $G$ and by $N_G(H)$ the
normalizer of $H$ in $G$. For two groups $M$ and $N$, $N\rtimes M$ denotes a semidirect product of $N$ by $M$.
Below, we restate some group-theoretic results, of which the first
is usually called the $N/C$-theorem.
\begin{prop} {\rm~\cite[Chapter 1, Theorem 4.5]{Huppert}}\label{NC}
Let $H$ be a subgroup of a group $G$. Then $C_G(H)$ is normal in $N_G(H)$, and
the quotient group $N_G(H)/C_G(H)$ is isomorphic to a subgroup of $\hbox{\rm Aut\,}(H)$.
\end{prop}
Now we give two results regarding metacyclic $p$-groups.
\begin{prop}{\rm~\cite[Lemma~2.4]{pcomplement}}\label{metap}
Let $P$ be a split metacyclic $p$-group:
\begin{center}
$P=\lg x,y\mid x^{p^m}=y^{p^n}=1, yxy^{-1}=x^{1+p^l}\rg$, where $0<l<m$, $m-l\leq n$.
\end{center}
Then the automorphism group $\hbox{\rm Aut\,}(P)$ of $P$ is a semidirect product of a normal $p$-subgroup and the cyclic subgroup $\lg\sigma\rg$ of order $p-1$,
where $\sigma(x)=x^r$ and $\sigma(y)=y$, $r$ is a primitive $(p-1)$th root of unity modulo $p^m$.
\end{prop}
\begin{prop}{\rm~\cite[{Proposition~2.3}]{pcomplement}}\label{pcom}
Let $G$ be a finite group with a non-abelian metacyclic Sylow $p$-subgroup $P$. If $P$ is nonsplit, then $G$ has a normal $p$-complement.
\end{prop}
Next, we give some results about graphs. Let $\Gamma$ be a connected graph with an edge-transitive group $G$ of automorphisms and let $N$ be a normal subgroup of $G$. The {\em quotient graph} $\Gamma_N$ of $\Gamma$ relative to $N$ is defined as the graph
with vertices the orbits of $N$ on $V(\Gamma)$ and with two orbits
adjacent if there exists an edge in $\Gamma$ between the vertices lying in
those two orbits. Below we introduce two propositions, of which the
first is a special case of \cite[Theorem~9]{VTgraph}.
\begin{prop}\label{3orbits}
Let $\Gamma$ be a cubic graph and let $G\leq \hbox{\rm Aut\,}(\Gamma)$ be arc-transitive on $\Gamma$.
Then $G$ is an $s$-arc-regular subgroup of $\hbox{\rm Aut\,}(\Gamma)$ for some integer $s$.
If $N\unlhd G$ has more than two orbits in $V(\Gamma)$, then $N$ is semiregular on $V(\Gamma)$,
$\Gamma_N$ is a cubic symmetric graph with $G/N$ as an $s$-arc-regular subgroup of automorphisms.
\end{prop}
The next proposition is a special case of \cite[Lemma~3.2]{6p2}.
\begin{prop}\label{intransitive}
Let $\Gamma$ be a cubic graph and let $G\leq \hbox{\rm Aut\,}(\Gamma)$ be transitive on $E(\Gamma)$ but intransitive on $V(\Gamma)$.
Then $\Gamma$ is a bipartite graph with two partition sets, say $V_0$ and $V_1$.
If $N \trianglelefteq G$ is intransitive on each of $V_0$ and $V_1$, then $N$ is semiregular on $V(\Gamma)$,
$\Gamma_N$ is a cubic graph with $G/N$ as an edge- but not vertex-transitive group of automorphisms.
\end{prop}
The next proposition is basic for bi-Cayley graphs.
\begin{prop}{\rm~\cite[Lemma~3.1]{Zhoucubic}}\label{bicayley}
Let $\Gamma={\rm BiCay}(H,R,L,S)$ be a connected bi-Cayley graph over a group $H$. Then the following hold:
\begin{enumerate}
\item[$(1)$] $H$ is generated by $R\cup L\cup S$.
\item[$(2)$] Up to graph isomorphism, $S$ can be chosen to contain the identity of $H$.
\item[$(3)$] For any automorphism $\alpha$ of $H$, ${\rm BiCay}(H, R, L, S)\cong {\rm BiCay}(H, R^{\alpha}, L^{\alpha}, S^{\alpha})$.
\item[$(4)$] ${\rm BiCay}(H, R, L, S) \cong {\rm BiCay}(H, L, R, S^{-1})$.
\end{enumerate}
\end{prop}
Next, we collect several results about the automorphisms of the bi-Cayley graph $\Gamma={\rm BiCay}(H, R, L, S)$.
Recall that for each $g\in H$, $R(g)$ is a permutation on $V(\Gamma)$ defined by the rule
\begin{equation}\label{1}
h_{i}^{R(g)}=(hg)_{i},~~~\forall i\in \mathbb{Z}_{2},~h,~g\in H,
\end{equation}
and $R(H)=\{R(g)\ |\ g\in H\}\leq\hbox{\rm Aut\,}(\Gamma)$.
For an automorphism $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ of $H$ and $x,y,g\in H$, define two permutations on $V(\Gamma)=H_0\cup H_1$ as following:
\begin{equation}\label{2}
\begin{array}{ll}
\d_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,x,y}:& h_0\mapsto (xh^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon})_1, ~h_1\mapsto (yh^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon})_0, ~\forall h\in H,\\
\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,g}:& h_0\mapsto (h^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon)_0, ~h_1\mapsto (gh^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon})_1, ~\forall h\in H.\\
\end{array}
\end{equation}
Set \begin{equation}\label{3}
\begin{array}{lll}
{\rm I}&=& \{\d_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,x,y}\ |\ \alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in\hbox{\rm Aut\,}(H)\ s.t.\ R^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=x^{-1}Lx, ~L^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=y^{-1}Ry, ~S^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=y^{-1}S^{-1}x\},\\
{\rm F} &=&\{ \s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,g}\ |\ \alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in\hbox{\rm Aut\,}(H)\ s.t.\ R^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=R, ~L^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=g^{-1}Lg, ~S^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=g^{-1}S\}.
\end{array}
\end{equation}
\begin{prop}{\rm~\cite[Theorem~3.4]{Zhouaut}}\label{bicayleyaut}
Let $\Gamma={\rm BiCay}(H, R, L, S)$ be a connected bi-Cayley graph over the group $H$.
Then $N_{\hbox{\rm Aut\,}(\Gamma)}(R(H))=R(H)\rtimes F$ if $I=\emptyset$ and
$N_{\hbox{\rm Aut\,}(\Gamma)}(R(H))=R(H)\langle F, \delta_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,x,y}\rangle$ if
$I\neq\emptyset$ and $\delta_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,x,y}\in I$. Furthermore, for any $\delta_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,x,y}\in I$, we have the following:
\begin{enumerate}
\item[$(1)$] $\langle R(H), \delta_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,x,y}\rangle$ acts transitively on $V(\Gamma)$;
\item[$(2)$] if $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ has order $2$ and $x=y=1$, then $\Gamma$ is isomorphic to the Cayley graph $\hbox{\rm Cay }(\bar{H},~R\cup \alpha S)$, where $\bar{H}=H\rtimes \langle\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\rangle$.
\end{enumerate}
\end{prop}
\begin{prop}{\rm~\cite[Proposition~5.2]{NET}}\label{abelian-biCayley}
Let $n, m$ be two positive integers such that $nm^2\geq 3$. Let $\ld=0$ if $n=1$, and let $\ld\in\mathbb{Z}_n^{*}$ be such that $\ld^2-\ld+1\equiv 0\ ({\rm mod}\ n)$ if $n> 1$.
Let
\[
\begin{array}{ll}
\mbox{$\mathcal{H}_{m,n}=\lg x\rg\times\lg y\rg\cong\mathbb{Z}_{nm}\times\mathbb{Z}_m$,} & \\
\mbox{$\Gamma_{m,n,\ld}={\rm BiCay}(\mathcal{H}_{m,n}, \emptyset, \emptyset, \{1,x,x^{\ld}y\})$.} &
\end{array}
\]
Let $\Gamma={\rm BiCay}(H,R,L,S)$ be a connected cubic normal edge-transitive bi-Cayley graph over an abelian group $H$. Then $\Gamma\cong\Gamma_{m,n,\ld}$ for some integers $m,n,\ld$.
\end{prop}
Finally, we give some results about cubic edge-transitive graphs.
\begin{prop}{\rm~\cite[Theorem~3.2]{2pn}}\label{Sym-P-normal}
Let $\Gamma$ be a connected cubic symmetric graph of order $2p^n$ with $p$ an odd prime and $n$ a positive integer.
If $p\neq 5,7$, then every Sylow $p$-subgroup of $\hbox{\rm Aut\,}(\Gamma)$ is normal.
\end{prop}
\begin{prop}{\rm~\cite[Proposition~8]{2p3}}\label{Av}
Let $\Gamma$ be a connected cubic edge-transitive graph and let $G\leq\hbox{\rm Aut\,}(\Gamma)$ be transitive on the edges of $\Gamma$. For any $v\in V(\Gamma)$, the stabilizer $G_v$ has order $2^r\cdot 3$ with $r\geq 0$.
\end{prop}
\section{A few technical lemmas}
In this section, we shall give two easily proved lemmas about metacyclic $p$-groups that are useful in this paper.
\begin{lem}\label{comput}
Let $p$ be an odd prime, and let $H$ be a metacyclic $p$-group generated by $a, b$ with the following defining relations:
\begin{center}
$a^{p^m}=b^{p^n}=1, b^{-1}ab=a^{1+p^r}$,
\end{center}
where $m, n, r$ are positive integers such that $r<m\leq n+r$. Then the following hold:
\begin{enumerate}
\item [{\rm (1)}]\ For any $i\in{\mathbb Z}_{p^m}, j\in{\mathbb Z}_{p^n}$, we have
$$a^ib^j=b^ja^{i(1+p^r)^j}.$$
\item [{\rm (2)}]\ For any positive integer $k$ and for any $i\in{\mathbb Z}_{p^m}, j\in{\mathbb Z}_{p^n}$, we have
$$(b^ja^i)^k=b^{kj}a^{i\sum_{s=0}^{k-1}(1+p^r)^{sj}}.$$
\item [{\rm (3)}]\ For any $i_1,i_2\in{\mathbb Z}_{p^m}$, $j_1,j_2\in{\mathbb Z}_{p^n}$, we have
$$(b^{j_1}a^{i_1})(b^{j_2}a^{i_2})=b^{j_1+j_2}a^{{i_1(1+p^r)}^{j_2}+i_2}.$$
\end{enumerate}
\end{lem}
{\bf Proof}\hskip10pt
For any $i\in{\mathbb Z}_{p^m}, j\in{\mathbb Z}_{p^n}$, since $b^{-1}ab=a^{1+p^r}$, we have $b^{-j}ab^j=a^{(1+p^r)^j}$, and then $b^{-j}a^ib^j=a^{i(1+p^r)^j}$.
It follows that $a^ib^j=b^ja^{i(1+p^r)^j}$, and so (1) holds.
For any positive integer $k$ and for any $i\in{\mathbb Z}_{p^m}, j\in{\mathbb Z}_{p^n}$, if $k=1$, then $(2)$ is clearly true.
Now we assume that $k>1$ and (2) holds for any positive integer less than $k$.
Then $(b^ja^i)^{k-1}=$$b^{(k-1)j}a^{i\sum_{s=0}^{k-2}(1+p^r)^{sj}}$, and
then
\[
\begin{array}{ll}
\mbox{$(b^ja^i)^k$} & \mbox{$=b^ja^i(b^ja^i)^{k-1}$}\\
& \mbox{$=b^ja^i[b^{(k-1)j}a^{i\sum_{s=0}^{k-2}(1+p^r)^{sj}}]$}\\
& \mbox{$=b^j[a^ib^{(k-1)j}]a^{i\sum_{s=0}^{k-2}(1+p^r)^{sj}}$}\\
& \mbox{$=b^j[b^{(k-1)j}a^{i(1+p^r)^{(k-1)j}}]a^{i\sum_{s=0}^{k-2}(1+p^r)^{sj}}$}\\
& \mbox{$=b^{kj}a^{i\sum_{s=0}^{k-1}(1+p^r)^{sj}}$.}
\end{array}
\]
By induction, we have (2) holds.
For any $i_1,i_2\in{\mathbb Z}_{p^m}$ and $j_1,j_2\in{\mathbb Z}_{p^n}$, from $(1)$ it follows that $$(b^{j_1}a^{i_1})(b^{j_2}a^{i_2})=b^{j_1}(a^{i_1}b^{j_2})a^{i_2}=b^{j_1}(b^{j_2}a^{i_1(1+p^r)^{j_2}})a^{i_2}=b^{j_1+j_2}a^{{i_1(1+p^r)}^{j_2}+i_2},$$
and so (3) holds.
\hfill\hskip10pt $\Box$\vspace{3mm}
\begin{lem}\label{comput2}
Let $p$ be an odd prime, and let $H$ be an inner-abelian metacyclic $p$-group generated by $a, b$ with the following defining relations:
\begin{center}
$a^{p^m}=b^{p^n}=1, b^{-1}ab=a^{1+p^r}$,
\end{center}
where $m, n, r$ are positive integers such that $m\geq 2, n\geq 1$ and $r=m-1$. Then the following hold:
\begin{enumerate}
\item [{\rm (1)}]\ For any positive integer $k$, we have
$$a^{(1+p^r)^k}=a^{1+kp^r}.$$
\item [{\rm (2)}]\ For any $i\in{\mathbb Z}_{p^m}, j\in{\mathbb Z}_{p^n}$, we have
$$(b^ja^i)^p=b^{jp}a^{ip}.$$
\item [{\rm (3)}]\ $H'\cong\mathbb{Z}_p$.
\end{enumerate}
\end{lem}
{\bf Proof}\hskip10pt For (1), the result is clearly true if $k=1$. In what follows, assume $k\geq 2$. Since $r=m-1$ and $m\geq 2$, we have $2r\geq m$. This implies that $a^{p^{2r}}=1$, and hence $a^{p^{\ell r}}=1$ for any $\ell\geq2$. It then follows that
\[
\begin{array}{ll}
\mbox{$a^{(1+p^r)^k}$} & \mbox{$=a^{[C_k^0\cdot 1^k\cdot (p^r)^0+C_k^1\cdot 1^{k-1}\cdot (p^r)^1+C_k^2\cdot 1^{k-2}\cdot (p^r)^2+...C_k^k\cdot 1^{0}\cdot (p^r)^k]}$}\\
& \mbox{$=a^{C_k^0\cdot (p^r)^0}\cdot a^{C_k^1\cdot (p^r)^1}\cdot a^{C_k^2\cdot (p^r)^2}\cdot ...\cdot a^{C_k^k\cdot (p^r)^k}$}\\
& \mbox{$=a\cdot (a^{p^{r}})^{C_k^1}\cdot (a^{p^{2r}})^{C_k^2}\cdot ...\cdot (a^{p^{kr}})^{C_k^k}$}\\
& \mbox{$=a\cdot a^{kp^r}$}\\
& \mbox{$=a^{1+kp^r}$,}
\end{array}
\]
and so (1) holds. (Here for any integers $N\geq l\geq 0$, we denote by $C_N^l$ the binomial coefficient, that is, $C_N^l=\frac{N!}{l!(N-l)!}$.)
For (2), for any positive integer $k$ and for any $i\in{\mathbb Z}_{p^m}, j\in{\mathbb Z}_{p^n}$, by Lemma~\ref{comput}~$(2)$, we have
\[
\begin{array}{ll}
\mbox{$(b^ja^i)^{p}$} & \mbox{$=b^{jp}a^{i[1+(1+p^r)^{j}+(1+p^r)^{2j}+...+(1+p^r)^{(p-1)j}]}$}\\
& \mbox{$=b^{jp}a^{i[1+(1+j\cdot p^r)+(1+2j\cdot p^r)+...+(1+(p-1)\cdot jp^r)]}$}\\
& \mbox{$=b^{jp}a^{i(p+\frac{1}{2}p(p-1)\cdot jp^r)}$}\\
& \mbox{$=b^{jp}a^{ip}$. }
\end{array}
\]
Hence $(2)$ holds.
From \cite{p-gp} we can obtain $(3)$.
\hfill\hskip10pt $\Box$\vspace{3mm}
\section{Proof of Theorem~\ref{5no}}
We shall prove Theorem~\ref{5no} by a series of lemmas. We first prove three lemmas regarding cubic edge-transitive graphs of order twice a prime power.
\begin{lem}\label{N}
Let $\Gamma$ be a connected cubic edge-transitive graph of order $2p^n$ with $p$ an odd prime and $n\geq 2$.
Let $G\leq\hbox{\rm Aut\,}(\Gamma)$ be transitive on the edges of $\Gamma$.
Then any minimal normal subgroup of $G$ is an elementary abelian $p$-group.
\end{lem}
\noindent{\bf Proof}\hskip10pt Let $N$ be a minimal normal subgroup of $G$. If $G$ is transitive on the arcs of $\Gamma$, then by {\rm~\cite[Lemma~3.1]{2pn}}, $N$ is an
elementary abelian $p$-group, as required.
In what follows, assume that $G$ is not transitive on the arcs of $\Gamma$. Then since $\Gamma$ has valency $3$, $\Gamma$ is semisymmetric and so it is bipartite. Let $B_0$ and $B_1$ be the two partition sets of $V(\Gamma)$. Then $B_0, B_1$ are just the two orbits $G$ on $V(\Gamma)$ and have size $p^n$. Recalling that $N\unlhd G$, each orbit of $N$ has size dividing $p^n$. So, if $N$ is solvable, then $N$ must be an elementary abelian $p$-group, as required.
Suppose that $N$ is non-solvable. By Proposition~\ref{Av}, we have $|G|=2^r\cdot 3\cdot p^n$, where $r\geq 0$. If $p=3$, then by Burnside $p^aq^b$-theorem, $G$ would be solvable, which is impossible because $N$ is non-solvable. Thus, $p>3$. Since $N$ is a minimal normal subgroup of $G$, $N$ is a product of some isomorphic non-abelian simple groups. Observing that $3^2\nmid |G|$, by \cite[pp.12-14]{Gorenstein}, we obtain that $N\cong A_5$ or $\hbox{\rm PSL}(2,7)$. Then $p=5$ or $7$, and $p^2\nmid |N|$. Since $n\geq 2$, it follows that $N$ is intransitive on each bipartition sets of $\Gamma$. By Proposition~\ref{intransitive}, $N$ is semiregular on $V(\Gamma)$, and so $|N|\ |\ p^n$, which is impossible. This completes the proof of our lemma.\hfill\hskip10pt $\Box$\vspace{3mm}
\begin{lem}\label{11}
Let $p\geq 5$ be a prime and let $\Gamma$ be a connected cubic edge-transitive graph of order $2p^n$ with $n\geq 1$.
Let $A=\hbox{\rm Aut\,}(\Gamma)$ and let $H$ be a Sylow $p$-subgroup of $A$. Then $\Gamma$ is a bi-Cayley graph over $H$, and moreover,
if $p\geq 11$, then $\Gamma$ is a normal bi-Cayley graph over $H$.
\end{lem}
\noindent{\bf Proof}\hskip10pt By Proposition~\ref{Av}, the stabilizer of any $v\in V(\Gamma)$ in $A$ has order dividing $2^r\cdot 3$ with $r\geq 0$. Recalling $H$ is a Sylow $p$-subgroup of $A$, $H$ must be semiregular on $V(\Gamma)$ since $p\geq 5$. Since $\Gamma$ is edge-transitive, $\Gamma$ is either arc-transitive or semisymmetric, and so $p^n\ |\ |A|$. It follows that $p^n\ |\ |H|$, and so $|H|=p^n$. Thus, $H$ has two orbits on $V(\Gamma)$, and hence $\Gamma$ is bi-Cayley graph over $H$.
Now suppose that $p\geq 11$. We shall prove the second assertion. It suffices to prove that $H\unlhd A$. Use induction on $n$. If $n=1$, then $\Gamma$ is symmetric by {\cite[Theorem~2]{2p}}, and then by~\cite[Theorem~1]{BiCayley.7} (see also \cite[Table~1]{Cheng87} or \cite[Proposition~2.8]{2pn}), we have $H\unlhd A$, as required.
Assume $n\geq 2$. Take $N$ to be a minimal normal subgroup of $A$. By Lemma~\ref{N}, $N$ is an elementary abelian $p$-group and $|N|\mid p^n$.
Consider the quotient graph $\Gamma_N$ of $\Gamma$ corresponding to the orbits of $N$. If $|N|=p^n$, then $H=N\unlhd A$, as required.
Suppose that $|N|<p^n$. Then each orbit of $N$ has size at most $p^{n-1}$, and by Propositions~\ref{intransitive} and \ref{3orbits}, $N$ is semiregular, and $\Gamma_N$ is of valency $3$ with $A/N$ as an edge-transitive group of automorphisms of $\Gamma_N$. Clearly, $\Gamma_N$ has order $2p^m$ with $m<n$. By induction, we have any Sylow $p$-subgroup of $\hbox{\rm Aut\,}(\Gamma_N)$ is normal. It follows that $H/N\unlhd A/N$ because $H/N$ is a Sylow $p$-subgroup of $A/N$. Therefore, $H\unlhd A$, as required.
\hfill\hskip10pt $\Box$\vspace{3mm}
\begin{lem}\label{Q}
Let $\Gamma$ be a connected cubic edge-transitive graph of order $2p^n$ with $p=5$ or $7$ and $n\geq 2$.
Let $Q=O_p(A)$ be the maximal normal $p$-subgroup of $A=\hbox{\rm Aut\,}(\Gamma)$. Then $|Q|=p^n$ or $p^{n-1}$.
\end{lem}
{\bf Proof}\hskip10pt Let $|Q|=p^m$ with $m\leq n$. Suppose that $n-m\geq 2$. Then by Propositions~\ref{3orbits} and \ref{intransitive},
the quotient graph $\Gamma_Q$ is a connected cubic graph of order $2p^{n-m}$ with $A/Q$ as an edge-transitive group of automorphisms.
Let $N/Q$ be a minimal normal subgroup of $A/Q$. By Lemma~\ref{N}, $N/Q$ is an elementary abelian $p$-group.
It follows that $N\unlhd A$ and $Q<N$, contrary to the maximality of $Q$.
Thus $n-m\leq 1$, and so $|Q|=p^n$ or $p^{n-1}$.\hfill\hskip10pt $\Box$\vspace{3mm}
Now we are ready to consider cubic edge-transitive bi-Cayley graphs over a metacyclic $p$-group. We first prove that $p=3$.
\begin{lem}\label{57}
Let $\Gamma$ be a connected cubic edge-transitive bi-Cayley graph over a non-abelian metacyclic $p$-group $H$ with $p$ an odd prime. Then $p=3$.
\end{lem}
\noindent{\bf Proof}\hskip10pt Suppose to the contrary that $p>3$. Let $A=\hbox{\rm Aut\,}(\Gamma)$. Then $R(H)$ is a Sylow $p$-subgroup of $A$. We shall first prove the following claim.\medskip
\noindent{\bf Claim}.\ $R(H)\unlhd A$.\medskip
Suppose to the contrary that $R(H)$ is not normal in $A$. By Lemma~\ref{11}, we have $p=5$ or $7$.
Let $N$ be the maximal normal $p$-subgroup of $A$. Then $N\leq R(H)$, and by Lemma~\ref{Q}, we have $|R(H): N|=p$. Then the quotient graph $\Gamma_N$ is a cubic graph of order $2p$ with $A/N$ as an edge-transitive automorphism group. By \cite{Sym768, Semisym768}, if $p=5$, then $\Gamma_N$ is the Petersen graph, and if $p=7$, then $\Gamma_N$ is the Heawood graph. Since $A/N$ is transitive on the edges of $\Gamma_N$ and $R(H)/N$ is non-normal in $A/N$, it follows that
$$
\begin{array}{ll}
A_5\lesssim A/N\lesssim S_5, & {\rm if }\ p=5;\\
\hbox{\rm PSL}(2,7)\lesssim A/N\lesssim \hbox{\rm PGL}(2,7), & {\rm if }\ p=7.
\end{array}
$$
Let $B/N$ be the socle of $A/N$. Then $B/N$ is also edge-transitive on $\Gamma_N$, and so $B$ is also edge-transitive on $\Gamma$.
Let $C=C_B(N)$. By Proposition~\ref{NC}, $B/C\lesssim\hbox{\rm Aut\,}(N)$. And $C/(C\cap N)\cong CN/N \unlhd B/N$. Since $B/N$ is non-abelian simple, one has $CN/N=1$ or $B/N$.
Suppose first that $CN/N=1$. Then $C\leq N$, and so $C=C\cap N=C_N(N)=Z(N)$. Then $B/Z(N)=B/C\lesssim\hbox{\rm Aut\,}(N)$.
Since $R(H)$ is a metacyclic $p$-group, $N$ is also a metacyclic $p$-group. If $N$ is non-abelian, then by Proposition~\ref{metap} and \cite[Lemma~2.6]{pcomplement}, $\hbox{\rm Aut\,}(N)$ is solvable. It follows that $B/Z(N)$ is solvable, and so $B$ is solvable. This is contrary to the fact that $B/N$ is non-abelian simple.
If $N$ is abelian, then $C=Z(N)=N$.
Let $$\hbox{\rm Aut\,}^{\Phi}(N)=\langle\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in\hbox{\rm Aut\,}(N)\mid g^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}\Phi(N)=g\Phi(N),\forall g\in N\rangle,$$ where $\Phi(N)$ is the Frattini subgroup of $N$.
Recall that $\hbox{\rm Aut\,}^{\Phi}(N)$ is a normal $p$-subgroup of $\hbox{\rm Aut\,}(N)$ and $\hbox{\rm Aut\,}(N)/\hbox{\rm Aut\,}^{\Phi}(N)\leq \hbox{\rm Aut\,}(N/\Phi(N))$ (see{~\cite{commutator}}).
Let $K/C=(B/C)\cap \hbox{\rm Aut\,}^{\Phi}(N)$.
Then $K/C\unlhd B/C$, and so $K\unlhd B$. It follows that
$$B/K\cong(B/C)/(K/C)\cong((B/C)\cdot \hbox{\rm Aut\,}^{\Phi}(N))/\hbox{\rm Aut\,}^{\Phi}(N)\leq\hbox{\rm Aut\,}(N/\Phi(N)).$$
Clearly, $K/C$ is a $p$-group. Since $C=N$, $K$ is also a $p$-group.
As $N$ is the maximal normal $p$-subgroup of $A$, $N$ is also the maximal normal $p$-subgroup of $B$. This implies that $K=N$.
If $N$ is cyclic, then $N/\Phi(N)\cong \mathbb{Z}_p$, and so $B/N=B/K\lesssim\hbox{\rm Aut\,}(N/\Phi(N))\cong\mathbb{Z}_{p-1}$, again contrary to the fact that $B/N$ is a non-abelian simple group. If $N$ is not cyclic, then $N/\Phi(N)\cong \mathbb{Z}_p\times\mathbb{Z}_p$.
It follows that $B/N=B/K\lesssim\hbox{\rm Aut\,}(N/\Phi(N))\cong\hbox{\rm GL}(2, p)$. This forces that either $A_5\leq \hbox{\rm GL}(2,5)$ with $p=5$, or $\hbox{\rm PSL}(2,7)\leq \hbox{\rm GL}(2,7)$ with $p=7$. However, each of these can not happen by Magma{~\cite{Magma}}, a contradiction.
Suppose now that $CN/N=B/N$. Since $C\cap N=Z(N)$, we have $1<C\cap N\leq Z(C)$.
Clearly, $Z(C)/(C\cap N) \unlhd C/(C\cap N)\cong CN/N$. Since $CN/N=B/N$ is non-abelian simple, $Z(C)/C\cap N$ must be trivial.
Thus $C\cap N=Z(C)$, and hence $B/N=CN/N\cong C/C\cap N=C/Z(C)$. If $C=C'$, then $Z(C)$ is a subgroup of the Schur multiplier of $B/N$. However, the Schur multiplier of $A_5$ or $\hbox{\rm PSL}(2,7)$ is ${\mathbb Z}_2$, a contradiction. Thus, $C\neq C'$.
Since $C/Z(C)$ is non-abelian simple, one has $C/Z(C)=(C/Z(C))'=C'Z(C)/Z(C)\cong C'/(C'\cap Z(C))$, and then we have $C=C'Z(C)$. It follows that $C''=C'$. Clearly, $C'\cap Z(C)\leq Z(C')$, and $Z(C')/(C'\cap Z(C)) \unlhd C'/(C'\cap Z(C))$. Since $C'/(C'\cap Z(C))\cong C/Z(C)$ and since $C/Z(C)$ is non-abelian simple, it follows that $Z(C')/(C'\cap Z(C))$ is trivial, and so $Z(C')=C'\cap Z(C)$.
As $C/(C\cap N)\cong CN/N$ is non-abelian, we have $C/(C\cap N)=(C/(C\cap N))'=(C/Z(C))'\cong C'/(C'\cap Z(C))=C'/Z(C')$.
Since $C'=C''$, $Z(C')$ is a subgroup of the Schur multiplier of $CN/N$. However, the Schur multiplier of $A_5$ or $\hbox{\rm PSL}(2,7)$ is ${\mathbb Z}_2$, forcing that $Z(C')\cong{\mathbb Z}_2$. This is impossible because $Z(C')=C'\cap Z(C)\leq C\cap N$ is a $p$-subgroup. Claim is proved.\medskip
If $H$ is non-split, then by Proposition~\ref{pcom}, $A$ has a normal $p$-complement $Q$. By Propositions~\ref{3orbits} and \ref{intransitive}, the quotient graph $\Gamma_Q$ would be cubic graph of odd order, a contradiction.
Thus, $H$ is split. Then we may assume that $$H=\lg a,b\ |\ a^{p^{m}}=b^{p^{n}}=1, a^b=a^{1+p^{r}}\rg,$$ where $m, n, r$ are positive integers such that $r<m\leq m+n$.
By Claim, $R(H)\unlhd A$. Since $\Gamma$ is edge-transitive, we assume that $\Gamma={\rm BiCay}(H,\emptyset,\emptyset,S)$. By Proposition~\ref{bicayley}, we may assume that $S=\{1, g, h\}$ with $g, h\in H$. By Proposition~\ref{bicayleyaut}, there exists $\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, x}\in\hbox{\rm Aut\,}(\Gamma)_{1_0}$, where $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in\hbox{\rm Aut\,}(H)$ and $x\in H$, such that $\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, x}$ cyclically permutates the three elements in $\Gamma(1_0)=\{1_1, g_1, h_1\}$. Without loss of generality, assume that $(\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, x})_{|\Gamma(1_0)}=(1_1\ g_1\ h_1)$. Then $g_1=(1_1)^{\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, x}}=x_1$, implying that $x=g$.
Furthermore, $h_1=(g_1)^{\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,x}}=(gg^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon})_1$ and $1_1=(h_1)^{\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, x}}=(gh^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon})_1$.
It follows that $g^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=g^{-1}h$, $h^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=g^{-1}$. This implies that $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ is an automorphism of $H$ order dividing $3$. If $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ is trivial, then $h=g^{-1}$ and $g=g^{-1}h=g^{-2}$, and then $g^3=1$. Since $p>3$, we must have $h=g=1$, a contradiction. Thus, $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ has order $3$. By Proposition~\ref{metap}, we must have $3\ |\ p-1$. Furthermore, $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ is conjugate to the following automorphism of $H$ induced by the following map:
$$\b: a\mapsto a^s, b\mapsto b,$$
where $s$ is an element of order $3$ of ${\mathbb Z}_{p^m}^*$.
Assume that $\b=\pi^{-1}\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\pi$ for $\pi\in\hbox{\rm Aut\,}(H)$. Consider the graph $\Gamma^\pi={\rm BiCay}(H, \emptyset, \emptyset, S^\pi)$. By Proposition~\ref{bicayleyaut}~(3), we have $\Gamma^\pi\cong \Gamma$, and $\s_{\b, g^{\pi}}$ cyclically permutates the three elements in $\Gamma^\pi(1_0)=\{1_1^{\pi}, g_1^{\pi}, h_1^{\pi}\}$. For convenience of the statement, we may assume that $\pi$ is trivial and $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=\b$.
Let $g=b^ja^i$, where $i\in\mathbb{Z}_{p^m}$, $j\in\mathbb{Z}_{p^n}$. Then $h=gg^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=b^ja^ib^ja^{is}$.
Since $\Gamma$ is connected, we have $H=\lg S\rg=\lg g,h\rg=$$\lg b^ja^i, b^ja^ib^ja^{is}\rg$$=\lg b^j, a^i, a^{is}\rg=\lg a^i, b^j\rg$, implying that
$i,j$ are coprime to $p$. Then there exists an integer $u$ such that $ui\equiv1\ (\hbox{\rm mod } p^m)$.
It is easy to check that the map $\g: a\mapsto a^u, b\mapsto b$ can induce an automorphism of $H$, and then $(a^i)^{\g}=a^{ui}=a$.
Again, by Proposition\ref{bicayleyaut}~(3), we have $\Gamma\cong {\rm BiCay}(H, \emptyset, \emptyset, S^{\g})$,
where $S^{\g}=\{1, b^ja, b^jab^ja^{s}\}$. Let $\Gamma'={\rm BiCay}(H, \emptyset, \emptyset, S^{\g})$. Then $\s_{\g^{-1}\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\g, g^\g}\in\hbox{\rm Aut\,}(\Gamma')$ cyclically permutates the elements in $\Gamma'(1_0)=\{1_1, (b^ja)_1, (b^jab^ja^s)_1\}$.
It is easy to check that $a^{\g^{-1}\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\g}=(a^i)^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\g}=(a^{is})^{\g}=a^s$ and $b^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon^{\g}}=b$.
It then follows that $1_1^{\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon^{\g}, b^ja}}=(b^ja)_1$, $(b^ja)_1^{\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon^{\g}, b^ja}}=(b^jab^ja^s)_1$,
and $(b^jab^ja^s)_1^{\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon^{\g},b^ja}}$$=(b^ja(b^jab^ja^s)^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon^{\g}})_1$$=(b^jab^ja^sb^ja^{s^2})_1=(b^{3j}a^{(1+p^r)^{2j}+s(1+p^r)^{j}+s^2})_1\neq 1_1$. This is a contradiction. Thus $p=3$. \hfill\hskip10pt $\Box$\vspace{3mm}
In what follows, we consider cubic edge-transitive bi-Cayley graph over the group $H$, where $H$ is a non-abelian metacyclic $3$-group.
\begin{lem}\label{HNormal}
Let $\Gamma={\rm BiCay}(H,R,L,S)$ be a connected cubic edge-transitive bi-Cayley graph over a non-abelian metacyclic $3$-group $H$ with $|H|=3^s$, where $s\geq 4$.
Then $\Gamma$ is a normal bi-Cayley graph over $H$.
\end{lem}
\noindent{\bf Proof}\hskip10pt Let $A=\hbox{\rm Aut\,}(\Gamma)$ and let $P$ be a Sylow $3$-subgroup of $A$ such that $R(H)\leq P$. By Proposition~\ref{Av}, we have $|A|=3^{s+1}\cdot 2^r$ with $r\geq 0$. This implies that $|P|=3|R(H)|$, and so $|P_{1_0}|=|P_{1_1}|=3$. Thus, $P$ is transitive on the edges of $\Gamma$. Clearly, $R(H)\unlhd P$. This implies that the two orbits $H_0, H_1$ of $R(H)$ do not contain the edges of $\Gamma$, and so $R=L=\emptyset$.\medskip
\noindent{\bf Claim}\ $P\unlhd A$.
Let $M\unlhd A$ be maximal subject to that $M$ is intransitive on both $H_0$ and $H_1$. By Proposition~\ref{3orbits} and Proposition~\ref{intransitive}, $M$ is semiregular on $V(\Gamma)$ and the quotient graph $\Gamma_M$ of $\Gamma$ relative to $M$ is a cubic graph with $A/M$ as an edge-transitive group of automorphisms. Assume that $|M|=3^t$. Then $|V(\Gamma_M)|=2\cdot 3^{s-t}$. If $s-t\leq 2$, then by \cite{Sym768, Semisym768}, $\Gamma_M$ is isomorphic to $\F006A$ or the Pappus graph $\F018A$, and then $\hbox{\rm Aut\,}(\Gamma_M)$ has a normal Sylow $3$-subgroup. It follows that $P/M\unlhd A/M$, and so $P\unlhd A$, as claimed.
Now assume that $s-t>2$. Take a minimal normal subgroup $N/M$ of $A/M$. By Lemma~\ref{N}, $N/M$ is an elementary abelian $3$-group. By the maximality of $M$, $N$ is transitive on at least one of $H_0$ and $H_1$, and so $3^{s}\ |\ |N|$. If $3^{s+1}\ |\ |N|$, then $P=N\unlhd A$, as claimed. Assume that $|N|=3^s$. If $N$ is transitive on both $H_0$ and $H_1$, then $N$ is semiregular on both $H_0$ and $H_1$, and then $\Gamma_M$ would be a cubic bi-Cayley graph on $N/M$. Since $\Gamma_M$ is connected, by Proposition~\ref{bicayley}, $N/M$ is generated by two elements, and so $N/M\cong{\mathbb Z}_3$ or ${\mathbb Z}_3\times{\mathbb Z}_3$. This implies that $|V(\Gamma_M)|=6$ or $18$, contrary to the assumption that $|V(\Gamma_M)|=2\cdot 3^{s-t}>18$. Thus, we may assume that $N$ is transitive on $H_0$ but intransitive on $H_1$. Then $N/M\neq R(H)M/M$, and so $NR(H)M/M=P/M$. Since $|P/M: R(H)M/M|\ |\ 3$, one has $|N/M: (N/M\cap R(H)M/M)|\ |\ 3$, and since $H$ is metacyclic, one has $N/M\cap R(H)M/M$ is also metacyclic and so is a two-generator group. This implies that $|N/M|\ |\ 3^3$, and so $|N/M|=3^3$ because $|N/M|=3^{s-t}>9$. Then $|V(\Gamma_M)|=2\cdot |N/M|=54$. Since $s\geq 4$, we have $|M|\geq 3$. If $M\nleq R(H)$, then $P=MR(H)$ and then $N/M\leq R(H)M/M$. As $H$ is metacyclic, $N/M$ is also metacyclic, and so $|N/M|=3$ or $9$, a contradiction. Thus, $M\leq R(H)$, and hence $M$ is metacyclic. Then $M/\Phi(M)\cong{\mathbb Z}_3$ or ${\mathbb Z}_3\times{\mathbb Z}_3$. Since $\Phi(M)$ is characteristic in $M$, one has $\Phi(M)\unlhd A$ because $M\unlhd A$. Then the quotient graph $\Gamma_{\Phi(M)}$ is a cubic graph of order $2\cdot 3^4$ or $2\cdot 3^5$ with $A/\Phi(M)$ as an edge-transitive group of automorphisms. By \cite{Sym768,Semisym768} and Magma~\cite{Magma}, we obtain that every Sylow $3$-subgroup of $\hbox{\rm Aut\,}(\Gamma_{\Phi (M)})$ is normal. This implies that $P/\Phi(M)\unlhd A/\Phi(M)$, and so $P\unlhd A$, completing the proof of our claim.\medskip
Now we are ready to finish the proof of our lemma. By Claim, we have $P\unlhd A$. Since $|P: R(H)|=3$, one has $\Phi(P)\leq R(H)$. As $H$ is non-abelian, one has $\Phi (P)<R(H)$ for otherwise, we would have $P$ is cyclic and so $H$ is cyclic which is impossible. Then $\Phi(P)$ is intransitive on both $H_0$ and $H_1$, the two orbits of $R(H)$ on $V(\Gamma)$. Since $\Phi(P)$ is characteristic in $P$, $P\unlhd A$ gives that $\Phi(P)\unlhd A$. By Propositions~\ref{3orbits} and \ref{intransitive}, the quotient graph $\Gamma_{\Phi(P)}$ of $\Gamma$ relative to $\Phi(P)$ is a cubic graph with $A/\Phi(P)$ an edge-transitive group of automorphisms. Furthermore, $P/\Phi(P)$ is transitive on the edges of $\Gamma_{\Phi(P)}$. Since $P/\Phi(P)$ is abelian, it is easy to see that $\Gamma_{\Phi(P)}\cong K_{3,3}$, and so $P/\Phi(P)\cong{\mathbb Z}_3\times{\mathbb Z}_3$. Since $|P|=3^{s+1}\geq 3^5$, one has $|\Phi(P)|=3^{s-1}\geq 3^3$.
Let $\Phi_2$ be the Frattini subgroup of $\Phi(P)$.
Then $\Phi_2\unlhd A$ because $\Phi_2$ is characteristic in $\Phi (P)$ and $\Phi(P)\unlhd A$.
Clearly, $\Phi_2\leq \Phi(P)<R(H)$, so $\Phi_2$ is intransitive on both $H_0$ and $H_1$.
Consider the quotient graph $\Gamma_{\Phi_2}$ of $\Gamma$ relative to $\Phi_2$. By Propositions~\ref{3orbits} and \ref{intransitive}, $\Gamma_{\Phi_2}$ is a cubic graph with $A/\Phi_2$ as an edge-transitive group of automorphisms. Furthermore, $\Gamma_{\Phi_2}$ is a bi-Cayley graph over the group $R(H)/\Phi_2$. Again, since $H$ is a metacyclic group, we have $\Phi(P)/\Phi_2\cong{\mathbb Z}_3$ or ${\mathbb Z}_3\times{\mathbb Z}_3$. If $\Phi(P)/\Phi_2\cong{\mathbb Z}_3$, then $\Phi(P)$ is a cyclic $3$-group, and so $\Gamma$ is an edge-transitive cyclic cover of $\Gamma_{\Phi(P)}\cong K_{3,3}$. By Feng et al.~\cite{K33Feng, K33Wang}, we have $\Gamma$ is isomorphic to either $K_{3,3}$ or the Pappus graph, a contradiction.
Thus, $\Phi(P)/\Phi_2\cong{\mathbb Z}_3\times{\mathbb Z}_3$. Since $|\Phi(P)|=3^{s-1}\geq 3^3$, one has $|\Phi_2|\geq 3$.
Let $\Phi_3$ be the Frattini subgroup of $\Phi_2$. Then $\Phi_3$ is characteristic in $\Phi_2$, and so normal in $A$ because $\Phi_2\unlhd A$.
As $\Phi_2\leq R(H)$, one has $\Phi_2/\Phi_3\cong\mathbb{Z}_3$ or $\mathbb{Z}_3\times\mathbb{Z}_3$, and so $|R(H)/\Phi_3|=3^4$ or $3^5$. Clearly, $\Phi_3$ is intransitive on both $H_0$ and $H_1$. Again, by Propositions~\ref{3orbits} and \ref{intransitive}, the quotient graph $\Gamma_{\Phi_3}$ is a cubic graph of order $162$ or $486$ with $A/\Phi_3$ as an edge-transitive group of automorphisms. Observe that $R(H)/\Phi_3$ is metacyclic semiregular on $V(\Gamma_{\Phi_3})$ with two orbits.
If $|\Gamma_{\Phi_3}|=486$, then by \cite{Sym768, Semisym768}, $\Gamma_{\Phi_3}$ is semisymmetric or symmetric. For the former, by Magma \cite{Magma}, all semiregular subgroups of $\hbox{\rm Aut\,}(\Gamma_{\Phi_2})$ of order $243$ are normal, and so $R(H)/\Phi_3\unlhd \hbox{\rm Aut\,}(\Gamma_{\Phi_3})$. It follows that $R(H)/\Phi_3\unlhd A/\Phi_3$, and so $R(H)\unlhd A$, as required. If $\Gamma_{\Phi_3}$ is symmetric, then by \cite{Sym768}, $\Gamma_{\Phi_3}\cong \F486A$, $\F486B$, $\F486C$ or $\F486D$. By Magma \cite{Magma}, if $\Gamma_{\Phi_3}\cong \F486B, \F486C$ or $\F486D$, then $\hbox{\rm Aut\,}(\Gamma_{\Phi_3})$ does not have a metacyclic semiregular subgroup of order $243$, a contradiction.
If $\Gamma_{\Phi_3}\cong \F486A$, then by Magma \cite{Magma}, all semiregular subgroups of $\hbox{\rm Aut\,}(\Gamma_{\Phi_3})$ of order $243$ are normal, and so $R(H)/\Phi_3\unlhd \hbox{\rm Aut\,}(\Gamma_{\Phi_3})$. It follows that $R(H)/\Phi_3\unlhd A/\Phi_3$, and so $R(H)\unlhd A$, as required.
If $|\Gamma_{\Phi_3}|=162$, then by \cite{Sym768, Semisym768}, $\Gamma_{\Phi_3}$ is symmetric, and is isomorphic to $\F162A$, $\F162B$ or $\F162C$. By Magma \cite{Magma}, if $\Gamma_{\Phi_3}\cong \F162C$, then $\hbox{\rm Aut\,}(\Gamma_{\Phi_3})$ does not have a metacyclic semiregular subgroup of order $81$, a contradiction. If $\Gamma_{\Phi_3}\cong \F162A$ or $\F162B$, then by Magma \cite{Magma}, all semiregular subgroups of $\hbox{\rm Aut\,}(\Gamma_{\Phi_3})$ of order $81$ are normal, and so $R(H)/\Phi_3\unlhd \hbox{\rm Aut\,}(\Gamma_{\Phi_3})$. It follows that $R(H)/\Phi_3\unlhd A/\Phi_3$, and so $R(H)\unlhd A$, as required.
\hfill\hskip10pt $\Box$\vspace{3mm}
\noindent{\bf Proof of Theorem~\ref{5no}}\ Let $\Gamma={\rm BiCay}(H,R,L,S)$ be a connected cubic edge-transitive bi-Cayley graph over a non-abelian metacyclic $p$-group $H$ with $p$ an odd prime. By Lemma~\ref{57}, we have $p=3$, and since $H$ is a non-abelian metacyclic $3$-group, we have $|H|=3^s$ with $s\geq 3$. If $s=3$, then $\Gamma$ has order $54$, and by \cite{Sym768, Semisym768}, $\Gamma$ is isomorphic to $\F054$ or the Gray graph. However, by Magma \cite{Magma}, $\hbox{\rm Aut\,}(\F054)$ does not have a non-abelian metacyclic $3$-subgroup which acts semiregularly on the vertex set of $\F054$ with tow orbits. It follows that $\Gamma$ is isomorphic to Gray graph. If $s>3$, then by Lemma~\ref{HNormal}, $R(H)\unlhd\hbox{\rm Aut\,}(\Gamma)$, as required.\hfill\hskip10pt $\Box$\vspace{3mm}
\section{A class of cubic edge-transitive bi-$3$-metacirculants}
In this section, we shall use Theorem~\ref{5no} to give a characterization of connected cubic edge-transitive bi-Cayley graphs over inner-abelian metacyclic $3$-groups.
\subsection{Construction}\label{sec-5}
We shall first construct two classes of connected cubic edge-transitive bi-Cayley graphs over inner-abelian metacyclic $3$-groups.
\begin{construction}\label{con-1}
Let $t$ be a positive integer, and let $${\mathcal G}=\lg a, b\ |\ a^{3^{t+1}}=b^{3^t}=1, b^{-1}ab=a^{1+3^t}\rg.$$
Let $S=\{1, a, a^{-1}b\}$, and let $\Gamma_{t}={\rm BiCay}({\mathcal G}, \emptyset, \emptyset, S)$.
\end{construction}
\begin{lem}\label{Example1}
For any integer $t$, the graph ${\Gamma_t}$ is semisymmetric.
\end{lem}
\noindent{\bf Proof}\hskip10pt We first prove the following four claims.\medskip
\noindent{\bf Claim 1.} ${\mathcal G}$ has an automorphism $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ mapping $a, b$ to $a^{-2}b, a^{3^t-3}b$, respectively.\medskip
Let $x=a^{-2}b$ and $y=a^{3^t-3}b$. Then,
$$\begin{array}{l}
(yx^{-1})^{3^t+1}=[(a^{3^t-3}b)(a^{-2}b)^{-1}]^{3^t+1}=(a^{3^t-1})^{3^t+1}=a^{-1},\\
((yx^{-1})^{3^t+1})^{-2}\cdot x=a^2\cdot a^{-2}b=b,
\end{array}
$$
and hence $\lg a, b\rg=\lg x, y\rg$.
By Lemma~\ref{comput2}~(2), we have $x^{3^{t+1}}=(a^{-2}b)^{3^{t+1}}=1$ and $y^{3^{t}}=(a^{3^t-3}b)^{3^{t}}=1$.
Furthermore, we have
$$x^{1+3^t}=(a^{-2}b)^{1+3^t}=(a^{-2}b)(a^{-2}b)^{3^t}=a^{-2}ba^{-2\cdot 3^t}=a^{-2-2\cdot 3^t}b=a^{3^t-2}b,$$
and
$$
\begin{array}{lll}
y^{-1}xy&=&(a^{3^t-3}b)^{-1}(a^{-2}b)(a^{3^t-3}b)\\
&=&(b^{-1}a^{3-3^t}a^{-2}b)a^{3^t-3}b\\
&=&(b^{-1}a^{1-3^t}b)a^{3^t-3}b\\
&=&a^{(1+3^t)(1-3^t)}a^{3^t-3}b\\
&=&a^{3^t-2}b\\
&=&x^{1+3^t}.
\end{array}$$
It follows that $x$ and $y$ have the same relations as do $a$ and $b$.
Thus, the map $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon: a\mapsto a^{-2}b, b\mapsto a^{3^t-3}b$ induces an automorphism of ${\mathcal G}$, as claimed.\medskip
\noindent{\bf Claim 2.}\ ${\mathcal G}$ has no automorphism mapping $a, b$ to $a^{-1}, a^{3t}b^{-1}$, respectively.\medskip
Suppose to the contrary that ${\mathcal G}$ has an automorphism, say $\b$, such that $a^\b=a^{-1}, b^\b=a^{3t}b^{-1}$.
Then $(b^{-1}ab)^\b=(a^{3^t+1})^\b$, and so
$$\begin{array}{lll}
a^{-3^t-1}&=&(a^{3^t+1})^\b=(b^{-1}ab)^\b\\
&=&(a^{3^t}b^{-1})^{-1}\cdot a^{-1}\cdot (a^{3^t}b^{-1})\\
&=&ba^{-1}b^{-1}=a^{-(1+3^t)^{3^t-1}}=a^{-1+3^t}.
\end{array}
$$
It follows that $a^{2\cdot 3^t}=1$, and so $3^{t+1}\ |\ 2\cdot 3^t$, a contradiction.\medskip
\noindent{\bf Claim 3.}\ ${\mathcal G}$ has no automorphism mapping $a, b$ to $b^{-1}a, b^{-1}$, respectively.\medskip
Suppose to the contrary that there exists $\g\in\hbox{\rm Aut\,}({\mathcal G})$ such that $a^\g=b^{-1}a, b^\g=b^{-1}$.
Then $(b^{-1}ab)^\g=(a^{1+3^t})^\g$, and then
$$
\begin{array}{lll}
b^{-1}a^{3^t+1}=(b^{-1}a)^{1+3^t}=(a^{1+3^t})^\g=(b^{-1}ab)^\g=b(b^{-1}a)b^{-1}=ab^{-1}.\\
\end{array}
$$
It follows that $b^{-1}a^{3^t+1}b=a$, and so $a^{3^{2t}+2\cdot 3^t+1}=a^{2\cdot 3^t+1}=a$, forcing that $3^{t+1}\ |\ 2\cdot 3^t$, a contradiction.\medskip
Now we are ready to finish the proof. By Claim~1, there exists $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in\hbox{\rm Aut\,}({\mathcal G})$ such that $a^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=a^{-2}b$ and $b^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=a^{3^t-3}b$.
Then $(a^{-1}b)^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=(a^{-2}b)^{-1}(a^{3^t-3}b)=b^{-1}a^{3^t-1}b=a^{-1}$. It then follows that
$$S^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=\{1^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon},a^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon},(a^{-1}b)^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}\}=\{1,a^{-2}b,a^{-1}\}=a^{-1}S.$$ By Proposition~\ref{bicayleyaut},
$\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, a}$ is an automorphism of $\Gamma_t$ fixing $1_0$ and cyclically permutating the three neighbors of $1_0$.
Set $B=R({\mathcal G})\rtimes\lg \s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,a}\rg$. Then $B$ acts regularly on the edges of ${\Gamma_t}$.
If $t=1$, then by Magma~\cite{Magma}, $\Gamma_1$ is isomorphic to the Gray graph, which is semisymmetric.
In what follows, assume that $t>1$. By Theorem~\ref{5no}, $\Gamma_t$ is a normal bi-Cayley graph over $R(H)$. Suppose that $\Gamma_t$ is vertex-transitive.
Then $\Gamma_t$ is also arc-transitive. So, there exist $f\in\hbox{\rm Aut\,}({\mathcal G}), g, h\in{\mathcal G}$ so that $\d_{f, g, h}$ is an automorphism of $\Gamma_t$ taking the arc $(1_0, 1_1)$ to $(1_1, 1_0)$. By the definition of $\d_{f, g, h}$, one may see that $g=h=1$ and $S^{f}=S^{-1}$, namely,
$$\{1, a, a^{-1}b\}^f=\{1, a^{-1}, b^{-1}a\}.$$
So, $f$ takes $(a, a^{-1}b)$ either to $(a^{-1}, b^{-1}a)$ or to $(b^{-1}a, a^{-1})$. However, this is impossible by Claims 2-3. Therefore, ${\Gamma_t}$ is semisymmetric.\hfill\hskip10pt $\Box$\vspace{3mm}
\begin{construction}\label{con-2}
Let $t$ be a positive integer, and let
$${\mathcal H}=\lg a, b\ |\ a^{3^{t+1}}=b^{3^{t+1}}=1, b^{-1}ab=a^{1+3^t}\rg.$$
Let $T=\{1,b,b^{-1}a\}$, and let $\Sigma_{t}={\rm BiCay}({\mathcal H},\emptyset, \emptyset, T)$.
\end{construction}
\begin{lem}\label{Example2}
For any positive integer $t$, the graph $\Sigma_t$ is symmetric.
\end{lem}
\noindent{\bf Proof}\hskip10pt We first prove the following two claims.\medskip
\noindent{\bf Claim 1} ${\mathcal H}$ has an automorphism $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ mapping $a, b$ to $a^{2\cdot 3^t+1}b^{-3}, a^{2\cdot 3^t+1}b^{-2}$, respectively.\medskip
Let $x=a^{2\cdot 3^t+1}b^{-3}$ and $y=a^{2\cdot 3^t+1}b^{-2}$. Note that $((y^{-1}x)^{-1})=b$ and $xb^3=a^{2\cdot 3^t+1}$. This implies that $\lg x, y\rg=\lg a, b\rg={\mathcal H}$.
By Lemma~\ref{comput2} (2), we have $x^{3^{t+1}}=(a^{-2}b)^{3^{t+1}}=1$ and $y^{3^{t+1}}=(a^{3^t-3}b)^{3^{t}}=1$. Furthermore, we have
$$
\begin{array}{lll}
y^{-1}xy&=&(a^{2\cdot 3^t+1}b^{-2})^{-1}(a^{2\cdot 3^t+1}b^{-3})(a^{2\cdot 3^t+1}b^{-2})\\
&=& b^{-1}a^{2\cdot 3^t+1}b^{-2}= b^{-1}a^{2\cdot 3^t+1}bb^{-3}\\
&=& a^{(2\cdot 3^t+1)(3^t+1)}b^{-3}=ab^{-3}=x^{3^t}x\\
&=& x^{3^t+1}.
\end{array}
$$
It follows that $x$ and $y$ have the same relations as do $a$ and $b$. Therefore, ${\mathcal H}$ has an automorphism taking $(a, b)$ to $(x, y)$, as claimed.\medskip
\noindent{\bf Claim 2.} $H_t$ has an automorphism $\b$ mapping $a,b$ to $a^{-1},a^{-1}b$.\medskip
Let $x=a^{-1}$ and $y=a^{-1}b$. Clearly, $\lg a,b\rg=\lg x,y\rg$. By Lemma~\ref{comput2} (2), we have that $x^{3^{t+1}}=(a^{-1})^{3^{t+1}}=1$ and $y^{3^{t+1}}=(a^{-1}b)^{3^{t+1}}=1$. Furthermore, we have
$$
\begin{array}{lll}
y^{-1}xy&=&(a^{-1}b)^{-1}(a^{-1})(a^{-1}b)=b^{-1}a^{-1}b=a^{-3^t-1}=x^{3^t+1}.
\end{array}
$$
It follows that $x$ and $y$ have the same relations as do $a$ and $b$. Therefore, ${\mathcal H}$ has an automorphism $\b$ which takes $(a, b)$ to $(a^{-1}, a^{-1}b)$, as claimed.\medskip
Now we are ready to finish the proof. By Claim~1, there exists $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in\hbox{\rm Aut\,}({\mathcal H})$ such that $a^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=a^{2\cdot 3^t+1}b^{-3}$ and $b^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=a^{2\cdot 3^t+1}b^{-2}$. Then
$$S^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=\{1, b, b^{-1}a\}^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon=\{1, a^{2\cdot 3^t+1}b^{-2}, b^{-1}\}.$$
By an easy computation, we have $a^{2\cdot 3^t+1}b^{-2}=a^{2\cdot 3^t+1}b^{-3}b=b^{-3}a^{2\cdot 3^t+1}b=b^{-2}b^{-1}a^{2\cdot 3^t+1}b=b^{-2}a^{(2\cdot 3^t+1)(3^t+1)}=b^{-2}a.$ It follows that $$b^{-1}S=b^{-1}\{1, b, b^{-1}a\}=\{b^{-1}, 1, b^{-2}a\}=S^\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon.$$
By Proposition~\ref{bicayleyaut}, $\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, b}$ is an automorphism of $\Sigma_t$ fixing $1_0$ and cyclically permutating the three neighbors of $1_0$.
Set $B=R({\mathcal H})\rtimes\lg \s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,b}\rg$. Then $B$ acts transitively on the edges of $\Sigma_t$.
By Claim~2, there exists $\b\in\hbox{\rm Aut\,}({\mathcal H})$ such that $a^{\b}=a^{-1}$ and $b^{\b}=a^{-1}b$.
Then $S^{\b}=\{1, b, b^{-1}a\}^\b=\{1, a^{-1}b, b^{-1}\}=S^{-1}$. By Proposition~\ref{bicayleyaut}, $\d_{\b,1,1}$ is an automorphism of $\Sigma_t$ swapping $1_0$ and $1_1$. Thus, $\Sigma_t$ is vertex-transitive, and so $\Sigma_t$ is symmetric.
\hfill\hskip10pt $\Box$\vspace{3mm}
\subsection{Classification}
In this section, we shall give a classification of cubic edge-transitive bi-Cayley graph over an inner-abelian metacyclic $3$-group.
\begin{lem}\label{classify}
Let $H$ be an inner-abelian metacyclic $3$-group, and let $\Gamma$ be a connected cubic edge-transitive bi-Cayley graph over $H$.
Then $\Gamma\cong \Gamma_t$ or $\Sigma_t$.
\end{lem}
\noindent{\bf Proof}\hskip10pt Since $H$ is an inner-abelian metacyclic $3$-group, it has order at least $3^3$. If $|H|=3^3$, then $|\Gamma|=54$ and by \cite{Sym768, Semisym768}, we know that $\Gamma$ is isomorphic to $\Gamma_1$. In what follows, assume that $|H|>3^3$. By Theorem~\ref{5no}, $\Gamma$ is a normal bi-Cayley graph over $H$. Let $\Gamma={\rm BiCay}(H, R, L, S)$. Since $\Gamma$ is edge-transitive, the two orbits $H_0, H_1$ of $R(H)$ on $V(\Gamma)$ do not contain edges, and so $R=L=\emptyset$. By Proposition~\ref{bicayley}, we may assume that $S=\{1, x, y\}$ for $x, y\in H$. Since $\Gamma$ is connected, by Proposition~\ref{bicayley}, we have $H=\lg S\rg=\lg x, y\rg$.
Let $A=\hbox{\rm Aut\,}(\Gamma)$, since $\Gamma$ is normal and since $\Gamma$ is edge-transitive, by Proposition~\ref{bicayleyaut}, there exists $\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, h}\in A_{1_0}$, where $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon\in\hbox{\rm Aut\,}(H)$ and $h\in H$, such that $\s_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, h}$ cyclically permutates the three elements in $\Gamma(1_0)=\{1_1, x_1, y_1\}$. Without loss of generality, assume that $(\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, h})_{|\Gamma(1_0)}=(1_1\ x_1\ y_1)$. Then $x_1=(1_1)^{\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, h}}=h_1$, implying that $x=h$.
Furthermore, $y_1=(x_1)^{\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, h}}=(xx^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon})_1$ and $1_1=(y_1)^{\sigma_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon, h}}=(xy^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon})_1$.
It follows that $x^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=x^{-1}y$ and $y^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=x^{-1}$. This implies that $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ is an automorphism of $H$ order dividing $3$. If $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ is trivial, then $x=y^{-1}$ and $x=x^{-1}y=y^2$, and then $y^3=1$ and $x^3=1$. This implies that $H\cong{\mathbb Z}_3$ or ${\mathbb Z}_3\times{\mathbb Z}_3$, contrary to the assumption that $|H|>3^3$. Thus, $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ has order $3$.
Since $H$ is an inner-abelian $3$-group, by elementary group theory (see also \cite{inner}), we may assume that
$$H=\lg a, b\ |\ a^{3^{t+1}}=b^{3^s}=1, b^{-1}ab=a^{3^t+1}\rg,$$
where $t\geq 2, s\geq 1$. We first prove the following claim.\medskip
\noindent{\bf Claim 1}\ $H/H'=\lg aH'\rg\times\lg bH'\rg\cong \mathbb{Z}_{3^{t}}\times\mathbb{Z}_{3^{t}}$, $\mathbb{Z}_{3^{t}}\times\mathbb{Z}_{3^{t-1}}$ or $\mathbb{Z}_{3^{t}}\times\mathbb{Z}_{3^{t+1}}$.\medskip
By Lemma~\ref{comput2}~(3), we have the derived subgroup $R(H)'$ of $R(H)$ is isomorphic to $\mathbb{Z}_3$. Since $R(H)'$ is characteristic in $R(H)$, $R(H)\unlhd A$ gives that $R(H)'\unlhd A$. Consider the quotient graph $\Gamma_{R(H)'}$ of $\Gamma$ relative to $R(H)'$. Clearly, $R(H)'$ is intransitive on both $H_0$ and $H_1$, the two orbits of $R(H)$ on $V(\Gamma)$. By Propositions~\ref{3orbits} and \ref{intransitive}, $\Gamma_{R(H)'}$ is a cubic graph with $A/R(H)'$ as an edge-transitive group of automorphisms. Clearly, $\Gamma_{R(H)'}$ is a bi-Cayley graph over the abelian group $R(H)/R(H)'$.
Since $R(H)/R(H)'\unlhd A/R(H)'$, by Proposition~\ref{abelian-biCayley}, we have $R(H)/R(H)'\cong\mathbb{Z}_{3^{m+n}}\times\mathbb{Z}_{3^{m}}$ for some integers $m, n$ satisfying the equality ${\ld}^2-\ld+1\equiv 0\ ({\rm mod}\ 3^n)$ with $\ld\in\mathbb{Z}_{3^n}^{*}$. This implies that $n=0$ or $1$, and so $R(H)/R(H)'\cong \mathbb{Z}_{3^m}\times\mathbb{Z}_{3^{m}}$ or $\mathbb{Z}_{3^{m+1}}\times\mathbb{Z}_{3^{m}}$.
Since $a^{3^t}=[a, b]$, one has $\lg aH'\rg\cong{\mathbb Z}_{3^t}$, and since $H'\cap\lg b\rg=1$, one has $H/H'=\lg aH'\rg\times\lg bH'\rg\cong{\mathbb Z}_{3^t}\times{\mathbb Z}_{3^s}$. So, if $R(H)/R(H)'\cong \mathbb{Z}_{3^m}\times\mathbb{Z}_{3^{m}}$, then we have $m=s=t$ , and if $R(H)/R(H)'\cong \mathbb{Z}_{3^{m+1}}\times\mathbb{Z}_{3^{m}}$, then $(t, s)=(m, m+1)$ or $(m+1, m)$. Claim~1 is proved. \medskip
For any $h\in H$, denote by $o(h)$ the order of $h$. Let $n={\rm Max}\{t+1, s\}$. By Lemma~\ref{comput2}~(2), it is easy to see that $3^n$ is the exponent of $H$. \medskip
\noindent{\bf Claim 2}\ $o(x)=o(y)=o(x^{-1}y)=3^{n}$ and $x^{3^{n-1}}\neq y^{3^{n-1}}$. \medskip
Observing that $x^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=x^{-1}y$ and $y^{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon}=x^{-1}$, we have $o(x)=o(y)=o(x^{-1}y)$. By Lemma~\ref{comput2}~(2), we must have $o(x)=o(y)=o(x^{-1}y)=3^{n}$. Then $(x^{-1}y)^{3^{n-1}}\neq 1$, and again by Lemma~\ref{comput2}~(2), we have $x^{-3^{n-1}}y^{3^{n-1}}\neq 1$, namely, $x^{3^{n-1}}\neq y^{3^{n-1}}$, as claimed.\medskip
By Claim~1, we shall consider the following three cases:\medskip
\noindent{\bf Case~1}\ $H/H'=\lg aH'\rg\times\lg bH'\rg\cong \mathbb{Z}_{3^{t}}\times\mathbb{Z}_{3^{t}}$.\medskip
In this case, we have $s=t$. By Claim~2, we have $o(x)=o(y)=o(x^{-1}y)=3^{t+1}$ and $x^{3^t}\neq y^{3^t}$.
As $H'\cong{\mathbb Z}_3$, we have $H'=\lg x^{3^t}\rg=\lg y^{3^t}\rg$, implying that $y^{3^t}=x^{-3^t}$.
Thus $(xy)^{3^t}=x^{3^t}y^{3^t}=x^{3^t}x^{-3^t}=1$.
Since $[x, y]\in H'$ and $H'=\lg x^{3^t}\rg$, we have $[x, y]=x^{3^t}$ or $x^{-3^t}$.
It follows that $(xy)^{-1}\cdot x\cdot(xy)=y^{-1}xy=x^{1+3^t}$ or $x^{1-3^t}$.
If $(xy)^{-1}\cdot x\cdot(xy)=y^{-1}xy=x^{1+3^t}$, then
$$H=\lg x, xy\ |\ x^{3^{t+1}}=(xy)^{3^t}=1, (xy)^{-1}\cdot x\cdot(xy)=x^{1+3^t}\rg, $$
and $S=\{1, x, y\}=\{1, x, x^{-1}(xy)\}$. So, $\Gamma$ is isomorphic to $\Gamma_t$ (see Construction~1).
If $(xy)^{-1}\cdot x\cdot(xy)=y^{-1}xy=x^{1-3^t}$, then
$$H=\lg x, (xy)^{-1}\ |\ x^{3^{t+1}}=[(xy)^{-1}]^{3^t}=1, (xy)^{}\cdot x\cdot(xy)^{-1}=x^{1+3^t}\rg, $$
and $S=\{1, x, y\}=\{1, x, x^{-1}[(xy)^{-1}]^{-1}\}$. By Proposition~\ref{bicayley}~(4), we have
$$\Gamma={\rm BiCay}(H, \emptyset, \emptyset, S)\cong {\rm BiCay}(H, \emptyset, \emptyset, S^{-1}).$$
Note that $S^{-1}=\{1, x^{-1}, y^{-1}\}=\{1, x^{-1}, (xy)^{-1}x\}$. It is easy to check that the map
$$f: x\mapsto x^{-1}, (xy)^{-1}\mapsto (xy)^{-1}x^{-3^t}$$
induces an automorphism of $H$ such that $\{1, x, x^{-1}(xy)^{-1}\}^f=S^{-1}$. By Proposition~\ref{bicayley}~(3), we have
$$\Gamma\cong{\rm BiCay}(H, \emptyset, \emptyset, S^{-1})\cong{\rm BiCay}(H, \emptyset, \emptyset, \{1, x, x^{-1}(xy)^{-1}\})\cong\Gamma_t,$$
as required.\medskip
\noindent{\bf Case 2}\ $H/H'=\lg aH'\rg\times\lg bH'\rg\cong \mathbb{Z}_{3^{t}}\times\mathbb{Z}_{3^{t-1}}$.\medskip
In this case, we have $s=t-1$. Let $T=\lg R(h)\ |\ h\in H, h^{3^{t-1}}=1\rg$. Then $T=\lg R(a)^9\rg\times\lg R(b)\rg$ and $T$ is characteristic in $R(H)$, and so normal in $A$ for $R(H)\unlhd A$. Furthermore, $R(H)/T\cong{\mathbb Z}_9$. By Propositions~\ref{3orbits} and \ref{intransitive}, the quotient graph $\Gamma_T$ of $\Gamma$ relative to $T$ is a cubic edge-transitive graph of order $18$. Clearly, $R(H)/T$ is semiregular on $V(\Gamma_{T})$ with two orbits, so $\Gamma_{T}$ is a bi-Cayley graph over the cyclic group $R(H)/T$ of order $9$. Since $R(H)/T\unlhd A/T$, by Proposition~\ref{abelian-biCayley}, there exists $\ld\in\mathbb{Z}_{3^2}^{*}$ such that ${\ld}^2-\ld+1\equiv 0\ ({\rm mod}\ 3^2)$, which is impossible.\medskip
\noindent{\bf Case 3} $H/H'=\lg aH'\rg\times\lg bH'\rg\cong \mathbb{Z}_{3^{t}}\times\mathbb{Z}_{3^{t+1}}$.\medskip
In this case, we have $s=t+1$. Let $N=\lg h\ |\ h\in H, h^{3}=1\rg$. Then $N=\lg a^{3^t}, b^{3^t}\rg\cong{\mathbb Z}_3\times{\mathbb Z}_3$.
By Claim~2, we have $o(x)=o(y)=3^{t+1}$. Since $H=\lg x, y\rg$, one has $N=\lg x^{3^t}, y^{3^t}\rg$. As $H'\cong{\mathbb Z}_3$, one has $H'\leq N$.
So, $H'=\lg x^{3^t}\rg$, $\lg y^{3^t}\rg$, $\lg (xy)^{3^t}\rg$ or $\lg (xy^{-1})^{3^t}\rg$.
Recall that $H$ has an automorphism $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ taking $(x, y)$ to $(x^{-1}y, x^{-1})$. Suppose that one of the three subgroups: $\lg x\rg, \lg y\rg, \lg x^{-1}y\rg$ is normal in $H$. Then all of them are normal in $H$. So $H=\lg x, y\rg=\lg x\rg\times\lg y\rg$ because $|H|=3^{2(t+1)}$. This is impossible because $H$ is non-abelian. Thus, all of the three subgroups: $\lg x\rg, \lg y\rg, \lg x^{-1}y\rg$ are not normal in $H$.
It then follows that $H'=\lg (xy)^{3^t}\rg$. Then either $x^{-1}(xy)x=(xy)^{1+3^t}$ or $x^{-1}(xy)x=(xy)^{1-3^t}$.
For the former, we have
$$H=\lg xy, x\ |\ (xy)^{3^{t+1}}=x^{3^{t+1}}=1, x^{-1}(xy)x=(xy)^{3^t+1}\rg,$$
and $S=\{1, x, y\}=\{1, x, x^{-1}(xy)\}$. Hence, $\Gamma\cong\Sigma_t$ (see Construction~2).
For the latter, we have
$$H=\lg xy, x^{-1}\ |\ (xy)^{3^{t+1}}=x^{-3^{t+1}}=1, x^{}(xy)x^{-1}=(xy)^{3^t+1}\rg,$$
and $S=\{1, x, y\}=\{1, (x^{-1})^{-1}, x^{-1}(xy)\}$. By Proposition~\ref{bicayley}~(4), we have
$$\Gamma={\rm BiCay}(H, \emptyset, \emptyset, S)\cong {\rm BiCay}(H, \emptyset, \emptyset, S^{-1}).$$
Note that $S^{-1}=\{1, x^{-1}, y^{-1}\}=\{1, x^{-1}, (xy)^{-1}x\}$. It is easy to check that the map
$$f': x^{-1}\mapsto x^{-1}, xy\mapsto (xy)^{3^t-1}$$
induces an automorphism of $H$ such that $\{1, x^{-1}, x(xy)\}^{f'}=S^{-1}$. By Proposition~\ref{bicayley}~(3), we have
$$\Gamma\cong{\rm BiCay}(H, \emptyset, \emptyset, S^{-1})\cong{\rm BiCay}(H, \emptyset, \emptyset, \{1, x^{-1}, x(xy)\})\cong\Sigma_t,$$
as required.\hfill\hskip10pt $\Box$\vspace{3mm}
\section{Proof of Corollary~\ref{cor2p3}}
Let $p$ be a prime, and let $\Gamma$ be a connected cubic edge-transitive graph of order $2p^3$. By \cite{2p}, the smallest semisymmetric graph has $20$ vertices. So, if $p=2$, then $\Gamma$ is vertex-transitive. If $p=3$, then by \cite{Sym768,Semisym768}, we know that $\Gamma$ is not vertex-transitive if and only if it isomorphic to the Gray graph.
Now assume that $p>3$. By Lemma~\ref{11}, $\Gamma$ is a bi-Cayley graph over a group $H$ of order $p^3$.
Suppose that $\Gamma$ is not vertex-transitive. Then $\Gamma$ is bipartite with the two orbits of $H$ as its two parts. So we may let $\Gamma={\rm BiCay}(H, \emptyset, \emptyset, S)$. By Proposition~\ref{bicayley}, we may assume that $S=\{1, a, b\}$ form $a,b\in H$. If $H$ is abelian, then $H$ has an automorphism $\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon$ which maps every element of $H$ to its inverse. By Proposition~\ref{bicayleyaut}, $\d_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,1,1}$ is an automorphism $\Gamma$ swapping the two parts of $\Gamma$, and so $\Gamma$ is vertex-transitive, a contradiction. If $H$ is non-abelian, then $H$ is either metacyclic or isomorphic to the following group:
$$J=\lg a,b,c\ |\ a^p=b^p=c^p=1, c=[a,b], [a,c]=[b,c]=1\rg.$$
By Theorem~\ref{5no}, $H$ is non-metacyclic. If $H\cong J$, then it is easy to see that $J$ has an automorphism taking $(a, b)$ to $(a^{-1}, b^{-1})$. Again, by Proposition~\ref{bicayleyaut}, $\d_{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} \def\e{\varepsilon,1,1}$ is an automorphism of $\Gamma$ swapping the two parts of $\Gamma$, and so $\Gamma$ is vertex-transitive, a contradiction. This completes the proof of Corollary~\ref{cor2p3}.
\medskip
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2110.15157
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\section{Achieving Low Latency}\label{sec:background}
We first look at the causes for long tail latency in DCN ($\S$\ref{sec:background:cause}). Then, we review the prior solutions to cut tail latency ($\S$\ref{sec:background:prior}). Finally, we overview CloudBurst\xspace ($\S$\ref{sec:background:lowl}).
\subsection{Causes of long tail latency}
\label{sec:background:cause}
The traffic characteristics of DC applications, current DCN configurations, and the usage of TCP in DCNs jointly contribute to the long tail latency for short flows
\textbf{High fan-in bursts (incast~\cite{incast}):} Tree-based applications emit synchronized high fan-in bursts. They are constructed with multiple layers, and each parent aggregates results from its children. The interactivity constraint (e.g. 300ms\cite{d2tcp}) is distributed to each layer (e.g. 40ms). Hence children of a same parent respond at almost the same time, causing fan-in burst at the parent. These synchronized fan-in bursts exceed the egress port capacity of switches, causing queue to build up and may lead to congestive packet drops.
\textbf{Shallow shared-buffer switch:} Shared buffer is usually configured to absorb bursts, i.e. a portion of buffer is shared among multiple ports, so if one port is experiencing bursts, it can take all available shared buffer. However, shallow buffer switches cannot absorb fan-in bursts from applications at high bandwidth. The expansion of buffer (from 4MB to 12MB~\cite{dc-buffer}) in commodity switches is not compatible with the 10X$+$ growth of per port bandwidth (from 1Gbps to 10/40Gbps~\cite{google-dcn, gemini}). Additionally, throughput-intensive flows requires a certain amount of buffering
($\alpha \times$Bandwidth Delay Product, $\alpha$ depends on the transport protocol~\cite{dctcp}). in the switch to achieve high throughput, requiring guaranteed buffer in each port and further reducing the shared buffer for burst tolerance. When bursts occurs, packets from latency-sensitive flows may be dropped due to lack of buffer~\cite{incast}.
\textbf{Error handling and retransmission timeout:} Applications deliver data reliably using TCP with Automatic Retransmission Request (ARQ) error handling (e.g., Go-back-N~\cite{gobackn}, Selective Repeat~\cite{tcpsack}), and timeout-based error discovery). For each error, ARQ needs at least one RTT to recover, thus it works well for long flows, because acknowledgement of received or missed packets are batched for large congestion windows. However, ARQ does not help for short flows, which may finish within the slow start phase. If a flow's first few packets coincide with congestion and get dropped, it can only discover the loss by RTO. Therefore, setting a proper RTO$_{min}$~\cite{detail,incast} is key to fast recovery of short flows. RTO$_{min}$ in DCN is usually set to 5ms~\cite{incast,fuso}, which is almost two orders of magnitude larger than the base RTT ($\sim100$us). A single timeout can easily lead to long latency tail. Reducing the timeout can surely benefit packet recovery, but may also increase the load on network and servers due to frequent retransmissions and require high-precision timers.
\textbf{Malfunctioning hardware:} Packet loss may also occur due to hardware failures, which happens even in well-engineered modern DCNs with lossless fabric~\cite{fuso,scalerdma,pingmesh}. Such failures may come from TCAM deficits, aging transceivers, etc. Particularly, silent packet drops or blackholes are discovered with network-wide diagnostic tools~\cite{pingmesh}. Thus, these failures are difficult for single-path transport to recover, and can take multiple RTTs for multipath transport to recover.
\textbf{Imperfect flow load balancing (LB):} Current load balancing in commodity DCNs usually depends on flow hashing, i.e., per-flow ECMP~\cite{ecmp}, to keep utilization of parallel paths even. However, hash collision may occur, which can temporarily overload some links and cause queue length to grow, prolonging per-packet latency and inducing packets drops. Randomized LB also cannot help with application bursts when all flows have the same destination port.
\subsection{Prior Solutions}\label{sec:background:prior}
We overview the prior solutions to the tail latency problem:
\textbf{Reducing queueing latency:} Generally, solutions in this category follow a few strategies such as fine-grained load balancing~\cite{conga,fastpass,detail,hermes}, rate control~\cite{dctcp,d2tcp,hpcc,dcqcn,timely,swift,dcn-transport,mcp,mqecn,tcn,ecn-sharp}, and traffic prioritization~\cite{pfabric,pias-nsdi,pase,detail,qjump,ras,karuna}.
\begin{itemize}
\item By load balancing traffic across multiple paths evenly, we can avoid excessive queue build-up that may result in queueing latency or loss. For better performance, congestion awareness is often required~\cite{conga,detail,fastpass,hermes}. For example, CONGA~\cite{conga} measures link utilization and directs flows to less congested paths, while DeTail~\cite{detail} detects and avoids congestion by monitoring queue lengths. In the extreme, Fastpass~\cite{fastpass} centrally schedules and routes every packet with complete knowledge of per-path load conditions. These solutions, however, require very complex network control or switch modifications.
\item By rate-throttling flows (especially the larger ones) based on ECN~\cite{ecn}, delay~\cite{ecn-rtt}, or in-band network telemetry (INT)~\cite{hpcc} signal, we can control the queue build-up so that latency-sensitive short flows see small queues at the switch.
While helpful, these solutions~\cite{dctcp, d2tcp, dcqcn, timely, hpcc, swift} still rely on load-balancing, and short flows may suffer long latency when traffic is unevenly distributed. Besides, the advanced INT signal may not even be available on switches and thus require customized hardware.
\item By giving latency-sensitive flows high priority, the switch dequeues them first, without regards to lower priority ones before them, thus achieving low latency~\cite{pias-nsdi, pfabric, pase, detail, qjump}. For example, pFabric~\cite{pfabric} assumes priority dequeueing (and dropping) to minimize flow completion time of short flows, and QJump~\cite{qjump} leverages prioritization to cut tail latency. However, they either assume infinite switch queues or rely on accurate configurations.
\end{itemize}
\textbf{Recovering from packet losses:} Fast in-network feedback for packet drops~\cite{cp,ndp} can accelerate the transition of TCP state machine, thus triggering retransmission earlier. As above, these solutions also require non-trivial hardware modifications. Retransmission can also be proactive: FUSO~\cite{fuso} augments MPTCP~\cite{mptcp,mptcp-dcn} by eagerly retransmitting on less congested paths. However, it needs to keep complex states of subflows.
Another line of work~\cite{dcqcn,timely,pcn} seeks help from lossless fabric~\cite{pfc}. However, at large scale, packet loss may still happen even on lossless fabric, due to mis-configuration or hardware failures~\cite{scalerdma}. In case of losses, they often rely on the NIC hardware for efficient recovery. This line of work is still under exploration and is beyond the scope of this paper.
\textbf{Proactive transport solutions:} Some recent works~\cite{expresspass,homa,ndp} use pre-allocated rate to avoid excessive packet delay and inaccurate self-adapted rate control. In these solutions, link capacities are {\em proactively} allocated by the receivers as ``credits'' to each active sender who then send ``scheduled packets'' at an optimal rate to ensure low queueing delay and near-zero packet loss. However, this approach requires at least one RTT to allocate credits to a new flow, which is unacceptable for short flows.
While recent augmentation scheme~\cite{aeolus} enables line rate start, issues still exist for short flows. For example, it is hard to assign a right amount of credits to senders before termination in the last RTT~\cite{flashpass}---too large for short flows will lead to link under-utilization due to credit wastage, while too small will introduce large delay.
\subsection{Cutting tail latency via FEC over multipath (Overview)}\label{sec:background:lowl}
CloudBurst\xspace aims to cut the long tail latency with a simple and readily deployable protocol. It applies FEC to multiple paths ($\S$\ref{sec:fec}) in commodity datacenters. By proactively spreading encoded packets over multiple paths in parallel, and decoding the first few arriving ones to recover the original message, CloudBurst\xspace obliviously exploits the uncongested paths to achieve persistent low latency, while not maintaining network state or performing complex control.
At the end-host, CloudBurst\xspace performs ``burst-until-received'' ($\S$\ref{sec:protocol}). The sender encodes short messages with FEC and proactively sends the encoded packets over multipaths until the messages are decoded at the receiver with the first few arriving packets. In this way, CloudBurst\xspace always exploits the best paths for low latency. If any uncongested path exists, CloudBurst\xspace uses it without signaling; if a path is experiencing congestion or failure, CloudBurst\xspace avoids it automatically by using encoded packets from other under-utilized paths.
At the switch, CloudBurst\xspace performs aggressive dropping ($\S$\ref{sec:aggdrop}) for CloudBurst\xspace flows, We limit the buffer usage of the CloudBurst\xspace flows to minimum, i.e., separating CloudBurst\xspace traffic and other traffic in different queues, and limiting the maximum depth of the CloudBurst\xspace queue to a small value. This ensures that the per-hop queueing latency is deterministically low. Besides, it protects other flows from being affected by the burst-until-received behavior of CloudBurst\xspace flows, making it friendly to co-existing traffic.
\iffalse
\textbf{Why CloudBurst\xspace works?} The key insight of CloudBurst\xspace is to discard all packets which observe a queue build-up, and construct the message with packets which experience no queueing.
Because, from the latency-sensitive application's point of view, delayed packets are equally useless as lost packets, and only the non-delayed packets are useful.
The tiny queue of CloudBurst\xspace filters out all delayed packets, and while doing so, CloudBurst\xspace maximizes the number of useful packets (i.e., non-delayed packets) delivered per unit time via multipath forwarding and proactive transmission. This approach differs drastically from today's de facto approach (i.e., TCP$+$buffering), which tries to minimize the number of packets needed to deliver a message.
Compared to the de-facto approach (TCP$+$buffering), which tries to minimize the number of packets needed to deliver a message, CloudBurst\xspace minimizes the queueing delay experienced when delivering a message.
CloudBurst\xspace works in practice due to the heavy-tailed DCN traffic~\cite{dctraffic09,dctraffic10}: most flows are small, but the majority of volume is from large flows~\cite{dctcp}. The network in DC is also underutilized~\cite{heracles}, at 80\% of the time the 95th percentile utilization for each link is less than 30\%~\cite{dctraffic09}. CloudBurst\xspace, by creating redundancy among packets using FEC, obliviously exploits the under-utilization, and automatically avoids congestion caused by a few long flows or failures caused by malfunctioning devices.
CloudBurst\xspace is also incrementally deployable as a user-space library in existing DCNs along side TCP traffic. Applications decide whether their short flows (messages) need low latency, and voluntarily use the library for the transmission.
\fi
\section{Discussion}\label{sec:discuss}
In this section, we address existing issues, and discuss concerns and further improvement of CloudBurst\xspace.
\textbf{Hardware en/decoding:} The overhead of CloudBurst\xspace can be further reduced if the en/decoding is offloaded to hardware (e.g., programmable NIC). The encoding is constant time (given degree $d$), but the decoding using Gaussian elimination has $O(k^2)$ time complexity ($k$ is the number of packets in original message). Although $k$ is always small for CloudBurst\xspace, offloading coding to dedicated hardware still significantly reduces the stress on end-hosts. In future high-speed networks, hardware offload is inevitable, because the computational load to en/decode at 100/400Gbps line rate will overburden the CPUs. We leave design of such hardware as future work.
\textbf{Using other rateless codes:}
Coding scheme is not closely coupled with CloudBurst\xspace, and other rateless codes can work as well (with modification to the header format in Figure~\ref{fig:header}).
LTC is chosen for its simplicity, so that we can quickly validate our idea of coding over multipath.
We note that more efficient coding schemes, such as Raptor codes~\cite{raptor} which have linear en/decoding complexity, are good alternatives.
\textbf{Fairness among CloudBurst\xspace flows:} CloudBurst\xspace achieves fairness with aggressive dropping with tiny buffers, because dropping is a form of congestion control~\cite{decongestion} for erasure-coded flows. It is shown that, if congested link drop packets in a fair manner~\cite{diffdrop}, each flow will receive its max-min fair throughput. For a network with purely erasure coded flows, the switches do not need deep buffers to keep network stable~\cite{nocc}, and the sources can greedily send as fast as possible~\cite{decongestion}.
\textbf{CloudBurst\xspace for long flows:}
Coding mechanisms provide reliability at the cost of redundancy, so we carefully limit CloudBurst\xspace to short messages. For decoding to work, the number of packets in the original message should be received first. The bit-set indicating which of the original packets are included in the received encoded packet is also necessary for the decoding process. We put all the necessary in the header (Figure~\ref{fig:header}), so the decoding process can start no matter which encoded packet arrives first. The header overhead increases with the number of packets in a message. Since the header size must be less than MTU, the MTU of the network therefore limits the number of packets in a message.
Using rateless codes for long flows, the coding rate can be unbounded, and the resulting redundancy can overload the network fabric. Consider the situation when long flows using CloudBurst\xspace collide with each other on multiple paths. The long flow will repeatedly send out redundancy packets trying to overcome the poor channel condition, which worsens congestion gravely.
Splitting a long flow into several small CloudBurst\xspace flows is a work-around. In this way, a long flow is segmented into a series of short flows, and CloudBurst\xspace delivers each segment.
\textbf{CloudBurst\xspace with programmable data plane:}
With advanced data plane, we envision the following improvements to CloudBurst\xspace: (1) \textit{True zero queueing delay} for CloudBurst\xspace packet can be achieved by dropping incoming CloudBurst\xspace packets whenever a CloudBurst\xspace packet is in buffer. Current buffer limit (2\%) is needed to maintain full port throughput, as setting it to 0 drops all packets.
(2) Instead of dropping based on each switch's local queue limit, we can make \textit{per-packet dropping decisions}, because packets themselves can accumulate the queueing latency values over multiple hops and let intermediate switches drop packets when the cumulative queueing latency (i.e., experienced path latency) exceeds a threshold.
Per-packet latency can be recorded in a field in the packet header, and the packet can be dropped if its experienced delay exceeds a threshold.
Dropping these packets will reduce the load on later switches on the path, and give more opportunity for other packets to go through.
\section{Evaluation}\label{sec:eval}\label{sec:eval:setting}
In this section, we evaluate CloudBurst\xspace with testbed experiments ($\S$\ref{sec:basicdesign}--$\S$\ref{sec:deepdive}), complemented by large-scale simulations ($\S$\ref{sec:simulations}).
We summarize the results below:
\begin{itemize}
\item $\S$\ref{sec:basicdesign}: we inspect CloudBurst\xspace's design choices and quantify their benefits.
With all choices combined, CloudBurst\xspace reduces the tail latency by $75.32$\% compared to DCTCP.
\item $\S$\ref{sec:eval:latency}: we compare CloudBurst\xspace with the prior practical schemes, and find that it achieves more than $60.06\%$ reduction in p99 flow completion time compared to DCTCP+ECMP~\cite{dctcp,ecmp} or PIAS~\cite{pias-nsdi}.
\item $\S$\ref{sec:deepdive}: we dive into CloudBurst\xspace and find that it is resilient to many critical cases including incast and failures.
\item $\S$\ref{sec:simulations}: we use large-scale simulations to show that CloudBurst\xspace achieves $24\%$ tail latency reduction compared to a near-optimal clean-slate design, pFabric~\cite{pfabric}.
\end{itemize}
\begin{figure}[t]
\centering
\vspace{-1em}
\includegraphics[width=\linewidth]{figs/testbed.png}\\
\vspace{-2em}
\caption{Testbed setup}\label{fig:testbed}
\vspace{-1em}
\end{figure}
\textbf{Traffic patterns:} Following related works~\cite{detail}, we emulate traffic of latency-sensitive applications: data retrieval (request/response) and page generation.
Each response message is triggered by a 1.5K-byte (MTU) request from the receiver to the senders. The senders reply with a message with variable size uniformly chosen from $\{5, 10, 20, 50, 93\}$KB.
The inter-arrival time of initiating requests follows exponential distribution with mean $4,5,10$ms (A Poisson random process with arrival rate $250,200,100$ requests per server per second).
Sender-receiver selection is as follows:
\begin{itemize}
\item \textit{All-to-all request/response (by default):} Each server randomly picks another server to send request.
\item \textit{Front/Back-end page generation:} The two racks are designated as Front-end and Back-end. Front-end servers request data from a randomly selected Back-end server.
\end{itemize}
\textbf{Testbed:} We built a spine-leaf (8 servers, 2 leaf or Top-of-rack \& 4 spine switches) testbed to create 4 paths between any pair of servers from 2 racks (Figure~\ref{fig:testbed}). Each path corresponds to a spine switch. We use Pronto-3295 switches and Dell PowerEdge R320 servers, each with a quad core Xeons E5-1410 CPU and 1GbE NIC, and with Debian 6.0 (kernel 2.6.32-5) installed. XPath~\cite{xpath} is enabled by default.
We generate background flows to create network congestion.
For the 4 servers on one rack, each randomly chooses another server in the other rack, and sends a flow of 10MB using DCTCP on a random path. When a flow finishes, the server will start another one.
In this way, each path has the same probability for different degrees of congestion ($1/256, 3/64, 27/128, 27/64, 81/256$ chance to have $0, 1, 2, 3, 4$ flows, respectively).
Unless specified otherwise, background flows are in a separate queue (totally 2 queues are used), and switches use WRR for these 2 queues.
\subsection{Inspecting Design Choices}\label{sec:basicdesign}
CloudBurst\xspace incorporates three design choices:
1) CloudBurst\xspace uses LTC to encode the message, and runs the "burst-until-received" protocol; 2) CloudBurst\xspace spreads the encoded packets on multiple paths, obliviously taking advantage of uncongested paths; 3) the switches perform aggressive dropping with tiny queues for CloudBurst\xspace flows. We now study the impact of each of these decisions progressively.
We run the all-to-all pattern with varying request rates (each for 10 minutes), and plot the p99 completion times for $\{5,20,93\}$KB flows in Figure~\ref{fig:pro:5},\ref{fig:pro:20}\&\ref{fig:pro:93}, respectively.
We take DCTCP as a reference (parameter setting follows $\S$\ref{sec:eval:latency}). We compare the following schemes:
\begin{itemize}
\item \textbf{A}: FEC (Design Choice 1). We encode the message into encoded packets, and send the packets at line rate using UDP on a randomly chosen single path.
\item \textbf{B}: \textbf{A} + Multipath (Design Choice 1\&2). Senders spread the encoded packets of each flow on multiple paths.
\item \textbf{C}: \textbf{A} + Aggressive Dropping (Design Choice 1\&3). Senders send encoded packets on a single path, and switches aggressively drop packets by limiting buffering.
\item \textbf{D}: CloudBurst\xspace (Design Choice 1,2,\&3). CloudBurst\xspace combines all three design choices.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/pro-all-to-all-5.eps}\\
\vspace{-1em}
\caption{p99 Completion Time (5KB)}\label{fig:pro:5}
\includegraphics[width=\linewidth]{figs/pro-all-to-all-20.eps}\\
\vspace{-1em}
\caption{p99 Completion Time (20KB)}\label{fig:pro:20}
\includegraphics[width=\linewidth]{figs/pro-all-to-all-93.eps}\\
\vspace{-1em}
\caption{p99 Completion Time (93KB)}\label{fig:pro:93}
\vspace{-1em}
\end{figure}
\textbf{Impact of FEC:}
CloudBurst\xspace uses FEC to perform loss recovery. Consider a simple mathematical model: message size is $M$ packets, and link capacity is $C$ packet/s. Assume the packet drop probability is $p_d$ on a path. If a packet is lost, a sender transmitting at link capacity without coding takes $\frac{M}{C}$s to retransmit it, and the expected latency is $E_D=(1-p_d)\frac{M}{C}\sum_{i=0}^{\infty} (i+1)p_d^i =\frac{M}{(1-p_d)C}$. Thus, with a larger $p_d$ (aggressive dropping), $D$ takes more time to recover a lost packet. Encoding essentially sends multiple packets' information at the same time. For degree $d$, $d$ original packets are XOR'd for each encoded packet. In this way, a packet do not have to wait for $\frac{M}{C}$ to be retransmitted. For each encoded packet, the sender randomly chooses $d$ packets, thus it takes $\frac{M}{dC}$ to deliver all the packet's information. Therefore, the time to receive information of all packets becomes $\frac{M}{d(1-p_d)C}$, $d$ times smaller than $E_D$.
However, FEC alone cannot work. Scheme \textbf{A} keeps generating encoded packets, and sends them on a single path. We see that, the tail latency is $51.12\%$ longer compared to DCTCP across all message sizes at $2000$ Request per second (rps). This is caused by the queueing created by the aggressive sending of encoded packets.
In contrast, DCTCP controls the queueing by enforcing a small queue at the switch, which allows the end-hosts to react to congestion quickly.
\textbf{Impact of Multipath:} Queueing is created when background flows choose the same path due to imperfect load balancing.
Good load balancing is difficult to achieve for TCP flows, because in-order packet delivery is expected.
However, the same is not true for erasure-coded flows, as there is no order between encoded packets and the original message is reconstructed as a whole. This allows for easy implementation of load balancing: the sender can simply spray encoded packets on multiple paths. Scheme \textbf{B} does exactly this. We observe that, with evenly load balanced FEC flows, the tail latency is improved by $44.52\%$ for $5$KB messages across all loads. This improvement is smaller with increasing message size ($23.88\%$ for $93$KB message), because larger messages take longer to decode (Figure~\ref{fig:coding-overhead}). Compared to DCTCP, we see the same trend: \textbf{B} outperforms DCTCP for short messages (tail latency is $16.24\%$ shorter for $5$KB) and sightly worse for long messages (tail latency is $5.89\%$ longer for $93$KB). This is because Scheme \textbf{B} spreads traffic to all available paths evenly, while DCTCP is vulnerable to imperfect load balancing.
\textbf{Impact of Aggressive Dropping:}\label{sec:eval:drop}
To counter \textbf{A}'s queueing, we can also limit the queue depth. Scheme \textbf{C} uses FEC and aggressive dropping on a single path. In Figure~\ref{fig:pro:20}\&\ref{fig:pro:93}, \textbf{C} is similar to DCTCP for $20$KB and $93$KB messages. For $5$KB short messages, \textbf{C}'s tail latency is $51.42\%$ shorter than that of DCTCP, because DCTCP relies on timeouts to discover packet loss for short messages. In contrast, \textbf{C} proactively retransmits encoded packets despite dropping.
\textbf{Summary:} With all three design points, we have CloudBurst\xspace. With erasure coding (Design Choice 1), each encoded packet can help recover any of the $d$ original packets. With multipath forwarding (Design Choice 2), packets are spread on all paths evenly, thus exploiting uncongested paths obliviously. Finally, aggressive dropping (Design Choice 3) ensures that packets arriving at the receiver experience deterministically low queueing delay, because packets that encounter any queue build-up are dropped. Compared to DCTCP, CloudBurst\xspace reduces the tail latency by $75.32\%$ (averaged over all sizes).
\subsection{Comparing with prior schemes}\label{sec:eval:latency}
We proceed to compare CloudBurst\xspace with existing schemes that are implementable in commodity DCNs:
\begin{itemize}
\item \textbf{DCTCP~\cite{dctcp} + ECMP:} RTO$_{min}$ is set to 10ms. ECN marking threshold is set to 65 packets. All flows share the same switch queue.
\item \textbf{PIAS~\cite{pias-nsdi}:} We implemented PIAS with 2-level feedback queue and set the first threshold to be 95KB, so that all short flows or messages have highest priority.
\item \textbf{Replicated DCTCP:} Transmitting flows with same content using DCTCP on 2-4 paths~\cite{redundancy} (paths are randomly chosen). All flows share the same switch queue.
\item \textbf{MPTCP~\cite{mptcp,mptcp-dcn}:} Using tc in Linux, MPTCP flows are tagged with DSCP value for the short flow queue.
\end{itemize}
We run the all-to-all request/response traffic pattern, and collect message completion time (MCT) for each scheme.
\textbf{Average Latency:} In Figure~\ref{fig:eval:avg}, despite the en/decoding overheads, CloudBurst\xspace performs similarly to PIAS for different request arrival rates in terms of average MCT. MPTCP shows the worst performance, because if any congested path exists, MPTCP is bound to experience congestion, prolonging MCT. Among the DCTCP-based schemes, for low request rate (800 r/s), DCTCP with the most duplicated flows (4) achieves the best performance, as it transmits on all paths, thus can always avoid congestion. However, as the request rate increases, the performance of Replicated DCTCP starts to degrade, because the replication essentially multiplies the load, leading to congestion and packet drops.
\textbf{Tail Latency:} In Figure~\ref{fig:eval:99}, for p99 MCT, CloudBurst\xspace outperforms all the other schemes. The p99 MCT captures the tail latency events (e.g. long queueing, packet loss). At $2000$ r/s, its tail latency is over $60.06\%$ ($63.69\%$) less than that of PIAS (DCTCP$+$ECMP). This is because CloudBurst\xspace packets are encoded with redundancy, thus flows need not wait for retransmission timeout, unlike TCP variants.
\begin{figure}[t]
\centering
\vspace{-0.5em}
\includegraphics[width=\linewidth]{figs/avg.eps}\\
\vspace{-1em}
\caption{Average Message Completion Time}\label{fig:eval:avg}
\includegraphics[width=\linewidth]{figs/t99.eps}\\
\vspace{-1em}
\caption{p99 Message Completion Time}\label{fig:eval:99}
\vspace{1em}
\resizebox{0.8\linewidth}{!}{
\centering
\begin{tabular}{l|lll|l}
& 800 r/s & 1600 r/s & 2000 r/s & Average \\ \hline
5KB & 1.221 & 1.241 & 1.381 & 1.281\\
10KB & 1.478 & 1.133 & 1.367 & 1.331\\
20KB & 1.398 & 1.972 & 1.875 & 1.748\\
50KB & 1.167 & 1.311 & 1.643 & 1.373\\
93KB & 1.542 & 1.928 & 1.676 & 1.715\\ \hline
Average & 1.361 & 1.517 & 1.591 & 1.490
\end{tabular}
}
\caption{Coding rates of experiments in Figure~\ref{fig:eval:avg},\ref{fig:eval:99}}\label{eval:table}
\vspace{-1.5em}
\end{figure}
\textbf{Coding Overhead:} In Table~\ref{eval:table}, we list the coding rates in Figure~\ref{fig:eval:avg}\&\ref{fig:eval:99}, which is the ratio of the number of sent packets over the number of packets in the original message. The average coding rate is $1.49$, i.e. CloudBurst\xspace adds $\sim 49\%$ more traffic load in the experiments.
\subsection{CloudBurst\xspace Deep Dive}\label{sec:deepdive}
In this section, we use a series of targeted experiments to further understand CloudBurst\xspace.
\textbf{Impact of incast:}\label{sec:eval:incast}
We first examine the incast scenario. We have 4 senders on one rack sending 90KB (60 pkts) messages to a receiver on the other rack, and set receiver's ToR switch buffer size to 100KB (with no traffic to other ports, the total switch buffer size for this port is the sum of shared and dedicated buffer). The flows start at the same time and are evenly distributed among the senders. We increase the number of concurrent flows, $N$, and measure the time from the start to the last flow.
Figure~\ref{fig:eval:incast} shows the average flow completion times (FCT) with the increase of $N$. We find that, as $N$ grows larger, the performances of different schemes start to diverge, and DCTCP-based schemes (including PIAS) start to have increasingly longer FCT. In contrast, CloudBurst\xspace shows consistently low FCT under incast, and its FCT grows almost linearly with $N$. The key reason is that, unlike TCP, CloudBurst\xspace's aggressive burst-until-received protocol does not require a timeout to discover packet loss, which is bound to happen in incast. A CloudBurst\xspace flow in an incast proactively retransmits without need to discover a packet loss.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{figs/incast.eps}\\
\vspace{-1em}
\caption{Incast: Average Completion Time}\label{fig:eval:incast}
\vspace{0.5em}
\includegraphics[width=\linewidth]{figs/imp.eps}\\
\vspace{-1em}
\caption{Failure: p99 Completion Time}\label{fig:eval:imp}
\vspace{-1em}
\end{figure}
\textbf{Impact of failures:}\label{sec:eval:multipath}
When link/switch failures happen, some paths may become unavailable.
We evaluate how CloudBurst\xspace handles such scenario. We vary the number of failed paths, and compare p99 FCT.
As shown in Figure~\ref{fig:eval:imp}, with increasing number of failed paths, the p99 FCT increases gradually: for $2000$rps, from $5412\mu$s to $6907\mu$s. As expected, the scenario with the most failed paths performs the worst. This is because, when there is only one path, the tiny buffer on the switch may drop many packets on this path, as CloudBurst\xspace senders continuously resend encoded packets on this path to recover the drops. This shows CloudBurst\xspace is sensitive to the existence of multipath, but not the number of available paths.
\textbf{Impact to other traffic:}\label{sec:eval:friend}
In this experiment, we assess the impact of CloudBurst\xspace flows to other traffic.
We run the Front/Back-end traffic pattern with CloudBurst\xspace for 5 minutes. We initiate 4 DCTCP background flows from Back-end rack to Front-end rack on the 4 paths using iperf, each flow between a different pair of servers. We then increase the bandwidth demand of the CloudBurst\xspace flows from $1.424$Mbps per server to $284.4$Mbps (by increasing the request arrival rate from $100$rps to $20000$rp
; For arrival rate $\geq 1000$rps, each request initiate $i$ responses, $i=1,4,7,10,15,18,20$.). On the switch, we configure WRR with weight 4 for background traffic, and 1 for CloudBurst\xspace traffic, so that the bandwidth limit for CloudBurst\xspace flows is $200$Mbps, smaller than the largest demands. We set the transmission timeout for CloudBurst\xspace to be $10$ms (the sender aborts the flow if it is not completed before timeout). We examine the average throughput of background and CloudBurst\xspace flows
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/friend.eps}
\vspace{-2em}
\caption{Friendliness: background flow throughput}\label{fig:eval:friend}
\vspace{2em}
\resizebox{\linewidth}{!}{
\centering
\begin{tabular}{l|l|l|l|l|l}
Demand (Mbps) & 99.68 & 142.4 & 213.6 & 256.32 & 284.8 \\ \hline
Completion \% & 100 & 100 & 43.11 & 29.24 & 12.83 \\
\end{tabular}
}\caption{Completion Percentages in Figure~\ref{fig:eval:friend}}\label{eval:friend:table}
\end{figure}
Figure~\ref{fig:eval:friend} shows that CloudBurst\xspace shares the bandwidth with DCTCP flows. The dotted line shows the demand of the traffic pattern, and the purple dash line is the bandwidth limit for short flows. We see that DCTCP flows are able to maintain high throughput in the presence of CloudBurst\xspace flows, owing to the minimal buffer usage of CloudBurst\xspace as well as the guaranteed bandwidth from WRR in the switch.
\textbf{Encoding/decoding complexity:}\label{sec:eval:overhead}
We measure the en/decoding latencies for varying message sizes by running CloudBurst\xspace on network without background flows.
In Figure~\ref{fig:eval:encdec}, we show that, with increasing number of packets, the encoding overhead remains constant, while the decoding time increases, which indicates that CloudBurst\xspace has most to gain by reducing decoding complexity.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/encdec-time.eps}\\
\caption{En/decoding time}\label{fig:eval:encdec}
\end{figure}
\subsection{Large-scale Simulations}\label{sec:simulations}
We complement the testbed experiments by simulating CloudBurst\xspace in a large DCN (Figure~\ref{fig:eval:simtopo}). We use a leaf-spine topology with 144 hosts, 9 leaf (ToR) switches, and 4 spine (Core) switches. Each leaf switch has $16\times$ 10Gbps downlinks and $4\times$ 40Gbps uplinks to the spine. The base RTT across the spine (4 hops) is 20$\mu$s. We generate the all-to-all traffic workload as above, and fix the traffic load to $2000$rps.
We implemented CloudBurst\xspace based on ns-2 simulation of QJump~\cite{qjumpns2}. The en/decoding times of CloudBurst\xspace are obtained from experiments (not shown due to space limitation), and are added to CloudBurst\xspace's FCT measurement. We compare CloudBurst\xspace with the following:
\begin{itemize}
\item \textbf{DCTCP~\cite{dctcp}:} We configure DCTCP with the recommended ECN marking threshold of 65 packets.
\item \textbf{pFabric~\cite{pfabric}:} Queue size is 34 packets ($2\times$BDP), initial window is 17 packets, and RTO$_{min}$ is 1ms.
\item \textbf{QJump~\cite{qjump}:} Based on topology, we configure the minimum bandwidth $R=10$Gbps, cumulative switching delay $\epsilon=4$us, $P=MTU=1.5$KB. For messages, we set the throughput factor $f=1$ (guaranteed latency); for background flows, we set $f=n$ (maximum throughput. $n=144$, the total number of end-hosts).
\item \textbf{Expresspass\cite{expresspass}:} For credit packet, the packet size is 84 bytes and the queue size is 10 packets; for data packet, the packet size is 1538 bytes and the queue size is 100 packets. The credit rate is 500Mbps.
\end{itemize}
\textbf{Tail latency:}
We first compare the tail latency. Figure~\ref{fig:eval:over} shows p99 FCT. For small message size ($5$KB), the difference is insignificant. As size grows larger, CloudBurst\xspace begins to show its advantage over the others. The main reason is that TCP-like schemes takes at least RTO$_{min}$ to recover loss. Since packet loss is captured by the tail latency, CloudBurst\xspace outperforms DCTCP, pFabric, and QJump by $40.12\%$, $24\%$, and $29.63\%$, respectively. To our surprise, our result shows that Expresspass performs the worst among all the schemes. We imagine there are two main reasons: 1) it requires one additional RTT for credit allocation, and 2) such credit-based algorithm is fundamentally not suitable for mice flows in a dynamically changing environment, because it is very hard to set up an appropriate amount of credits for short-live flows.
\begin{figure}[t]
\centering
\vspace{-0.5em}
\includegraphics[width=\linewidth]{figs/simtopo.png}\\
\vspace{-2em}
\caption{Simulation Topology}\label{fig:eval:simtopo}
\vspace{-1em}
\end{figure}
\textbf{Impact of over-subscription:}\label{sec:eval:oversub}
We further examine how CloudBurst\xspace performs in over-subscribed networks. We reduce all ToR-to-Core links' capacity to 20Gbps, creating a network with 2:1 over-subscription. We run this experiment for DCTCP and CloudBurst\xspace. For CloudBurst\xspace,
we vary $r$, the rate limit in Algorithm~\ref{algo:send}, from $0.1$ (sending at $1$Gbps) to $1$ ($10$Gbps), and collect MCTs.
We summarize the results in Figure~\ref{fig:eval:r-over} normalized to DCTCP.
We find that, for varying sizes, choosing $r$ close to the over-subscription ratio results in higher tail latency reduction. If $r$ is too small, the sender is not sending enough to compensate for the aggressive dropping, and the message can take longer to finish. If $r$ is too large, the senders may overload the network and cause more frequent packet drops, which is bad for longer messages.
Overall, when $r$ is chosen appropriately ($\sim0.5$), the MCT reduction is $>12.88\%$ for all message sizes.
Therefore, we suggest setting the sending rate $r$ (in Algorithm~\ref{algo:send}) to the over-subscription ratio.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/ns2-compare.eps}\\
\vspace{-1em}
\caption{p99 Completion Time}\label{fig:eval:over}
\vspace{1em}
\includegraphics[width=\linewidth]{figs/r-oversub.png}\\
\vspace{-1em}
\caption{Choosing $r$ in 2:1 over-subscribed network}\label{fig:eval:r-over}
\vspace{1em}
\includegraphics[width=\linewidth]{figs/fail.eps}\\
\vspace{-1em}
\caption{Two failure scenarios}\label{fig:eval:fail}
\end{figure}
\textbf{Impact of failures:}\label{sec:eval:fail}
Finally, we examine two failure scenarios at large scale.
We run all-to-all pattern. For link failure scenarios, we randomly remove one of the Core-ToR links.
For link degradation scenarios, we randomly decrease one Core-ToR link's capacity to 1Gbps.
Figure~\ref{fig:eval:fail} shows both scenarios, and the metric is also reduction percentage compared to DCTCP.
We observe that, for both scenarios, CloudBurst\xspace is tolerant of failures, and can achieve reduction similar to that in Figure~\ref{fig:eval:over}. This is due to CloudBurst\xspace's inherent randomness. As discussed in $\S$\ref{sec:eval:multipath}, CloudBurst\xspace benefits from multipath, but is not sensitive to the number of available paths.
\section{Conclusion}\label{sec:conclusion}
We present the design, implementation, and evaluation of CloudBurst\xspace ~--- a simple scheme to cut long tail latency of message delivery by proactively sending FEC-coded packets generated from the messages on multipath in parallel, thus obliviously exploiting the uncongested paths without complexities like prior solutions. CloudBurst\xspace is readily deployable in today's commodity DCNs. We implemented a CloudBurst\xspace prototype, and validated its performance with extensive testbed experiments as well as large-scale simulations.
\section{Design}\label{sec:design}
We proceed to describe the details of CloudBurst\xspace design.
\subsection{Choosing FEC for encoding}\label{sec:fec}
Figure~\ref{fig:blockcode} is a simplified illustration of how FEC works. There are 3 paths from source to destination, and past experience indicates only one of them has congestion at any given time, but the sender does not know which until it sends.
Suppose the sender has 2 packets to send, say $A$ and $B$. A suitable FEC scheme is to create another packet $C=A\otimes B$ (bitwise exclusive-OR). After encoding, the sender sends $A$, $B$, $C$ on three paths in parallel. The receiver can recover the message as soon as it collects any 2 out of 3 packets, thus avoiding congestion at the cost of sending 1.5x more traffic.
\begin{figure}[t]
\centering
\vspace{-1em}
\includegraphics[width=\linewidth]{figs/parallelism-block.png}\\
\vspace{-1em}
\caption{Example of a 3-path FEC}\label{fig:blockcode}
\vspace{-1.5em}
\end{figure}
Fixed-rate block codes, as shown in the example in Figure~\ref{fig:blockcode}, are simple to understand and implement, but have the fundamental difficulty of choosing a right coding rate.
In Figure~\ref{fig:blockcode}, the suitable code rate is $\frac{3}{2}$. Fixed rate codes like this may suffer if the path condition deteriorates: if two of the paths suddenly become congested instead of one in Figure~\ref{fig:blockcode}, then the message will take much longer time to recover, as the first two arrivals determine the transmission time.
Rateless erasure codes, or fountain codes~\cite{rateless}, are better suited for dynamic traffic characteristics in DCNs, as its code rate is adaptable to dynamic channel conditions.
The key property of fountain codes is that they can generate a potentially limitless sequence of encoded symbols from a given set of source symbols (the source symbols can be decoded with a large enough subset of encoded symbols).
Specifically, we adopt LT Code (LTC)~\cite{ltc} in our prototype due to its low complexity, where an encoded packet is generated by random linear combination of original packets.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/fountaincode.png}\\
\vspace{-1em}
\caption{Random linear encoding in one round of transmission}\label{fig:fountain}
\vspace{-0.5em}
\includegraphics[width=\linewidth]{figs/header.png}\\
\vspace{-2.5em}
\caption{CloudBurst\xspace header format}\label{fig:header}
\vspace{0.5em}
\includegraphics[width=\linewidth]{figs/dec.png}\\
\vspace{-1em}
\caption{Decoding example}\label{fig:dec}
\vspace{-1em}
\end{figure}
\textbf{Encoding:}
We assume the message is given before the transmission.
As shown in Figure~\ref{fig:fountain}, we first packetize the original message into equal-length parts.
If necessary, we pad zero bits to packets so that every part from the original message has the same size: $MTU-H_{IP}-H_{UDP}-H_{cbrst}$, where MTU is Maximum Transmission Unit (1.5KB), $H_{IP}$ is IP header size (20B), $H_{UDP}$ is UDP header size (8B), and $H_{cbrst}$ is CloudBurst\xspace header size.
For each encoded packet, its degree ($d$) is defined as the number of un-decoded original packets encoded in it. We first chose the value of $d$ from a certain probability distribution. Then, we randomly choose $d$ packets from the original message $\{x^{(1)},\ldots,$ $x^{(d)}\}$. The encoded packet in the $i$th iteration is then:
$y_i = x_i^{(1)}\oplus x_i^{(2)} \oplus \ldots \oplus x_i^{(d)}$.
The header of encoded packet (Figure~\ref{fig:header}) includes the number of packets ($n$) in the message, the message size in unit of bytes, an $8$-bit message ID, and the list of indices $(1), \ldots ,(d)$.
\textbf{Decoding:} The decoding algorithm uses standard Gaussian elimination method~\cite{rfc6330}.
We illustrate the decoding with an example in Figure~\ref{fig:dec}. In step 1, $y_1$ is received, and its degree is 1 (Degree of a symbol is defined as the number of un-decoded original packets encoded in it), so that the corresponding $x_1$ is decoded. In step 2, $y_2$ is received, and its degree is 2. But since $x_1$ is already decoded, $y_2$'s degree is reduced by 1, so $x_2$ is decoded. By iteratively finding symbols with degree 1, the original message is recovered.
\subsection{Burst-until-Received at End-host}\label{sec:protocol}
The sequence diagram of the end-host operations of CloudBurst\xspace is shown in Figure~\ref{fig:timeseq}. We elaborate on the sender and receiver operations below.
\begin{figure}[t]
\centering
\vspace{1em}
\includegraphics[width=\linewidth]{figs/timeseq.png}\\
\vspace{-2em}
\caption{Sequence Diagram}\label{fig:timeseq}
\vspace{-1em}
\end{figure}
\textbf{Sender operations:} As a variant of fountain code, LT coding can generate endless sequence of encoded packets for a given message. LT coding enables the CloudBurst\xspace sender to continuously generate encoded packets and sending them on multiple paths, as described in Algorithm~\ref{algo:send}. It stops only after it receives a "STOP" signal from the receiver.
This burst-until-received behavior may consume all bandwidth of the network.
So we add a rate control mechanism to allow users to adjust the sending rate. The user/application specifies rate $r$, which is the share for CloudBurst\xspace traffic. We define a round of transmission as the period of time the sender puts an encoded packet on each of the paths (Line\#6-9 in Algorithm~\ref{algo:send}), and the rate is adjusted by adding delay between rounds of transmission.
For example, if NIC capacity is 10Gbps, $r=0.5$, and 5 paths between source and destination. Then the sending rate of the message is 5Gbps (1Gbps per path). Pumping 5 1.5KB encoded packets on 5 paths takes 12$\mu$s, thus, a 12$\mu$s inter-round delay is added.
\begin{algorithm}[t]
\scriptsize
\DontPrintSemicolon
\KwIn{Receiver IP \& Port, Message $m$, Transmission timeout $t$, Ratio $r$}
\KwOut{Success/Failure}
\uIf{sizeof($m$) $>$ MAX\_MSG\_SIZE}{
\Return{Failure}\;
}
Set up count down timer of $t$\;
\While{STOP not received}{
Wait for ($\frac{1}{r}$ * NUM\_PATHS * PKT\_SIZE / LINK\_CAP)\;
\ForEach{path $p$ to Receiver}{
Get symbol $y$ from encoding thread and send it on $p$\;
\uIf{timer expires}{
\Return{Failure}\;
}
}
}
\Return{Success}
\caption{\texttt{cbrst\_send($\cdot$)}}
\label{algo:send}
\end{algorithm}
\textbf{Receiver operations:}
All encoded packets are received by the listening thread, and forwarded to their corresponding decoder (each decoder is instantiated for a message).
Decoder buffers all the encoded packets that have unrecovered packets for decoding. It returns the message to the receiver listening thread after full recovery, or expires if there is no more incoming encoded packets after a pre-defined timeout. Then the receiver signals a STOP to the sender. In case that the STOP signal is lost, the receiver will send a new STOP for each received encoded packet after the first STOP is sent.
\subsection{Aggressive Dropping in Network}\label{sec:aggdrop}
A major concern is that the open-loop rate control of CloudBurst\xspace is dangerous to other flows and may overflow switch buffers in the network. Thus, it seems that a congestion control mechanism for CloudBurst\xspace is needed.
However, dropping is also a form of congestion control~\cite{decongestion} for high-bandwidth erasure coded flows. When packets are erasure encoded, dropping some is not an issue as the message can be recovered with a subset of the packets sent from the source.
Thus, for a network with purely erasure coded flows, the switches do not need deep buffers to keep network stable~\cite{nocc}, and the sources can burst as fast as possible~\cite{decongestion}.
This is also true for TCP-like transport. For example, pFabric~\cite{pfabric} features minimum TCP, a line rate transport, with shallow buffer and priority dropping at the switch. The flows with same priority achieves the max-min fairness despite dropping due to shallow buffer.
In fact, replacing congestion control with erasure coding has been discussed for future Internet~\cite{geni}, but there are fundamental difficulties. First, in public Internet, many transport protocols coexist, and bursts of a few erasure coded flows may take all the buffer and bandwidth, hurting others. Second, on the path of an erasure coded flow, if a packet gets dropped after it passes a few switches, the bandwidth it consumed is wasted. These ``dead packets''~\cite{decongestion} may induce congestion collapse: packet loss requires additional packets (more redundancy) to recover, which incurs more load in the network and more packet drops, forming a positive feedback loop.
By leveraging the properties of DCNs, targeting only the short flows, and limiting buffer usage, we avoid the drawbacks of erasure coding transport.
\begin{itemize}
\item DCNs often have abundant multipath~\cite{fattree, google-dcn, facebook-dcn}. With encoded symbols on all paths, the network core with CloudBurst\xspace is load balanced for short flows, and the only bottleneck is the egress switch. With just one bottleneck, ``dead packets'' are not likely to occur.
\item For long flows, using erasure coding is dangerous due to the positive feedback loop described above. However, CloudBurst\xspace serves only short flows, which do not persist: because of the small sizes and the high bandwidth network, short flows dissipates quickly.
\item We enforce aggressive dropping by separating the buffer for CloudBurst\xspace and other flows, and limiting the buffer usage of the CloudBurst\xspace flows to minimum. This protects other flows from CloudBurst\xspace traffic. Thus, fan-in bursts of CloudBurst\xspace flows no longer affect others on a shared shallow-buffer switch; meanwhile, TCP flows can also safely use remaining buffer for high throughput.
\end{itemize}
\subsection{Discussion: two potential issues}\label{sec:design_discussion}
\textbf{Infiniteness of flow sending:} The termination condition of our algorithm is that the receiver can decode all of the original packets. However, since we do not use the ACK mechanism, the senders do not know which packets have already arrived, they just randomly chose packets, which may cause the failure of flow completion. In fact, the mathematical property of LTC~\cite{ltc} guarantees the flow completion. $k$ original packets can be recovered by $k+\mathcal{O}(\sqrt{k}\ln^2{k/\delta})$ encoding packets with probability of $1-\delta$. That means, we can bound the total number of encoding packets with a certain probability. Meanwhile, we can also set a packet sending upper bound to avoid the tail latency caused by unlimited sending. When a certain number of encoding packets are sent, the sender will request the receiver for the information of received packets.
\textbf{Aggravate last hop congestion:} In fact, our algorithm focuses on the tail latency caused by the different congestion condition of sub-paths. In DCN, the last hop may be the bottleneck and sending FEC-coded packets only aggravates the congestion. This problem is still caused by the agnostic of packets' receiving in senders. When the last hop congestion (e.g., incast) happens, even the slow flow may not be finished in one RTT, it gives us the opportunity and time to respond the packet receiving information to the senders. More specifically, when a flow does not finish within a certain time, the sender will ask the receiver for the arrival information. Even though our algorithm is not designed for solving incast problem, it eliminates the retransmission cost, thus the experiment result shows that CloudBurst\xspace still performs well in incast scenario.
\section{Introduction}
Low latency message\footnote{Message and short flow are used interchangeably in this paper, both referring to small network flows.} delivery is critical for many applications in datacenter networks (DCN), such as web search~\cite{dctcp,pias-nsdi}, page creation~\cite{detail}, recommendation systems, stream processing, and online advertising~\cite{d2tcp}. These applications are usually user-facing, and even a very small delay in flow completion time (FCT) can reduce application performance, degrading user experience~\cite{dctcp,pias-nsdi,pias-hotnets} and causing financial loss~\cite{d2tcp}.
However, the long tail latency problem is particularly pronounced in production DCNs for multiple reasons ($\S$\ref{sec:background:cause}): (1)~applications emit synchronized high fan-in bursts (incast~\cite{incast, pac}); (2)~shared-buffer switches are too shallow~\cite{dc-buffer,dc-buffer-apnet} to absorb bursts; (3)~transport protocols use Automatic Repeat Request (ARQ)~\cite{gobackn,tcpsack} and retransmission timeouts for packet recovery; (4)~coarse-grained load balancing (e.g., ECMP~\cite{ecmp}); and (5)~hardware malfunctioning (e.g., packet black-hole, silent random packet drops, etc.~\cite{pingmesh,fuso,scalerdma}) is unpredictable.
In order to tackle such long tail latency problem, prior works ($\S$\ref{sec:background:prior}) adopt a variety of strategies from fine-grained load balancing~\cite{detail,conga,fastpass,hermes}, rate control~\cite{dctcp,d2tcp,dcqcn,timely,swift,hpcc,dcn-transport,mcp,mqecn,tcn,ecn-sharp}, prioritization~\cite{pias-nsdi,pfabric,pase,detail,qjump,ras,karuna}, to fast loss recovery~\cite{cp,fuso}. However, most of these proposals introduce non-trivial implementation difficulties, such as global state monitoring~\cite{conga,fastpass}, complex network control~\cite{fastpass,pase}, and switch modifications~\cite{cp,pfabric,conga,hpcc}. While achieving superior performance, these solutions are hard to deploy in existing commodity datacenters.
Therefore, we ask a pragmatic question: \emph{is there a simple scheme for commodity datacenters to cut tail latency without the above complexities, while still delivering similar or better performance?} In this paper, we answer the question affirmatively by presenting CloudBurst\xspace, a simple and deployable solution to cut tail latency of short messages in datacenters.
\begin{figure}[t]
\centering
\vspace{-1.2em}
\includegraphics[width=1\linewidth]{figs/cbrst-ano}\\
\vspace{-2em}
\caption{CloudBurst\xspace Concept}\label{fig:sys}
\vspace{-1em}
\end{figure}
CloudBurst\xspace works as follows (Figure~\ref{fig:sys} \& $\S$\ref{sec:design}):
\begin{itemize}
\item \textit{Encoding messages with FEC:} Forward error correction (FEC) has been deployed in many applications~\cite{ffc, oec}, it uses proactive and redundant approach to tolerate errors. In DCN, the error tolerant feature can be adopted to handle packet losses. That is, we do not need to retransmit the lost packets while still guaranteeing the reliable transmission.
CloudBurst\xspace explores the coding dimension of the transport-layer design. It employs FEC with proactive and oblivious redundant transmission. Each encoded packet contains information of multiple original packets. In this way, even if some of the encoded packets are lost, the original packets can still be reconstructed at the receiver.
\item \textit{Bursting over multiple paths:} CloudBurst\xspace spreads encoded packets over multiple paths, which obliviously exploits the rich path diversity in modern DCNs~\cite{fattree, facebook-dcn, google-dcn}, as well as the temporary network under-utilization~\cite{dctraffic10}. If any congestion-free path exists, CloudBurst\xspace will take advantage of them without extra signalling overhead.
\item \textit{Separated and size-limited switch queue:} We limit the buffer usage of the CloudBurst\xspace short flows to minimum, so that the per-hop queueing latency of a CloudBurst\xspace packet is deterministically low (less than a few $\mu s$). This is achieved by separating CloudBurst\xspace short flows and other traffic in different queues, and limiting the maximum depth of the CloudBurst\xspace queue to a tiny value (a few packets). Such a limitation on buffer usage of CloudBurst\xspace traffic also makes it friendly to other traffic.
\end{itemize}
Our design choices result in a very simple transport that works drastically different from TCP. CloudBurst\xspace performs neither congestion control\footnote{In fact, for erasure coded flows, dropping can be considered as a form of congestion control~\cite{geni}, or ``decongestion control''~\cite{decongestion}.} nor reactive retransmission (i.e., no per-packet ACK, no congestion feedback, no loss detection, and no retransmission timeout (RTO)).
Instead, it simply transmits at line-rate and unilaterally keeps generating and sending the encoded packets until the receiver acknowledges the reception of the message. Meanwhile, the in-network requirement of CloudBurst\xspace is also simple: limiting queue size is readily configurable in commodity switches\cite{pica8}.
To be deployable with existing commodity datacenters, CloudBurst\xspace is now implemented as a user-space library ($\S$\ref{sec:implement}), so that applications can voluntarily use it for latency-sensitive short flows.
We built a small 2-tier leaf-spine testbed and deployed CloudBurst\xspace on it. The testbed consists of 6 Pronto-3295 (Broadcom) Ethernet switches and 8 Dell R320 servers with quad core Xeons E5-1410 CPU and 1GbE NIC. Our implementation experience shows that CloudBurst\xspace is readily-deployable with existing commodity switches.
Our testbed experiments ($\S$\ref{sec:basicdesign}--$\S$\ref{sec:deepdive}) show that CloudBurst\xspace outperforms prior practical schemes. For example, it achieves more than $60.06\%$ and $63.69\%$ reduction in 99th percentile (p99) flow completion time compared to PIAS~\cite{pias-nsdi} and DCTCP~\cite{dctcp}.
Furthermore, our results also verify that it is resilient to incast,
and handles failures gracefully.
We complement testbed experiments with large-scale simulations ($\S$\ref{sec:simulations}) in 10/40G environments. We find that CloudBurst\xspace achieves comparable or better performance than prior solutions. For example, it achieves $24\%$ tail latency reduction compared to a clean-slate design, pFabric~\cite{pfabric}.
\section{Analysis}\label{sec:math}
In this section we show that CloudBurst\xspace is complementary to existing low latency proposals. Given any per-packet latency distribution, CloudBurst\xspace always reduce the 99th percentile (tail) latency.
Consider the following scenario: an message of $k$ packets is to be delivered from a sender, $S$, to a receiver, $D$. Using FEC code of rate $s$, $(s>1)$, the $k$ packets generated $k\cdot s$ encoded symbols. For rateless codes, we define $s$ as the eventual coding rate, which is the number of packets received before full recovery over $k$.
The cumulative density function (CDF) for the per-packet latency of the paths in DCN is $F_{P}(\cdot)$.
The 99th percentile latency is denoted as $T_{99}$, where $F_P(T_{99}) = 0.99$.
We compare the average per packet latency for CloudBurst\xspace, and the 99th percentile tail packet latency for CloudBurst\xspace.
Assume $F_X$ is absolutely continuous, it has a density such that $d F_X(x)=f_X(x) dx$, and we can use the substitutions
$u=F_X(x)$
and
$du=f_X(x)\,dx$
to derive the following probability density functions (pdfs) for the order statistics of a sample of size $k\cdot s$ drawn from the distribution of $X$:
\[f_{X_{(k)}}(x) =\frac{n!}{(k-1)!(n-k)!}F_X(x)^{k-1}[1-F_X(x)]^{n-k} f_X(x)\]
\section{Related work}\label{sec:related}
In addition to prior solutions ($\S$\ref{sec:background:prior}),
we briefly survey related work in applications of coding on multipaths.
Coding techniques over multipath are extensively employed to improve transport layer in various settings, including wireless environments~\cite{huang2008tcp, sundararajan2009network, hmtp}, vehicular networks~\cite{oec}, and information dissemination in overlay networks~\cite{bullet}.
They use FEC to mitigate impact of packet loss, and reduce retransmission. A similar work that employs fountain coding over multipath is HMTP~\cite{hmtp}, which solves the receiver packet reordering problem in multi-homing wireless networks.
Another example of FEC over multipath is OEC in vehicular networks~\cite{oec}, which has a streaming (receiver-feedback) setting and a greedy encoding scheme to maximize new information at the receiver.
Relative to them, CloudBurst\xspace applies FEC over multipath for DCN tail latency reduction.
More recently, FMTCP~\cite{fmtcp} combines fountain code with MPTCP~\cite{mptcp}, in order to mitigate MPTCP's problem of being straggled by the slowest subflow. FMTCP employs a data allocation algorithm to coordinate the coding and transfer of many subflows.
QUIC~\cite{quic} also uses FEC to mask packet drops in order to reduce web latency. Specifically, it uses proactive speculative retransmission (sending duplicated copies of more important packets) to optimize streaming applications. In contrast to them, CloudBurst\xspace employs aggressive dropping to reduce per-hop delay, and focuses on reducing delivery latency of short flows in DCNs.
\section{Implementation}\label{sec:implement}
In this section, we describe implementation and parameter settings of CloudBurst\xspace.
\subsection{Enabling CloudBurst\xspace at End-host}\label{sec:implement:endhost}
For the end-host, we build a prototype CloudBurst\xspace with Rust 1.6~\cite{rust} as a user-space library. The implementation is multi-threaded: sending (receiving) and encoding (decoding) are handled by different threads at the sender (receiver).
We discuss the settings of coding-related parameters as follows.
\textbf{Choosing message size $n$ \& degree $d$:}\label{sec:implement:size}
A larger header size supports larger message size at the cost of more header overhead and possibly higher load on the network (with the same code rate, a larger message generates more encoded symbols).
We consider header overhead (percentage of packet used for header), computation overhead (number of decoding operations), and coding overhead (number of encoded symbols to reconstruct the message) when choosing $n$ \& $d$.
Header overhead is as follows: $x$ is the length of the field representing the maximum number of packets in a message ($n=2^x$ packets). The message size is $log_2(MTU\times 2^x)=x+log_2(MTU)$ (bytes). The bit-set representing the packets included in the symbol is $2^x$ (bits).
Thus, assuming 1.5KB MTU, the header overhead for $x=9$ is less than $7\%$, which suggests that, choosing $x<10$ is efficient.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figs/nonsys.png}\\
\vspace{-1em}
\caption{Computation and coding overhead}\label{fig:coding-overhead}
\vspace{-1em}
\end{figure}
Computation overhead for different encoding degree $d$ and message size $n$ is plotted in Figure~\ref{fig:coding-overhead}(a). For different $(d,n)$ pairs, we perform the en/decoding process 100 times, and calculate the average number of XOR operations necessary to reconstruct the message. We see that the number of operations increases with $n$. For each $n$, the number of operations increases with $d$, but slows down for $d\geq 5$.
Coding overhead is measured by the ratio between the number of packets needed to recovery the message and the number of original packets in the message. We collect this ratio by running the same experiment as above, and plot the results in Figure~\ref{fig:coding-overhead}(b). The coding overhead drops for all sizes with respect to $d$, but beyond $d\geq5$, increasing $d$ does not lead to lower coding overhead.
With the above results, we recommend to set $d=5, n=64=2^6$ for the current prototype. It thus supports message size of at most 93.44KB (i.e., $2^6 \times 1.46KB$).
We note that typical latency-sensitive applications in DCNs exhibit multi-tier partition/aggregation patterns, and the queries are often less than 10KB in size~\cite{dctcp,d2tcp}, which suggests the current header design should work well in practice.
\textbf{Choosing $r$:} $r$ is the expected throughput of CloudBurst\xspace flows in the DCN. For oversubscribed DCN, we can adjust the parameter $r$ in Algorithm~\ref{algo:send} to the oversubscription ratio. For network with full bisection bandwidth, $r=1$.
In the experiments and simulations, we choose $r$ to be exactly the oversubscription ratio of the network, so that CloudBurst\xspace flows will not overload the network.
\subsection{Enabling CloudBurst\xspace in Network}\label{sec:impl:network}
\textbf{Configuring switch buffer:} To enforce aggressive dropping, we set the packets of CloudBurst\xspace flows to the same DSCP value, so that they are carried in the same switch queue, separated from other non-CloudBurst\xspace traffic. We then set the dedicated buffer for the CloudBurst\xspace queue to a small value.
On our testbed, we set it to be $1$\% of total buffer size, and we disable the shared buffer of this queue to avoid affecting other flows. This is done by setting ``\texttt{buffer queue-limit}'' in our Pronto-3295 switches~\cite{pica8ref}. Other switches also support such configurations. For example, in Cisco switches, we can set the depth of a traffic class queue~\cite{ciscoref}.
\textbf{Multipath routing}
A key issue for CloudBurst\xspace is how to spread packets on different paths. The implementation is dependent on how multipath is supported in the network.
\noindent\textit{Sub-optimal multipath:}
The network may use multipath implicitly, so that load balancing over multipath is transparent to the applications. ECMP~\cite{ecmp} is an good example: for each flow, an ECMP-enabled switch picks an outgoing port at random based on the hash of the flow's source and destination IPs and ports. To use CloudBurst\xspace on ECMP, the sender and receiver need to maintain a pool of ports, and each encoded symbol will be given a header with a random combination of sender ports and receiver ports, which implicitly asks ECMP to hash packets on different paths. While this may not fully utilize all paths, we show that CloudBurst\xspace still works well in $\S$\ref{sec:eval:multipath}.
\noindent\textit{Explicit multipath routing:}
Multiprotocol label switching~\cite{mpls} (MPLS) can provide explicit routing for each packet, but this requires support from the network fabric. Also, setting up multiple path labels for each short message requires signaling the switches on multipath, which is undesirable for low latency delivery.
To attain the same efficiency as implicit multipath with ECMP, we turn to a DCN routing scheme---XPath~\cite{xpath}, which enables explicit path-based routing in DCNs. It compresses and pre-installs end-to-end paths into forwarding tables of commodity switches, and packets are routed based on the path ID in their headers.
With XPath's explicit path control, CloudBurst\xspace adds path IDs to the headers of the encoded symbols, which will place them on different paths.
|
hep-th/9607042
|
\section{Introduction}
Non--trivial fixed points are a highly challenging aspect of renormalization
theory. Much of what is known about non--trivial fixed point is due to the
$\epsilon$--expansion of Wilson and Fisher \cite{WF72,WK74}, which is an
interpolation from a critical dimension to the one of interest. In this
paper we present another interpolation scheme where the dimension of the
underlying (Euclidean) space--time is kept fixed.
A prototype of a non--trivial fixed point is the infrared fixed point of
massless $\phi^4$--theory in three dimensions, also called Wilson fixed
point \cite{WK74}. We choose it as an example for our method. Although it
has been investigated by various other means, for instance by Monte Carlo
simulations of the three dimensional Ising model close to criticality,
hopping parameter expansion, field theoretic perturbation theory for its
scaling limit, and numerical integration of renormalization group flows
in a number of setups, our knowledge of it is far from satisfactory.
Accurate data for its spectrum of anomalous dimensions is lacking, its
functional form is largely unknown, in particular its locality properties,
and its mathematical construction remains an outstanding difficult problem.
We mention \cite{ZJ89} and references therein as a guide to the extensive
literature. We mention further that all this has been accomplished to a
very satisfactory status in the hierarchical approximation by Koch and
Wittwer \cite{KW91}. Our interpolation is a brick in the analysis of the
full model.
As starting point we choose a functional differential equation from the
infinitesimal renormalization group of Wilson \cite{WK74}. Specifically
we choose a normal ordered and rescaled representation for the fixed point
interaction, expressed in terms of a scalar field with non--anomalous
scaling dimension. It contains a bilinear renormalization form. This
bilinear form is continuously turned on with an auxiliary parameter such
that zero gives a linear theory and one restores the full equation. The
linear theory is arranged such that the $\phi^4$--interaction acquires
the scaling dimension zero in it. We then expand the fixed point interaction
into a power series in the interpolation parameter. This part is similar
to the $\epsilon$--expansion. In order to perform the expansion to high
orders on the computer, the interaction is written in a basis of
interactions which includes a general two point interaction in
derivative expansion together with local higher interactions. We compute
both the fixed point interaction and the eigenvalue associated with a
massive perturbation. The resulting power series are evaluated by means of
Pad\'{e} , Dlog, Borel--Pad\'{e} , and Borel--conformal--Pad\'{e} resummation.
The interpolation idea applies also to other renormalization schemes.
The infinitesimal renormalization group is a particularly convenient one
because it involves a minimal set of Feynman integrals. Interpolations
in fixed dimensions can also be formulated for discrete renormalization
group transformations both in continuum regularization and on the lattice,
at the expense of dealing with general non--linear rather than quadratic
equations. It is conceivable that our interpolation can be given a meaning
beyond perturbation theory.
The paper is organized as follows. In section two we explain the structure
of our particular functional differential equation. It is taken from
\cite{W96} and is the Wilson equation \cite{WK74} in a kind
of interaction picture. In section three we present our interpolation
scheme. It is compared to a naive interpolation which has only a
trivial solution. We solve our equations to lowest order to explain
their recursive treatment. In section four we discuss their form in
a coordinate representation. The result is a set of algebraic recursion
relations for the fixed point interaction. They involve a set of structure
constants whose computation again involves certain Feynman integrals
and multiplicities. We devote section five to this issue.
In section six and seven the eigenvalue problem for the scaling fields
of the non--trivial fixed point and their anomalous dimensions is
treated along the same lines. We restrict our attention to a mass
perturbation of the fixed point and the associated critical index $\nu$.
In chapter eight and nine the resulting recursions are studied
by means of computer algebra. We conclude with a brief discussion of
our results on the value $\nu$.
\section{Renormalization group fixed point}
We consider a real scalar field $\phi$ on three dimensional Euclidean
space. We use a momentum space renormalization group built from the
decomposition of a massless propagator $v$ with exponential ultraviolet
regulator. The renormalization group will be formulated in terms of an
interaction $V(\phi)$. Concerning the general background, we refer to
the work of Wilson \cite{W71}, Wilson and Kogut \cite{WK74}, and also
to Gallavotti \cite{G85}. Our setup will be identical with that in
\cite{W96}.
We study the non--trivial infrared fixed point in three dimensions as
solution to the functional differential equation
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phiV(\phi)=\bra{V(\phi)}{V(\phi)},
\label{fixed}
\end{equation}
which was derived in \cite{W96}.
Eq. \Ref{fixed} is a normal ordered and rescaled variant of the
infinitesimal renormalization group due to Wilson \cite{WK74}.
Its origin is a flow equation governing the behaviour of interactions
upon the infinitesimal change of a floating cutoff. Eq. \Ref{fixed}
gives stationary flows modulo the rescaling of units.
The left hand side of
\Ref{fixed} is a generator of dilatations
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phiV(\phi)=\frac{{\rm d}}{{\rm d}L}V(\phi_L)
\biggr\vert_{L=1},\quad \phi_L(x)=L^{-1/2}\phi\left(\frac{x}{L}
\right),
\end{equation}
acting on the interaction. The field $\phi$ is here rescaled
non--anomalously with its canonical dimension at the trivial fixed
point. It should be distinguished from the scaling fields of the
infrared fixed point which have non--zero anomalous dimensions.
The right hand side of \Ref{fixed} is a bilinear renormalization
group form
\begin{equation}
\bra{V(\phi)}{V(\phi)}=\lap{\chi}\exp\left\{
\lap{v}\right\}V(\phi_1)V(\phi_2)\biggr\vert_{\phi_1=\phi_2=\phi}.
\label{bilinear}
\end{equation}
It can be visualized as a sum of contractions between two copies
of the interaction. Each contraction is made of one {\sl hard}
propagator $\chi$ and any number of {\sl soft} propagators $v$.
The propagators are here given by
\begin{equation}
\widetilde{\chi}(p)=e^{-p^2},\quad
\widetilde{v}(p)=\frac{e^{-p^2}}{p^2},
\label{propagators}
\end{equation}
as in \cite{W96}. Eq. \Ref{fixed} is the main dynamical equation
in this investigation.
Being a differential equation, it has to be
supplied with further data to select a particular solution. In a
rigorous theory in the sense of Glimm and Jaffe \cite{GJ87}, the
infrared fixed point should come as a global ${\bf Z}_2$--symmetric
solution, where {\sl global} refers to some criterion of finiteness.
Our point of view in this approach will be more modest. An
interaction $V(\phi)$ will stand for a power series
\begin{equation}
V(\phi)=\sum_{n=1}^{\infty}\int{\rm d}^3x_1
\cdots{\rm d}^3x_{2n}\;\phi(x_1)\cdots\phi(x_{2n})\;
V_{2n}(x_1,\ldots,x_{2n})
\label{power}
\end{equation}
in the field, with symmetric Euclidean invariant distributional
kernels given by Fourier integrals
\begin{eqnarray}
V_{2n}(x_1,\ldots,x_{2n})&=&\int
\frac{{\rm d}^3p_1}{(2\pi)^3}\cdots
\frac{{\rm d}^3p_{2n}}{(2\pi)^3}\;e^{i(p_1x_1+\cdots+p_{2n}x_{2n})}
\nonumber\\&&(2\pi)^3\delta^{(3)}(p_1+\cdots+p_{2n})
\widetilde{V}_{2n}(p_1,\ldots,p_{2n})
\end{eqnarray}
of smooth momentum space kernels. I.e., we identify an interaction
with its collection of momentum space kernels. The question of
convergence of the expansion \Ref{power} in powers of fields will
not be addressed. It is conceivable that it could be tackled
with a suitable norm on the collection of momentum space kernels
as a whole.
In the iterative approach to be defined below we will
meet at finite order no more than polynomial expressions in the
field. We will understand \Ref{fixed} as a system of differential
equations for the momentum space kernels. Its explicit form
can be looked up in \cite{W96}. Boundary data is substituted for
by the condition of regularity. Homogeneous functions give particular
kernels, which correspond to scaling fields of the trivial fixed
point. Expanding a kernel in powers of momentum derivatives, we can
always express it in terms of such scaling fields. To distinguish
them from the scaling fields of the non--trivial fixed point and
also because we will use perturbation theory, we will speak of them
as {\sl vertices}.
\section{Interpolation parameter and expansion}
Our strategy to solve \Ref{fixed} is to interpolate to a solvable
situation. If the interpolation is smooth it can be performed by
means of perturbation theory. A natural candidate
is
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phi V(\phi,z)=z\bra{V(\phi,z)}{V(\phi,z)}
\label{naive}
\end{equation}
with an interpolation parameter $z=0\ldots 1$. It can be thought
to turn on continuously the bilinear form, which is identified as
the source of troubles. The interpolation \Ref{naive} is inappropriate
for the following reason, when the dimension parameter is fixed
to three. Expand the interpolated interaction as a function of the
interpolation parameter in a power series
\begin{equation}
V(\phi,z)=\sum_{r=0}^\infty z^r\;V^{(r)}(\phi).
\label{pertur}
\end{equation}
Unfortunately there is little hope that \Ref{pertur} has a finite
radius of convergence both in the case of \Ref{naive} and the
interpolation \Ref{better} considered below. To be cautious we will
therefore view \Ref{pertur} as a formal power series and interpret
all equations below in this sense. It will however be argued that
non--perturbative information can be extracted by Borel
resummation. In order to solve \Ref{naive}, the expansion
\Ref{pertur} has to satisfy
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phi\fix{r}=\sum_{s=0}^{r-1}\bra{\fix{s}}{\fix{r-1-s}}
\label{perfix}
\end{equation}
to every order $r\in{\bf N}$, with the understanding $\fix{-1}=0$.
In particular it requires the interaction to satisfy
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phi\fix{0}=0
\label{marginal}
\end{equation}
to zeroth order. In other words, the zeroth order has to be
a marginal scaling field of the trivial fixed point. In three
dimensions we have two marginal scaling fields, a wave
function term and a $\phi^6$--vertex, to be abbreviated as
\begin{equation}
{\mathcal O}_{1,1}(\phi)=\int{\rm d}^3x\;\phi(x)
(-\bigtriangleup )\phi(x),
\quad {\mathcal O}_{3,0}(\phi)=\int{\rm d}^3x
\;\phi(x)^6.
\label{vertices}
\end{equation}
We emphasize that vertices should be understood as momentum space
kernels at zero momentum and their Taylor expansions. Each of
them comes with a formal orthogonal projector $\pro{1}{1}$ and
$\pro{3}{0}$, selecting the corresponding vertex from a general
interaction \Ref{power}. The zeroth order has to be a linear
combination
\begin{equation}
\fix{0}=V^{(0)}_{1,1}\;{\mathcal O}_{1,1}(\phi)+
V^{(0)}_{3,0}\;{\mathcal O}_{3,0}(\phi).
\label{zero}
\end{equation}
The coupling constants are not determined by the zeroth order
equation \Ref{marginal}. To first order, \Ref{perfix} reads
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phi\fix{1}=\bra{\fix{0}}{\fix{0}}.
\label{problem}
\end{equation}
But eqs. \Ref{zero} and \Ref{problem} together have only a
trivial solution. The left hand side of \Ref{problem} cannot
contain the vertices \Ref{vertices} because the dilatation generator
$\left({\mathcal D}\phi,\frac{\delta}{\delta\phi$ has no marginal image. Therefore, \Ref{problem} requires
that
\begin{equation}
\pro{1}{1}\bra{\fix{0}}{\fix{0}}=
\pro{3}{0}\bra{\fix{0}}{\fix{0}}=0.
\end{equation}
Computing the bilinear form with two copies of \Ref{zero}
inevitably gives $V^{(0)}_{1,1}=V^{(0)}_{3,0}=0$.
Eq. \Ref{naive} is thus an inappropriate interpolation and has
to be given up.
A way around the obstacle is to interpolate
simultaneously the dimensionality of the theory. This is the
strategy of the $\epsilon$--expansion of Wilson and Fisher
\cite{WF72} in a field theoretic setup. Another way is to
interpolate the scaling dimension, remaining firmly in three
dimensions. We choose this second route and replace \Ref{naive}
by
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi-1\right] V(\phi,z)=
z\;\bra{V(\phi,z)}{V(\phi,z)}-z\;V(\phi,z).
\label{better}
\end{equation}
The power series expansion \Ref{pertur} solves \Ref{better}
if the coefficients satisfy the system of differential equations
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi-1\right]\fix{r}=
\sum_{s=0}^{r-1}\bra{\fix{s}}{\fix{r-1-s}}-\fix{r-1}
\label{system}
\end{equation}
to all orders $r\in{\bf N}$. To order zero, \Ref{system} requires
now
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi-1\right]\fix{0}=1
\label{unit}
\end{equation}
in contrast to \Ref{marginal}. The zeroth order interaction is
now a scaling field with unit scaling dimension. In three dimensions
we have only one candidate, the $\phi^4$--vertex
\begin{equation}
{\mathcal O}_{2,0}(\phi)=\int{\rm d}^3x\;\phi(x)^4.
\label{phifour}
\end{equation}
The zeroth order interaction thus has to be proportional to
\Ref{phifour}. The proportionality factor is the $\phi^4$--coupling.
It is not determined by the zeroth order equation \Ref{unit}.
We conclude that
\begin{equation}
\fix{0}=\cou{2}{0}{0}\;{\mathcal O}_{2,0}(\phi).
\end{equation}
This expansion proves to have indeed a non--trivial solution. To
see this, consider the first order equation in \Ref{system}.
It reads
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi-1\right]\fix{1}=
\bra{\fix{0}}{\fix{0}}-\fix{0}.
\label{first}
\end{equation}
Eq. \Ref{first} cannot have a $\phi^4$--vertex on its left hand
side. Therefore it is required that
\begin{equation}
\pro{2}{0}\biggl\{\bra{\fix{0}}{\fix{0}}-\fix{0}\biggr\}=0.
\end{equation}
Computing the bilinear form, this condition reads explicitely
\begin{equation}
\cou{2}{0}{0}\biggl\{144\;\widetilde{\chi}\star\widetilde{v}(0)
\;\cou{2}{0}{0}-1\biggr\}=0,
\end{equation}
where $\star$ means convolution times $(2\pi)^{-3}$.
Besides the trivial solution $\cou{2}{0}{0}=0$ it has a
non--trivial solution
\begin{equation}
\cou{2}{0}{0}=
\frac{1}{144\widetilde{\chi}\star\widetilde{v}(0)}=
\frac{(2\pi)^{3/2}}{72}=0.21874445\ldots
\label{nontrivial}
\end{equation}
The value of the $\phi^4$--coupling at any given order will in fact
be determined by the equations at the next order, a feature of this
particular interpolation expansion. To first order, the interaction
can be split into
\begin{equation}
\fix{1}=\cou{2}{0}{1}\obs{2}{0}+
\pro{2}{0}^\perp\fix{1}
\end{equation}
with $\pro{2}{0}^\perp=1-\pro{2}{0}$ the projector
on the formal orthogonal complement. Eq. \Ref{first} defines a
system of first order differential equations for the momentum space
kernels therein. They have a unique integral in the space of smooth
functions of momenta, see \cite{W96}. We denote this integral by
\begin{equation}
\pro{2}{0}^{\perp}\fix{1}=
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi-1\right]^{-1}
\pro{2}{0}^{\perp}\bra{\fix{0}}{\fix{0}}.
\label{integralone}
\end{equation}
This iterative scheme carries on to every order of interpolation
expansion. Consider eq. \Ref{system} at order $r\geq 2$.
The first step of the iteration is to compute $\cou{2}{0}{r-1}$
at order $r-1$. Making use of \Ref{nontrivial}, its value follows
from
\begin{eqnarray}
\cou{2}{0}{r-1}\;\obs{2}{0}&=&
-2\pro{2}{0}\bra{\fix{0}}{\pro{2}{0}^\perp\fix{r-1}}-
\nonumber\\& &\quad
\sum_{s=1}^{r-2}\pro{2}{0}\bra{\fix{s}}{\fix{r-1-s}}.
\label{coupiteration}
\end{eqnarray}
Again one splits the order $r$ interaction into
\begin{equation}
\fix{r}=\cou{2}{0}{r}\;\obs{2}{0}+
\pro{2}{0}^{\perp}\fix{r}
\end{equation}
and computes the formal orthogonal complement to the
$\phi^4$--vertex by integrating the first order differential
equations \Ref{system}. The result can be written as
\begin{eqnarray}
\pro{2}{0}^{\perp}\fix{r}&=&
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi-1\right]^{-1}
\Biggl\{\sum_{s=0}^{r-1}\pro{2}{0}^{\perp}
\bra{\fix{s}}{\fix{r-1-s}}-
\nonumber\\& &\quad
\pro{2}{0}^{\perp}\fix{r-1}\Biggr\}.
\label{otheriteration}
\end{eqnarray}
Thereafter it is time to proceed to the equations at order
$r+1$. For this scheme to work as above it is important
that the kernel of $\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -1$ be one dimensional. Otherwise we
would have to compute further order $r-1$ data from the
equations to order $r$. An example where this happens is the
$\phi^4$--trajectory in four dimensions \cite{W96}.
Although being in principle doable, the computation of this
scheme to very high orders of interpolation expansion is a
tedious enterprise. The main work is the computation of a wealth
of Feynman kernels generated in the course of iteration.
A low order analysis of this program will be presented
elsewhere. In this paper we choose to evaluate the expansion
to high orders for a sub--class of contributions in the
iteration. For this purpose we reformulate the fixed point
equation \Ref{better} into an algebraic system of equations for
a set of coupling constants. We find it interesting by its own.
It also allows to perform the expansion on a computer.
\section{Coordinate representation}
We choose a system of vertices $\Obs{i}$ labelled by elements $i$
of an index set ${\mathcal I}$. The vertices will be required to be
${\bf Z}_2$--symmetric, Euclidean invariant, and linearly independent.
They will also be required to be regular in the sense that they are
given by smooth momentum space kernels. We choose the system such
that the dilatation generator acts linearly on it through a scaling
dimension matrix
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phi\Obs{i}=\sum_{j\in{\mathcal I}}\Obs{j}\;\Sigma^{j}_{i}.
\label{scalematrix}
\end{equation}
We restrict our attention to systems with the property that the
scaling dimension matrix is diagonalizable. Non--diagonalizable
matrices will not be considered here. In this case, we can
arrange the system to consist of eigenvectors
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phi\Obs{i}=\sigma_{i}\Obs{i}.
\label{scalefield}
\end{equation}
In other words, we take $\Obs{i}$ to be a scaling field of the
trivial fixed point with scaling dimension $\sigma_i$. Recall that
such vertices are given by homogeneous momentum space kernels.
Eq. \Ref{scalematrix} says that the system closes under the action
of an infinitesimal dilatation. We also require it to close under
the action of the bilinear renormalization group form. For any two
vertices $\Obs{i}$ and $\Obs{j}$ the bilinear form \Ref{bilinear}
will be assumed to be a linear combination
\begin{equation}
\bra{\Obs{i}}{\Obs{j}}=
\sum_{k\in{\mathcal I}}\Obs{k}F^{k}_{i,j}
\label{structure}
\end{equation}
with a set of structure constants $F^{k}_{i,j}$. The scaling
dimensions $\sigma_i$ and the structure constants $F^{k}_{i,j}$
comprise all the information needed in the following about the
system of vertices. We remark that the structure constants are
well defined through
\begin{equation}
F^{k}_{i,j}\;\Obs{k}=
\Pro{k}\bra{\Obs{i}}{\Obs{j}}
\end{equation}
even when the system does not close under the bilinear form.
In this case \Ref{structure} holds only up to an error term.
Below we will indeed work with an approximation of this kind
and argue that the error term is small.
We define a coordinate representation for the interpolated
interaction in terms of a given system of vertices as
\begin{equation}
V(\phi,z)=\sum_{i\in{\mathcal I}}\Obs{i}\; V^{i}(z).
\label{coordinates}
\end{equation}
The idea is then to investigate the interpolation
\Ref{better} for the infrared fixed point by means of the
parameter dependent coordinates \Ref{coordinates}. Eq.
\Ref{better} becomes a system of algebraic equations
\begin{equation}
(\sigma_{k}-1)\; V^{k}(z) =
z\sum_{i,j\in{\mathcal I}} F^{k}_{i,j}\;V^{i}(z)\;V^{j}(z)-
z\;V^{k}(z)
\label{algebra}
\end{equation}
in the coordinate representation. The advantage of \Ref{algebra}
as compared to \Ref{better} is that we are no longer dealing with
differential equations for momentum space kernels. Their
integration is hidden in the structure constants. If the
interpolation is smooth, we can expand the coordinate functions
into power series
\begin{equation}
V^{k}(z)=\sum_{r=0}^{\infty}\;z^r\;\Cou{k}{r}.
\label{coeffs}
\end{equation}
By standard arguments \Ref{coeffs} is expected to be singular but
Borel summable. Our below evaluation of \Ref{coeffs} supports this
expectation. Eq. \Ref{coeffs} yields a solution to \Ref{algebra}
in the sense of a formal power series in $z$ if the coefficients
obey
\begin{equation}
(\sigma_{k}-1)\;\Cou{k}{r} =
\sum_{s=0}^{r-1}\sum_{i,j\in{\mathcal I}} F^{k}_{i,j}\;
\Cou{i}{s}\;\Cou{j}{r-1-s}-
\Cou{k}{r-1}
\label{computer}
\end{equation}
holds for all couplings $k\in{\mathcal I}$ to all orders $r\in{\bf N}$ of
interpolation expansion. We organize \Ref{computer} into a
recursion relation which can be solved on the computer.
To zeroth order \Ref{computer} simplifies to the linear
equation
\begin{equation}
(\sigma_{k}-1)\;\Cou{k}{0}=0.
\label{orderzero}
\end{equation}
We assume that our system of vertices contains only
one element labelled by $k=\underline{2}=(2,0)$ such that
$\sigma_{\underline{2}}=1$. This element is of course the
$\phi^4$--vertex \Ref{phifour}. All other elements are
assumed to have scaling dimensions different from one.
Then \Ref{orderzero} implies that
\begin{equation}
\Cou{k}{0}=\Cou{\underline{2}}{0}
\;\delta_{\underline{2},k}.
\end{equation}
The value of $\Cou{\underline{2}}{0}$ is as above determined
by \Ref{computer} to order one,
\begin{equation}
(\sigma_{k}-1)\;\Cou{k}{1}=
\Cou{\underline{2}}{0}
\left(F^{\underline{2}}_{\underline{2},\underline{2}}\;
\Cou{\underline{2}}{0}-\delta_{\underline{2},k}\right).
\label{orderone}
\end{equation}
Evaluating \Ref{orderone} for $k=\underline{2}$ it follows
immediately that we have
\begin{equation}
\Cou{\underline{2}}{0}=
\frac{1}{F^{\underline{2}}_{\underline{2},\underline{2}}},
\label{coupzero}
\end{equation}
besides the trivial solution $\Cou{\underline{2}}{0}=0$.
Eq. \Ref{orderone} does not tell the value of
$\Cou{\underline{2}}{1}$. But for $k\in{\mathcal I}\setminus
\{\underline{2}\}$ it gives
\begin{equation}
\Cou{k}{1}=
\frac{F^{k}_{\underline{2},\underline{2}}
\;\left(\Cou{\underline{2}}{0}\right)^2}
{\sigma_{k}-1}.
\label{otherone}
\end{equation}
Eq. \Ref{coupzero} and \Ref{otherone} are of course the
coordinate versions of \Ref{nontrivial} and \Ref{integralone}.
The strategy to any order $r>1$ is again to first compute
$\Cou{\underline{2}}{r-1}$ and thereafter
$\Cou{k}{r}$ for $k\in{\mathcal I}\setminus\{\underline{2}\}$.
The explicit formulas are
\begin{equation}
\Cou{\underline{2}}{r-1}=
-2\sum_{i\in{\mathcal I}\setminus\{\underline{2}\}}
F^{\underline{2}}_{i,\underline{2}}
\;\Cou{\underline{2}}{0}\;\Cou{i}{r-1}-
\sum_{s=1}^{r-2}\sum_{i,j\in{\mathcal I}}
F^{\underline{2}}_{i,j}\;
\Cou{i}{s}\;\Cou{j}{r-1-s}
\label{couprecursion}
\end{equation}
and
\begin{equation}
\Cou{k}{r}=
\frac{1}{\sigma_{k}-1}\left\{
\sum_{s=0}^{r-1}\sum_{i,j\in{\mathcal I}}
F^{k}_{i,j}\;\Cou{i}{s}\;\Cou{j}{r-1-s}-
\Cou{k}{r-1}\right\}
\label{otherrecursion}
\end{equation}
in complete analogy to \Ref{coupiteration} and \Ref{otheriteration}.
Thus once we know the scaling dimensions and the
structure constants, the iteration proceeds by means of purely
algebraic operations. We remark that the sums in \Ref{couprecursion}
and \Ref{otherrecursion} will be finite in the system of vertices
considered below. The reason is that the outcome of the
bilinear form of two monomials in the field is a polynomial in
the field of finite order, and consists only of connected
vertices. A very interesting question is whether it is possible
to find finite systems of vertices that close under both
\Ref{scalematrix} and \Ref{structure}. It is clear that this
cannot be achieved in terms of polynomial vertices. Unfortunately
no such system is known in three dimensions.
\section{Structure constants}
We consider the following system of vertices. First we include
a full two point vertex in derivative expansion. A convenient
notation for it is
\begin{equation}
\obs{1}{\alpha}=
\int{\rm d}^3x\;\phi(x)\;
(-\bigtriangleup)^{\alpha}\;\phi(x),
\label{twovertex}
\end{equation}
where $\alpha =0,1,2,\ldots$. Second we include local $(2n)$--point
vertices with arbitrary many external legs. They will be abbreviated
as
\begin{equation}
\obs{n}{0}=\int{\rm d}^3x\;\phi(x)^{2n},
\label{anyvertex}
\end{equation}
where $n=2,3,4,\ldots$. Notice that both \Ref{twovertex} and
\Ref{anyvertex} meet the demands stated at the beginning of the
previous section. More general interactions include also
momentum dependent higher vertices \Ref{anyvertex}. They will
not be considered here. Our index set is thus
\begin{equation}
{\mathcal I} =\{1\}\times\{\alpha\in{\bf N}\vert\alpha\geq 0\}\cup
\{n\in{\bf N}\vert n\geq 2\}\times\{0\}
\end{equation}
and $\underline{2}=(2,0)$. The bilinear form does {\sl not}
close under this set of vertices. For instance two local
vertices \Ref{anyvertex} contract in general to a bilocal
vertex. Thus if we perform an iteration \Ref{couprecursion} and
\Ref{otherrecursion} with this system of vertices, we make
a systematic error due to the truncation of the system. Our
ansatz rests upon the assumption that non--local higher vertices
are small compared to their local parts.
The scaling dimensions of \Ref{twovertex} and \Ref{anyvertex} come
out as
\begin{equation}
\sigma_{1,\alpha}=2-2\alpha ,\quad
\sigma_{n,0}=3-n.
\end{equation}
The structure constants for this set of vertices come out as follows.
Two quadratic vertices always contract again to a quadratic vertex.
The associated structure constants are computed to
\begin{equation}
F^{(1,\alpha)}_{(1,\beta),(1,\gamma)}=
4\;\frac{(-1)^{\alpha-\beta-\gamma}}{(\alpha-\beta-\gamma)!}
\;\Theta_{\alpha,\beta+\gamma},
\label{structureone}
\end{equation}
where $\Theta_{a,b}=1$ for $a\geq b$ and zero else. A quadratic
vertex and a higher vertex return upon pairing both a quadratic
vertex and a higher vertex. First we have
\begin{equation}
F^{(1,\gamma)}_{(1,\alpha),(2,0)}=
24\;\widetilde{K}_{1,\alpha}(0)\;\delta_{\gamma,0}.
\label{structuretwo}
\end{equation}
The structure constant \Ref{structuretwo} involves the one
loop integral
\begin{equation}
\widetilde{K}_{1,\alpha}(p)=
\int\frac{{\rm d}^3q}{(2\pi)^3}\;
\widetilde{v}(q)\;(q^2)^{\alpha}\widetilde{\chi}(p-q)
\label{oneloopintegral}
\end{equation}
at zero momentum. Recall that the propagators are given by
\Ref{propagators}. The exponential regulator gives a convergent
integral which is evaluated in \Ref{Oneloopintegral}. Second we
have a one loop contribution
\begin{equation}
F^{(m-1,0)}_{(1,\alpha),(m,0)}=
4m(2m-1)\;\widetilde{K}_{1,\alpha}(0)
\label{structurethree}
\end{equation}
as well as a zero loop contribution
\begin{equation}
F^{(m,0)}_{(1,\alpha),(m,0)}=
4m\;\delta_{\alpha,0}.
\label{structurefour}
\end{equation}
This last pairing also contributes to momentum dependent
higher vertices which we neglect. Two higher vertices
yield upon pairing both a quadratic vertex and higher vertices.
One quadratic term is
\begin{equation}
F^{(1,\gamma)}_{(n,0),(n,0)}=
2n(2n-1)(2n)!\;
\widetilde{K}^{(\gamma)}_{2n-2}(0)
\label{structurefive}
\end{equation}
with the $(2n-2)$--loop (the number of soft propagators $v$)
integral
\begin{equation}
\widetilde{K}_{2n-2}(p)=
\widetilde{v}\star\cdots\star\widetilde{v}\star
\widetilde{\chi}(p)
\label{loopintegral}
\end{equation}
expanded into
\begin{equation}
\widetilde{K}_{2n-2}(p)=
\sum_{\alpha=0}^{\infty} (p^2)^{\alpha}
\widetilde{K}^{\alpha}_{2n-2}(0)
\end{equation}
at zero momentum. A second quadratic term is
\begin{equation}
F^{(1,\gamma)}_{(n,0),(n-1,0)}=
(n-1)(2n)!\;\widetilde{K}_{2n-2}(0)\;\delta_{\gamma,0}.
\label{structuresix}
\end{equation}
This second term is exactly local. The general higher
vertex content of the pairing of two higher vertices is
summarized in
\begin{eqnarray}
F^{(l,0)}_{(n,0),(m,0)}&=&
\frac{(2n)!\;(2m)!}{(n+m-l-1)!(m+l-n)!(l+n-m)!}\times
\nonumber\\& &\quad
\widetilde{K}_{n+m-l-1}(0)\;
\Theta_{n+m,l+1}\;\Theta_{m+l,n}\;\Theta_{l+n,m}.
\label{structureseven}
\end{eqnarray}
This last set of structure constants \Ref{structureseven}
alone defines a local approximation for the renormalization
group fixed point. As mentioned above, the general outcome
of the pairing of two higher vertices also contains momentum
dependent vertices which are not encorporated in
\Ref{structureseven}. All other structure constants
between vertices in ${\mathcal I}$ are zero.
The one loop integral \Ref{oneloopintegral} is evaluated to
\begin{equation}
\widetilde{K}_{1,\alpha}(0)=
\frac{1}{(8\pi)^2}\;2^{1/2-\alpha}\;
\Gamma (\alpha+1/2).
\label{Oneloopintegral}
\end{equation}
The $l$--loop integral \Ref{loopintegral} is computed as
a function of the external momentum squared to
\begin{equation}
\widetilde{K}_l(p)=
(4\pi)^{-3l/2}\int_{1}^{\infty}{\rm d}\alpha_1
\cdots{\rm d}\alpha_l\;
A^{-3/2}\;\exp\left(\frac{-B}{A}p^2\right)
\end{equation}
with the abbreviations
\begin{equation}
A=\sum_{m=1}^{l+1}\prod_{n\neq m}\alpha_n,\quad
B=\prod_{m=1}^{l}\alpha_m,
\end{equation}
where $\alpha_{l+1}=1$. Its momentum derivatives at zero
can be reduced further to a one dimensional integral
\begin{equation}
\widetilde{K}_{l}^{(\alpha)}(0)=
\frac{1}{(8\pi)^l}\;
\frac{(-1)^\alpha}{\alpha !\Gamma (\alpha+3/2)}\;
\int_{0}^{\infty}{\rm d}x\;
x^{\alpha+1/2}\;e^{-x}
\left\{\frac{{\rm erf}(\sqrt{x})}{\sqrt{x}}\right\}^{l}.
\end{equation}
This remaining integral can be done explicitely at least
in the one--loop case. We evaluated it in the general case
numerically to high accuracy (45 digits) on the computer.
\section{Eigenvalue problem for critical indices}
\label{ev1}
The fixed point equation \Ref{fixed} comes together with an
eigenvalue problem
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -\lambda\right]\; W(\phi)=
2\;\bra{V(\phi)}{W(\phi)},
\label{eigenvalueproblem}
\end{equation}
defining scaling fields $W(\phi)$ and their anomalous dimensions
$\lambda$. We emphasize that $W(\phi)$ is a composite field of $\phi$.
The spectrum of anomalous dimensions is the object of principle
interest associated with a fixed point. It directly determines
the critical exponents, see Wilson and Kogut \cite{WK74}.
The interpolation \Ref{better} is accompanied by an interpolation
of \Ref{eigenvalueproblem}, given by
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -\lambda (z)\right]\; W(\phi,z)=
2\;z\;\bra{V(\phi,z)}{W(\phi,z)}.
\label{intereigen}
\end{equation}
Eq. \Ref{intereigen} can be solved by means of perturbation theory.
We expand not only the interaction \Ref{pertur}, but also the
scaling field and its anomalous dimension into power series in the
interpolation parameter,
\begin{equation}
W(\phi,z)=\sum_{r=0}^{\infty}\; z^r\;\eig{r},\quad
\lambda(z)=\sum_{r=0}^{\infty}\; z^r\;\spe{r}.
\label{eigenseries}
\end{equation}
We interpret \Ref{eigenseries} in the
sense of a formal power series. It yields a solution to
\Ref{intereigen} if the coefficients satisfy the system of
differential equations
\begin{equation}
\left({\mathcal D}\phi,\frac{\delta}{\delta\phi\eig{r}-\sum_{s=0}^{r}\spe{s}\eig{r-s}=
2\sum_{s=0}^{r-1}\bra{\fix{s}}{\eig{r-1-s}}.
\label{spectrumseries}
\end{equation}
This system can be integrated iteratively. To order zero,
\Ref{spectrumseries} becomes the eigenvalue problem
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -\spe{0}\right]\; \eig{0}=0.
\end{equation}
The zeroth order $\eig{0}$ thus has to be a scaling field of the
trivial fixed point, and $\spe{0}$ has to be its scaling dimension.
With each scaling field of the trivial fixed point is therefore
associated in perturbation theory a scaling field of the non--trivial
fixed point.
Let us consider for definiteness the perturbation associated with
a mass term
\begin{equation}
\eig{0}=\obs{1}{0},\quad\obs{1}{0}=\int{\rm d}^3x\;\phi(x)^2.
\label{massterm}
\end{equation}
Then the zeroth order eigenvalue is of course $\spe{0}=\sigma_{1,0}=2$.
As a perturbation of the non--trivial fixed point, \Ref{massterm} turns
out to be relevant. The associated non--trivial renormalized trajectory
in the sense of \cite{W96} describes the renormalization group flow of a
non--trivial massive field theory. Associated with it is the critical
exponent
\begin{equation}
\nu =\frac{1}{\lambda}.
\label{nu}
\end{equation}
The mass perturbation \Ref{massterm} is non--degenerate in the sense
that the kernel of $\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -2$ is one dimensional. The formal orthogonal
projector on this one dimensional kernel is $\pro{1}{0}$. Another
non--degenerate perturbation is the scaling field associated with
$\obs{2}{0}$. The ones associated with $\obs{1}{1}$ and $\obs{3}{0}$
on the other hand form a degenerate marginal duplet. We will
restrict our attention to the non--degenerate case for the sake
of notational economy. The kernel of $\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -\spe{0}$ will thus be
assumed to be one dimensional. The formal orthogonal projector on
this rank one kernel will be denoted by ${\mathcal P}$. To first order
\Ref{spectrumseries} becomes the differential equation
\begin{equation}
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -\spe{0}\right]\;\eig{1}-
\spe{1}\eig{0}=2\bra{\fix{0}}{\eig{0}}.
\label{firstspectrum}
\end{equation}
The first order correction to the eigenvalue follows from
\Ref{firstspectrum} by projection with ${\mathcal P}$. We have
that
\begin{equation}
\spe{1}\;\eig{0}=
-2{\mathcal P}\bra{\fix{0}}{\eig{0}}.
\label{sigmaone}
\end{equation}
Spelled out explicitely for the mass perturbation \Ref{massterm},
eq. \Ref{sigmaone} says that
\begin{equation}
\spe{1}=-48\;\widetilde{\chi}\star\widetilde{v}(0)\;
\cou{2}{0}{0}=\frac{-1}{3}.
\label{firstnu}
\end{equation}
It is amusing that this first order correction can be inferred
without having to compute the convolution integral, because the
convolution integral in \Ref{firstnu} is canceled exactly by the one in
\Ref{nontrivial}. Next we impose the normalization condition
that
\begin{equation}
{\mathcal P}\eig{1}=0.
\label{spectrumnormalization}
\end{equation}
This condition is appropriate in the non--degenerate case because
\Ref{sigmaone} already takes care of the ${\mathcal P}$--information
contained in \Ref{firstspectrum}. The orthogonal complement is
then integrated as in the case of the interaction. The outcome is
\begin{equation}
{\mathcal P}^{\perp}\eig{1}=
2\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -\spe{0}\right]^{-1}
{\mathcal P}^{\perp}\bra{\fix{0}}{\eig{0}}.
\end{equation}
This scheme carries on immediately to every order of interpolation
expansion. Projecting \Ref{spectrumseries} to ${\mathcal P}$, we first
deduce that
\begin{equation}
\spe{r}\eig{0}=
2\sum_{s=0}^{r-1}{\mathcal P}\bra{\fix{s}}{\eig{r-1-s}}.
\end{equation}
This equation determines the order $r$ eigenvalue in terms of lower
order data. Generalizing \Ref{spectrumnormalization}, we impose the
normalization condition
\begin{equation}
{\mathcal P}\eig{r}=0
\end{equation}
for $r\geq 1$. Then to order $r$ we are left with the computation
of
\begin{eqnarray}
{\mathcal P}^{\perp}\eig{r}&=&
\left[\left({\mathcal D}\phi,\frac{\delta}{\delta\phi -\spe{0}\right]^{-1}
\Biggl\{\sum_{s=1}^{r-1}\spe{s}{\mathcal P}^{\perp}\eig{r-1}+
\nonumber\\& &\quad
2\sum_{s=0}^{r-1}{\mathcal P}^{\perp}\bra{\fix{s}}{\eig{r-1-s}}.
\Biggr\}
\label{spectrumintegral}
\end{eqnarray}
This scheme iterates to every order of interpolation expansion.
Recall that the inverse of the dilatation generator in
\Ref{spectrumintegral} involves the integration of a first order
partial differential equation. As in the case of the fixed point,
the explicit computation of this program to very high orders
requires considerable computational resources. In this paper we
restrict our attention to a partial resummation by means of our
coordinate representation.
\section{Eigenvalue problem in coordinates}
\label{ev2}
In this section we perform the interpolation expansion for
the eigenvalue problem \Ref{intereigen} in the coordinate
representation. The coordinate representation for the scaling
fields reads
\begin{equation}
W(\phi,z)=\sum_{i\in{\mathcal I}}\Obs{i}\;W^{i}(z).
\label{cooeig}
\end{equation}
In the coordinate representation, the eigenvalue problem
\Ref{intereigen} becomes a set of algebraic equations
\begin{equation}
\left(\sigma_{k}-\lambda (z)\right)
W^{k}(z)=2z\sum_{i,j\in{\mathcal I}}F^{k}_{i,j}\;V^{i}(z)\;
W^{j}(z).
\label{eigenalgebra}
\end{equation}
It can be solved recursively in an interpolation expansion
\begin{equation}
W^{k}(z)=\sum_{r=0}^{\infty}\;z^r\;\Coo{k}{r}.
\label{scalepert}
\end{equation}
The power series \Ref{coeffs}, \Ref{eigenseries}, and
\Ref{scalepert} yield a solution to \Ref{eigenalgebra} provided
that the coefficients satisfy
\begin{equation}
\left(\sigma_k-\spe{0}\right)\;\Coo{k}{r}-
\spe{r}\;\Coo{k}{0}=
\sum_{s=1}^{r-1}\spe{s}\;\Coo{k}{r-s}+
2\sum_{s=0}^{r-1}\sum_{i,j\in{\mathcal I}}
F^{k}_{i,j}\;\Cou{i}{s}\;\Coo{j}{r-1-s}.
\label{scaleeigen}
\end{equation}
As in the case of the interaction, the system of equations
\Ref{scaleeigen} can be organized into a recursion relation. To
order zero \Ref{scaleeigen} reads
\begin{equation}
\left(\sigma_k-\spe{0}\right)\Coo{k}{0}=0.
\end{equation}
It tells us that we should select one of the $k\in{\mathcal I}$ as
zeroth order eigenvector. We choose $\underline{1}=(1,0)$
for definiteness. Then the zeroth order is
\begin{equation}
\Coo{k}{0}=\delta_{\underline{1},k},\quad
\spe{0}=\sigma_{\underline{1}}=2.
\end{equation}
The only $k$ with $\sigma_k=2$ is $k=\underline{1}$. We will
again restrict our attention to this non--degenerate case. The
below recursion relation is valid for general non--degenerate
perturbations, with minor notational changes. The first order
equation in the system \Ref{scaleeigen} is given by
\begin{equation}
\left(\sigma_k-\spe{0}\right)\;\Coo{k}{1}-
\spe{1}\Coo{k}{0}=2\;F^{k}_{\underline{2},\underline{1}}
\Cou{\underline{2}}{0}.
\end{equation}
Therefrom it follows that the first order correction to the
eigenvalue is in the coordinate representation
\begin{equation}
\spe{1}=-2\;F^{\underline{1}}_{\underline{2},\underline{1}}
\Cou{\underline{2}}{0}.
\end{equation}
We remark that in the degenerate case, the degeneracy is
typically lifted by the first order correction to the
eigenvalue. The other coefficients to first order are
\begin{equation}
\Coo{\underline{1}}{1}=0
\end{equation}
and, for $k\in{\mathcal I}\setminus\{\underline{1}\}$,
\begin{equation}
\Coo{k}{1}=
\frac{2\;F^{k}_{\underline{2},\underline{1}}}{\sigma_k-\spe{0}}.
\end{equation}
This computation generalizes immediately to higher orders. The
formula for the order $r$ eigenvalue in terms of lower order data
is
\begin{equation}
\spe{r}=-2\sum_{s=0}^{r-1}\sum_{i,j\in{\mathcal I}}
F^{\underline{1}}_{i,j}\;\Cou{i}{s}\;\Coo{j}{r-1-s}.
\label{recursion1}
\end{equation}
The order $r$ eigenvector is then given by
\begin{equation}
\Coo{\underline{1}}{r}=0,
\label{recursion2}
\end{equation}
for $r\geq 1$, together with
\begin{equation}
\Coo{k}{r}=\frac{1}{\sigma_{k}-\spe{0}}
\Biggl\{\sum_{s=1}^{r-1}\;\spe{s}\;\Coo{k}{r-s}+
2\sum_{s=0}^{r-1}\sum_{i,j\in{\mathcal I}}F^{k}_{i,j}\;
\Cou{i}{s}\;\Coo{j}{r-1-s}\Biggr\},
\label{recursion3}
\end{equation}
for $k\in{\mathcal I}\setminus\{\underline{1}\}$. Eq. \Ref{recursion1},
\Ref{recursion2}, and \Ref{recursion3} define a recursive
perturbation expansion for the critical indices of the non--trivial
fixed point.
\section{Computation of the recursions}
We computed the $z$--expansion for the potential recursively by
means of (\ref{couprecursion}) and (\ref{otherrecursion}), and
for the eigenvalue problem by means of (\ref{recursion1}) and
(\ref{recursion3}) using computer algebra. We restricted our
attention to the case of three dimensions. It turned out to be
crucial to compute the structure coefficients to high accuracy.
We calculated them to an accuracy of 45 digits with Maple V.
The perturbation expansion was performed up to a maximal order
of 25. The derivative expansion was performed up to $\alpha_{m\! a\! x}=20$
orders of $p^2$ in the 2-point vertex. Table \ref{table1} shows
the series for the $\phi^4$--coupling both in the ultra--local
approximation $\alpha_{m\! a\! x}=0$ and for $\alpha_{m\! a\! x}=4$ up to the order $z^{11}$.
\begin{table}[htpb]
\begin{center}
\leavevmode
\begin{tabular}[r]{|r|r@{$\cdot$}l|r@{$\cdot$}l|}\hline
\multicolumn{1}{|c|}{$n$} &
\multicolumn{2}{c|}{$\cou{2}{0}{n}$, $\alpha_{m\! a\! x}=0$} &
\multicolumn{2}{c|}{$\cou{2}{0}{n}$, $\alpha_{m\! a\! x}=4$} \\ \hline
0 & 2.1874 & $10^{-1}$ & 2.1874 & $10^{-1}$ \\
1 & 4.5814 & $10^{-1 }$& 4.5814 & $10^{-1}$ \\
2 & -8.7171 & $10^{-1 }$& -8.6761 & $10^{-1}$ \\
3 & 4.6575 & $10^{0 }$& 4.6815 & $10^{0}$ \\
4 & -4.0553 & $10^{1 }$& -4.0546 & $10^{1}$ \\
5 & 4.2980 & $10^{2 }$& 4.2992 & $10^{2}$ \\
6 & -5.2117 & $10^{3 }$& -5.2130 & $10^{3}$ \\
7 & 7.0118 & $10^{4 }$& 7.0133 & $10^{4}$ \\
8 & -1.0267 & $10^{6 }$& -1.0269 & $10^{6}$ \\
9 & 1.6155 & $10^{7 }$& 1.6158 & $10^{7}$ \\
10 & -2.7080 & $10^{8 }$& -2.7084 & $10^{8}$ \\
11 & 4.8059 & $10^{9 }$& 4.8066 & $10^{9}$ \\ \hline
\end{tabular}
\parbox[t]{\textwidth}
{
\caption[]{\label{table1}
\sl Examples for the behaviour of the expansion coefficients.}
}
\end{center}
\end{table}
The coefficients prove to increase in absolute value proportional
to $C^n n!$ with some constant $C$. Their signs alternate. From
this behavior we conclude that the series does not converge but
is Borel summable. A proof of local Borel summability will be
presented elsewhere. The constant $C$ is related to an instanton
singularity of the Borel transform on the negative real axis. It
can be seen as an accumulation point of poles when the series is
converted into various Pade approximants. The derivative expansion
on the other hand proves to converge. This is illustrated in
table \ref{table2} for two values of $\alpha_{m\! a\! x}$.
\begin{table}[htpb]
\begin{center}
\leavevmode
\begin{tabular}[r]{|r|r@{$\cdot$}l|r@{$\cdot$}l|}\hline
\multicolumn{1}{|c|}{$\alpha$} &
\multicolumn{2}{c|}{$\cou{2}{\alpha}{10}$, $\alpha_{m\! a\! x}=5$} &
\multicolumn{2}{c|}{$\cou{2}{\alpha}{10}$, $\alpha_{m\! a\! x}=10$} \\ \hline
0 & -1.08730107 & $10^{5}$ & -1.08730280 & $10^{5 }$\\
1 & -7.30254527 & $10^{4 }$& -7.30254464 & $10^{4 }$\\
2 & 9.42673139 & $10^{3 }$& 9.42673053 & $10^{3 }$\\
3 & -1.47739400 & $10^{3 }$& -1.47739385 & $10^{3 }$\\
4 & 2.14057814 & $10^{2 }$& 2.14057791 & $10^{2 }$\\
5 & -2.79810089 & $10^{1 }$& -2.79810056 & $10^{1 }$\\
6 & 0.00000000 & $10^{0 }$& 3.33201984 & $10^{0 }$\\
7 & 0.00000000 & $10^{0 }$& -3.70314164 & $10^{-1 }$\\
8 & 0.00000000 & $10^{0 }$& 3.99772841 & $10^{-2 }$\\
9 & 0.00000000 & $10^{0 }$& -4.43094961 & $10^{-3 }$\\
10 & 0.00000000 & $10^{0 }$& 5.30798357 & $10^{-4 }$\\ \hline
\end{tabular}
\parbox[t]{\textwidth}
{
\caption[]{\label{table2} \sl
Examples for the behaviour of the mass coefficients at order
$z^{10}$ for $\alpha_{m\! a\! x}=5$ and $\alpha_{m\! a\! x}=10$.}
}
\end{center}
\end{table}
We note in passing that the difference between $\alpha_{m\! a\! x}=5$ and $\alpha_{m\! a\! x}=10$
is small.
The spectrum of the non--trivial fixed is computed along the strategy
explained in section \ref{ev1} and \ref{ev2}. It requires as an input
the fixed point interaction in $z$--expansion. We evaluated it for
all values of $\alpha_{m\! a\! x}$ inbetween zero and twenty. In the following we
will concentrate on an estimate of the critical index $\nu$ (\ref{nu})
by resummation of the series for all these twenty one approximations.
We computed the series by means of (\ref{recursion1}) and
(\ref{recursion3}) to order twenty five of $z$--expansion.
Table \ref{table3} shows as an example the series for the eigenvalue
$\lambda$ in the ultra--local case $\alpha_{m\! a\! x}=0$ and in the case of $\alpha_{m\! a\! x}=4$
up to the order twelve of perturbation theory.
\begin{table}[htpb]
\begin{center}
\leavevmode
\begin{tabular}[r]{|r|r@{$\cdot$}l|r@{$\cdot$}l|}\hline
\multicolumn{1}{|c|}{$n$} &
\multicolumn{2}{c|}{$\la{n}$, $\alpha_{m\! a\! x}=0$} &
\multicolumn{2}{c|}{$\la{n}$, $\alpha_{m\! a\! x}=4$} \\ \hline
0 & 2.000000 &$10^{0}$ & 2.000000 &$10^{0}$ \\
1 & -3.333333 &$10^{-1}$ & -3.333333 &$10^{-1}$ \\
2 & -3.490659 &$10^{-1}$ & -3.490659 &$10^{-1}$ \\
3 & 1.148993 &$10^{0}$ & 1.159189 &$10^{0}$ \\
4 & -7.414413 &$10^{0}$ & -7.369227 &$10^{0}$ \\
5 & 6.358855 &$10^{1}$ & 6.358630 &$10^{1}$ \\
6 & -6.649232 &$10^{2}$ & -6.646081 &$10^{2}$ \\
7 & 7.999490 &$10^{3}$ & 7.996744 &$10^{3}$ \\
8 & -1.070838 &$10^{5}$ & -1.070532 &$10^{5}$ \\
9 & 1.562548 &$10^{6}$ & 1.562166 &$10^{6}$ \\
10 & -2.452524 &$10^{7}$ & -2.452002 &$10^{7}$ \\
11 & 4.103373 &$10^{8}$ & 4.102601 &$10^{8}$ \\
12 & -7.271917 &$10^{9}$ & -7.270698 &$10^{9}$ \\ \hline
\end{tabular}
\parbox[t]{\textwidth}
{
\caption[]{\label{table3}\sl
Series coefficients for $\lambda$ up to
order 12 for $\alpha_{m\! a\! x}=0$ and $\alpha_{m\! a\! x}=4$.}
}
\end{center}
\end{table}
Again the series alternate, and the coefficients grow in absolute
value as $C^n n!$. The series are therefore not expected to converge.
We remark that a proof thereof is however missing. The Borel transform
of a series with this asymptotics has a finite radius of analyticity
$R_{\alpha_{m\! a\! x}}$. It is determined by an instanton
singularity on the negative real axes of the complex Borel plane.
This radius of analyticity is an interesting quantity. It can be
investigated by a number of methods, see \cite{DI89} and references
therein. One of them is the
Pad\'{e} method. Recall that the Pad\'{e} approximant of order $(l,m)$ for
a function $f$ is a rational function $f_{l,m}(z)=\frac{P_l(z)}{Q_m(z)}$.
Here $P_l$ and $Q_m$ are polynomials of degree $l$ and $m$ respectively,
determined such that the taylor expansions of $f$ and $f_{l,m}$ agree up
to order $z^{l+m}$. One then observes that the poles of the various
possible Pad\'{e} approximants accumulate around a cut or a singularity of
$f$. With the Pad\'{e} method
we found
\begin{equation}
\label{conv}
R_{\alpha_{m\! a\! x}} = 0.88 \pm 0.02 \, ,
\end{equation}
with no significant dependence of $\alpha_{m\! a\! x}$. Figure \ref{fig1} shows a
plot of all poles of all Pad\'{e} approximants $(B\lambda)_{l,m}$ with
$l+m=25$ in the complex Borel plane for the two cases $\alpha_{m\! a\! x}=0$ and
$\alpha_{m\! a\! x}=4$ respectively.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfxsize=12cm
\epsffile{radius.ps}
\caption{\sl Radius of convergence of $B\lambda$ by the Pad\'{e} method for
$\alpha_{m\! a\! x}=0$ and $\alpha_{m\! a\! x}=4$.}
\label{fig1}
\end{center}
\end{figure}
Here $B\lambda$ denotes the Borel transform of $\lambda$. As expected, the
poles accumulate on the negative real axes. Notice however that there are
many spurious singularities on and nearby the positive real axes. These
spurious poles endanger the inverse Borel transform as a contour integral
along the positive real axis. The pictures for $\alpha_{m\! a\! x}=0$ and $\alpha_{m\! a\! x}=4$ show
tiny differences. For instance, the poles on the positive real axis are
not on fixed locations and can therefore be regarded as spurious.
\section{Determination of $\nu$}
\label{sec:nu}
To compute the value of the critical index $\nu$, we have to evaluate
the $z$--expansion at $z=1$. Naive evaluation does not give a meaningful
answer since the expansion does not converge. Therefore we had to rely
on resummation technology. A review of series resummation and references
to the original literature is given in \cite{ZJ89} and \cite{DI89}.
We tried four standard
methods and compared the results.
First we computed (for all values of $\alpha_{m\! a\! x}$ between zero and twenty) all
Pad\'{e} approximants $(\lambda)_{l,m}$ with $l+m \leq 25$, and evaluated
them at $z=1$. These values are conveniently displayed in a Pad\'{e} table
($l$-$m$ grid). To get an idea for the value of $\lambda$ and
an estimate for the error we computed the mean value and deviation
for the lines of fixed order in $z$ ($l+m=\mbox{const}$) in these
diagrams after having discarded all values below a
lower value $\lambda_{min}$ and above an upper value $\lambda_{max}$.
The idea thereof is that large
deviations come from spurious singularities. We were careful not to choose
the window too narrow. Our error estimate should be regarded as rather
pessimistic.
If these mean values converge with increasing order
in $z$ we use them and
an inspection of the whole table to find an estimate
for the value of $\nu$.
In the second method (Dlog) one computes the Pad\'{e} approximants
for the logarithmic derivative $\frac{\lambda'(z)}{\lambda(z)}$.
$\lambda$ is then reconstructed as the exponential of an
integral
\begin{equation}
\label{dlog}
\lambda_{l,m} = \lambda(0) e^{\int_0^1 dz
\left( \frac{d}{dz} \log\lambda(z)\right)_{l,m}} \, .
\end{equation}
The integration can be performed numerically to high accuracy.
The Dlog method is particularly efficient when the singularity is of
the type $\lambda(z) = \frac{A}{(x-x_c)^{\gamma}}$ with a nonintegral
exponent $\gamma$.
The third proposal is to use a Pad\'{e} approximants for the Borel
transform of the series. The Borel transform of a
power series $f(x) = \sum_{n\geq 0} f_n x^n$ is defined by
\begin{equation}
\label{bp}
(B f)(z) = \sum_{n\geq 0} \frac{f_n}{n!} z^n \, .
\end{equation}
The Borel transform of power series with
finite radius of convergence defines an analytic continuation of
the function to a maximal simplex through the integral
\begin{equation}
\label{bpback}
f(x) = \int_0^{\infty} dt e^{-t} (B f)(x t).
\end{equation}
Again we get Pad\'{e} tables of approximants for $\lambda$ by numerically
integrating this back transformation for various Pad\'{e} approximants of
$B\lambda$.
The off diagonal estimates in these tables can be improved by using
information on the analyticity properties of the Borel transform.
$B f$ could for instance have a cut along $(-\infty, -R]$ on the negative
real axes. Let us assume that this is indeed the case (with $R=0.88\pm
0.02$). Then the cut plane can be mapped conformally via
\begin{equation}
\label{conf}
u(z) = \frac{\sqrt{z/R+1}-1}{\sqrt{z/R+1}+1} \, ;
\quad
z = \frac{4Ru}{(1-u)^2}
\end{equation}
onto the unit circle. Under this mapping $(Bf)(z)$ transforms to
$(\tilde{B f})(u)$. We then use Pad\'{e} approximants for the mapped series.
A Pad\'{e} table for $\lambda$ is obtained via the inverse transformation
\begin{equation}
\label{bpconfback}
\lambda_{l,m} = 4R\int_0^1 du \frac{1+u}{(1-u)^3}
e^{-\frac{4Ru}{(1-u)^2}}(\tilde{B f})_{l,m}(u) \, .
\end{equation}
The outcome of this method relies on a careful estimate of the radius of
convergence of the Borel transform. For each $\alpha_{m\! a\! x}$, we calculated three
estimates for $\lambda$, one for our estimated value of $R$ and one for
$R+\Delta R$ and $R-\Delta R$ respectively, where $\Delta R$ means the
error in our estimate for the error of the radius. The inspection of all
three Pad\'{e} tables yields $\lambda$ and an error estimate.
We also tried out inhomogenous differential approximants, but we
could see no improvement as compared with Pad\'{e} or Dlog Pad\'{e}
approximants. The integration of the differential equations in this
method turned out to be both time consuming and fragile due to
the poles close to the origin.
In table \ref{table4} we summarize our results for $\nu$ for the different
values of $\alpha_{m\! a\! x}$. The errors refer as usually to the last digit.
\begin{table}[htpb]
\begin{center}
\leavevmode
\begin{tabular}[r]{|r|l|l|l|l|}\hline
\multicolumn{1}{|c|}{$\alpha_{m\! a\! x}$} &
\multicolumn{1}{c|}{$\nu$ Pad\'{e} } &
\multicolumn{1}{c|}{$\nu$ Dlog} &
\multicolumn{1}{c|}{$\nu$ BP} &
\multicolumn{1}{c|}{$\nu$ BPconf} \\ \hline
0 & 0.6630(20) & 0.6640(10) & 0.6599(30) & 0.6630(30) \\
1 & 0.6200(150) & 0.6180(50) & 0.6150(70) & 0.6100(10) \\
2 & 0.6340(130) & 0.6290(40) & 0.6264(18) & 0.6300(10) \\
3 & 0.6300(110) & 0.6220(40) & 0.6220(70) & 0.6200(40) \\
4 & 0.6320(90) & 0.6260(30) & 0.6286(68) & 0.6260(10) \\
5 & 0.6300(100) & 0.6260(40) & 0.6240(50) & 0.6256(10) \\
6 & 0.6330(100) & 0.6260(40) & 0.6290(40) & 0.6270(10) \\
7 & 0.6310(110) & 0.6280(70) & 0.6220(60) & 0.6260(20) \\
8 & 0.6330(90) & 0.6230(70) & 0.6310(60) & 0.6266(4) \\
9 & 0.6290(90) & 0.6260(80) & 0.6230(60) & 0.6220(40) \\
10 & 0.6330(90) & 0.6260(40) & 0.6320(70) & 0.6286(3) \\
11 & 0.6300(110) & 0.6260(50) & 0.6200(150) & 0.6310(50) \\
12 & 0.6310(150) & 0.6260(60) & 0.6340(60) & 0.6305(25) \\
13 & 0.6280(120) & 0.6260(80) & 0.6200(200) & 0.6350(60) \\
14 & 0.6360(110) & 0.6270(50) & 0.6410(80) & 0.6440(30) \\
15 & 0.6330(100) & 0.6250(60) & 0.6200(180) & 0.6440(30) \\
16 & 0.6340(200) & 0.6310(40) & 0.6520(50) & 0.6300(50) \\
17 & 0.6250(150) & 0.6270(200) & 0.6320(200) & 0.6259(40) \\
18 & 0.6380(60) & 0.6380(110) & 0.6549(58) & 0.6590(160) \\
19 & 0.6420(150) & 0.6000(400) & 0.6060(130) & 0.6240(70) \\
20 & 0.6420(80) & 0.6420(200) & 0.6550(40) & 0.6430(160) \\ \hline
\end{tabular}
\parbox[t]{\textwidth}
{
\caption[]{\label{table4}\sl
Results for the critical exponent $\nu$ with the
Pad\'{e} method, the Dlog Pad\'{e} method (Dlog), the Borel Pad\'{e} method (BP)
and the Borel Pad\'{e} method with conformal mapping (BPconf) for
various orders of the derivative expansion of the 2-point vertex.}
}
\end{center}
\end{table}
We come to the following conclusions. The Pad\'{e} method is the least precise
one with an error of about 0.01. From it we can get an idea about the value
of $\nu$, but no accurate estimate. The errors of the Dlog Pad\'{e} method
(Dlog) and the Borel Pad\'{e} method (BP) are of comparable size. The Borel
Pad\'{e} method with conformal mapping (BPconf) has the least errorbars. At higher
orders of the derivative expansion of the 2-point vertex the errors increase
significantly. To display this effect, we have plotted the data of table
\ref{table4} in figure \ref{fig2}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfxsize=14cm
\epsffile{nu.ps}
\caption{\sl The critical index $\nu$ as a function of
the order of the derivative expansion of the
2-point function $\alpha_{m\! a\! x}$ for the four extrapolation
methods.}
\label{fig2}
\end{center}
\end{figure}
One can see that the values for $\nu$ oscillate around a mean value.
Up to a certain order this sequence seems to converge. Thereafter, the
difference between $\nu(\alpha_{m\! a\! x})$ and $\nu(\alpha_{m\! a\! x} + 1)$ and the error grows.
We believe this effect to be the consequence of a numerical instability.
In high orders of derivative expansion and high orders of perturbation
theory one is dealing with numbers of enormously varying magnitudes
(in our case one hundred orders). In practice we computed our series
to an accuracy of 45 digits, and a problem arises in the cancellation
of large numbers in the course of the recursion. A more destructive
explanation would be that the resummation fails to produce a convergent
derivative expansion, or even more desastrous that the non--perturbative
kernels are not analytic functions of the momenta. The final answer to
this question can only be given on the basis of a non--perturbative
construction of the fixed point and is outside the scope of this paper.
Our insight comes from the evaluation of various approximants to
different orders of accuracy.
We confine our further discussion to those values of $\alpha_{m\! a\! x}$ which lie
before the onset of instability. The Dlog method and the BPconf method
both yield nearly constant values for $\nu$ at orders between $\alpha_{m\! a\! x}=4$
and $\alpha_{m\! a\! x}=12$ and between $\alpha_{m\! a\! x}=4$ and $\alpha_{m\! a\! x}=8$ respectively. We propose
this value to be the limit of $\nu$ at arbitrary order of the derivative
expansion.
Consider the data for the ultra--local case $\alpha_{m\! a\! x}=0$ and to first order
$\alpha_{m\! a\! x}=1$ of derivative expansion. For the ultra--local case,
which can be compared with the hierarchical model ($\nu=0.6501625$,
\cite{KW88}) we find
$\nu=0.6625(33)$ which is bigger than the full critical index.
Disregarding the pure Pad\'{e} estimate, we get for $\alpha_{m\! a\! x}=1$
the result $\nu = 0.6144(62)$. This value is considerably lower than
the value at $\alpha_{m\! a\! x}=0$ and even lower than the full critical index.
In view of the tiny differences between the fixed point coefficients
at $\alpha_{m\! a\! x}=0$ and $\alpha_{m\! a\! x} \neq 0$, we find this surprising. Compare for
example the coefficients in table \ref{table1}.
At higher orders of derivative expansion, the values for $\nu$ oscillate
and converge to a mean value. The limit value has been determined
as the mean values of $\nu$ over the nearly constant plateaus.
As best estimate for the BPconf method we get
$\nu = 0.6262(13)$. The Dlog method yields $\nu=0.6259(57)$. These results
should be compared with the critical index $\nu$ of the three dimensional
Ising model in the literature. In table \ref{table5} we list a few
results for $\nu$.
\begin{table}[htbp]
\begin{center}
\leavevmode
\begin{tabular}{|l|l|c|}\hline
\multicolumn{1}{|c|}{$\nu$} &
\multicolumn{1}{c|}{Method} &
\multicolumn{1}{c|}{Literature} \\ \hline
0.6300(15) & three dimensional renormalization group & \cite{GZJ80}\\
0.6298(7) & & \cite{BB85}\\
0.630 & & \cite{N91}\\ \hline
0.6305(25) & renormalization group, $\epsilon$-expansion & \cite{GZJ85}\\ \hline
0.6301 & high temperature series & \cite{R95}\\
0.6300(15) & high temperature series for bcc-grid & \cite{NR90}\\ \hline
0.6289(8) & Monte-Carlo methods & \cite{FL91}\\
0.6301(8) & & \cite{BLH95}\\ \hline
0.625(1) & Monte-Carlo renormalization group & \cite{GT96}\\ \hline
0.626(9) & Scaling-field method & \cite{NR84}\\ \hline
\end{tabular}
\caption{\sl Results for the critical exponent $\nu$ of the full model.}
\label{table5}
\end{center}
\end{table}
A comprehensive article on this issue is \cite{BLH95}. It also contains
an overview of experimental data. With series expansion and Monte-Carlo
methods one gets $\nu=0.630$. On the other hand the Monte-Carlo
renormalization group suggests $\nu = 0.625$. This gap is object of
current discussions. Our value is closest to the value of
\cite{NR84} and \cite{GT96}.
\section{Summary and discussion}
\label{sec:disc}
In this article we investigated a form of Wilsons infinitesimal
renormalization group. The starting point was equation \Ref{fixed}.
We found a practical way to solve the equation in a systematic manner.
The central idea was to introduce an interpolating parameter $z$,
which continuously turns on the non--linear term in \Ref{fixed}.
Everything was expanded in this parameter. The interpolation was
arranged such that the zeroth order is a $\phi^4$--vertex.
The expansion was presented both in a coordinate free representation
and in coordinate form, where the interaction is expanded in a basis of
vertices. As a basis we advocated the use of a full two point interaction
in derivative expansion together with local vertices of any power of
fields. Derivative interactions of higher powers were neglected. The
basis of interactions came encoded in a system of scaling dimensions and
structure constants. Their evaluation was reduced to a one dimensional
Feynman integral which we evaluated numerically. We reformulated our
expansion into recursive equations for the fixed point interaction,
its scaling fields, and their anomalous dimensions. We performed a
detailed analysis of the series for the critical exponent associated
with a massive perturbation of the fixed point. The result is a new
and independent calculation of the critical index $\nu$ of the
the three dimensional Ising model. We solved the recursion relations
for the eigenvalue problem up to high orders and analyzed the resulting
series by means of four different extrapolation methods. Our best
estimator for the critical index $\nu$ is $\nu = 0.6262(13)$. We
compared our results with values for the critical exponent $\nu$ known
in the literature.
The results encourage us to further investigations. On the menu of
open problems we have the inclusion of momentum dependent higher
vertices for the scalar model, theoretical estimates on the
$z$--expansion, and the generalization to vector and matrix models.
We hope to return with accurate data on their critical properties
by means of $z$--expansion in the near future.
|
2205.13475
|
\section{Introduction}\label{sec-intro}
\input{introduction}
\section{The miniDAQ system}
\input{miniDAQ}
\section{Results from cosmic-ray studies}
\input{cosmic_studies}
\section{Conclusions}
\input{conclusions}
\section{Acknowledgement}
The work is supported by the US National Science Foundation (NSF) and the US Department of Energy (DOE) under contracts PHY1948993 (NSF) and DE-SC007859 (DOE).
\bibliographystyle{model1-num-names}
\subsection{Introduction of the MDT front-end and back-end electronics}
Each MDT tube has a diameter of 3 cm and a wall thickness of 400 $\mu$m. The central tungsten wire has a diameter of 50 $\mu$m. Tubes are operated with a gas mixture of Ar/CO$_2$ (93\%/7\%) at 3 bar absolute pressure. For each track, the electrons from primary ionization clusters drift to the central wire along radial lines. The induced signal propagates along the wire where it is read out by the MDT front-end electronics. The difference between the earliest arrival time of the signal at the wire and the reference time provided by trigger chambers gives the drift time of the muon hit, and this drift time is used to determine the drift radius.
The signal from each tube is first processed by a custom-designed Amplifier/Shaper/Discriminator (ASD) ASIC~\cite{Kroha:2016fid, DeMatteis:2017xky}. A discriminator is used to determine the signal arrival time, the time when the signal crosses a predefined threshold. This time depends on the signal pulse height which results in a degradation of the time resolution. The resolution degradation can partially be recovered by applying a time skew correction using the integrated charge of the signal pulse. A Wilkinson analog-to-digital converter (ADC) is introduced inside the ASD to integrate the signal pulse over a predefined integration window of $\sim 20$ ns. The total collected charge is measured by the discharge time of a capacitor by a rundown current. The signal arrival time (also called the leading edge time) and the discharge time (also called the trailing edge time) are converted into ADC counts using a time-to-digital converter (TDC) ASIC with a bin size of 0.78 ns~\cite{Wang:2017jnd, Liang:2019weg, Guo:2020zyb}.
To avoid multiple hits from multiple threshold crossings of a single signal, the ASD ASIC can be programmed with a dead time of $\sim 1~\mu$s. After the detection of the earliest arrival signal, there are no additional time measurements performed within this dead time.
Each ASD ASIC can handle 8 tubes and each TDC can handle discriminated signals from three ASDs. A mezzanine card with three ASDs and one TDC thus handles 24 tubes. A Chamber Service Module (CSM) multiplexes data from up to 18 mezzanine cards and sends these data via an optical module (VTRx+)~\cite{Soos:2017stv} to the MDT Trigger Processor~\cite{Cieri:2020bfv}, where the relevant hits are extracted out of the raw data stream. Pattern recognition, segment-finding, and track-fitting algorithms are then applied to determine the muon momentum at the first trigger level. Hit data are stored for transmission to a network called Front End LInk eXchange (FELIX)~\cite{Wu:2018rnc} after receiving the first-level trigger acceptance signal.
\subsection{Introduction of the miniDAQ system}
Due to the new MDT trigger and readout scheme, all front-end and back-end electronics need to be redesigned. Both ASD and TDC designs have been finished. All ASD chips have been produced, while all TDC chips are expected to be produced in 2022. The designs for both mezzanine cards and CSM are close to be final and minor modifications to the current prototypes are expected. The MDT Data Processor is still under development.
It is critical to design a miniDAQ system to integrate these prototype ASICs and boards together and to demonstrate that the new front-end electronics can run in the triggerless mode to send out all hits. The miniDAQ system is a lightweight version of the MDT Data Processor. It will send out all matched hits to a PC for storage and pattern recognition, segment-finding, and track-fitting algorithms will be performed offline. As a result, a low-cost FPGA can be used and a FELIX system is not needed. The miniDAQ system is expected to be mobile and can be used to study the performance of new sMDT chambers. It is also expected to be used for the integration and commissioning of new front-end electronics on the present MDT chambers inside the ATLAS collision hall.
Figure~\ref{fig:MDTDAQ_MiniDAQ} shows the overall miniDAQ readout system planned for a single (s)MDT chamber. Due to the smaller tube radius, an sMDT chamber can have more than 500 tubes, thus two CSMs are needed to read out all tubes. The miniDAQ system is designed to handle at least 108 ASD and 36 TDC ASICs on 36 mezzanine cards. All TDCs can be configured to run in either the triggerless or the trigger mode, and the default mode is the triggerless mode~\cite{Liang:2019weg}. The miniDAQ system also receives data from at least two CSMs. Matched hits with the arrival time within the trigger time window are sent to a PC using an ethernet port.
The requirements of the miniDAQ system are the following: 1) configure all ASICs and boards connected; 2) distribute Trigger, Timing, and Control (TTC) signals to each CSM via a downlink fiber at a line rate of 2.56 Gbps; 3) receive the CSM output optical signals, and each CSM has two serial optical uplink fibers running at a line rate of 10.24 Gbps; 4) receive the trigger signal (can be either the external coincidence trigger signal or a programmed trigger logic signal) and perform trigger matching to only send matched hit data for offline storage; (5) monitor voltages and temperatures of all mezzanine cards connected; and 6) monitor detector data in real time.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{figures/phase2_readout_with_miniDAQ.pdf}
\caption{The miniDAQ readout system used to read out data from two CSMs. Each CSM will handle up to 18 mezzanine cards, and each mezzanine card has three ASD ASICs and one TDC ASIC. Trigger signals can be either the external coincidence trigger signals or programmed trigger logic signals. The output data will be sent to a PC for storage.}
\label{fig:MDTDAQ_MiniDAQ}
\end{figure}
\subsection{The miniDAQ hardware}
Figure~\ref{fig:MiniDAQ_diagram} shows the connections of the miniDAQ system. A miniDAQ board is designed to communicate between front-end electronics and a PC. Figure~\ref{fig:MiniDAQ_board} shows a picture of the actual miniDAQ board.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\textwidth]{figures/MiniDAQ_board_diagram.pdf}
\caption{Connections between the front-end boards, the miniDAQ board, and a PC. The red box indicates major components on the miniDAQ board. The central part of the board is a Xilinx FPGA.}
\label{fig:MiniDAQ_diagram}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figures/MiniDAQ_board.pdf}
\caption{Picture of an miniDAQ board. Major components and connectors are indicated.}
\label{fig:MiniDAQ_board}
\end{figure}
The central part of the miniDAQ board is a Xilinx Ultrascale KU035-1FBVA676C FPGA. It contains about 444k logic cells, 19 Mb Block RAM, and 16 GTH transceivers with a maximum line rate of 12.5 Gbps each. In addition, the FPGA has 10 Clock Management tiles, 104 high-range I/Os, and 208 high-performance I/Os.
Since each CSM uses two uplink fibers and one downlink fiber, in total four SFP+ modules are needed to handle two CSMs. Four additional SFP+ modules are reserved to handle two additional CSMs if needed.
A 400 I/O FPGA Mezzanine Card (FMC) High-pin count (HPC) connector is placed on the lower left corner of the board and is connected to four FPGA GTH transceivers. This connector provides flexibility for extra devices (such as an additional SFP+ adapter board to integrate two extra CSMs).
Four SMA connectors are used to receive trigger signals, as shown in Fig.~\ref{fig:MiniDAQ_trigger}. Each channel contains a high-speed comparator (ADCMP604) and the comparator outputs are connected to the on-board FPGA. The trigger signal can be either the analog signals from photomultipliers (PMTs) connnected to plastic scintillators or the final coincidence signal of these analog signals. When multiple PMTs are used, the coincidence can be made inside the FPGA firmware. The coincidence signal is then digitized with a TDC implemented in the FPGA firmware (FPGA-TDC). After a programmable delay, the digitized signal queues in the first-in first-out buffer (FIFO) to start the trigger matching process.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{figures/MiniDAQ_trigger2.pdf}
\caption{Trigger option of the miniDAQ board. An adjustable threshold is applied to all comparators. Threshold-crossing time after the coincidence logic is digitized by the FPGA-TDC and then used in the trigger matching process after a programmable delay.}
\label{fig:MiniDAQ_trigger}
\end{figure}
The miniDAQ board features an on-board 320 MHz oscillator, providing the reference clock for the GTH transceivers. External clock can also be fed into for synchronization if multiple miniDAQ boards are used. A flash memory is used to boot the firmware into the FPGA when the board is powered on.
The board exchanges commands and front-end electronics monitoring information with a PC via a USB-UART interface, and the detector data is sent out for offline storage through the gigabit ethernet interface. Four miniSAS connectors are reserved for joint test with a TDC test board. DC-DC converters and low-dropout regulators are used to provide power to the FPGA and all other on-board devices.
\subsection{The miniDAQ firmware}
Figure~\ref{fig:MiniDAQ_function_diagram} shows the functional block diagram of the miniDAQ system. The FPGA firmware can handle uplink and downlink data of the CSMs, digitize the external trigger from scintillators, and communicate with a PC for offline data storage and command exchange. The uplink data is passed through the multi-stage decoding, the trigger matching, and sent out in packets to a PC via the ethernet interface. The front-end configuration commands are exchanged with the PC, then encoded and serialized as the downlink data. A JTAG debug core is also used for online debugging.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{figures/MiniDAQ_function_diagram.pdf}
\caption{Functional block diagram of the miniDAQ system. Main FPGA firmware modules are shown in the red box and the arrows indicate data-flow directions.}
\label{fig:MiniDAQ_function_diagram}
\end{figure}
The key to successfully store the detector timing data for offline analysis is to reduce the overall data bandwidth and select events of interest. When two CSMs are connected to the miniDAQ system (shown in Fig.~\ref{fig:MDTDAQ_MiniDAQ}), the line rate of the uplink data adds up to 40.96 Gbps. While this rate suits the maximum data rate requirement of the HL-LHC upgrade, data filtering and trigger matching are a must for the offline data storage as either background noises or idle words occupy more than 90\% of the uplink bandwidth. The miniDAQ firmware runs data decoding and trigger matching in parallel for each uplink.
Multi-stage data decoding for one uplink is processed as illustrated in Fig.~\ref{fig:MiniDAQ_decode_logic_en}. The 10.24 Gbps serial data is firstly sampled and de-serialized by the Xilinx GTH data interface, yielding a 32-bit word at 320 MHz. Secondly, the lpGBT-FPGA IP further de-interleaves, decodes, and de-scrambles the uplink data to a 230-bit word at 40 MHz according to the GBT protocol. This 230-bit word contains 160-bit TDC data from 10 mezzanine cards and 70-bit data from voltage and temperature sensors. The TDC data are then grouped into 10 slots corresponding to its original mezzanine number and are decoded individually.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{figures/MiniDAQ_decode_logic_en.pdf}
\caption{Decoding scheme of the miniDAQ system for one uplink. Serial data from one uplink goes through different de-serial and decoding stages to form data for each individual TDC and data for voltage/temperature monitoring.}
\label{fig:MiniDAQ_decode_logic_en}
\end{figure}
The MDT TDC utilizes two e-links running at 320 Mbps for its 8b/10b-encoded output data, one for even bits and one for odd bits. When there are no hits to be sent out, the TDC generates idle words continuously, which are useful for data alignment when decoding at the receiver end. Even and odd bits are aligned, concatenated and decoded by the miniDAQ system. At this stage, idle words are thrown out and hits are buffered in on-chip random-access memories (RAMs). While the throughput data rate is reduced by removing idle words, a trigger matching process is implemented to select interesting hits from the background noise.
The arrival time of a trigger signal is digitized by the FPGA-TDC. The FPGA-TDC utilizes multiple phases of the reference clock, with the help of the FPGA built-in clock manager modules to achieve the same time bin size as the front-end TDC ASIC. After a pre-set latency during when all expected hits are being collected in the RAMs, the trigger matching process compares the leading edge time information between the trigger signal and the hit data stored in the RAMs. Hits within a fixed time window relative to the trigger signal are packed with the trigger data as an valid event and sent out through the gigabit ethernet. An internal counter monitors the hits in the RAM and rejects the outdated ones, with a programmable rejection window set based on the trigger latency and trigger matching window.
The miniDAQ system sends front-end ASIC configuration bits and the encoded control (ENC) signals for bunch count reset and system reset through the downlink data, as indicated in Fig.~\ref{fig:front_control}. The two lpGBT ASICs~\cite{lpGBT2}, which work in the master/slave mode, are configured directly through their serial control interfaces. The configuration of the lpGBT ASICs enables the clock distribution and the data sampling of the mazzanine cards.
The configuration of the mezzanine cards is achieved by utilizing the JTAG master in the GBT-SCA ASIC~\cite{GBT-SCA}, which converts the serial downlink data into JTAG signals and provides a configurable clock rate. The converted JTAG signals in the GBT-SCA, along with the ENC signals directly from the downlink, are distributed to all mezzanine cards through the fan-out FPGA on the CSM board.
\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{figures/front_control2.pdf}
\caption{The uplink (red) and downlink (blue) data flow of the front-end electronics. Chamber data, as well as voltage/temperature monitoring data, are sent by uplinks. Configuration and TTC signals are sent by downlinks. }
\label{fig:front_control}
\end{figure}
|
2112.10441
|
\section{Introduction}
\label{sec:introduction}
\noindent
Critical care units increasingly often adopt machine learning to improve the outcome of patients \cite{maslove2021}, \cite{flechet2016informatics}. Critical care patients and patients undergoing surgery are connected to equipment used by physicians to monitor, such signals as ECG, blood pressure or blood oxygenation. These signals can also be used to predict upcoming adverse events and to decrease the time needed to detect these events, thereby reducing the number of invasive tests or increasing the certainty of the diagnosis \cite{wijnberge2020effect}, \cite{carra2020data}.
One of the challenges when using ML models in critical care is how to use a model trained on one patient for diagnosing another patient \cite{komorowski2019artificial}. From the medical perspective, the demographics of the patients can differ greatly, even if they undergo the same procedure or suffer from the same disease. From the technical perspective, the signals collected during the procedures can be collected differently (e.g. variations in how electrodes are attached to the patient). Therefore, in order to trust the ML models, it is important to identify the variability, understand the source of it and then choose the right strategy to handle it.
In practice, in most cases, ML engineers and scientists adopt the strategy to collect more data in the hope to create models which are robust to these variations \cite{smiti2020machine}. This means that the models' predictions will be of a better quality, but, at the same time, more data can lead to lower explainability and thus trustworthiness.
In this paper, we set off to investigate how much the results vary from when we train the model on all patients, and when we train the model on one patient at a time. The goal was to explore the variability of models and therefore assess how robust the models are to different patients.
The data was collected from a pipeline where we designed and conducted a clinical study of patients undergoing CEA \cite{staron2021robust}, \cite{block2020cerebral}.
We used a RandomForest classifier, which has been used in many studies analyzing signals from patients \cite{qi2012random}.
Our results show that there is a large variability between how we train and use the model. For the same model, when trained on the data from all patients, the accuracy was 0.86. However, for the same algorithm, but trained on one patient at a time, the accuracy varied from 0.07 to 0.63. The model performed best for the pair of patients who were the most similar (same gender, similar age, similar initial status). Although the results were better for the model trained on all patients, we could observe that the model is more trustworthy if we can trace the training model to a specific patient rather than a large dataset with high variability.
\section{Context -- CEA and Collected Signals}
\label{sec:context}
\noindent
Carotid endarterectomy (CEA) is recommended in symptomatic patients with $\geq$ 50\% stenosis and in asymptomatic patients with $\geq$ 60\% stenosis \cite{eckstein2018european}. CEA poses an increased risk of cerebral ischemia due to embolization and hemodynamic derangement during clamping of the carotid arteries. In our centre, all carotid endarterectomies are performed with patients under general anaesthesia. All intraoperative monitoring, including invasive arterial blood pressure (ABP) and neuromonitoring, is connected to patients and monitoring starts prior to induction of anaesthesia. Baseline values before induction of anesthesia are recorded. Noradrenaline infusion is titrated as needed to maintain ABP close to baseline values. After the skin incision and surgical exposure but before plaque removal, the surgeon cross-clamps the common carotid artery , the external carotid artery , and the internal carotid artery (ICA), so that the stenotic bifurcation is excluded from the circulation. At this moment, the surgeon, taking into account information from available neuromonitoring and measured blood pressure in the distal stump of the ICA, evaluates the risk of cerebral ischemia due to hypoperfusion and decides if a shunt is to be placed to maintain cerebral perfusion during clamping. After plaque removal the artery is sutured and the clamp is released to restore blood flow. After emergence from anaesthesia, the patient undergoes frequent bedside neurological examinations.
Since the patients are connected to monitors, we can collect the following signals, and transform them into ML features:
\begin{itemize}
\item EEG: frequency bands (alpha, beta, delta, gamma, and theta) for each channel (F3-Cz, F4-Cz, C3-Cz, C4-Cz, P3-Cz, and P4-Cz).
\item ECG: Inter-beat intervals (IBI), BPM (Heartbeats per minute), SDNN (Standard deviation of the NN (R-R) intervals), SDSD (standard deviation of successive differences), RMSSD (Root mean square of the successive differences), PNN50 (proportion of NN50\footnote{The number of pairs of successive NN (R-R) intervals that differ by more than 50 ms.} divided by the total number of NN (R-R) intervals), PNN20 (The proportion of NN50\footnote{The number of pairs of successive NN (R-R) intervals that differ by more than 20 ms.} divided by the total number of NN (R-R) intervals), HR\_MAD (median absolute deviation of RR intervals), SD1 (standard deviation perpendicular to identity line), SD2 (standard deviation along identity line), S (Area of ellipse described by SD1 and SD2), SD1/SD2 (SD1/SD2 ratio), Breathing rate (estimated from the ECG signal),
\item ABP (Arterial Blood Pressure): ABP mean,
\item NIRS (Near-Infrared Spectroscopy): rSO2 left (side of the brain), and rSO2 right (side of the brain), and
\item SpO\textsubscript{2} (Oxygen Saturation): SpO\textsubscript{2} value.
\end{itemize}
The data set contains also a decision class, which is the set of events which we want to recognize/diagnose during CEA. We use the following events:
\begin{itemize}
\item Pre-clamp anaesthesia -- induction of anaesthesia- when the patient’s anesthesia starts.
\item Pre-clamp surgery -- opening the cartoid artery.
\item Clamped artery -- when the carotid artery is closed.
\item Shunt -- when the surgeons make a shunt around the closed artery to restore blood flow (in case the patient's blood oxygenation decreases to a pre-defined level).
\item HRV window anasthesia pre-clamp -- six breaths during one minute in the ventilator, to establish a baseline for the analysis of HRV signals under controlled conditions. This is done both before the clamping of the artery and after the clamping of the artery.
\item Post-operative anesthesia -- when the patient is regaining consciousness.
\item Post-operative care -- when the patient is awake and under post-operative care in the neurointensive care unit.
\end{itemize}
The dataset in our study consists of 48 feature columns and one class column. It
We experiment with several ML algorithms -- RandomForest, SVM, AdaBoost, Decision Trees (CART) and artifical neural networks. We chose RandomForest for our comparisons due to its robustness and ability to handle large data sets \cite{qi2012random}. In this study, we follow the procedure:
\begin{enumerate}
\item Collect the data from all patients in one table.
\item Train and validate the classifier based on all data, using 0.33 train-test split.
\item Train the classifier on one patient at a time and validate on all other patients.
\end{enumerate}
This procedure let us compare how the performance of the classifier varies depending on the way in which it is trained and evaluated.
\section{Results and interpretation}
\label{sec:results}
\noindent
To understand the data, we use the t-SNE diagram as a means for visualization of all data points (Figure \ref{fig:tsne}). The diagram shows that there are areas where different events/decision classes overlap, but in the majority of cases the data points for specific events are grouped together (e.g. the purple dots, corresponding to the clamping of the artery, forms several distinct areas in the figure).
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{t-SNE.png}
\caption{t-SNE diagram to illustrate the data set. Each data point represents one minute of a CEA procedure. Each color represents the type of event/decision class.}
\label{fig:tsne}
\end{figure}
We train the classifier with the following parameters: number of trees: 64, number of leaves: 128. The accuracy for that model is 0.86. Figure \ref{fig:confusion} presents the confusion matrix for this model. The confusion matrix shows that most of the errors are for these classes which have the fewest data points (e.g. Pre-induction); this is despite balancing classes using weights.
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{confusion_matrix.png}
\caption{Confusion matrix for the cross-patient model for the validation data (0.33 of the entire data set).}
\label{fig:confusion}
\end{figure}
Figure \ref{fig:cross_patient} presents a diagram of accuracy for a cross-patient trained model. Each model has been trained on one patient and applied on another. The lowest accuracy is 0.07 and the highest is 0.63. Although the diagram shows a significant variability in accuracy, we can understand it better if we observe the difference between the feature importance for the cross-patient model and the model for the best pair of patients (C009 and C001).
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{cross_patient.png}
\caption{Accuracy of the models trained for each pair of patients. The model is trained on one patient and then validated on another patient (train patient --\textgreater validation patient).}
\label{fig:cross_patient}
\end{figure}
From the perspective of explainability and, thus, trust in the model, accuracy is not the best validation metric. In our study, we use feature importance charts as a means to understand how good the models are and whether they can be trusted. For the cross-patient model, Figure \ref{fig:all_patients} presents the feature importance chart for the most important 15 features. The chart shows that it is the arterial blood pressure (ABP) which is the most important feature.
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{feature_importance_all_patients.png}
\caption{Feature importance for the model trained for all patients. The most important feature is the Arterial Blood Pressure (ABP).}
\label{fig:all_patients}
\end{figure}
For the model trained on individual patients, however, the feature importance chart is different -- see Figure \ref{fig:best_patients}. The most important feature is the NIRS signal from the left side of the head. Although both ABP and NIRS can have a good explanation, there is a medical reason for this difference. Patients C001 and C009, are similar in terms of: which artery was clamped (right), condition for admittance (Amaurois fugax dx; Paresthesia left hand vs. Hemiparesis (left); Amaurois fugax sin.), gender and age. Patients C006 and C004 (the worst match) are of different gender and arterial clamp was performed on different sides. As NIRS has been found useful for same-side arterial clamp \cite{lewis2018cerebral}, it confuses the algorithm for different-side arterial clamping, thus rendering this signal untrustworthy in this context.
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{feature_importance_best_patients.png}
\caption{Feature importance for the model trained for patient C009 and applied on the patient C001 (the pair with the highest accuracy or 0.63). The most important feature is the NIRS signal.}
\label{fig:best_patients}
\end{figure}
Considering both of these charts together, our interpretation is that the similarity of the patients condition is important for the trustworthiness of the model -- clamping on the same side strengths the models accuracy, compare to other pairs of patients. We observe also that the number of data points in the data set increases the accuracy of the model in general.
We combine these two approaches: set aside one patient for validation while training the model on all other patients. The results are shown in Figure \ref{fig:combined}.
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{accuracy_combined_and_single.png}
\caption{Accuracy of the combined model -- one patient is set aside for validation while the model is trained for the data from all other patients; and models trained on the same patient -- setting aside 0.33 points for validation.}
\label{fig:combined}
\end{figure}
The highest accuracy is for patient C001 -- 0.65. The lowest accuracy is for patient C006 -- 0.16. There is an improvement across all patients, although not a large one. For patient C001, we also include the feature importance chart -- Figure \ref{fig:features_combined}.
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{features_combined.png}
\caption{Feature importance chart for the model validated for patient C001.}
\label{fig:features_combined}
\end{figure}
Our recommendation, therefore, is to decide early whether we want more explainability or higher accuracy. For a reference, the accuracy if we train the model for 0.67 of the data for one patient and validate on the remaining 0.33, the accuracy is above 0.82.
\section{Conclusions}
\label{sec:conclusions}
\noindent
In the context of using machine learning models in medical practice, trustworthiness is one of the crucial components in the adoption of this technology in practice. In this paper, we present results of a study on how the performance of a Random Forest classifier changes when using the same data in two different ways -- training the model on the entire data set and training it per patient.
The results show that the model trained on all data is more accurate, but has lower explainability and thus trustworthiness. The model trained on a single patient has lower accuracy, but higher explainability. The accuracy depends on the selection of the patients, which is, therefore, a required feature for this algorithm.
Our conclusions are that we need to combine both approaches -- selecting the most relevant patients and training the model on a larger data set. We plan to evaluate these conclusions based on new patients groups.
|
0909.2162
|
\section{Introduction}
Correlations in the full sky or large enough surveys contain clues to the early Universe and its present structure. The acoustic peaks of the cosmic microwave background (CMB) power spectrum revealed a set of cosmological parameters with particular accuracy \cite{dB,WMAP5}. The baryon acoustic oscillations (see Percival et al 2009) are crucial for studies of the formation of the large-scale structure, including the role of dark matter and dark energy.
Below, we construct the power spectrum of a novel type of full sky map, those representing the distribution of the Kolmogorov stochasticity parameter of the CMB temperature maps. Kolmogorov's parameter is a descriptor for a degree of randomness \cite{Kolm,Arnold} and when applied to the CMB temperature datasets results in a map (K-map) \cite{GK2009} that has both features resembling the temperature maps, like the outlined Galactic disk, but also ones with different contents. The Cold Spot \cite{Cruz}, the non-Gaussian structure of negative mean temperature, was noticed thanks to the excess of the K-parameter with respect to its mean value over the sky. Moreover, the behavior of the K-parameter, i.e. of the degree of the randomness was increasing towards the boundary of the Cold Spot (Gurzadyan and Kocharyan 2008, 2009). Both features are compatible to the void nature of the Cold Spot. Other spots and regions have been noticed in the K-map, which are studied with other descriptors as well (Rossmanith et al 2009), and other noticed non-Gausianities can also be among the applications (Gurzadyan et al 2005, 2008).
If the Kolmogorov CMB map is able to reflect the features in the matter distribution, it is therefore natural to study the large-scale correlations in such a map, along with the above-mentioned small-scale features. We used the latest
available full sky maps, i.e. those of the Wilkinson Microwave Anisotropy Probe (WMAP) of W, Q, V-bands, and the foreground cleaning procedure elaborated by Tegmark et al. (2003). The power spectra obtained for them have common structures that are, however, absent in the simulated maps based on the CMB temperature power spectrum. This is the first attempt, and more detailed analysis of the K-parameter's power spectra can be performed when higher resolution CMB maps are available.
\section{Kolmogorov's stochasticity parameter map}
The Kolmogorov map can be constructed by estimating the stochasticity parameter for the CMB temperature dataset sequence. Kolmogorov's stochasticity parameter \cite{Kolm,Arnold} is defined for the sequence $\{X_1,X_2,\dots,X_n\}$ in increasing order. The cumulative distribution function is
$F(x) = P\{X\le x\}\ $, and the empirical distribution function is defined as
\begin{equation}
F_n(x)=
\begin{cases}
0\ , & x<X_1\ ;\\
k/n\ , & X_k\le x<X_{k+1},\ \ k=1,2,\dots,n-1\ ;\\
1\ , & X_n\le x\ .
\end{cases}
\end{equation}
The stochasticity parameter is
\begin{equation}\label{KSP}
\lambda_n=\sqrt{n}\ \sup_x|F_n(x)-F(x)|\ .
\end{equation}
The universality of this definition stems from how for any continuous $F$, the convergence
$
\lim_{n\to\infty}P\{\lambda_n\le\lambda\}=\Phi(\lambda)\ ,
$
where
\begin{equation}
\Phi(\lambda)=\sum_{k=-\infty}^{+\infty}\ (-1)^k\ e^{-2k^2\lambda^2}\ ,\ \Phi(0)=0,\, \ \lambda>0\ ,\label{Phi}
\end{equation}
is uniform, and $\Phi$ is independent on $F$ \cite{Kolm}.
More specifically, to obtain the degree of randomness for a given sequence, one must compute the Kolmogorov stochasticity parameter $\lambda_n$, and then the estimated Kolmogorov's distribution $\Phi$ will provide information on the degree of randomness in the sequence for the $\lambda_n$ within the interval of their probable values, i.e. approximately, 0.4-1.8 \cite{Arnold_ICTP}. The mean value of $\lambda_n$
given by Kolmogorov distribution is
\begin{equation}
\lambda_{mean}=\int{\lambda\Phi'(\lambda)d\lambda}\approx 0.875029.
\end{equation}
The behavior of $\lambda_n$ and $\Phi$ for a set of sequences that model the CMB as composition of signals, is studied in Ghahramanyan et al. (2009).
The Kolmogorov map obtained based on this definition exhibits, as mentioned above, that structures are linked not only to those noticed by other descriptors but also to those indicating voids \cite{GK2009}.
\section{Power spectrum}
Once the Kolmogorov statistic $\Phi$ is represented on a map, then one can define
a correlation function on a sphere in spherical coordinates, as for the temperature
data,
\begin{equation}
C(\theta )=<\Phi (\vec{n}_{1})\Phi (\vec{n}_{2})>,\,\,\,\vec{n}_{2}\vec{n}%
_{2}=\cos \theta,
\end{equation}
and expand $\Phi$ via spherical harmonics,
\begin{equation}
\Phi (\theta ,\varphi )=\sum_{l,m}a_{lm}Y_{lm}(\theta ,\varphi ),
\end{equation}%
where the coefficients $a_{lm}$, as usual, are found from
\begin{equation}
a_{lm}=\int \Phi (\theta ,\varphi )Y^{\ast}_{lm}(\theta ,\varphi )\sin \theta
d\theta d\varphi .
\end{equation}%
Then
\begin{equation}
C(\theta )=\frac{1}{4\pi }\sum_{l,m}(2l+1)C_{l}P_{l}(\cos \theta )
\end{equation}%
and
\begin{equation}
C_{l}=<a_{lm}^{\ast}a_{lm}> \label{Clr}
\end{equation}%
or
\begin{equation}
C_{l}=\frac{1}{2l+1}\sum_{m=-l}^{l}|a_{lm}|^{2} \label{Cl}.
\end{equation}%
However, for our purposes, i.e. when the $\Phi$ is averaged within certain numbers of pixels with noise, the cross-power spectra $\widetilde{C}_{l}^{ij}$ of various bands are more efficient than those of autocorrelations; i.e., then one may get more cleaner power spectrum for correlations, we study the cross power spectra for $\Phi$
\begin{equation}
\widetilde{C}_{l}^{ij}=\frac{1}{2l+1}\sum_{m=-l}^{l}a_{lm}^{i}a_{lm}^{j\ast }
\label{Ccross}
\end{equation}%
where $i\neq j, i,j=1,...,8$ for Q1, Q2, V1,V2, and W1-W4 bands.
In our analysis we used the eight of WMAP's maps, of W, V, Q-bands, in the usual HEALPix format \cite{HP},
of the resolution parameter $n_s=512$, of a total number of pixels $%
N_{pix}=12n_s^2=3145728$. For each $n_s=512$ temperature map,
we constructed Kolmogorov's stochasticity parameter map for $%
n_s=32$, $N_{pix}=12288$, since for the Kolmogorov map one needs about 100
temperature pixels. To obtain the $n_s=32$ K-map from the
$n_s=512$ CMB map, each $\Phi$ pixel is calculated from 64 temperature pixels.
Then, for the HEALPix map of given $n_{side}$, the maximum $l$ in the obtained power spectrum will be
\begin{equation}
l_{max}=\sqrt{3\pi}n_{s}. \label{lm}
\end{equation}
This corresponds to $l_{max}=96$ for $n_{s}=32$ map and $l_{max}=1536$ for $%
n_{s}=512$.
The procedure for getting $\Phi$ cross-power spectra included:
1. calculation of $i$-th $a_{lm}$ for each K-map,
2. obtaining of all possible combinations of cross-power spectra,
3. estimation of the mean and the error bars for the set of spectra:
\begin{figure}[ht]
\begin{center}
\centerline{\epsfig{file=15mean-s1.jpg,width=0.5\textwidth}} \vspace*{8pt}
\end{center}
\caption{The mean for 15 cross-power
spectra for Kolmogorov CMB maps for WMAP's 6 frequency bands, V1,V2,W1,W2,W3,W4.}
\label{15mean}
\end{figure}
\begin{eqnarray}
\widetilde{C}_{l(mean)} &=&<\widetilde{C}_{l}^{ij}>,\,\,i\neq j,\,0\leq l\leq 96; \notag \\
\epsilon &=&\sqrt{<(\widetilde{C}_{l(mean)}-\widetilde{C}_{l}^{ij})^{2}>}. \label{cross}
\end{eqnarray}
Note that $a_{lm}$-s are complex variables, making the correlation function complex as well.
However, since the noise differs from map to map, the resulting complex part
due to noise is vanishing at cross correlations.
The calculations were performed for $a_{lm}$ without a Galactic disk region within $\pm
20^{\circ}$, for 6 and 8 K-maps, and we get 15 and 28
cross-power spectra, respectively, once their mean and error bars were obtained. The
results are shown in Fig. 1. We see that, for the 28 cross-power spectra, the mean
is the same as for 15, but the estimated errors are bigger because of using the noisier Q1, Q2 maps.
The mean power spectrum is similar to the CMB pseudo-power spectrum
discussed in Hinshaw et al. (2003), so one may think to use the Peebles weighting
method \cite{Peebles73,Hivon2002} to find the power spectrum with the Galactic disk. However, this causes
two types of difficulties. First, we do not have enough pixels
($n_s=32, N_{pix}=12288$) to calculate the $a_{lm}$ up to $l=250$, which is needed
for calculating the precise weighting. Second, even if we keep the Galactic
disk region where we have approximately $\Phi=1$, it differs very little from the situation
if we a priori adopt $\Phi=1$. The reasonable solution seems not to use
the Galactic region at all and to construct the power spectrum only for odd $l$, which
are not affected by the Galactic disk cut.
\section{Foreground cleaned $\Phi$ map}
We then obtained the power spectrum of $\Phi$ using the foreground cleaning method
developed for CMB maps by Tegmark et al. (2003) and the linear combination method of (Saha et al. 2006, 2008).
This is based on the use of a linear combination of different maps with weighting of $w_l^i$, not only depending
on $i$-th map but also on the multipole $l$.
\begin{figure}[ht]
\begin{center}
\centerline{\epsfig{file=tegmark1.jpg,width=0.5\textwidth}} \vspace*{8pt}
\end{center}
\par
\caption{The power spectrum for foreground cleaned Kolmogorov maps.}
\label{ps-final}
\end{figure}
For constructing of a cleaned $\Phi$ map, we first calculated all $a_{lm}$ and then
calculated the cleaned $a_{lm}$ using the relation
\begin{equation}
a_{lm}^{(clean)}=\sum_{i=1}^{8}w_{l}^{i}a_{lm}^{i}
\end{equation}%
where $w_{l}^{i}$ is
\begin{eqnarray}
w_{l}^{i} &=&\frac{\sum_{k=1}^{8}e_{k}(C_{l}^{-1})^{ki}}{%
\sum_{i,k=1}^{8}e_{k}(C_{l}^{-1})^{ki}e^{i}}, \notag \\
\sum_{i=1}^{8}{w_{l}^{i}} &=&1. \label{wl}
\end{eqnarray}%
Here $C_{l}$ is an $8\times 8$ dimensional matrix constructed by all possible
auto and cross-power spectra from all maps (see eq.(\ref{Ccross})), so that $%
C_{l}^{-1}$ refers to an inverse matrix, $e^{i}$ and $e_{i}$ are
8-dimensional unit vector and its transponated vector, respectively. For the covariance of
this representation see (Tegmark et al. 2003). We get all power spectra
in Eq.\ref{wl} smoothed by $\Delta {l}=10$ interval to avoid the
singular $C_{l}$ matrix. For example, we get triplets of different maps from different
bands Q1,V1,W1 for $w_{l}^{i}$.
We thus get 16 different triplets. For any triplet, a linearly superposed $a_{lm}$-s is
constructed. The last step is to find all possible cross-power spectra
from those linearly superposed ones, whose initial map components are different. For
example, $(Q1+V1+W1)\otimes (Q2+V2+W2)$ complies to this restriction, but $%
(Q1+V1+W1)\otimes (Q2+V2+W1)$ does not, so only 3 maps of eq.(\ref{wl}) were used. In this way we obtain three cross-power spectra from triplets. The mean power spectra from these
cross-power spectra is shown in Fig. 2.
\section{Simulations}
We repeated the estimations described above for simulated maps.
Four different types of simulations were constructed from:
a. the real maps' $a_{lm}$-s (T maps),
b. real maps with added Gaussian noise
of the same parameters as the noise in WMAP CMB maps (T+N maps),
c. Gaussian maps of the distribution
parameters $T,\sigma$ from WMAP W band real map (G maps), and
d. Gaussian maps with added Gaussian noise, both
from the parameters $T,\sigma$ of WMAP W band map
(G+N maps).
\begin{figure}[ht]
\begin{center}
\centerline{\epsfig{file=SimTG1.jpg,width=0.5\textwidth}} \vspace*{8pt}
\end{center}
\par
\caption{The Kolmogorov power spectra for simulated CMB temperature (T+N) and Gaussian (G+N) maps with superimposed WMAP's noise, averaged over 190 cross-power spectra each; the smoothed error bars are shown.}
\label{SimT}
\end{figure}
For each group of $n_s=32$ $\Phi$ simulated map we obtain the mean cross power spectra as described above. For 20
different maps one has 190 cross power spectra. Similarly, 190 cross-power spectra were computed for the Gaussian maps generated with the WMAP's $\sigma$ and mean temperature and with superimposed noise of WMAP.
Although the number of the cross spectra for simulations is more than those we used for calculating the power spectra for real K-maps, neither of the resulted spectra shows the features found for real maps with 0.6 and 2.7-$\sigma$ level for W and foreground cleaned maps, respectively, as shown in Fig. 3. Even more important than the $\sigma$-level, however, seems that the features only appear at cross and not at auto correlations, thus indicating that they do not come from the noise in the maps. The principal limitation in the above analysis is the angular resolution, since $\Phi$ reflects the statistical properties of the signal, the efficiency of the method will increase with higher resolution data.
\section{Conclusion}
We have obtained the first power spectrum of Kolmogorov stochasticity parameter map of CMB temperature data.
The WMAP W,Q,V-band datasets were used to compute the Kolmogorov's CMB maps. The foreground cleaning method of Tegmark et al. (2003) was also applied while computing the $\Phi$ maps. The mean for the set of cross-correlated maps
was computed. They show features, particularly at around $l=25$, that are absent in the maps simulated either for the WMAP's temperature power spectrum parameters or in the Gaussian maps with superimposed noise, i.e. additional effects to those usually included in the simulated models.
Although the accuracy of the present analysis is limited by the WMAP's angular resolution and signal-to-noise ratio, it shows the principal possibility of obtaining such crucial information from CMB, and even the already obtained behaviors can affect the development of scenarios for the void correlations at the large-scale structure formation.
Higher angular resolution maps expected soon at Planck and other experiments will enable the finer structure analysis of structures in the power spectra of Kolmogorov CMB maps.
|
0909.1481
|
\section{Introduction}
Gaussian disorder model (GDM)[1-4] is a widely accepted model that could explain most of the charge transport behavior observed in disordered organic systems. According to GDM the charge transport in disordered organic materials occur by hopping among transport sites that are subjected to energetic and positional disorder [1-4]. Since the model assumes Gaussian density of states [1-4] a complete analytical solution of the hopping transport is therefore difficult, especially in 3-D. Hence the predictions of the GDM are made on the basis of Monte Carlo simulation of hopping charge transport[1-4]. Monte Carlo simulation is considered as an idealized experiment with which one can study the charge transport in disorder system as function of several parameters. Generally charge transport is simulated for a sample length of few microns. This is to make sure that the carrier has attained a dynamic equilibrium during its transit and also to have a better comparison with the experiment, like Time of flight (TOF), which is generally performed on micron size thick samples [1-4]. In order to simulate the charge transport for a sample length of several microns periodic boundary condition (PBC) is frequently employed [1a,5-10]. The advantage of using the PBC is that the simulation can be performed on a sample length of several microns using an array of smaller size. If PBC is not employed then an array of bigger sizes, in the all the three Cartesian directions, is required which demands large computational resources. Even though the PBC have been used frequently in Monte Carlo simulation of charge transport in disordered organic systems a clear explanation of the method of implementing PBC in these simulation is scarce. Earlier literature [1a, relevant references therein] reports that PBC have been implemented in the preliminary form as explained below. Implementing PBC in the preliminary form may lead to serious artifact which is the subject of this paper.
The preliminary form of PBC is a set of boundary conditions that are used to simulate the properties of bulk system by simulating a part of it [11]. In principle PBC generates an infinitely large system with help of a smaller array, that represent only a part of the bulk system, with the assumption that the small array will replicate periodically in all the three directions to form the bulk system. When we implement the PBC along one Cartesian direction, we take the carrier to the first plane of the array (lattice) in that direction when carrier reaches the final plane of the array in same direction. In this process the carrier's energy and other Cartesian coordinates remain same as at the boundary. When simulating the charge transport for several microns the carrier will encounter several such boundaries. Since the carrier is taken to the first plane when it reaches the final plane, in this process some of the hops that carrier may make in the absence of such a boundary get neglected, see Fig. 1(a). The neglected hops help the carrier to cover the required thickness in short time, which effectively provides a bias in addition to the applied field. Number of such neglected hops at boundary can be significant particularly at smaller field strengths, higher temperature and higher energetic disorder. It is possible that these neglected hops can lead to an artifact which can seriously affect the charge transport simulation studies using PBC. This artifact, due to the neglected hops, can be significant especially at low electric field regime and is very critical as far as the investigations related to the mechanism of charge transport and the operation of devices like organic photovoltaic devices is concerned.
In this article, we present our Monte Carlo simulation studies of charge transport on the basis of GDM [1-4] which unambiguously proves the artifact due to conventional use of PBC. At low field regime, field dependence of mobility simulated using PBC with zero positional disorder shows negative field dependence of mobility (NFDM) for all values of energetic disorder while a clear saturation of mobility with field was observed when simulated for the same inputs without using PBC. The observed NFDM in the absence of positional disorder contradicts with GDM [1-4,12] which asserts that the NFDM at low field regime can occur only in the presence of high positional disorder. Thus, the observed NFDM in the absence of any positional disorder is an artifact due to the conventional method of implementing the PBC. The origin of the artifact is attributed to the neglect of hops that the carrier may make in the absence of boundary which is created upon implementing the PBC. These neglected hops gives an extra bias to the carrier and there by enhancing the mobility. In concise, this study not only highlights an artifact, which can mislead interpretation and modeling of charge transport, but also cautions on the use of Monte Carlo simulation along with PBC for investigating the origin of NFDM / charge transport at low field regime. On the basis of the origin of artifact an alternative simulation approach for implementing the PBC which is free from the observed artifact is also proposed.
\section{Details of Monte Carlo simulation}
The Monte Carlo simulation is based on the commonly used algorithm reported by Sch\"{o}nherr et al [13]. A 3D array was considered as the lattice with size 70x70 along \textit{x} and \textit{y} direction. Along \textit{z} direction, the direction of the applied field, various sizes were used to implement different simulation approaches adopted for covering the required sample thickness along this direction. Three different approaches are:
\textbf{\textit{Case 1}}: In this case a lattice of size 70x70x70 along \textit{x, y} and \textit{z} direction was used for simulation (Fig.1(b)). PBC was implemented along all the three directions. PBC along the applied field direction was implemented by taking the carrier to the plane \textit{z}=1 when the carrier reaches \textit{z}=70$^{th}$ plane, keeping the other coordinates and the carrier energy same as at the boundary. PBC was also implemented along \textit{x} and \textit{y} direction in a similar manner. The disadvantage of implementing the PBC in this manner is that some of the hops that the carrier may make in the absence of boundaries, which it encounters in the process of transit, along \textit{x, y} and \textit{z} directions are neglected. Simulations were also performed in a similar way with bigger lattice size along \textit{z} direction (70x70x300, 70x70x1600, etc.). Size of the lattices was always chosen after considering the available computational resources.
\textbf{\textit{Case 2}}: In this case the simulation was performed without using PBC along \textit{z} direction (Fig.1(a)). Here the size of lattice along \textit{z} direction was taken to be the sample length, which requires array of bigger size. However, PBC along \textit{x} and \textit{y} direction was implemented as explained above. In this case, carrier does not see any boundary along z direction and hence neglected hops at such boundaries in \textit{case 1} are taken into consideration. This simulation approach requires large computational resources.
\textbf{\textit{\textit{Case 3}}}: In this case we show an effective way to implement the PBC for simulating the charge transport (Fig.1(c)). In order to show an effective way to use PBC, a array of size 70x70x150 along \textit{x, y} and \textit{z} direction was used. Justification for the size of array is given later. Carrier is first injected into \textit{z}=1 plane and is allowed to move in the direction of applied electric field. The carrier was taken into z=70$^{th}$ plane when it reaches \textit{z}=140$^{th}$ plane, keeping the \textit{x }and \textit{y} coordinates same. The energy of the carrier was kept same as at \textit{z}=140$^{th}$ plane. PBC along \textit{x} and \textit{y} direction was implemented as mentioned before. Compared to \textit{case 1}, this approach considers all those hops that are neglected at the boundaries perpendicular to \textit{z} direction. Moreover, it requires less computational resources compared to \textit{case 2}. In this case, when the PBC is implemented the carrier can perform all the hops around \textit{z}=70$^{th}$ plane and hence the artifact due to neglected hops can be removed.
Due to the PBC along x and y directions, in all the three cases, some hops may be still neglected when the carrier encounters a boundary perpendicular to \textit{x} / \textit{y} directions. In order to understand the influence of those neglected hops on charge transport simulation was performed, for all the three cases, by taking the carrier to the middle of the same plane (same \textit{z} coordinate) when the carrier encounters a boundary either perpendicular to \textit{x }/ \textit{y} direction. Carrier energy was kept same as that at the boundary. Here the carrier is allowed to make all the neglected hops around the middle of lattice. We found that the hops that may be neglected at the boundaries perpendicular to \textit{x} / \textit{y} direction upon implementing the PBC have negligible effect on charge transport. Data presented in this article, for all the three cases, was simulated using PBC along \textit{x} / \textit{y} direction.
In this study simulation was always performed for a sample length of 4$\mu$m along field direction. The lattice constant \textit{a} = 6\r{A} was used for the whole set of simulation [1]. The site energies of lattice were taken randomly from a Gaussian distribution with a known standard deviation ($\sigma$). Through out the simulation the positional disorder was neglected ($\Sigma$=0). This is to avoid the huge computer time required for simulating the charge transport with non-zero positional disorder. More over, we believe that the outcome of this study can be unambiguously conveyed and justified even with present simulation with zero positional disorder. Simulation was always performed on this energetically disordered lattice with the assumption that the carrier hops among the lattice sites following Miller-Abrahams equation [14]. Throughout the simulation we took $2\gamma a=10$ [1,13]. Transit time of a carrier was calculated by adding all the hopping times and averaging over few hundred carriers. The mobility was calculated using drift mobility equation. The electric field range ($\sim$$>10^4$ V/cm) over which the simulations were performed is well above the field range ($\sim$10$^1$-10$^2$V/cm) over which the pure diffusion dominates the charge transport [12,15]. Hence the use of drift mobility equation over the field range of this study is appropriate. Simulation was preformed for various values of energetic disorder ($\sigma$) and electric field strengths. All the data presented in this article was simulated with uncorrelated site energies.
\section{Results and Discussions}
Fig. 2 compares the simulated field dependence of mobility, at T=300K, for various values of energetic disorder, for \textit{\textit{case1}} and \textit{\textit{case 2}}. For both \textit{\textit{case1}} and \textit{\textit{case 2}}, field dependence of mobility for all the values of energetic disorder under study is similar except at low electric field strengths ($\sim$$<$3.6x10$^5$ V/cm). In \textit{case1}, the field dependence of mobility at lower electric field strengths first decreases with increase of electric field and reaches a minimum value of mobility (represented by arrows in Fig. 2) before it shows positive dependence in a $\log\mu Vs. E^{1/2}$ fashion as predicted by GDM [1-4]. In \textit{case 2}, for all the values of energetic disorder under study, the field dependence of mobility at lower electric field strengths show a clear saturation of mobility before it show positive dependence in a $\log\mu Vs. E^{1/2}$ fashion as predicted by GDM [1-4]. Here the important point to highlight is that in \textit{case1} we have observed the NFDM, at lower electric field strengths, even in the absence of positional disorder. This is against the GDM which strongly asserts that NFDM can occur only in the presence of high positional disorder [1-4,12]. When the energetic disorder decreases the NFDM at lower electric field strengths become remarkable. The strength of NFDM is assigned as the difference between the mobility value for the lowest electric field strength under study and the observed minima of the mobility at low field regime (shown by arrows in Fig. 2). The strength of NFDM increases with decrease of disorder as shown in Fig. 3. Here we concentrate only on the origin of the difference in field dependence of mobility at lower electric field strengths between \textit{case1} and \textit{case 2}. Simulation similar to \textit{case 1} reported by H. B\"{a}ssler [1a] shows a saturation of mobility at low field regime. Unfortunately, in the reported data presents very few number of data points at low field regime which is not all sufficient to draw a clear conclusion about the nature of field dependence of mobility. Thus the authors might have missed the very small NFDM present at high value of energetic disorder ($\sim$0.1eV) which is generally used for simulation.
In both \textit{case 1} and \textit{case 2} the simulation was performed for same input parameters. The only different is that in \textit{case 1} the required sample length (4$\mu$m) is covered with use of PBC and in \textit{case 2} without using PBC. So the observed difference in field dependence of mobility at low field regime between \textit{case 1} and \textit{case 2} can be an artifact of PBC. When PBC is implemented as in \textit{case 1}, the carrier is taken to the first plane (\textit{z}=1 or \textit{x}=1 or \textit{y}=1, depending on the direction) when it reaches the final plane (\textit{z}=70 or \textit{x}=70 or \textit{y}=70). In that process some hops that the carrier may make in the absence of a boundary can be neglected. In the absence of a boundary the carrier may wander more before proceeding further in the field direction (Fig. 1a). So in \textit{case 1}, some hops are neglected while implementing the PBC. Therefore, the number of hops made by the carrier in \textit{case 2} must be higher than in \textit{case 1}. Compared to high electric field strength, carrier wanders more in the lattice at low electric field strength during its transit along the field direction. Thus the number of hops neglected at the boundaries upon implementing the PBC is expected to be high at low field and vice versa. This predicts a remarkable difference between the total number of hops made by the carrier in \textit{case 1} and \textit{2} and this difference is expected to be high at lower electric field strength. This difference should gradually decreases with increase of electric field strength and become negligibly small at higher electric field strengths. Compared to low value of energetic disorder the carrier wanders more in the lattice at high value of energetic disorder during its transit. At low value of energetic disorder carrier is forced to move mostly in the direction of the applied field. Similar to the above explanation, for a constant field strength (for example the lowest field strength used in the study, 4x10$^4$ V/cm) the difference between the total number of hops, made by the carrier in \textit{case 1} and \textit{2}, should increase with the increase of energetic disorder. We have confirmed these conjectures in our simulation.
Fig. 4 shows the variation of the total number of hops made by the carrier in \textit{case 1} and \textit{2}, at T=300K, with electric field for various values of energetic disorder. Fig. 4 clearly shows that at low electric field strengths, for all the values of energetic disorder under study, the total number of hops made by the carrier in \textit{case 2} is higher than in \textit{case 1}. This difference between the total number of hops for the two cases (\textit{case 1} and \textit{case 2}) is higher for the lowest electric field strength used in the study (4x10$^4$ V/cm). This difference decreases with increase of electric field strength and become negligible at very high electric field strengths. It also shows that the difference in the total numbers of hops (hops), made by the carrier in two cases (\textit{case 1} and \textit{2}), is higher for higher value of energetic disorder (inset of Fig. 4). Thus, Fig. 4 unambiguously establishes the fact that upon implementing the PBC substantial number of hops that the carrier may make in the absence of boundary is neglected especially at low electric field strengths as well as at high value of energetic disorder.
The neglect of hops, in \textit{case 1}, upon implementing the PBC certainly influences the charge transport especially the transit time. It is confirmed through simulation that in both cases the charge carriers have attained dynamic equilibrium [8,16] in a similar fashion while covering the required sample length. So the neglected hops do not affect the energy relaxation of carrier during the hopping in Gaussian DOS. Fig. 5 shows the field dependence of the difference between the carrier transit time ($\Delta\tau$) for \textit{case 2} and \textit{case 1} for various value of energetic disorder. For all values of energetic disorder under study, the maximum difference in carrier transit time between \textit{case 2} and \textit{case 1} is observed for the lowest electric field strength. This difference gradually decreases with increase of electric field strength and become negligibly small at higher electric field strengths. In \textit{case 1}, the neglect of hops effectively gives an extra bias for the carrier to move in the applied field direction. In principle, carrier covers the required sample length in less number of hops. Thus the transit time of the carrier in \textit{case 1} must be less than that of \textit{case 2}. The maximum difference in transit time between \textit{case 1} and \textit{case 2} must occur for the case with maximum difference in the average number of hops. Hence the maximum difference in transit time must occur at low field strengths and this difference should gradually decreases with increase of electric field strength which ultimately become negligibly small at high electric field strengths (Fig. 5). For the same reason the difference in transit time between \textit{case 1} and \textit{case 2} must be higher for higher value of energetic disorder (Fig. 5) and this difference should decrease with decrease of energetic disorder. Having shown how the difference in transit time, for \textit{case 1} and \textit{case 2}, varies as a function of electric field and energetic disorder, we would like to relate the difference in transit time to the difference in mobility,
\begin{equation}
\Delta \mu = \mu _1 - \mu _2 = \frac{L}{E}\left[ {\frac{1}{{\tau _1 }} - \frac{1}{{\tau _2 }}} \right] = \frac{L}{E}\xi
\label{eq1}
\end{equation}
where \textit{L} is the thickness of sample, \textit{E} is the applied electric field, $\xi=\frac{\Delta\tau}{\tau_{1}\tau_{2}}$, $\Delta\tau$ is the difference in transit time between \textit{case 1} and \textit{case 2}, $\tau_{2}$ is transit time in \textit{case 2} and $\tau_{1}$ is the transit time in \textit{case 1}. Equation 1 suggests that the difference in mobility, for a constant electric field and thickness, depends on the value of $\xi$ rather than on $\Delta \tau$ alone. In Fig. 6 electric field is limited to the range where the value of $\Delta \tau$ is significant. For any value of energetic disorder, $\xi$ decreases with increase of electric field strengths (Fig. 6(a)). This supports the observed NFDM at low field regime for all values of energetic disorder under study. It is clear from Fig. 6(b) that the value of $\xi$ also decreases with increase of energetic disorder. This suggests a strong NFDM for low value of energetic disorder and vice versa as shown in Fig. 3.
The above presented data and its explanation clearly showed that an artifact can occur at low electric field regime upon implementing the PBC in conventional way. The observed artifact is reduced when simulated in a lattice with large size along the field direction. When the lattice size along the field direction is large, carrier encounters fewer boundaries while covering the required sample length. This reduces the number of neglected hops and hence the artifact (data not shown). This observation stresses the influence of neglected hops and supports the explanation of observed artifact in \textit{case 1}. This method is inconvenient because the artifact cannot be completely removed and bigger arrays require large computations resources as well. More over, the size of the array required depends on chosen sample length and energetic disorder. In \textit{case 3} a different method of simulation is adopted with which the PBC can be implemented without the observed artifact. Hence the simulation was performed on a lattice of size 70x70x150 along \textit{x, y} and \textit{ z} direction. The carrier was injected into the first plane (\textit{z}=1) and allowed to move till \textit{z}=140$^{th}$ plane. Once it reaches the \textit{z}=140$^{th}$ plane the carrier was taken to \textit{z}=70 plane, keeping the energy of carrier and \textit{x} and \textit{y} co-ordinates the same as at z=140$^{th}$ plane. In \textit{case 3}, the carrier is allowed to make all the required number of hops around \textit{z}=70 plane when the PBC is used. Thus the hops that are neglected at the boundaries in \textit{case 1} are properly accounted for, there by eliminating the observed artifact. Therefore, the field dependence of mobility and the total number of hops made by the carrier in \textit{case 3} is expected to be similar as in \textit{case 2} for all the values of energetic disorder. This is clearly shown in from Fig. 7 where the data in both cases (\textit{case 2} and \textit{case 3}) super impose each other. Fig. 7 asserts the fact that NFDM observed in \textit{case 1} is due to extra bias gained by the carrier due to the neglected hops. Care should be taken while choosing the dimension of the array because carrier in any case should not reach the \textit{z}=1 plane from the plane it has taken (\textit{z}=70 in our case) after the use of PBC. If carrier move back and touches \textit{z}=1 plane, then some hops may be neglected and method become less effective which in turn may result in NFDM. Hence a sufficient buffer lattice must be provided for the wandering of the carrier around the plane where it has taken after the use of PBC. In this study, the optimum buffer size (70x70x70) is obtained by comparing the field dependence of mobility for different buffer sizes with the field dependence of mobility for similar inputs using \textit{case 2} (data not shown). The presence of small buffer between z=140$^{th}$ and z=150$^{th}$ plane makes sure that the probability of jumping is always calculated for the same number of sites.
Our simulation studies also showed that PBC applied along \textit{x} and \textit{y} direction has negligible effect on the field dependence of mobility (data not shown). This suggests that the observed artifact is due to the substantial number of hops that are neglected at the boundaries perpendicular to the field direction (\textit{z} direction) upon implementing the PBC. The reason may be that the number of boundaries carrier encounters perpendicular to \textit{x} and \textit{y} direction is less compared to the number of boundaries the carrier encounters perpendicular to z direction, which is inevitable while covering the required sample length.
\section{Conclusion}
This study exposes an artifact in conventional method of implementing the PBC in Monte Carlo simulation of charge transport in disordered organic systems. This manifests in NFDM at low field regime for all values of energetic disorder under study even in the absence of positional disorder. Upon implementing the PBC a substantial number of hops that the carrier may make in the absence of boundary, which it encounters perpendicular to the field direction, are neglected. Effectively, the carrier covers the required sample length in less number of hops resulting in a shorter transit time. Thus the carrier gains an extra bias along the field direction. Hence the origin of artifact is rationalized on the basis of this extra bias gained by the carrier to move in the applied field direction. The simualtions confirmed that the boundary condition applied in directions other than applied field directions have negligible effect on charge transport. An alternative approach for simulation with which one can implement the PBC without the observed artifact is also proposed. In cocise this study cautions the researchers who adopt periodic boundary condition in Monte Carlo simulation for studying the charge transport mechanism especially at low field regime.
\section{Acknowledgments}
Authors are grateful to S.C. Mehendale for his guidance and the critical reading of the manuscript. S. Raj Mohan is grateful to the BRNS, India, for providing Dr. K.S. Krishnan Research Associateship.
\newpage
\section{References}
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\item P. M. Borsenberger and D. S. Weiss, Organic Photoreceptors for Xerography,Vol. 59 of Optical engineering series, Marcel Dekker, New York, 1998.
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\end{enumerate}
\newpage
\begin{center}
\title{Figure Captions}
\end{center}
\textbf{Figure 1.} (a) Schematic diagram showing a typical hopping motion of carrier inside the lattice. Shaded region shows the hops that are neglected upon implementing the PBC. (b) Schematic diagram showing \textit{case 1 }(c) schematic diagram showing \textit{case 3}. Schematic diagram of \textit{case 2} is same as shown in (a).\newline
\textbf{Figure 2.} Comparison of field dependence of mobility, at T=300K, for various values of disorder for ($\circ$) \textit{case 1 }and ($\bullet$) \textit{case 2}. Arrow shows the minima of mobility occurred at low field regime in \textit{case 1}. Inset shows the magnified view of low field regime for the respective cases.\newline
\textbf{Figure 3.} Variation of the strength of negative field dependence of mobility, observed in \textit{ case 1}, with energetic disorder, at T=300K. Solid line shows a guide to eye.\newline
\textbf{Figure 4.} Field dependence of the total number of hops made by the carrier in two cases (\textit{case 1} (solid line) and \textit{case 2} (dashed line)), for various values of energetic disorder, at T=300K. Inset shows the dependence of difference in total number of hops (hops), made by the carrier in two cases (\textit{case 1} and \textit{case 2}), on energetic disorder at 4x10$^4$V/cm and T=300K.\newline
\textbf{Figure 5.} Field dependence of the difference between the carrier transit time ($\Delta\tau$), for \textit{case 1} and \textit{case 2}, for various values of energetic disorder at T=300K. \newline
\textbf{Figure 6.} Variation of $\xi$ (a) with electric field strength for the values energetic disorder where reasonably strong NFDM is observed (b) with energetic disorder for the various value of electric field strength at low field regime. Solid line in (b) is a guide to eye. Averaging is carried over 500 carriers at T=300K.
\textbf{Figure 7.} Comparison of the field dependence of mobility, at T=300K, simulated for ($\bullet$) \textit{case 2} and ($\circ$) \textit{case 3}. Inset shows the comparison of total number of hops made by carrier in ($\blacktriangle$) \textit{case 2} and ($\circ$) \textit{case 3.}
\newpage
\newpage
\begin{figure}
\includegraphics*[width=15cm]{Figure1.eps}
\label{Fig1}
\end{figure}
\newpage
\begin{figure}
\includegraphics*[width=15cm]{Figure2.eps}
\label{Fig2}
\end{figure}
\newpage
\begin{figure}
\includegraphics*[width=15cm]{Figure3.eps}
\label{Fig3}
\end{figure}
\newpage
\begin{figure}
\includegraphics*[width=15cm]{Figure4.eps}
\label{Fig4}
\end{figure}
\newpage
\begin{figure}
\includegraphics*[width=15cm]{Figure5.eps}
\label{Fig5}
\end{figure}
\newpage
\begin{figure}
\includegraphics*[width=15cm]{Figure6a.eps}
\label{Fig6}
\end{figure}
\newpage
\begin{figure}
\includegraphics*[width=15cm]{Figure6b.eps}
\label{Fig6b}
\end{figure}
\newpage
\begin{figure}[hbtp]
\includegraphics*[width=15cm]{Figure7.eps}
\label{Fig7}
\end{figure}
\end{document}
|
0909.2235
|
\section{Introduction}
\label{sec:int}
Various dynamic phenomena in the solar corona, as for example soft
X-ray jets and specific flares, have been associated with magnetic
reconnection occurring in a three-dimensional (3D) magnetic nullpoint
topology consisting of a dome-like fan separatrix surface located
below the nullpoint and a spine field line above it \citep[e.g.][]
{lau90,ant98,pri02}. Observational evidence of 3D nullpoint topologies
in the corona is provided by, e.g., ``saddle-like'' loop structures
\citep{fil99}, ``ellipsoidal flare ribbons'' \citep{mas09}, and
``anemone'' active regions within coronal holes \citep[][]{shi92}.
The latter are characterized by a full or partial ring of radially
aligned bright loops which connect the opposite polarities of the
region and the surrounding coronal hole \citep[e.g.][]{asa08}. Anemone
active regions are often associated with soft X-ray jets \citep[e.g.]
[]{shi94}, which are a strong indication of 3D nullpoint reconnection
occurring in the corona. Further evidence for coronal nullpoint
topologies comes from subphotospheric source models of the coronal
magnetic field \citep{dem93}, and from potential and linear force-free
field extrapolations of flare and jet regions \citep[e.g.][]{aul00,
fle01,uga07,mor08}. Consequently, 3D fan-spine configurations are
increasingly used as the initial condition in numerical simulations
of 3D reconnection, jets, and flares \citep{pon07,par09,mas09}. The
reconnection is triggered by boundary motions in these simulations.
The {\em formation} of 3D fan-spine configurations in the corona,
however, has not yet been studied in much detail. In potential or
near-potential magnetic fields, a 3D nullpoint configuration with
a fan and a spine naturally occurs if a magnetic bipole is embedded
into one polarity region of a large-scale bipolar ambient field
\citep[e.g.][]{ant98}. Therefore it is expected to form when magnetic
flux emerges into regions of essentially unipolar fields, such as a
coronal hole. The fan-spine configuration then results from the
relaxation of the coronal field after its reconnection with the
emerging flux. 2D numerical simulations of coronal soft X-ray jets,
inspired by the flux emergence model by \cite{hey77}, have
demonstrated that reconnection between emerging bipolar flux and a
vertical or oblique coronal field yields the formation of hot loops
connecting the ambient field with the opposite polarity flux of the
emerging bipole \citep[e.g.][]{yok96,nis08}.
\cite{mor08} recently performed a 3D simulation of the emergence of a
twisted flux tube into an oblique unipolar coronal field. As in the 2D
cases, they found the launch of a jet and the formation of a growing
system of hot reconnected loops connecting the ambient field with the
emerging flux of opposite polarity. The resulting fan surface extends
on one side of the emerging region, while on the other side it consists
of non-reconnected emerged loops only. The latter are not strongly
heated and would hence unlikely be seen in soft X-ray observations,
therefore only one half of an anemone loop pattern should be visible.
\cite{par09} used an alternative approach to produce a jet, by starting
from a 3D nullpoint topology and driving the jet by reconnection between
open and closed field lines, after the latter have been significantly
twisted by line-tied boundary motions. One outcome of this calculation
is that, as a result of reconnection of twisted fields, the nullpoint
moved around the axis of the spine, thus
allowing reconnection of field lines from all sides of the fan. While
this evolution may allow for the brightening of the whole fan in soft
X-rays, it still does not explain how the fan-spine topology was formed
in the first place.
These models, although nicely reproducing coronal jets, their associated
inverse-\textsf{Y} ``Eiffel tower'' shape, and the field line geometry
obtained from a linear force-free field extrapolation of a jet region
\citep{mor08}, do therefore not yet provide a satisfying scenario for the
formation of anemone active regions. As mentioned above, these regions
typically show a ring of bright loops below the jet that is reminiscent
of a fan dome, which in some cases might well extend the area of emerged
flux.
A two-dimensional effect may also play a role in the brightening of both
sides of fan surfaces which form during flux emergence. This effect is
the reconnection of emerging field lines back and forth with the ambient
fields on both sides of the emerging flux. This is exactly what happens
in the 2.5D simulation of \cite{mur09}, where an ``oscillatory reconnection''
pattern \citep{cra91} occurs, which the authors attributed to thermal
pressure effects around the reconnection layer. This process was also found
in 2.5D simulations of a quadrupolar closed field configuration, being driven
by a non line-tied chromospheric ad-hoc monotonic force \citep{kar98}.
Reconnection back and forth was there attributed to coronal relaxation, as a
response to an ``overshoot'' due to a chromospheric driving which was faster
in the simulation than on the real Sun. In the simulation of \cite{mur09},
the photospheric emergence velocities were small compared to the coronal
Alfv\'en speeds (Murray, private communication, 2009), so that such an ``overshoot''
probably did not occur. However, the existence of (non-wave driven) oscillatory
coronal reconnection in fully 3D configurations is yet to be established.
An interesting observational feature of coronal jets is their frequent
transverse oscillation \citep{cir07}. Even though 2D oscillatory reconnection
could account for such perturbations \citep{mur09}, they could also be caused
by non-steady reconnection in a turbulent current sheet, where magnetic islands
are gradually formed and destroyed \citep[e.g.][]{yok96,kli00,arc06}, as well
as by upward propagating Alfv\'en waves being launched from the reconnection
point \citep{cir07,mor08}, or by the propagation of a torsional Alfv\'en wave
resulting from the reconnection of kinking twisted field lines with their ambient
field \citep{par09,fil09}. The latter mechanism provides an explanation also for
the frequently observed helical patterns traveling along jets
\citep{shi92a,pat08,nis09}.
In this paper, we propose a fully 3D two-step reconnection model for
the formation of broad fan-spine configurations in the corona. The
model results from a zero-$\beta$ line-tied MHD simulation, in which
the evolution of the coronal magnetic field is driven by twisted flux
emergence prescribed at the photospheric boundary. The simulation was
initially developed for the interpretation of a puzzling event observed
at the solar limb by the {\it Hinode} X-ray telescope (XRT). In this
event, two distinct small coronal loop systems developed one after the
other beside the edge of a prominence cavity, the first one apparently
``feeding'' the second one, while a swaying jet-like brightening was
propagating along the cavity edge.
Our model does not only reproduce the shape and timing of the main features
observed in this particular event, but also accounts for the formation of
full (and not half) anemone bright features, as generally observed in soft
X-rays within coronal holes. It finds (in accordance with some past
observations and models of coronal jets) that a large fraction of the
emerged magnetic twist reconnects and is evacuated upward in the form of
a torsional Alfv\'en wave. It furthermore shows that nullpoint reconnection
can be accompanied by slipping reconnection \citep[e.g.][]{aul06}, which is
supported by apparently slipping cavity loops observed by XRT in the event.
It finally predicts that a fraction of the twist eventually remains around
the inner spine beneath the fan/anemone surface, which therefore does not
fully relax to a potential field configuration, even though it looks
potential at first order.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{fg1.jpg}
\caption
{
{\bf (a):} MSDP $H_{\alpha}$ observation of the prominence on 2007 April 24,
at 13:38 UT.
{\bf (b):} XRT observation of the bright loop systems on 2007 April 24, at
18:35 UT (compare with Fig.~\ref{fig:xrt}), overlaid with $H_{\alpha}$
intensity contours of (a).
}
\label{fig:prom}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.97\linewidth]{fg2.jpg}
\caption
{
Snapshots from the XRT observations ($300^{\prime\prime} \times
212^{\prime\prime}$) at different times. The bright loop systems
are denoted by L1, L2, and K. Asterisks and horizontal arrows are
plotted at the same positions in all images to outline the oscillatory
transverse motion of K and the displacement of its elbow. The vertical
arrow points at loops which are seen to ``slip'' toward the prominence
at the later phase of the evolution. See the text for details. (An mpeg
animation of this figure is available in the online journal.) The
animation uses a different intensity scaling, which outlines the cavity
loops in more clarity.
}
\label{fig:xrt}
\end{figure*}
\section{Observations}
\label{sec:obs}
\subsection{Instrumentation}
\label{subsec:obs_ins}
The observations presented here were obtained during a coordinated campaign of
prominence observations, involving several space and ground-based instruments.
They were performed in the JOP 178 frame on 2007 April 23--29, during the first
SUMER-{\it Hinode} observing campaign. JOP 178 has been running successfully many
times in the past (see \url{http://bass2000.bagn.obs-mip.fr/jop178/}). A
prominence surrounded by a cavity on the west limb, at S30 degree, was extensively
studied during the campaign \citep[e.g.][]{hei08}. Fig.~\ref{fig:prom} shows an
$\mathrm{H}_{\alpha}$ observation of the prominence obtained by the Multi channel
subtractive double pass spectrograph (MSDP) operating at the Solar Tower of Meudon,
as well as an overlay of the corresponding intensity contours with the XRT
observations described below. The prominence was located in a quiet Sun region,
apparently along the polarity inversion line (PIL) of an extended bipolar area of
weak magnetic field, without visible strong field concentrations. A corresponding
filament channel could be observed days before in EUV. Here we focus on XRT
observations of a small dynamic event which took place at the edge of the cavity
on 24 April.
The {\it Hinode} mission has been operating since 2006 October \citep{kos07}.
XRT is a high resolution grazing incidence telescope which consists of X-ray
and visible light optics and a 2k x 2k CCD camera. A set of filters and a
broad range of exposure times enable the telescope to observe hot plasma in
the range 10$^5$--10$^8$ K \citep[for more details see][]{gol07}. The
observations presented here were obtained with a resolution of $512 \times 512$
pixels, each pixel having a size of $1.03'' \times 1.03''$. The filter combination
used was Al\_poly/Open. The exposure time was $8.19$ s or $16.38$ s, the cadence
was $60$ s.
\subsection{Event description}
\label{subsec:obs_eve}
The brightening of several loop systems at the edge of the prominence
cavity was observed by XRT for about two hours on 2007 April 24,
between about 18:30 and 20:30 UT. The dynamic evolution described in
the following lasted about 45 minutes, ending at 19:15 UT.
Fig.~\ref{fig:xrt} shows snapshots outlining the main features of the
evolution. At about 18:30 UT, two bright loop systems become visible,
one set of small closed loops at the left cavity edge (denoted as L1),
and one large loop arching over the prominence (denoted as K), outlining
the cavity edge (Fig.~\ref{fig:xrt}a). K exhibits an elbow (indicated
by the horizontal arrow), apparently located above L1. A very small loop
system is observed at the apparent footpoint of K (denoted as L2 in
Fig.~\ref{fig:xrt}a). In the following minutes, L1 continuously grows
and the elbow slowly moves to the left. At about 18:39 UT, a brightening
begins to propagate along K, starting from below the elbow.
At about 18:45 UT, the evolution becomes more dynamic: L1 seems to
expand toward L2, loops apparently connecting L1 and L2 become visible,
and L2 starts to grow (Fig.~\ref{fig:xrt}b,c). At the same time,
the propagating brightening has arrived at the elbow and now moves
further along K, exhibiting a jet-like appearance. The upper part of K
moves leftward (see asterisks) and after a while the elbow cannot be
observed anymore. Shortly after, the part of K on the opposite side of
the prominence starts to brighten weakly, indicating that the loop has
been filled with hot plasma, ejected from below the elbow (the
propagation of the jet-like brightening along K is better visible in
the online movie accompanying Fig.~\ref{fig:xrt}).
At 18:52 UT, L1 and L2 are roughly of the same size. They appear to have
a common footpoint region and they collectively exhibit an anemone-like
shape (Fig.~\ref{fig:xrt}d,e). About this time, the upper part of K
moves back toward the right. At 19:01 UT L1 has started to fade away, and
L2 has stopped growing (Fig.~\ref{fig:xrt}e). From about 19:10 UT on, L1
is no longer visible. At 19:19 UT, the upper part of K has returned to its
initial position (Fig.~\ref{fig:xrt}f). The transverse oscillatory motion
of K is suggestive of an Alfv\'en wave traveling along it \citep{cir07}.
Between about 19:00 UT and 19:20 UT, a successive leftward displacement
of loops starting from the footpoint of K on the opposite edge of the
cavity is observed (see the vertical arrow in Fig.~\ref{fig:xrt}f). After
that, L2 slowly fades away in about an hour and no further significant
dynamic evolution is observed.
The observations suggest that the dynamic evolution was caused by the
interaction of newly emerging flux with the arcade-like field overlying
the prominence. The inspection of SoHO/MDI magnetogram data from the days
leading up to the event did not reveal significant long-lived bipolar field
concentrations in the vicinity of the prominence, which would have indicated
the presence of a 3D nullpoint topology before the event \citep[as in the
simulation of][]{par09}. Since the prominence-cavity system was located at
the limb during our event, direct observations of emerging flux are not
available. However, the brightening of the elbowed loop system K, as well
as the jet-like brightening propagating along it, can be understood by means
of models of magnetic reconnection between a small emerging bipole and a
predominantly vertical coronal field (see Sect.~\ref{sec:int}). This suggests
that the loop system L1 was not outlining emerging fields, but rather a
reconnected arcade that formed along with the elbow in K. The fact that L1
has a significant height when it becomes visible supports this interpretation.
Still, the transverse oscillation of K, the growth of the loop system L2,
and the ``slipping''-like motion of loops at the opposite edge of the
cavity cannot be understood straightforwardly within this scenario.
We note that the observed dynamics did not seem to have a noticeable
effect on the stability of the prominence-cavity system. The prominence
was still observed on April 25 \citep{hei08} and on April 26 by the Meudon
Solar Tower. It hence appears that this is a case where emerging flux in
the vicinity of a filament or prominence does not result in the eruption
of the latter \citep[see, e.g.,][]{fey95}.
\section{Numerical simulation}
\label{sec:simu}
In order to understand the full dynamics observed, we perform a 3D MHD
simulation of the interaction of emerging flux with an arcade-like
potential field overlying a coronal flux rope. The choice of such a
coronal topology is supported by the presence of the prominence-cavity
system. Magnetic flux ropes have been successfully used to model
prominences and cavities \citep[e.g.][]{low95,aul98,van04,gib06a}. We
use the analytical flux rope model by \citet[][hereafter TD]{tit99} as
the initial condition in the simulation. The model consists of a toroidal
current ring with major radius $R$ and minor radius $a$, which is partly
submerged below a ``photosphere'' and is held in equilibrium by an external
potential field created by two subphotospheric magnetic charges $\pm q$,
which are placed at the axis of the torus, at distances $\pm L$ from the
current ring. The coronal part of the model yields an arched, line-tied,
and twisted flux rope which is embedded into an arcade-like potential field
(see Fig.~2 in TD). The depth $d$ of the torus axis (and hence of the magnetic
charges) below the photospheric layer determines the compactness and strength
of the magnetic flux distribution in the photospheric plane. Here we choose
a relatively large depth (see below), in order to account for the observed
extended area of weak field above which the prominence was located (see
Sect.~\ref{subsec:obs_ins}).
Previous simulations \citep[][]{toe04,toe05,toe07,sch08} and analytical
calculations \citep{ise07} have shown that the TD flux rope can be subject
to the ideal MHD helical kink and torus instabilities. Therefore we use a
weakly twisted rope here, with the field lines winding on average only once
about the rope axis (in a right-handed sense), and we choose the potential
field such that the rope is stable with respect to the torus instability
\citep{bat78,kli06}.
As in these previous simulations we integrate the $\beta=0$ compressible
ideal MHD equations:
\begin{eqnarray}
\partial_t\rho&=&
-\nabla\cdot(\rho\,\boldsymbol{u})\,, \label{eq_rho}\\
\rho\,\partial_{t}\boldsymbol{u}&=&
-\rho\,(\,\boldsymbol{u}\cdot\nabla\,)\,\boldsymbol{u}
+\boldsymbol{j}\mbox{\boldmath$\times$}\boldsymbol{B}
+\nabla\,\cdot\bf{T}\,,
\label{eq_mot}\\
\partial_{t}\boldsymbol{B}&=&
\nabla\mbox{\boldmath$\times$}(\,\boldsymbol{u}\mbox{\boldmath$\times$}
\boldsymbol{B}\,)\,, \label{eq_ind}
\end{eqnarray}
where $\boldsymbol{B}$, $\boldsymbol{u}$, and $\rho$ are the magnetic
field, velocity, and mass density, respectively. The current density
is given by
$\boldsymbol{j}=\mu_0^{\,-1}\,\nabla\mbox{\boldmath$\times$}\boldsymbol{B}$.
The term {\bf T} is the viscous stress tensor, included to improve
the numerical stability \citep{toe03}. We neglect thermal pressure
and gravity since we are interested in the qualitative evolution of
the magnetic field only.
The MHD equations are normalized by quantities derived from a characteristic
length, taken here to be the initial apex height of the TD flux rope axis
above the photospheric plane, $R-d$, the initial magnetic field strength,
$B_0$, and Alfv\'en velocity, $v_{a0}$, at the apex of the axis, and derived
quantities. We use a nonuniform cartesian grid of size $[-4,4] \times [-5,5]
\times [0,8]$, resolved by $261 \times 301 \times 208$ grid points (including
two layers of ghost points in each direction which are used to implement the
boundary conditions), with a resolution of $\Delta x = \Delta y = 2\Delta z
= 0.02$ in the box center. The resolution is nearly constant in the subdomain
$[-1,1] \times [-1.5,1.5] \times [0,1]$, and increases exponentially toward
the boundaries, to maximum values $\Delta x_{\mathrm{max}}=0.14$,
$\Delta y_{\mathrm{max}}=0.10$, and $\Delta z_{\mathrm{max}}=0.40$. The plane
\{$z=0$\} corresponds to the photosphere.
The TD flux rope axis is oriented along the $y$ direction, with the positive
polarity rope footpoint rooted in the half-plane \{$y>0$\} (see Fig.~\ref{fig:reco1}).
The normalized geometrical flux rope parameters used are: $R=2.75$, $a=1.$,
$L=2.5$, and $d=1.75$. The top left panel in Fig.~\ref{fig:reco1} essentially
shows the initial TD configuration, except for the small parasitic bipole and
the blue field lines on the left-hand side of the TD flux rope.
We employ a modified two-step Lax-Wendroff method for the integration,
and we additionally stabilize the calculation by artificial smoothing of
all integration variables \citep[see][for details]{sat79,toe03}. The
reconnection occurring in our simulation is due to the resulting numerical
diffusion. The initial density distribution is
$\rho_0(z)=2.6\,\mathrm{exp}\,(-[z+\Delta z]/1.1)$, chosen such that the
Alfv\'en velocity, $v_a$, slowly decreases with height above the TD flux
rope. The system is at rest at $t=0$.
We first perform a numerical relaxation of the system for $37$ Alfv\'en times
and reset the time to zero afterwards. We then model the emergence of a second
twisted flux rope in the vicinity of the TD rope, following the boundary-driven
method by \citet[][hereafter FG]{fan03}. In their model, a toroidal twisted
flux rope is rigidly emerged from a fictitious solar interior into a coronal
magnetic field by successively changing the boundary conditions in the
photospheric layer of the simulation box. We refer to Fig.~1 in FG for a sketch
of the model. In our simulation, we choose the FG torus to be about one order of
magnitude smaller in size than the TD torus (the major and minor radius of the
FG torus are 0.3 and 0.2, respectively), in order to account (within the limitations
given by the finite number of grid points) for the typically large difference in
size between quiescent filaments and small bipoles that emerge in their vicinity
\citep[see, e.g.,][]{fey95}, which was also suggested by the relative scale-lengths
of the coronal loops observed in the event described in Sect.~\ref{subsec:obs_eve}.
The FG flux rope is uniformly twisted along its cross-section. The twist is
chosen right-handed, with the field lines winding $\sim 4.5$ times along the
whole torus. We position the rope such that the emergence region is centered
at $(x,y)=(-1.0,-0.5)$, within the large-scale negative polarity of the TD
potential field. As the TD flux rope, it is oriented along the $y$ direction,
but with the opposite orientation of the axial magnetic field (see
Fig.~\ref{fig:reco1}). The magnetic field strength within the FG torus varies
along its cross-section, being $\sim 3\,B_0$ at the outer torus surface,
$4 B_0$ at the axis, and $\sim 13 B_0$ at the inner torus surface, for the
parameters used in the simulation. Although the emergence is driven until the
apex of the inner surface approaches the bottom boundary of the box (see below),
the field strength which effectively enters the corona during the simulation
does typically not become larger than $\sim 6 B_0$. The field strength of the
large-scale TD field in the small volume above the emergence area is approximately
constant, $\sim 0.7\,B_0$. Within the TD flux rope, the field strengths vary
between $\sim 0.6$ and $1.0\,B_0$. We discuss the rationale for our choice of
the orientation and strength of the magnetic field within the FG flux rope in
Sect.~\ref{sec:dis}.
The boundary-driven emergence is imposed in the layer \{$z=-\Delta z$\}.
Within the emergence area in this layer, we overwrite the pre-existing
TD field by the respective FG flux rope field, and we set the vertical
velocity equal to the respective driving velocity, while keeping the
horizontal velocities at zero. Outside this area, the TD field and the
density in \{$z=-\Delta z$\} are kept at their initial values, and the
velocities are set to zero, at all times. These settings lead to
significant jumps in strength and orientation of the magnetic field (i.e.
to the formation of large values of $\nabla \cdot \bf{B}$) at the interface
between the TD and FG fields at and close to the bottom plane. Since our
code does not conserve $\nabla \cdot \bf{B}=0$ to rounding error, we use
a diffusive $\nabla \cdot \bf{B}$ cleaner \citep[][]{kep03}, as well as
Powell's source term method \citep[][]{gom94}, to minimize unphysical
effects resulting from $\nabla \cdot \bf{B}$ errors. Furthermore, our
overspecified boundary conditions (see above) trigger spurious oscillations,
which after some time lead to numerical instabilities close to the bottom
plane, in particular at the interface between the TD and FG fields. In order
to prevent these instabilities, we apply an enhanced smoothing of all
variables close to the boundary \citep[as in][]{toe03}, and we set the
Lorentz force densities at \{${z=0}$\} to zero at all times. We find
that these settings result in the formation of a rising twisted flux tube
above the emergence area, as desired.
The emergence is driven quasi-statically with a maximum velocity of
$0.01 v_{a0}$. The driving velocity is linearly increased and decreased
for $10$ Alfv\'en times before and after the main emergence phase (which
lasts for $30$ Alfv\'en times), respectively. The emergence is stopped at
$t=50$, when the apex of the inner surface of the FG torus has reached
the layer \{${z=-\Delta z}$\}. The total twist transported into the corona
by the emerged section of the FG torus corresponds to $\sim 1.7$ field
line turns.
\begin{figure*}[t]
\centering
\includegraphics[width=0.97\linewidth]{fg3.jpg}
\caption
{
Selected magnetic field lines outlining the first reconnection which occurs
between the emerging flux rope and the outer coronal arcade. The bottom panels
show a zoom into the emergence region. Contours of $B_z>0$ ($B_z<0$) at the
bottom plane \{$z=0$\} are shown in white (black). Polarity inversion lines (PILs)
are drawn in yellow. The negative polarity of the large-scale potential field is
located on the left-hand side of the main PIL. All field lines are calculated
from fixed footpoints on the left-hand side of the main PIL in all panels.
From left to right: at onset of ($t=22$), during ($t=34$), and after ($t=43$)
the reconnection. Blue field lines the negative
polarity of the emerging flux rope, green field lines show the core of the
prominence flux rope. Red field lines outline the outer coronal arcade at early
times and the newly formed small loop system later on. Pink field lines show the
inner coronal arcade. Arrowheads mark the direction of the
magnetic field. The black circles located above the main PIL
mark field line dips, which are assumed to carry prominence material.
(An mpeg animation of this figure is available in the online journal.)
}
\label{fig:reco1}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.97\linewidth]{fg4.jpg}
\caption
{
Snapshots of the configuration outlining the second reconnection, which
occurs between the newly formed small loop system and the inner coronal
arcade. Field lines, contours, and black circles are as in
Fig.~\ref{fig:reco1}. The bottom panels show a zoom into the emergence
region. From left to right: at onset of ($t=46$) and during ($t=49$, $57$)
the reconnection. Pink field lines show the inner arcade initially, and
the second newly formed loop system later on. The propagation of an
Alfv\'en wave along the blue field lines and the slipping motion of the
footpoints of the red field lines on the right-hand side of the prominence
flux rope are visible. (An mpeg animation of this figure is available in
the online journal.)
}
\label{fig:reco2}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.95\linewidth]{fg5.jpg}
\caption
{
{\bf (a--c):} Side view on the flux emergence region in the simulation at
$t=17$, $34$, and $51$. The contours at the bottom plane are as in
Figs.~\ref{fig:reco1} and \ref{fig:reco2}. Selected magnetic field lines
are shown in order to outline the topological evolution.
{\bf (d):} Top view on the center of the anemone-shaped loop pattern shown
in (c), outlining the twist of the field lines.
{\bf (e)} and {\bf (f):} Top and side view on the emergence region at $t=51$,
showing field lines starting from a circle around the spine (black line)
with twice the radius as used in (c) and (d). These field lines approximately
outline the fan surface. The respective positions of the nullpoint are
indicated by circles.
}
\label{fig:topo}
\end{figure*}
\section{Simulation results and comparison with XRT observations}
\label{sec:simu_result}
In this section, we describe our simulation results and compare them with
the XRT observations described in Sect.~\ref{subsec:obs_eve}. The interaction
between the emerging FG flux rope and the pre-existing TD coronal field
results in two distinct reconnection phases, which are described in
Sects.~\ref{subsec:firststep} and \ref{subsec:secondstep}, respectively.
In Sect.~\ref{subsec:geom}, we discuss the magnetic field geometry resulting
from the reconnections, and in Sect.~\ref{subsec:promresponse} we describe the
response of the TD flux rope and its surrounding arcade to the dynamics
accompanying the reconnection. In order to emphasize that the evolution
found in the simulation does not rely on the specific flux rope models
used, we refer to the FG rope as ``emerging flux rope'', and to the TD
rope as ``prominence flux rope'' throughout this section.
Figs.~\ref{fig:reco1}--\ref{fig:topo} display magnetic field lines which
outline the main features of the magnetic field evolution in the coronal
domain. Figs.~\ref{fig:reco1} and \ref{fig:reco2} focus on the dynamics,
showing the first and second reconnection phase, respectively.
Fig.~\ref{fig:topo} shows the evolution of the magnetic topology. The blue
field lines in Figs.~\ref{fig:reco1}--\ref{fig:topo} are integrated starting
from the negative polarity of the emerging flux rope, green field lines
in Figs.~\ref{fig:reco1} and \ref{fig:reco2} outline the core of the
prominence flux rope (black circles mark field line dips). Red (pink) field
lines in Figs.~\ref{fig:reco1} and \ref{fig:reco2} show the outer (inner)
arcade overlying the prominence flux rope initially, and reconnected field
lines later on. Note that all field lines are calculated from the same
positions on the left-hand side of the prominence flux rope in all panels.
\subsection{First step: formation of one half of the anemone and jet acceleration}
\label{subsec:firststep}
As the emerging flux rope (closed blue field lines in the bottom
panels of Fig.~\ref{fig:reco1}) enters the coronal domain, it
starts to expand and a current sheet forms above the outer edge
of its positive polarity, at the interface of the rope and the
outer coronal arcade that surrounds the prominence flux rope.
Since the outermost emerging fields and the outer arcade fields
are oppositely directed at the location of the current sheet, the
two flux systems readily start to reconnect, forming a new small
loop system below the current sheet and strongly bent, elbow-shaped
field lines above it (cusp-shaped red field lines and blue field
lines in the central panels of Fig.~\ref{fig:reco1}, respectively).
Note that the not yet reconnected part of the emerging flux rope
continues to expand in the corona after the reconnection has started.
The shape of the reconnected field lines agrees very well with the
shape of the bright loops observed by XRT in the early phase of the
event described in Sect.~\ref{subsec:obs_eve} (see L1 and K in
Fig.~\ref{fig:xrt}a), which indicates that the observed loops have
been formed in an analogous reconnection process.
The field line shapes are a signature of a fan-spine configuration in
a 3D nullpoint topology (see also Fig.~\ref{fig:topo}b). A magnetic
nullpoint is indeed formed within the current sheet in the simulation,
right after the reconnection has started. It forms as the system tends
to relax to a lower energy state\footnote{The lowest possible energy
state for any photospheric magnetic field distribution which develops
during the emergence process would locally (i.e. above the emergence
region) correspond to a potential field that has to contain one single
nullpoint, owing to the presence of a closed PIL embedded in a region
of nearly vertical field \citep[e.g.][see Introduction]{ant98}.}. The
reconnection continues as time evolves, thus the size of the reconnected
red loop system increases (Fig.~\ref{fig:reco1}), in agreement with the
observed growth of L1 (Fig.~\ref{fig:xrt}a,b). The ongoing expansion of
the emerging flux rope initially slowly pushes the nullpoint away from
the prominence flux rope, in agreement with the observed leftward
displacement of the elbow indicated by the horizontal arrow in
Fig.~\ref{fig:xrt}.
As the emergence continues, the emerging flux rope field lines become
increasingly sheared with respect to the surrounding coronal arcade.
This is due to the fact that the twist within the FG flux rope is nearly
uniform along its cross-section (as in the well-known Gold-Hoyle model).
While field lines far away from the flux rope axis are strongly inclined
with respect to the axis, field lines close to the axis are almost aligned
with it. Hence, as the flux rope emerges, its outer field lines resemble
a nearly potential coronal arcade that is oriented almost orthogonal to
the local PIL, whereas its inner field lines (i.e. those close to its axis)
resemble a small sheared coronal arcade (see Fig.~\ref{fig:twist}). The
first flux rope field lines which reconnect with the large-scale coronal
arcade are thus almost unsheared with respect to the arcade (see the
bottom left panel in Fig.~\ref{fig:reco1}). As the evolution continues,
progressively more sheared loops are reconnected. As a result, the system
of new reconnected (red) field lines eventually develops a shear distribution
that is opposite to the one of the emerging flux rope: the field lines are
sheared at the edges of the system, and almost unsheared close to the local
PIL (see the bottom right panel in Fig.~\ref{fig:reco1} and the corresponding
online animation).
The reconnection does not only yield the transfer of twist from the emerging
flux rope into the newly formed red loop system. Part of the flux rope twist
is also transferred into the lower parts of the reconnected overlying blue
field lines that are now rooted in the negative polarity of the rope. Since
the upper parts of these field lines are nearly potential, whereas their
lower parts experience a sudden injection of twist, they are far from being
force-free. Their relaxation is ensured through the launch of a torsional
Alfv\'en wave which travels from low altitudes all along the arcade (see the
evolution of the blue field lines in Figs.~\ref{fig:reco1} and \ref{fig:reco2}).
Such reconnection between twisted and untwisted coronal fields has been suggested
by several authors as a driving mechanism for jets \citep[see, e.g.,][]{sch95,shi97}.
If the reconnection is sufficiently impulsive, it can launch a shear (in 2.5D) or
torsional (in 3D) Alfv\'en wave, which can accelerate the plasma upward, as shown
in numerical simulations by \cite{yok99} and \cite{par09}, respectively. The
impulsive nature and large wavenumber of the wave in our simulation is a priori
not expected, since the transition from nearly unsheared to highly sheared emerging
flux rope fields lines involved in the reconnection is continuous. Also, our code
does not incorporate any time-varying resistivity. Therefore, the impulsive launch
of the wave must result from some perturbation of the system which yields a strong
increase of the reconnection rate. Indeed, we find a strong expansion of the not
yet reconnected central part of the emerging flux rope at $t \approx 40$ (i.e.
between the stages shown in the middle and right panels of Fig.~\ref{fig:reco1};
see the corresponding online animation). At this time, a flux rope twist of
$\sim 1.5$ turns has entered the corona, indicating that this sudden increase
in expansion might be related to the onset of a kink instability
\citep[as in][]{fan03,fan07}. The reconnection-driven torsional Alfv\'en wave
in our simulation suggests an explanation for the jet-like brightening traveling
along the cavity loops observed by XRT in our event, as well as for the observed
transverse oscillation of the upper parts of the cavity loops
(see Sect.~\ref{subsec:obs_eve}).
The transverse deformation of the blue field lines in our simulation during
the passage of the wave (Fig.~\ref{fig:reco2}) is, however, obviously much
larger than what is observed in our event and typically in coronal jet-like
events \citep{cir07}. This might simply be due to the fact that, although we
have chosen the emerging flux rope to be as small as possible as compared to
the prominence flux rope (see Sect.~\ref{sec:simu}), the difference in size
between the emerging flux and the prominence-cavity system might still be
significantly larger. Also, to some extent the unrealistic reconnection time
scales in our simulation might play a role. They are mostly constrained by the
intrinsic diffusivity of the numerical scheme and by the prescribed magnetic
field smoothing, and do neither correspond to fast reconnection nor to the
seminal Sweet-Parker regime. Still, the qualitative agreement which we find
here with other coronal jets simulations performed with different codes (see
Sect.~\ref{sec:int}) suggests that we can believe in the overall mechanism for
jet acceleration which our simulation finds.
Note that the nullpoint-related elbow in the lower part of the blue field
lines apparently disappears during the evolution in our simulation (as it
does in the observation too; see Fig.~\ref{fig:xrt}). In the simulation,
this is merely due to projection effects and the motion of the nullpoint.
From a different viewing angle, an elbow at low heights remains visible.
Up to this point, the evolution is as expected from the classical model
for coronal jets and previous 2.5D and 3D simulations of this model (see
Sect.~\ref{sec:int} and Fig.~\ref{fig:topo}a,b). It results in the
formation of a 3D nullpoint topology, but actually half of its fan surface
still consists of not reconnected emerged field lines, so that only
the other half is expected to brighten in soft X-rays. In other words,
only a {\em half} anemone has been formed at this stage in the simulation.
\subsection{Second step: formation of the second half of the anemone}
\label{subsec:secondstep}
As the reconnection-driven transfer of twist/shear from the emerging
flux rope into the ambient coronal field progresses, the footpoints
of the newly formed red field lines located within the positive polarity
of the emerging flux rope are continuously displaced in the negative
$y$ direction (see Fig.\,\ref{fig:reco1} and the online animation),
toward inner arcade field lines which also overlie the prominence flux
rope and are yet unaffected by the reconnection (shown in pink in
Fig.\,\ref{fig:reco1}). Meanwhile, following the magnetic field
reorientation at high altitude, the nullpoint (and hence the reconnection
site) undergoes a counterclockwise horizontal rotation, from the leftmost
edge of the emerging rope toward its center, thereby approaching the pink
arcade field lines (see Fig.~\ref{fig:topo}). The displacement of the
footpoints of the small red loop system corresponds to the apparent
expansion of L1 toward L2 in the observation, which starts at about
18:45 UT (see Sect.~\ref{subsec:obs_eve}).
Eventually a second reconnection starts, now between the previously
reconnected sheared red field lines and the pink arcade field lines
(Fig.~\ref{fig:reco2}). It must, and indeed does, take place at the
nullpoint, which has gradually moved toward the pink arcade during
the first reconnection episode (see above). This second reconnection
leads to the formation and growth of a second small loop system
(cusp-shaped pink field lines in Fig.\,\ref{fig:reco2}), which
corresponds to the growth of the observed bright loop system L2 from
about 18:50 UT on (Fig.~\ref{fig:xrt}). The reconnected red and pink
loop systems both have footpoints within the positive polarity of the
emerging flux, and they collectively display an anemone-like shape
which is significantly wider than this polarity (Fig.~\ref{fig:topo}c;
compare also with the collective shape of L1 and L2 in
Fig.~\ref{fig:xrt}d,e).
In the course of the second reconnection, the sheared red loop system
``feeds'' the newly formed small pink loop system with magnetic flux,
hence the former shrinks while the latter grows (Fig.~\ref{fig:reco2}).
Although, at first glance, the bottom right panel gives the impression
that the red loop system has almost disappeared, the elbow visible in
the rightmost red arcade field line shows that there still exist
relatively extended, not reconnected red loop field lines at this late
phase of the second reconnection. These field lines are rooted between
the elbow-shaped field line and the rightmost, very flat red field line.
The final relative size of the red and pink loop systems will depend on
how long reconnection persists, which, in turn, will depend on a number
of factors, as for example on the ratio of the respective magnetic fluxes
present in the reconnecting flux systems and on the lowest state of
magnetic energy the system can reach by means of its relaxation. A
shrinking of L1 during the growth of L2 is also indicated by the
observations (compare Fig.~\ref{fig:xrt}d and e). However, this
``shrinking'' might be merely due to a relatively fast cooling of L1.
Studying the issues of lifetime and visibility in soft X-ray of the
reconnected loops in our simulation would require the inclusion of proper
thermodynamics, which is beyond the scope of this paper.
\subsection{Geometrical properties of the reconnected field}
\label{subsec:geom}
The geometrical properties of these new loop systems in the simulation
are different from what has been found in the simulations mentioned in
Sect.~\ref{sec:int}. They arise from a two-step and fully 3D transfer
of sheared flux, first from the sheared core of the emerging flux
rope into the new small loop system formed at its side (as the emerging
blue field lines reconnect with ambient [red] arcade field lines that are
anchored on the left-hand side of the emerging rope), and second from the
very same loop system into a second generation of small loops, which form
at the other side of the emerging flux rope (as the sheared small red
field lines reconnect with other ambient [pink] arcade field lines).
As a result, a {\em full} anemone forms around the parasitic polarity
of the emerging bipole. A large fraction of the nullpoint associated
fan now consists of once or twice reconnected field lines, and not to
a large extent of emerged field lines as in most of the models described
in Sect.~\ref{sec:int}. Since both sides of the fan have been formed
through magnetic reconnection, they can a priori be both visible as hot
loops in soft X-rays. The center of the anemone structure contains
significant twist once it has fully formed, although the structure appears
to be potential when viewed from some distance (Fig.~\ref{fig:topo}c).
The twist is concentrated around the inner spine of the nullpoint (see
Fig.~\ref{fig:topo}d), but some of it is also present along the fan. This
twist is the remnant of magnetic shear that has not been ejected in the
form of a torsional Alfv\'en wave along the large-scale reconnected
arcades that overlie the prominence flux rope.
It is worth noting that, even though a fully 3D anemone has been formed,
it still locally possesses a quasi, but not exact, translational invariance
along the $y$ axis, i.e. along the axis of the emerging flux rope, around
the nullpoint. This can be seen in Fig.~\ref{fig:topo}c,d: almost all
of the red and pink field lines fan out roughly along the $x$ direction, i.e.
perpendicular to the rope axis, even though they have been integrated from
equidistant footpoint positions along a small circle centered at the inner
spine of the nullpoint, within the positive polarity of the emerging flux,
where $B_z$ is roughly constant (as shown in Fig.~\ref{fig:topo}d). This
strong departure from axisymmetry around the nullpoint, which does not exist
in all of the 3D nullpoint models \citep[e.g.][]{ant98,par09}, can here be
attributed to the elongation of the emerging flux, and possibly to the
different inclinations of the ambient field lines. The few red and pink
field lines in Fig.~\ref{fig:topo}d which fan out from the nullpoint region
along the positive $y$ direction, clearly show that the anemone does not
contain a true null line, but one single nullpoint. The quasi-translational
invariance is then due to fan-related eigenvalues of the single nullpoint,
which have very different amplitudes \citep{lau90}. This property, which
has already been identified in the simulation of \citet{mas09} with a
different MHD code, may be present also in other simulations, such as those
which let emerge a very long flux rope \citep[e.g.][]{mor08}.
Fig.~\ref{fig:topo}e and f show that, despite the strong departure from
axisymmetry, the fan surface also extends in $y$-direction, i.e., along the
axis of the emerging flux rope. The field lines shown in the two panels are
located very close to the fan surface and outline a ``heart-shaped'' form
of the fan. We expect the fan to develop a more uniform radial distribution
as the system relaxes to a force-free field, or if less elongated parasitic
polarities are considered \citep{ant98}.
\subsection{Response of the prominence cavity to the anemone/jet formation}
\label{subsec:promresponse}
While the two-step reconnection described in the previous subsections
persists beside the prominence flux rope, the torsional Alfv\'en wave
triggered during the first reconnection phase travels upward (see the
evolution of the blue field lines in the top panels of Fig.\,\ref{fig:reco2}).
Even though the upper part of the prominence flux rope significantly bends
to the side while the wave passes by, the simulated ``prominence material''
located in the flux rope dips \citep[computed as in ][and indicated by the
black circles in Figs.~\ref{fig:reco1} and \ref{fig:reco2}]{aul98,van04,gib06a,dud08}
does not show a significant motion. This difference is due to the relatively
small twist of the rope combined with its non-negligible curvature, which
result in the occurrence of magnetic dips only far below the rope axis,
at altitudes low enough not to be significantly affected by the wave
(see also the discussion on the strength of the perturbation caused by
the wave in Sect.~\ref{subsec:firststep}).
A perturbation of the cavity is observed by XRT on the right-hand side of
the prominence in our event. Between 19:01 and 19:19 UT, after the
formation of L1, while the jet-like brightening is still propagating
and L2 is still growing, a continuous motion (``slipping'') of the lowermost
parts of several loops, from the outer edge of the cavity toward the prominence,
becomes visible (see Fig.~\ref{fig:xrt}e, Fig.~\ref{fig:xrt}f, and the
corresponding XRT movie). Estimating their slipping velocities by following
them individually is not straightforward, as all these loops are not very much
contrasted with respect to the background corona. The only sure number we could
derive is a minimum drift velocity of $35\,\mathrm{km\,s^{-1}}$ for the ensemble
of these loops, since they all have moved by $51^{\prime\prime}$ along the solar
limb during a time interval of 18 minutes. Still, we cannot rule out that the
drift velocity of individual loops could be much larger.
\begin{figure}[t]
\centering
\includegraphics[width=0.70\linewidth,clip=]{fg6.jpg}
\caption
{
Top and oblique view on two field lines within the emerging flux rope
at $t=31$, outlining the change of field line inclination during the
emergence. See the text for details.}
\label{fig:twist}
\end{figure}
The very same phenomenon takes place in the simulation. The online animation of
the simulation shows a slipping motion of the footpoints of the red large-scale
arcade field lines on the right-hand side of the prominence flux rope (see also
the top panels in Fig.~\ref{fig:reco2}). This slipping occurs along the footpoints
of arcade field lines (of which some are shown by the pink field lines in the left
panels of Fig.~\ref{fig:reco2}). It starts right after the second reconnection
sets in and begins to exchange the connectivities between the small (and already
once reconnected) sheared red field lines and large-scale arcade field lines. This
slipping of field lines, after they have reconnected at the nullpoint, is not a
numerical artifact, caused by insufficient accuracy of the field line integration
or by a too strong smoothing of the magnetic field during the MHD simulation: the
slipping occurs over many grid points in the large-scale positive polarity beside
the prominence flux rope, and it does not occur outside of flux regions which
reconnect at the nullpoint. A very mild slipping is noticeable also during the
first reconnection, at the footpoints of the reconnected large-scale blue lines
along which the wave travels (see the online animation).
The explanation of this slipping is very probably the same as first put
forward by \cite{mas09} in their nullpoint associated confined flare model:
the asymmetry of the fan-spine configuration, manifesting as a local
quasi-translational invariance of the magnetic field around a nullpoint,
which has one fan-related eigenvalue of very small, but finite value (see
Sect.~\ref{subsec:secondstep} and Figs.~\ref{fig:topo}c,d), results in the
appearance of a narrow halo of finite width around the nullpoint separatrices,
in which field lines have strong squashing degrees, i.e. which constitutes
quasi-separatrix layers \citep[QSLs;][]{dem96}. It has been recently shown
that current sheet formation and diffusion at QSLs result in
slipping/slip-running magnetic reconnection, manifesting as apparent field
line motions at sub/super Alfv\'enic speeds \citep{aul05,aul06}. In the case
of a 3D nullpoint topology, the presence of a QSL halo surrounding both the
fan surface and the spine field line results in a complex pattern for magnetic
reconnection, in which a given field line slips and slip-runs, both before and
after its connectivity jumps at the nullpoint. To the best of our knowledge,
the present simulation is the second one after \cite{mas09} for which this
sequential nullpoint and slipping reconnection behavior is reported. Based on
these simulations, the slipping loops observed by XRT in our event can be
regarded as a new direct evidence for slipping reconnection in the corona,
as previously observed by \cite{aul07}.
\section{Discussion}
\label{sec:dis}
We presented a $\beta=0$ 3D MHD simulation of the interaction of a small
emerging bipole with a large-scale arcade-like coronal field overlying a
weakly twisted coronal flux rope. The simulation was developed in order
to understand {\it Hinode}/XRT observations of a small limb event which
took place at the edge of a quiet Sun prominence cavity. The event showed
a number of puzzling features, which could satisfactorily be explained by
the magnetic field evolution in the simulation
(see Sect.~\ref{sec:simu_result} for details).
The main results of the simulation can be summarized as follows:
\begin{enumerate}
\item The emergence of a twisted flux rope into one large-scale polarity
of an arcade-like coronal field yields the formation of a fully 3D single
nullpoint topology in the corona, consisting of a fan-spine configuration
in which the fan surface {\em significantly extends} over the parasitic
polarity of the emerged flux.
\item The configuration forms in a {\em two-step reconnection} process at
the nullpoint, which yields the successive formation of two small loop
systems below the fan surface, on opposing sides of the parasitic polarity.
The first loop system hereby ``feeds'' the second one with magnetic flux.
Since the two loop systems (and field lines surrounding them) have common
footpoints in the parasitic polarity, they collectively display an
{\em anemone} shape.
\item The reconnection facilitates the transfer of twist/shear from the
emerging flux rope into the coronal arcades by means of a {\em torsional
Alfv\'en wave} which travels along the arcades. A fraction of the twist
remains below the fan surface, where therefore the resulting magnetic field
is nonpotential.
\item The wave is launched by reconnection, as the expansion velocities of
the emerging fields in the corona suddenly increase, possibly due to the
onset of a ideal MHD kink instability in the not yet reconnected part of
the emerging flux rope, after sufficient twist has entered the corona.
\item The 3D nullpoint reconnection is accompanied by {\em slipping
reconnection} of arcade field lines on the opposite side of the
pre-existing coronal flux rope. This can probably be explained by the
presence of quasi-separatrix layers around the nullpoint.
\end{enumerate}
Our simulation combines dynamic and topological elements of flux emergence
into unipolar coronal fields and coronal jet formation which have previously
not been found altogether in a single simulation. The two-step reconnection
process found in our simulation provides a new model for the formation of
3D nullpoint topologies with extended fan surfaces in the corona. By means
of reconnection between twisted and untwisted fields, the model can also
account for coronal jet activity \citep[e.g.][]{sch95,shi97,par09}.
Finally, since the fields below the fan surface are all formed by reconnection,
and can therefore a priori assumed to brighten in soft X-rays, our simulation
also suggests a mechanism for the formation of (full) anemone active regions.
The fact that some of the emerging twist remains below the fan surface in the
final configuration supports models of jet formation which assume a twisted
null point configuration prior to the jet \citep[e.g.][]{par09}.
Our model can, however, not account for the {\em long-lived evolution} of anemone
active regions. Such regions are observed to consist of bright loops on both sides
of a parasitic polarity over time-periods of days \citep[e.g.][]{asa08}, whereas
the reconnection in our simulation is expected to produce rather short-lived
brightenings, which would occur successively rather than simultaneously (as in the
event described in this paper). It seems likely that the long-lived dynamics of
anemone regions is due to continuous dynamic perturbations of the configuration
after its formation by, for instance, ongoing flux emergence and photospheric flows.
Our model might serve as a starting point for a numerical study of anemone region
dynamics.
Let us now briefly discuss the robustness of our results with respect to the variation
of various model parameters. A detailed investigation of this question will require an
extensive parametric study, but some aspects may be discussed here.
First of all, the model relies on fully 3D effects. The most important ingredient
is the twist brought up by the emerging flux rope. For the two-step reconnection
as described in this paper to work, the emerging field lines must significantly
change their inclination as they rise into the corona, such that the first
reconnected loop system can evolve toward yet unreconnected remote ambient fields
during the reconnection. If the twist in the emerging flux rope (more precisely:
the change of field line inclination along the cross-section of the rope) is chosen
too small, the first reconnection will occur (as in the flux emergence simulations
mentioned in Sect.~\ref{sec:int}), but the second one will not set in. Our adopted
boundary-driven ``kinematic'' flux emergence technique models the rigid emergence
of a twisted flux rope into the corona, and does therefore not capture the dynamics
of real flux emergence \citep[for a discussion, see][]{fan03}. In particular, it
assures that the inclination of the emerging field lines is continuously changing.
It would be, of course, desirable to test our model in a ``dynamic'' flux emergence
simulation, as the one by \cite{mor08}. Here we have chosen the ``kinematic''
approach since, at present, ``dynamic'' flux emergence simulations do not allow to
consider complex coronal magnetic field configurations as the one used here, which
was chosen to model the observed prominence-cavity system.
Apart from the twist profile of the emerging flux rope, there are several other
model parameters which had to be chosen carefully in order to match the observations
(but not necessarily to produce a 3D nullpoint configuration, which is more general).
First, of course, the size of the emerging flux rope and its distance from the coronal
flux rope had to be adjusted as suggested by the observations. Second, the choice of
the magnetic field strength within the emerging rope (and therefore its magnetic flux )
is important. The ratio of the field strength (and flux) between the emerging rope and
the ambient corona will influence, for instance, the amplitude of the torsional
Alfv\'en wave and the final size of the reconnected small loops, i.e. the size of the
anemone. In the simulation presented here, the field strength within the coronal part
of the emerging flux rope was typically about five times larger than in the neighboring
arcade fields throughout most of the evolution, so that the flux within the (small)
rope was not negligible compared to the flux contained in the large-scale arcades.
Third, the inclination of the initial coronal field might play a role, although we
expect a similar evolution if the flux rope would be emerged into a purely vertical
coronal field, mimicking ``open'' field lines in coronal holes. Finally, the magnetic
orientation of the emerging rope with respect to the pre-existing coronal configuration
plays an important role. If this quantity would be reversed in the simulation, and if
the evolution would be seen from the same viewing angle as in Figs.~\ref{fig:reco1} and
\ref{fig:reco2}, we expect the second reconnected (pink) loop system to form on the
left-hand side of the first (red) one, and the blue reconnected field lines above the
nullpoint to develop an elbow which bends away from the prominence flux rope rather
than toward it. It would therefore be interesting to study how the system behaves if
intermediate orientations of the emerging rope are chosen.
Our results underline the importance of a precise examination of
the magnetic topology (and of its formation) for the understanding
of many dynamic events in the solar corona. Without a detailed
consideration of the topology, it would have been very difficult
to understand the complex sequence of dynamic features occurring
in the simulation and in the observed event.
\acknowledgments
We thank the referee for detailed comments and suggestions which helped
very much to improve the quality of this paper.
Financial support by the European Commission through the SOLAIRE Network
(MTRN-CT-2006-035484) and through the FP7 SOTERIA project (Grant Agreement
n$^o$ 218816) are gratefully acknowledged. {\it Hinode} is a Japanese
mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner
and NASA and STFC (UK) as international partners. It is operated by these
agencies in co-operation with ESA and the NSC (Norway). LG and KKR are
supported by NASA under contract NNM07AB07C to SAO.
We thank all the people who collaborate actively within the
JOP 178 observations campaign, particularly T. Berger and the {\it Hinode}
team, G. Molodij and the Meudon Solar Tower observers, and P. Mein for
reducing the MSDP data, and T. Roudier
and N. Labrosse for the coordination and the pointing of the observations.
TT thanks Y. Fan for several discussions on ``kinematic flux emergence''.
We also thank P. D\'emoulin, B. Kliem, and E. Pariat for very helpful
comments on this work. Our MHD calculations were done on the dual-core
quadri-Opteron computers of the Service Informatique de l'Observatoire de
Paris (SIO).
\bibliographystyle{apj}
|
0909.1962
|
\section{Introduction and Previous Work}
\label{intro}
Game Theory~\cite{vega-redondo-03} is the study of how social or economical agents take decisions
in situations of conflict. Some games such as the celebrated Prisoner's Dilemma have a high metaphorical value for society in spite of their simplicity and abstractness. Hawks-Doves, also known as Chicken, is one such socially significant game.
Hawks-Doves is a two-person, symmetric game with the generic payoff bi-matrix of Table~\ref{payoffs}.
\begin{table}[hbt]
\vspace{-0.1cm}
\begin{center}
{\normalsize
\begin{tabular}{c|cc}
& C & D\\
\hline
C & (R,R) & (S,T)\\
D & (T,S) & (P,P)
\end{tabular}
}
\end{center}
\caption{Payoff matrix for a symmetric two person game.}
\label{payoffs}
\vspace{-0.1cm}
\end{table}
\noindent In this matrix, D stands for the defecting strategy ``hawk'', and C stands for the
cooperating strategy ``dove''.
The ``row'' strategies correspond to player 1 and the ``column'' strategies to player 2.
An entry of the table such as (T,S) means that if player 1 chooses strategy D and
player 2 chooses strategy C, then the payoff or utility to player 1 is T, while the payoff
of player 2 is S.
Metaphorically, a hawkish behavior means a strategy of fighting, while a dove, when facing a confrontation, will always yield.
R is the \textit{reward}
the two players receive if they
both cooperate, P is the \textit{punishment} for bilateral defection, and T is the
\textit{temptation}, i.e. the payoff that a player receives if it defects, while the
other cooperates. In this case, the cooperator gets the \textit{sucker's payoff} S.
The game has a structure similar to that of the \textit{Prisoner's Dilemma}~\cite{axe84}.
However, the ordering of payoffs for the Prisoner's
Dilemma is $T > R > P > S$ rendering defection the best rational individual choice, while
in the Hawks-Doves game studied here the ordering is $T > R > S > P$ thus making
mutual defection, i.e. result (D,D), the worst possible outcome. Note that in game theory, as long as the
above orderings are respected, the actual numerical payoff values do not change the nature and number
of equilibria~\cite{vega-redondo-03}.\\
In contrast to the Prisoner's Dilemma which has a unique Nash equilibrium that corresponds to
both players defecting,
the strategy pairs (C,D) and (D,C) are both Nash equilibria of the Hawks-Doves game in pure strategies, and there is
a third equilibrium in mixed strategies in which strategy D is played
with probability $p$, and strategy C with probability $1-p$, where $0 < p < 1$ depends on the actual
payoff values. We recall that a Nash equilibrium is a combination of strategies (pure or mixed)
of the different players such that any unilateral deviation by any agent from this combination can only decrease her expected payoff~\cite{vega-redondo-03}.\\
As it is the case for the
Prisoner's Dilemma (see for example~\cite{axe84,lindnor94a} for the iterated case, among a vast literature), Hawks-Doves, for all its simplicity, appears to
capture some important features of social interactions. In this sense, it applies
in many situations in which ``parading'', ``retreating'', and ``escalating'' are common.
One striking example of a situation
that has been thought to lead to a Hawks-Doves dilemma is the Cuban missile crisis in 1962
~\cite{poundstone92}. Territorial threats at the border between nations are another case in point
as well as bullying in teenage gangs.
Other well known applications are found in the animal kingdom during ritualized fights
~\cite{maynard82}.\\
In this article, we shall present our methods and results in the framework of
\textit{evolutionary game theory}~\cite{hofb-sigm-book-98}. In evolutionary game theory a
very large mixing population of players is considered, and
randomly chosen pairs of individuals play a sequence of one-shot two-person games. In the Hawks-Doves game, the theory
prescribes that the only \textit{Evolutionary
Stable Strategy} (ESS) of the population is the mixed strategy, giving rise, at equilibrium,
to a polymorphic population composed of hawks and doves in which the frequency of hawks equals $p$, the probability with which strategy
hawk would be played in the NE mixed strategy.\\
In the case of the Prisoner's Dilemma, one finds a unique ESS with all the individuals defecting.
However, Nowak and May~\cite{nowakmay92} showed
that cooperation in the population is sustainable under certain conditions,
provided that the network of the interactions between players has a lattice spatial structure. Killingback and
Doebeli~\cite{KD-96} extended the
spatial approach to the Hawks-Doves game and found that a planar lattice structure
with only nearest-neighbor interactions may favor cooperation, i.e. the fraction of doves in
the population is often higher than what is predicted by evolutionary game theory. In a more recent work however, Hauert and Doebeli~\cite{hauer-doeb-2004} were led to a different conclusion, namely that
spatial structure does not seem to favor cooperation in the Hawks-Doves game.\\
Further studies extended the structured population approach to other graph structures
representing small worlds (for an excellent review, see~\cite{Szabo-Fath-07}). Small-world networks are produced by randomly rewiring a few links in
an otherwise regular lattice such as a ring or a grid~\cite{watts-strogatz-98}. These ``shortcuts'', as they are called,
give rise to graphs that have short path lengths between any two nodes in the average as in random graphs, but in contrast to the latter,
also have a great deal of local structure as conventionally measured by the \textit{clustering
coefficient}\footnote{The clustering coefficient
${\cal C}_{i}$ of a node $i$ is defined as ${\cal C}_i=2E_i/k_i(k_i-1)$, where $E_i$ is the number of edges in the
neighborhood of $i$. Thus ${\cal C}_i$ measures the amount of ``cliquishness'' of the
neighborhood of node $i$ and it characterizes the extent to which nodes adjacent to node $i$ are
connected to each other. The clustering coefficient of the graph is simply the average over all nodes:
${\cal C} = \frac{1}{N} \sum_{i=1}^{N}{ \cal C}_i$~\cite{newman-03}.}. These structures are much more typical of the networks that have been analyzed
in technology, society, and biology than regular lattices or random graphs~\cite{newman-03}.
In~\cite{tom-luth-giac-06} it was found that cooperation in Hawks-Doves may be either enhanced or inhibited in small-world
networks depending on the gain-to-cost ratio $r={R}/{(R-P)}$, and on the strategy update rule using standard
local evolutionary
dynamics with one-shot bilateral encounters. However, Watts--Strogatz
small-world networks, although more realistic than lattices or random graphs, are not good representations of typical social networks.
Santos and Pacheco~\cite{santos-pach-05} extended the study of the Hawks-Doves game to scale-free
networks, i.e.~to networks having a power-law distribution of the connectivity degree~\cite{newman-03}. They found that
cooperation is remarkably enhanced in them with respect to previously described population
structures through the existence of highly connected cooperator hubs.
Scale-free networks are much closer than Watts--Strogatz ones to
the typical socio-economic networks that have been investigated, but they are relatively uncommon
in their ``pure'' form due to finite cutoffs and other real-world effects (for example,
see~\cite{newman-03,am-scala-etc-2000,newman-collab-01-1,jordano03}),
with the notable exception of sexual contact networks~\cite{lil-et-al-01}.
Using real and
model static social networks, Luthi et al.~\cite{luthi-pest-tom-physa07} also found that cooperation is
enhanced in Hawks-Doves, although to a lesser degree than in the scale-free case,
thanks to the existence of tight clusters of cooperators that reinforce each other.\\
Static networks resulting from the analysis of actual social networks or good models of the latter are a good starting point;
however, the static approach ignores
fluctuations and non-equilibrium phenomena. As a matter of fact, in many
real networks nodes
may join the network forming new links, and old nodes may leave it as social actors
come and go. Furthermore, new links between agents already in the network may also form or be
dismissed. Often the speed of these network changes is comparable to that of the
agent's behavioral adaptation, thus making it necessary to study how they interact.
Examples of slowly-changing social networks are scientific collaborations, friendships, firm networks
among others. A static network appears to be a good approximation in these cases.
On the other hand, in our Internet times, there exist many social or pseudo-social
networks in which topology changes are faster. For example, e-mail networks~\cite{kossi-watts-06},
web-based networks for friendship and entertainment, such as Facebook, or professional purposes such as LinkedIn, and many others. Furthermore,
as it is not socially credible that people will keep for a long time unsatisfying relationships,
addition and dismissal of partners are an extremely common phenomenon, also due to natural causes
such as moving, changing fields, or interests.
We note at this point that some previous work has focused on the possibility of allowing players to choose
or refuse social partners in game interactions~\cite{batali,sherratt}, which has been shown to potentially promote
cooperation. However, this work does not consider an explicit underlying
interaction network of agents, nor does it use the social link strengths as indicators of partner's suitability as
we do here.\\
In light of what has been said above, the motivation of the present work is to study the co-evolution of strategy and
network structure and to investigate under which conditions cooperative behavior may emerge and be stable in the Hawks-Doves game.
A related goal is to study the topological structures of the emergent networks and their relationships with the strategic choices of the agents.
Some previous work has been done on evolutionary games on dynamic networks
~\cite{skyrms-pem-00,eguiluz-et-al-05,zimmer-pre-05,lut-giac-tom-al-06,santos-plos-06} almost all of them dealing with
the Prisoner's Dilemma. The only one briefly describing results for the Hawks-Doves
game is~\cite{santos-plos-06} but our model differs in several important respects and we obtain
new results on the structure of the cooperating clusters. The main novelty is the use
of pairwise interactions that are dynamically weighted according to mutual satisfaction.
The new contributions and the differences with
previous work will be described at the appropriate points in the article.
An early preliminary version of this study has been presented at the conference~\cite{pest-tom-hd-08}.\\
The paper is organized as follows. In the next section we present our coevolutionary model. This is
followed by an exhaustive numerical study of the game's parameter space. After that we present
our results on cooperation and we describe
and discuss the structure of the emerging networks. Finally we give our conclusions and suggestions
for possible future work.
\section{The Model and its Dynamics}
\label{mod-dyn}
The model is strictly local as no player uses information other than the one concerning the player itself and
the players it is directly connected to. In particular, each agent knows its own current strategy and payoff.
Moreover, as the model is an evolutionary one, no rationality, in the sense of
game theory, is needed~\cite{vega-redondo-03}. Players just adapt their behavior such that they imitate more successful strategies in their environment with higher probability. Furthermore, they are able to locally
assess the worthiness of an interaction and possibly dismiss a relationship that does not pay off enough.
The model has been introduced and fully explained in~\cite{pest-tom-lut-08}, where we study the Prisoner's Dilemma and
the Stag-Hunt games; it is reported here in some detail in order to make the
paper self-contained.
\subsection{Agent-Agent and Network Interaction Structure}
\label{net}
The network of agents is represented by a directed graph $G(V,E)$, where the
set of vertices $V$ represents the agents, while the set of oriented edges (or links) $E$ represents their unsymmetric interactions. The
population size $N$ is the cardinality of $V$. A neighbor of an agent $i$ is any other agent $j$ such that there is a pair of oriented
edges $\vec {ij}$ and $\vec {ji} \in E$.
The set of neighbors of $i$ is called $V_i$. For network structure description purposes, we shall also use an unoriented version $G^{'}$
of $G$ having exactly the same set of vertices $V$ but only a single unoriented edge $ij$ between any pair of connected vertices
$i$ and $j$ of $G$. For $G{'}$ we shall define the degree $k_i$ of vertex $i \in V$ as the number of neighbors of $i$. The average
degree of the network $G^{'}$ will be called $\bar k$.\\
A pair of directed links between vertices $i$ and $j$ in $G$ is schematically depicted in Fig.~\ref{force}. Each
link has a weight or ``force'' $f_{ij}$ (respectively $f_{ji}$). This weight, say $f_{ij}$, represents in an indirect way the
``trust'' player $i$ attributes to player $j$. This weight may take any value in $[0,1]$ and its
variation is dictated by the payoff earned by $i$ in each encounter with $j$, as explained below.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=4.5cm] {force} \protect \\
\caption{Schematic representation of mutual trust between two agents through the strengths
of their links.\label{force}}
\end{center}
\end{figure}
The idea behind the introduction of the forces $f_{ij}$ is loosely inspired by the potentiation/depotentiation of
connections between neurons in neural networks, an effect known as the \textit{Hebb rule} \cite{hebb}. In our context, it can be seen
as a kind of ``memory'' of previous encounters. However, it must be distinguished from the memory used in
iterated games, in which
players ``remember'' a certain number of previous moves and can thus conform their future strategy on the
analysis of those past encounters~\cite{vega-redondo-03}. Our interactions are strictly one-shot, i.e.~players ``forget'' the results of
previous rounds and cannot recognize previous partners and their possible playing patterns. However, a certain amount of past history is implicitly
contained in the numbers $f_{ij}$ and this information may be used by an agent when it will come to decide
whether or not an interaction should be dismissed (see below). \\
We also define a quantity $s_i$ called \textit{satisfaction} of an agent $i$ which is the sum of all the weights
of the links between $i$ and its neighbors $V_i$ divided by the total number of links $k_i$:
$$ s_i = \frac{\sum_{j \in V_i} f_{ij} } {k_i}. $$
\noindent We clearly have $0 \le s_i \le 1$. Note that the term satisfaction is sometimes used in game-theoretical
work to mean the amount of utility gained by a given player. Instead, here satisfaction is related to the average willingness
of a player to maintain the current relationships in the player's neighborhood.
\subsection{Initialization}
\label{init}
The network is of constant size $N=1000$; this allows a simpler yet significant model of network
dynamics in which social links may be broken and formed but agents do not disappear and new
agents may not join the network. The initial graph is generated randomly with
a mean degree $\bar k=10$ which is of the order of those actually found in many social networks such
as collaboration, association, or friendship networks in which relations are generally rather long-lived and there is a cost to maintain a large number; see, for
instance,~\cite{newman-collab-01-1,newman-03,moody-04,TLGL-GPEM-07}.
Players are distributed uniformly at random over the graph vertices with 50\% cooperators. Forces of links
between any pair of neighboring players are initialized at $0.5$.\\
We use a parameter $q$ which is akin to a
``temperature'' or noise level; $q$ is a real number in $[0,1]$ and
it represents the frequency with which an agent wishes to dismiss a link with one of its neighbors. The higher
$q$, the faster the link reorganization in the network. This parameter has been first introduced in~\cite{zimmer-pre-05}
and it controls the speed at which topological changes occur in the network, i.e. the time scale of the strategy-topology
co-evolution. It is
an important consideration, as social networks may structurally evolve at widely different speeds, depending
on the kind of interaction between agents. For example, e-mail networks change their structure at a faster pace
than, say, scientific collaboration networks.
\subsection{Strategy and Link Dynamics}
\label{strat-link-dyn}
Here we describe in detail how individual strategies, links, and link weights are updated. The node update sequence is chosen at
random with replacement as in many previous works~\cite{hubglance93,hauer-doeb-2004,lut-giac-tom-al-06}. Once a given
node $i$ of $G$ is chosen to be activated, it goes through the following steps:
\begin{itemize}
\item if the degree of agent $i$, $k_i = 0$ then
player $i$ is an isolated node. In this case a link with strength $0.5$ is created
from $i$ to a player $j$ chosen uniformly at random among the other $N-1$ players in the network.
\item otherwise,
\begin{itemize}
\item either agent $i$ updates its strategy according to a local \textit{replicator dynamics} rule with probability $1-q$ or, with probability $q$, agent $i$ may delete a link with a given neighbor $j$ and creates a new $0.5$ force link with another node $k$ ;
\item the forces between $i$ and its neighbors $V_i$ are updated
\end{itemize}
\end{itemize}
Let us now describe each step in more detail.
\subsection{Strategy Evolution}
We use a local version of replicator dynamics (RD) for regular graphs~\cite{hauer-doeb-2004} but modified as described
in~\cite{luthi-tomas-pest-09} to take into account the fact that the number of neighbors in a degree-inhomogeneous
network can be different for different agents. Indeed, it has been analytically shown that using straight accumulated payoff
in degree-inhomogeneous networks leads to a loss of invariance with respect to affine transformations of the payoff
matrix under RD~\cite{luthi-tomas-pest-09}.
The local dynamics of a player $i$ only depends on its own strategy and on the strategies of the $k_i$ players in its neighborhood
$V_i \in G^{'}$.
Let us call $\pi_{ij}$ the payoff player $i$ receives when interacting with neighbor $j$. This payoff is defined as
$$
\pi_{ij} = \sigma_i(t)\; M\; \sigma_{j}^T(t),
$$
\noindent where $M$ is the payoff matrix of the game and $\sigma_i(t)$ and $\sigma_j(t)$ are the strategies played by $i$ and $j$ at time $t$.
The quantity
$$
\widehat{\Pi}_i(t) = \sum _{j \in V_i}\pi_{ij}(t)
$$
\noindent is the weighted accumulated payoff defined in~\cite{luthi-tomas-pest-09} collected by player $i$ at time step $t$.
The rule according to which agents update their strategies is the conventional RD in which strategies that do
better than the average increase their share in the population, while those that fare worse than average
decrease.
To update the strategy of player $i$, another player $j$ is drawn at random from the neighborhood $V_i$.
It is assumed that the probability of switching strategy is a function $\phi$ of the payoff difference;
$\phi$ is
required to be monotonic increasing; here it has been taken linear~\cite{hofb-sigm-book-98}. Strategy $\sigma_i$ is replaced by $\sigma_j$ with probability
$$
p_i = \phi(\widehat{ \Pi}_j - \widehat{\Pi}_i),
$$
where
$$
\phi(\widehat{\Pi}_j -\widehat{ \Pi}_i) =
\begin{cases} \dfrac{\widehat{\Pi}_j - \widehat{\Pi}_i}{\widehat{\Pi}_{j,\textrm{max}} - \widehat{\Pi}_{i,\textrm{min}}} & \textrm{if $\widehat{\Pi}_j -
\widehat{\Pi}_i > 0$}\\\\
0 & \textrm{otherwise.}
\end{cases}
$$
In the last expression, $\widehat{\Pi}_{x,\textrm{max}}$ (resp.\ $\widehat{\Pi}_{x,\textrm{min}}$) is the maximum (resp.\ minimum) payoff a player $x$ can get (see ref.~\cite{luthi-tomas-pest-09} for more details).
The major differences with standard RD is that two-person encounters between players are only possible among neighbors, instead of being drawn from the whole population, and the latter is of finite size in our case.
Other commonly used strategy update rules include imitating the best in the
neighborhood~\cite{nowakmay92,zimmer-pre-05}, or replicating in proportion to the payoff ~\cite{hauer-doeb-2004,tom-luth-giac-06}.
\subsection{Link Evolution}
The active agent $i$, which has $k_i \ne 0$ neighbors will, with probability $q$, attempt to dismiss an interaction with one of its neighbors in the following way. In the description we focus on the outgoing links from $i$ in $G$, the incoming links play a subsidiary role.
Player $i$ first looks at its satisfaction $s_i$. The higher $s_i$, the more satisfied the player, since a high satisfaction
is a consequence of successful strategic interactions
with the neighbors. Thus, the natural tendency is to try to dismiss a link when $s_i$ is low. This is simulated by drawing a uniform pseudo-random number $r \in [0,1]$ and breaking a link when $r \ge s_i$.
Assuming that the decision is taken to cut a link, which one, among the possible $k_i$, should be chosen?
Our solution is based on the strength of the relevant links. First a neighbor $j$ is chosen with probability proportional to $1-f_{ij}$, i.e. the stronger the link, the less likely it is that it will be selected. This intuitively corresponds to $i$'s observation that it is preferable to dismiss an interaction with a neighbor $j$ that has contributed little to $i$'s payoff over several rounds of play. However, dismissing a link is not free: $j$ may
``object'' to the decision. The intuitive idea is that, in real social situations, it is seldom possible to take unilateral
decisions: often there is a cost associated, and we represent this hidden cost by a probability
$1 - (f_{ij} + f_{ji})/2$
with which $j$ may refuse to be cut away. In other words, the link is less likely to be deleted if $j$ appreciates $i$, i.e. when $f_{ji}$ is high.\\
Assuming that the $\vec {ij}$ and $\vec {ji}$ links are finally cut, how is a new interaction to be formed?
The solution adopted here is inspired by the observation that, in social settings, links are
usually created more easily between people who have a mutual acquaintance than those who do not.
First, a neighbor $k$ is chosen in $V_i \setminus \{j\}$ with probability proportional to $f_{ik}$, thus favoring neighbors $i$ trusts.
Next, $k$ in turn chooses player $l$ in his neighborhood $V_k$ using the same principle, i.e. with probability proportional to $f_{kl}$. If $i$ and $l$ are not connected,
two links $\vec {il}$ and $\vec {li}$ are created, otherwise the process is repeated in $V_l$. Again, if the selected node, say $m$, is not connected to $i$, an interaction between $i$ and $m$ is established by creating two new links $\vec {im}$ and $\vec {mi}$. If this also fails, new links between $i$ and a randomly chosen node are created.
In all cases the new links are initialized with a strength of $0.5$ in each direction.
This rewiring process is schematically depicted in Fig.~\ref{rewire} for the case in which a link can be successfully established between players $i$ and $l$ thanks to their mutual acquaintance $k$.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=6cm] {rewire} \protect \\
\caption{Illustration of the rewiring of link $\{ij\}$ to $\{il\}$. Agent $k$ is
chosen to introduce player $l$ to $i$ (see text). Only outgoing links are shown for clarity. \label{rewire}}
\end{center}
\end{figure}
At this point, we would like to stress several important differences with previous work in which links can be
dismissed and rewired in a constant-size network in evolutionary games. First of all, in all these works the
interaction graph is undirected with a single link between any pair of agents. In~\cite{zimmer-pre-05}, only links between
defectors are allowed to be cut unilaterally and the study is restricted to the Prisoner's Dilemma. Instead,
in our case, any interaction has a finite probability to be abandoned, even a profitable one between cooperators if it is
recent, although
links that are more stable, i.e. have high strengths, are less likely to be rewired. This smoother situation is made possible thanks to
our bilateral view of a link. It also allows for a moderate amount of ``noise'', which could
reflect to some extent the uncertainties in the system. The present link rewiring process is also different from
the one adopted in~\cite{santos-plos-06}, where the Fermi function is used to decide whether to cut a link or not and
also from their new version of it which has appeared in~\cite{vanSegbroeck}.
Finally, in~\cite{lut-giac-tom-al-06} links are cut according to a threshold decision rule and are rewired randomly anywhere in the network.
\subsection{Updating the Link Strengths}
Once the chosen agents have gone through their strategy or link update steps, the strengths of the
links are updated accordingly in the following way:
$$ f_{ij}(t+1) = f_{ij}(t) + \frac {\pi_{ij} - \bar\pi_{ij}} {k_i(\pi_{max} - \pi_{min}) }, $$
\noindent where $\pi_{ij}$ is the payoff of $i$ when interacting with $j$, $\bar\pi_{ij}$ is the payoff earned by
$i$ playing with $j$, if $j$ were to play his other strategy, and $\pi_{max}$ ($\pi_{min}$) is the maximal (minimal) possible payoff obtainable in a single interaction.
If $f_{ij}(t+1)$ falls outside the $[0,1]$ interval then it is reset to $0$ if it is negative, and to $1$ if it is larger than $1$.
This update is performed in both
directions, i.e. both $f_{ij}$ and $f_{ji}$ are updated $\forall j \in V_i$ because
both $i$ and $j$ get a payoff out of their encounter.
\section{Numerical Simulations and Discussion}
\label{simul}
\subsection{Simulation Parameters}
\label{sim-par}
We simulated the Hawks-Doves game with the dynamics described above exploring the game space by limiting our study to the variation
of only two game parameters.
We set $R=1$ and $P=0$ and the two parameters are $1 \leq T \leq 2$ and $0 \leq S \leq 1$.
Setting $R=1$ and $P=0$ determines the range of $S$ (since $T>R>S>P$)
and gives an upper bound of 2 for $T$, due to the $2R > T+S$ constraint, which ensures that
mutual cooperation is preferred over an equal probability of unilateral cooperation and defection.
Note however, that the only valid value pairs of $(T,S)$ are those that satisfy the latter constraint.
\begin{figure*} [!ht]
\begin{center}
\includegraphics[width=14.5cm] {hd_coop3} \protect \\
\caption{Average cooperation values for the Hawks-Doves game for three values of $q$ at steady-state.
Results are the average of $50$ runs.
\label{hd_coop}}
\end{center}
\end{figure*}
We simulated networks of size $N=1000$, randomly generated with an average degree $\bar k=10$ and randomly initialized with 50\% cooperators and 50\% defectors.
In all cases, parameters $T$ and $S$ are varied between their two bounds in steps of 0.1.
For each set of values, we carry out 50 runs of at most 10000 steps each, using
a fresh graph realization in each run. Each step consists in the update of a full population.
A run is stopped when all agents are using the same strategy, in order to
be able to measure statistics for the population and for the structural parameters of the graphs. After an initial transient period, the system is considered to have reached
a pseudo-equilibrium strategy state when the strategy of the agents (C or D) does not change over 150 further
time steps, which means $15 \times 10^4$ individual updates. It is worth mentioning that equilibrium is
always attained well before the allowed $10000$ time steps, in most cases, less than $1'000$ steps are enough.
We speak of pseudo-equilibria or steady states and not
of true evolutionary equilibria because there is no analog of equilibrium conditions in the dynamical systems sense.\\
To check whether scalability is an issue for the system, we have run several simulations with larger graphs namely,
$N=3000$ and $N=10000$. The overall result is that, although the simulations take a little longer and transient times
are also slightly longer, at quasi-equilibrium all the measures explored in the next sections follow the same trend
and the dynamics give rise to comparable topologies and strategy relative abundance.
\subsection{Emergence of Cooperation}
\label{coop}
Cooperation results in contour plot form are shown in Fig.~\ref{hd_coop}. We remark that,
as observed in other structured populations, cooperation is achieved in almost the whole configuration
space. Thus, the added degree of freedom represented by the possibility of refusing a partner and choosing
a new one does indeed help to find player's arrangements that help cooperation. When considering the dependence on the
parameter $q$, one sees in Fig.~\ref{hd_coop} that the higher $q$, the higher the cooperation level, although the differences are small, since full cooperation prevails already at $q=0.2$. This is a somewhat expected result, since
being able to break ties more often clearly gives cooperators more possibilities for finding and keeping
fellow cooperators to interact with. The results reported in the figures are for populations starting with $50\%$
cooperators randomly distributed. We have also tried other proportions with less cooperators, starting at $30\%$.
The results, not reported here for reasons of space, are very similar, the only difference being that
it takes more simulation time to reach the final quasi-stable state.
Finally, one could ask whether cooperation would still spread starting with very few cooperators. Numerical simulations
show that cooperation could indeed prevail even starting from as low as $1\%$ cooperators,
except on the far left border of the configuration space where cooperation is severely disadvantaged.\\
Compared with the level of cooperation observed in simulations in
static networks, we can say that results are consistently better for co-evolving networks. For all values of
$q$ (Fig.~\ref{hd_coop}) there is significantly more cooperation than what was found in model and real social networks~\cite{luthi-pest-tom-physa07} where the same local replicator dynamics
was used but with the constraints imposed by the
invariant network structure.
A comparable high cooperation level has only been found in static
scale-free networks~\cite{santos-pach-05,santos-pach-06} which are not as realistic as a social network structures. \\
The above considerations are all the more interesting when one observes that the standard RD result is
that the only asymptotically stable state for the game is a polymorphic population in which there
is a fraction $\alpha$ of doves and a fraction $1-\alpha$ of hawks, with $\alpha$ depending on the actual numerical
payoff matrix values.
To see the positive influence of making and breaking ties we can compare our results with what is prescribed by the standard RD solution. Referring to the payoff table~\ref{payoffs}, let's assume that the column player plays C with
probability $\alpha$ and D with probability $1-\alpha$. In this case, the expected payoffs of the row player
are:
$$
E_r[C] =\alpha R + (1-\alpha)S$$
and
$$E_r[D] =\alpha T + (1-\alpha)P
$$
\noindent The row player is indifferent to the choice of $\alpha$ when $E_r[C] = E_r[D]$. Solving for $\alpha$ gives:
\begin{equation}
\alpha = \frac{P-S}{R-S-T+P}.
\label{alpha}
\end{equation}
Since the game is symmetric, the result for the column player is the same and $(\alpha C, (1-\alpha) D)$
is a NE in mixed strategies.
We have numerically solved the equation for all the sampled points in the game's parameter space. Let us now use the following payoff values
in order to bring them within the explored game space (remember that NEs
are invariant w.r.t. such an affine transformation):
\begin{table}[hbt]
\begin{center}
{\normalsize
\begin{tabular}{c|cc}
& C & D\\
\hline
C & ($1,1$) & ($2/3,4/3$)\\
D & ($4/3,2/3$) & ($0,0$)
\end{tabular}
}
\end{center}
\end{table}
Substituting in equation~\ref{alpha} gives $\alpha=2/3$, i.e. the dynamically stable polymorphic population should be
composed by about $2/3$ cooperators and $1/3$ defectors. Now, if one looks at Fig.~\ref{hd_coop} at the
points where $S=2/3$ and $T=4/3$, one can see that the point, and the region around it, is
one of full cooperation instead.
Even within the limits of the approximations caused by the finite population size and the local dynamics, the non-homogeneous graph structure and an increased level of tie rewiring has allowed cooperation to be greatly enhanced with respect to the theoretical predictions of standard RD.
\subsection{Evolution of Agents' Satisfaction}
According to the model, unsatisfied agents are more likely to try to cut links in an attempt to improve
their satisfaction level, which could be simply described as an average value of the strengths of their
links with neighbors. Satisfaction should thus tend to increase during evolution. In effect, this is what happens,
as can be seen in Fig.~\ref{sat}. The figure refers to a particular run that ends in all agents cooperating,
but it is absolutely typical.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=10cm] {evo_sat} \protect \\
\caption{Fraction of agents having a given satisfaction level as a function of evolution time.\label{sat}}
\end{center}
\end{figure}
One can remark the ``spike'' at time $0$. This is clearly due to the fact that all links are initialized
with a weight of $0.5$. As the simulation advances, the satisfaction increases steadily and for the case
of the figure, in which all agents cooperate at the end, it reaches its maximum value of $1$ for almost all
players.
\subsection{Stability of Cooperation}
\label{stability}
Evolutionary game theory provides a dynamical view of conflicting decision-making in populations.
Therefore, it is important to assess the \textit{stability} of the equilibrium configurations.
This is even more important in the case of numerical simulation where the steady-state
finite population configurations are not really equilibria in the mathematical sense. In other words,
one has to be reasonably confident that the steady-states are not easily destabilized
by perturbations. To this end, we have performed a numerical study of the robustness of
final cooperators' configurations by introducing a variable amount of random noise into
the system. A strategy is said to be \textit{evolutionarily stable} when it cannot be invaded
by a small amount of players using another strategy~\cite{hofb-sigm-book-98}. We have chosen
to switch the strategy of
an increasing number of highly connected cooperators to defection, and to observe whether
the perturbation propagates in the population, leading to total defection, or if it stays
localized and disappears after a transient time.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=14cm,height=4cm] {hd_16_04} \protect \\
\caption{Cooperation percentage as a function of simulated time when the strategy of the $30\%$ most connected nodes is switched
from cooperation to defection. $T=1.6$, $S=0.4$ and, from left to right, $q=0.2, 0.5, 0.8$.\label{hd_noise_16}}
\end{center}
\end{figure}
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=14cm,height=4cm] {hd_19_01} \protect \\
\caption{Cooperation percentage when the strategy of the $30\%$ most connected nodes is switched
from cooperation to defection. $T=1.9$, $S=0.1$ and, from left to right, $q=0.2, 0.5, 0.8$.\label{hd_noise_19}}
\end{center}
\end{figure}
Figs.~\ref{hd_noise_16} and~\ref{hd_noise_19} show how the system recovers when the most highly
connected $30$\% of the cooperators are suddenly and simultaneously switched to defection. In Fig.
\ref{hd_noise_16} the value chosen in the game's configuration space is $T=1.6$ and $S=0.4$. This
point lies approximately on the diagonal in Fig.~\ref{hd_coop} and corresponds to an all-cooperate
situation. As one can see, after the perturbation is applied, there is a sizable loss of cooperation but,
after a while, the system recovers full cooperation in all cases (only $10$ curves are shown in each figure
for clarity, but the phenomenon is qualitatively identical in all the $50$ independent runs tried). From left to right,
three values of $q=0.2,0.5,0.8$ are used. It is seen that, as the rewiring frequency $q$ increases, recovering from the
perturbation becomes easier as defection has less time to spread around before cooperators are
able to dismiss links toward defectors. Switching the strategy of the $30$ \% most connected nodes
is rather extreme since they include most cooperator clusters but, nonetheless, cooperation is rather stable in the whole cooperating region. In Fig.~\ref{hd_noise_19} we have done the same this time with
$T=1.9$ and $S=0,1$. This point is in a frontier region in which defection may often prevail, at least
for low $q$ (see Fig.~\ref{hd_coop}) and thus it represents one of the hardest cases for
cooperation to remain stable. Nevertheless, except in the leftmost case ($q=0.2$) where half of the runs
permanently switch to all-defect, in all the other cases the population is seen to recover after
cooperation has fallen down to less than $10\%$. Note that the opposite case is
also possible in this region that is, in a full defect situation, switching of $30\%$ highly connected defectors
to cooperation can lead the system to one of full cooperation.
In conclusion, the above numerical experiments have empirically shown that cooperation is
extremely stable after cooperator networks have emerged. Although we are using here an artificial
society of agents, this can hopefully be seen as an encouraging result for cooperation in real societies.
\subsection{Structure of the Emerging Networks}
\label{topo}
In this section we present a statistical analysis of the global and local properties of the networks that
emerge when the pseudo-equilibrium states of the dynamics are attained. Note that in the following sections
the graph we refer to is the unoriented, unweighted one that we called $G^{'}$ in Sect.~\ref{net}. In other words, for
the structural properties of interest, we only take into account the fact that two agents interact and not
the weights of their directed interactions.
\subsubsection{Small-World Nature}
Small-world networks are characterized by a small mean path length and by a high clustering coefficient~\cite{watts-strogatz-98}. Our graphs start random, and thus have short path lengths by construction since
their mean path length $\bar l= O(log N)$ scales logarithmically with the number of vertices $N$~\cite{newman-03}.
It is interesting to notice that they maintain short diameters at equilibrium too, after rewiring has taken
place. We took the average $\bar L = \sum_{k=1}^{660} \bar l$ of the mean path length of $660$ evolved graphs, which represent ten graphs for each $T,S$ pair. This average
is 3.18, which is of the order of $log(1000)$, while its initial random graph average value is $3.25$. This fact, together with the remarkable increase of
the clustering coefficients with respect to the random graph (see below), shows that the evolved networks have the small-world property.
Of course, this behavior was expected, since the rewiring mechanism favors close partners in the network and thus
tends to increase the clustering and to shorten the distances.
\subsubsection{Average Degree} In contrast to other models~\cite{zimmer-pre-05,santos-plos-06}, the mean degree $\bar k$ can
vary during the course of the simulation. We found that $\bar k$ increases only slightly and tends to stabilize around $\bar k=11$. This is in qualitative agreement with observations made on real dynamical
social networks~\cite{kossi-watts-06,barab-collab-02,tom-leslie-evol-net-07} with the only difference that the network does not grow in our model.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=14.5cm] {hd_cc3} \protect \\
\caption{Average values of the clustering coefficient over $50$ runs for three
values of $q$. \label{hd_clust}}
\end{center}
\end{figure}
\subsubsection{Clustering Coefficients}
The clustering coefficient $\cal C$ of a graph has been defined in the Introduction section.
Random graphs are locally homogeneous in the average and for them $\cal C$ is simply equal to
the probability of having an edge between any pair of nodes independently. In contrast, real networks have
local structures and thus higher values of $\cal C$. Fig.~\ref{hd_clust} gives the average clustering coefficient
$\bar {\cal C}=\frac{1}{50} \sum_{i=1}^{50} {\cal C}$ for
each sampled point in the Hawks-Doves configuration space, where $50$ is the number of network realizations
used for each simulation.
The networks self-organize through dismissal of partners and choice of new ones and they
acquire local structure, since the clustering coefficients
are higher than that of a random graph
with the same number of edges and nodes, which is $\bar k/N=10/1000=0.01$.
The clustering tends to increase with $q$ (i.e.~from left to right in Fig.~\ref{hd_clust}). It is clear that
the increase in clustering and the formation of cliques is due to the fact that, when dismissing
an unprofitable relation and searching for a new one, individuals that are relationally at a short distance
are statistically favored. But this has a close correspondence in the way in which new
acquaintances are made in society: they are not random, rather people often get to interact with each
other through common acquaintances, or ``friends of friends'' and so on.
\subsubsection{Degree Distributions}
The \textit{degree distribution function} (DDF) $p(k)$ of a graph represents the probability that a randomly
chosen node has degree $k$. Random graphs are characterized by DDF of Poissonian
form $p(k) = {\bar k}^k e^{-\bar k}/k!$, while social and technological real networks often show long tails to the right, i.e. there are nodes
that have an unusually large number of neighbors~\cite{newman-03}. In some extreme cases
the DDF has a power-law form $p(k) \propto k^{-\gamma}$; the
tail is particularly extended and there is no characteristic degree. The \textit{cumulative
degree distribution function} (CDDF) is just the probability that the degree is greater than or equal to $k$
and has the advantage of being less noisy for high degrees.
Fig.~\ref{hd_df} shows the CDDFs for the Hawks-Doves for three values of $T$, $S=0.2$, and $q=0.5$ with
a logarithmic scale on the y-axis. A Poisson and an exponential distribution are also included for comparison. The Poisson curve actually represents the
initial degree distribution of the (random) population graph. The distributions at equilibrium are
far from the Poissonian that would apply if the networks would remain essentially random. However, they
are also far from the power-law type, which would appear as a straight line in the log-log plot of
Fig~\ref{cddf-ll}.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=8cm] {cddf_3} \protect \\
\caption{Empirical cumulative degree distribution functions for three different values of the
temptation $T$. A Poissonian and an exponential distribution are also plotted for
comparison. Distributions are discrete, the continuous lines are only a guide for
the eye. Lin-log scales.\label{hd_df}}
\end{center}
\end{figure}
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=8cm] {cddf_II_3} \protect \\
\caption{Empirical cumulative degree distribution functions for three different values
of the parameter $T$. Log-log scales.\label{cddf-ll}}
\end{center}
\end{figure}
Although a reasonable fit with a single law appears to be difficult, these empirical distributions are closer
to exponentials, in particular the curve for $T=1.7$, for which such a fit has been drawn.
It can be observed that the
distribution is broader the higher $T$ (The higher $T$, the more agents gain by defecting). In fact, although cooperation is attained nearly
everywhere in the game's configuration space, higher values of the temptation $T$ mean that
agents have to rewire their links more extensively, which results in a higher number of neighbors
for some players, and thus it leads to a longer tail in the CDDF.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=8cm] {q_cddf3} \protect \\
\caption{Empirical cumulative degree distribution functions for three different values of the
temptation $q$. Lin-log scales.\label{hd-cddf-q}}
\end{center}
\end{figure}
The influence of the $q$ parameter on the shape of the degree
distribution functions is shown in Fig.~\ref{hd-cddf-q} where average curves for three values of $q$, $T=1.7$, and $S=0.2$, are
reported. For high $q$, the cooperating steady-state is reached faster, which gives the network less time to rearrange its links. For lower values of $q$ the distributions become broader, despite the fact that
rewiring occurs less often, because cooperation in this region is harder to attain and more simulation time
is needed. In conclusion, emerging network structures at steady states have DDFs that are similar to those found
in actual social networks~\cite{newman-03,am-scala-etc-2000,newman-collab-01-1,jordano03,TLGL-GPEM-07}, with tails that are fatter the higher the temptation $T$ and the lower $q$. Topologies closer to scale-free would probably be obtained if
the model allowed for growth, since preferential attachment is already present to some extent due to the
nature of the rewiring process~\cite{poncela08}.
\subsubsection{Degree Correlations}
Besides the degree distribution function of a network, it is also sometimes useful to
investigate the empirical joint degree-degree distribution of neighboring vertices. However,
it is difficult to obtain reliable statistics because the data set is usually too small (if a network
has $L$ edges, with $L \ll N^2$ where $N$ is the number of vertices for the usually
relatively sparse networks we deal with, one then has only $L$ pairs of data to work with).
Approximate statistics can readily be obtained by using the average degree of the
nearest neighbors of a vertex $i$ as a function of the degree of this vertex, $\bar k_{V_i}(k_i)$~\cite{PS-VAZ-VESP}.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=7.5cm] {Knn} \protect \\
\caption{Average degree of the direct neighbors of a vertex Vs. the vertex degree. The relation is
disassortative. Log-lin scales.\label{ddc}}
\end{center}
\end{figure}
From Fig.~\ref{ddc} one can see that the correlation is slightly negative, or disassortative. This is at odds with
what is reported about real social networks, in which usually this correlation is positive instead,
i.e. high-degree nodes tend to connect to high-degree nodes and vice-versa~\cite{newman-03}. However,
real social networks establish and grow because of common interests, collaboration work, friendship
and so on. Here this is not the case, since the network is not a growing one, and the game played by the
agents is antagonistic and causes segregation of highly connected cooperators into clusters in which they
are surrounded by less highly connected fellows. This will be seen more pictorially in the following section.
\subsection{Cooperator Clusters}
\label{comm}
From the results of the previous sections, it appears that a much higher amount of cooperation
than what is predicted by the standard theory for mixing populations can be reached when
ties can be broken and rewired. We have seen that this dynamics causes the graph to
acquire local structure, and thus to loose its initial randomness. In other words,
the network self-organizes in order to allow players to cooperate as much as possible.
At the microscopic, i.e.~agent level, this happens
through the formation of clusters of
players using the same strategy. Fig.~\ref{hd_cluster} shows one typical cooperator cluster.
\begin{figure} [!ht]
\begin{center}
\includegraphics[width=8cm] {cluster_hd} \protect \\
\caption{A typical cooperator cluster. Links to the rest of the network have been suppressed
for clarity. The size of a node is proportional to its connectivity in the whole graph. The most connected central cooperator is shown as a square.\label{hd_cluster}}
\end{center}
\end{figure}
In the figure one can clearly see that the central cooperator
is a highly connected node and there are many links also between the other neighbors. Such
tightly packed structures have emerged to protect cooperators from defectors that, at earlier times, were
trying to link to cooperators to exploit them. These observations help understand why the degree distributions are long-tailed (see previous section), and also the higher values of the clustering coefficient.\\
Further studies of the emerging networks would imply investigating the communities and the way in which
strategies are distributed in them. There
are many ways to reveal the modular structure of networks~\cite{Arenas05} but we leave this study for further work.
\section{Conclusions}
\label{concl}
In this paper we have introduced a new dynamical population structure for agents playing
a series of two-person Hawks and Doves game. The most novel feature of the model is the
adoption of a variable strength of the bi-directional social ties between pairs of players. These strengths
change dynamically and independently as a function of the relative satisfaction of the two end points
when playing with their immediate neighbors in the network.
A player may wish to break a tie to a neighbor and the probability of cutting the link is higher
the weaker the directed link strength is. The ensemble of weighted links implicitly represents
a kind of memory of past encounters although, technically speaking, the game is not iterated.
While in previous work the rewiring parameters where ad hoc, unspecified probabilities,
we have made an effort to relate them to the agent's propensity to gauge the perceived
quality of a relationship during time. \\
The model takes into account recent knowledge coming from the analysis of the structure and
of the evolution of social networks and, as such, should be a better approximation of real
social conflicting situations than static graphs such as regular grids. In particular, new links are not created
at random but rather taking into account the ``trust'' a player may have on her relationally close social
environment as reflected by the current strengths of its links. This, of course, is at the origin of the de-randomization and self-organization of the network,
with the formation of stable clusters of cooperators. The main result concerning
the nature of the pseudo-equilibrium states of the dynamics is that cooperation is greatly
enhanced in such a dynamical artificial society and, furthermore, it is quite robust with respect to
large strategy perturbations. Although our model is but a simplified and incomplete
representation of social reality, this is encouraging, as the Hawks-Doves game
is a paradigm for a number of social and political situations in which aggressivity plays an
important role. The standard result is that bold behavior does not disappear at evolutionary
equilibrium. However, we have seen here that a certain amount of plasticity of the networked
society allows for full cooperation to be consistently attained. Although the model is an extremely abstract one,
it shows that there is place for peaceful resolution of conflict. In future work we would like to investigate
other stochastic strategy evolution models based on more refined forms of learning than
simple imitation and study the global modular structure of the equilibrium networks.\\
{\bf Acknowledgements.}
This work is funded by the Swiss National Science Foundation under grant number 200020-119719.
We gratefully acknowledge this financial support.
\bibliographystyle{elsart-num}
|
1503.01640
|
\section{Introduction}
In the past few months, tremendous progress has been made in the field of semantic segmentation \cite{hariharan2014simultaneous,Long2015,Hariharan2015,Dai2015,Chen2015,mostajabi2014feedforward}. Deep convolutional neural networks (CNNs) \cite{lecun1989backpropagation,krizhevsky2012imagenet} that play as rich hierarchical feature extractors are a key to these methods. These networks are trained on large-scale datasets \cite{deng2009imagenet,Russakovsky2014} as classifiers, and transferred to the semantic segmentation tasks based on the annotated segmentation masks as supervision.
\begin{figure*}[t]
\centering
\includegraphics[width=.9\linewidth]{fig/outline}
\caption{Overview of our training approach supervised by bounding boxes.}
\label{fig:outline}
\end{figure*}
But pixel-level mask annotations are time-consuming, frustrating, and in the end commercially expensive to obtain.
According to the annotation report of the large-scale Microsoft COCO dataset \cite{Lin2014}, the workload of labeling segmentation masks is more than 15 times heavier than that of spotting object locations. Further, the crowdsourcing annotators need to be specially trained for the tedious and difficult task of labeling per-pixel masks. These facts limit the amount of available segmentation mask annotations, and thus hinder the performance of CNNs that in general desire large-scale data for training. On the contrary, bounding box annotations are more economical than masks. There have already existed a large number of available box-level annotations in datasets like PASCAL VOC 2007\footnote{The PASCAL VOC 2007 dataset only has bounding box annotations.} \cite{everingham2010pascal} and ImageNet \cite{Russakovsky2014}. Though these box-level annotations are less precise than pixel-level masks, their amount may help improve training deep networks for semantic segmentation.
In addition, current leading approaches have not fully utilized the detailed pixel-level annotations. For example, in the Convolutional Feature Masking (CFM) method \cite{Dai2015}, the fine-resolution masks are used to generate very low-resolution (\eg, $6 \times 6$) masks on the feature maps. In the Fully Convolutional Network (FCN) method \cite{Long2015}, the network predictions are regressed to the ground-truth masks using a large stride (\eg, 8 pixels). These methods yield competitive results without explicitly harnessing the finer masks. If we consider the box-level annotations as very coarse masks, can we still retain comparably good results without using the segmentation masks?
In this work, we investigate bounding box annotations as an alternative or extra source of supervision to train convolutional networks for semantic segmentation\footnote{The idea of using bounding box annotations for CNN-based semantic segmentation is developed concurrently and independently in \cite{papandreou2015weakly}. We also compare with the results of \cite{papandreou2015weakly}.}.
We resort to unsupervised region proposal methods \cite{uijlings2013selective,arbelaez2014multiscale} to generate candidate segmentation masks. The convolutional network is trained under the supervision of these approximate masks. The updated network in turn improves the estimated masks used for training. This process is iterated. Although the masks are coarse at the beginning, they are gradually improved and then provide useful information for network training. Fig.~\ref{fig:outline} illustrates our training algorithm.
We extensively evaluate our method, called ``BoxSup'', on the PASCAL segmentation benchmarks \cite{everingham2010pascal,mottaghi2014role}. Our box-supervised (\ie, using bounding box annotations) method shows a graceful degradation compared with its mask-supervised (\ie, using mask annotations) counterpart. As such, our method waives the requirement of pixel-level masks for training.
Further, our semi-supervised variant in which 9/10 mask annotations are replaced with bounding box annotations yields comparable accuracy with the fully mask-supervised counterpart.
This suggests that we may save expensive labeling effort by using bounding box annotations dominantly.
Moreover, our method makes it possible to harness the large number of available box annotations to improve the mask-supervised results.
Using the limited provided mask annotations and extra large-scale bounding box annotations,
our method achieves state-of-the-art results on both PASCAL VOC 2012 and PASCAL-CONTEXT \cite{mottaghi2014role} benchmarks.
Why can a large amount of bounding boxes help improve convolutional networks? Our error analysis reveals that a BoxSup model trained with a large set of boxes effectively increases the object \emph{recognition} accuracy (the accuracy in the middle of an object), and its improvement on object boundaries is secondary. Though a box is too coarse to contain detailed segmentation information, it provides an instance for learning to distinguish object categories.
The large-scale object instances improve the feature quality of the learned convolutional networks, and thus impact the overall performance for semantic segmentation.
\section{Related Work}
Deep convolutional networks in general have better accuracy with the growing size of training data, as is evidenced in \cite{krizhevsky2012imagenet,Zeiler2014}.
The ImageNet classification dataset \cite{Russakovsky2014} is one of the largest datasets with quality labels, but the current available datasets for object detection, semantic segmentation, and many other vision tasks mostly have orders of magnitudes fewer labeled samples. The milestone work of R-CNN \cite{Girshick2014} proposes to pre-train deep networks as classifiers on the large-scale ImageNet dataset and go on training (fine-tuning) them for other tasks that have limited number of training data. This transfer learning strategy is widely adopted for object detection \cite{Girshick2014,He2014,Szegedy2015}, semantic segmentation \cite{hariharan2014simultaneous,Long2015,Hariharan2015,Dai2015,Chen2015,mostajabi2014feedforward},
visual tracking \cite{Wang2015}, and other visual recognition tasks. With the continuously improving deep convolutional models \cite{Zeiler2014,Sermanet2014,Chatfield2014,He2014,Simonyan2015,Szegedy2015,He2015}, the accuracy of these vision tasks also improves thanks to the more powerful generic features learned from large-scale datasets.
Although pre-training partially relieves the problem of limited data, the amount of the task-specific data for fine-tuning still matters. In \cite{agrawal2014analyzing}, it has been found that augmenting the object detection training set by combining the VOC 2007 and VOC 2012 sets improves object detection accuracy compared with using VOC 2007 only. In \cite{Liang2014}, the training set for object detection is augmented by visual tracking results obtained from videos and improves detection accuracy.
These experiments demonstrate the importance of dataset sizes for task-specific network training.
For semantic segmentation, there have been existing papers \cite{xia2013semantic,guillaumin2014imagenet} that investigate exploiting bounding box annotations instead of masks. But the box-level annotations have not been used to supervised deep convolutional networks in those works.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\linewidth]{fig/masks}
\caption{Segmentation masks used as supervision. (a) A training image. (b) Ground-truth. (c) Each box is na\"{\i}vely considered as a rectangle mask. (d) A segmentation mask is generated by GrabCut \cite{rother2004grabcut}. (e) For our method, the supervision is estimated from region proposals (MCG \cite{arbelaez2014multiscale}) by considering bounding box annotations and network feedbacks.}
\label{fig:masks}
\end{figure*}
\section{Baseline}
\label{sec:baseline}
Our BoxSup method is in general applicable for many existing CNN-based mask-supervised semantic segmentation methods, such as FCN \cite{Long2015}, improvements on FCN \cite{Chen2015,zheng2015conditional}, and others
\cite{Hariharan2015,Dai2015,mostajabi2014feedforward}. In this paper, we adopt our implementation of the FCN method \cite{Long2015} refined by CRF \cite{Chen2015} as the mask-supervised baseline, which we briefly introduce as follows.
The network training of FCN \cite{Long2015} is formulated as a per-pixel regression problem to the ground-truth segmentation masks. Formally, the objective function can be written as:
\begin{equation}
\mathcal{E}(\theta) = \sum_{p} e(X_{\theta}(p), l(p)),
\label{eq:fcn_loss}
\end{equation}
where $p$ is a pixel index, $l(p)$ is the ground-truth semantic label at a pixel, and $X_{\theta}(p)$ is the per-pixel labeling produced by the fully convolutional network with parameters $\theta$.
$e(X_{\theta}(p), l(p))$ is the per-pixel loss function. The network parameters $\theta$ are updated by back-propagation and stochastic gradient descent (SGD). A CRF is used to post-process the FCN results \cite{Chen2015}.
The objective function in Eqn.(\ref{eq:fcn_loss}) demands pixel-level segmentation masks $l(p)$ as supervision. It is not directly applicable if only bounding box annotations are given as supervision. Next we introduce our method for addressing this problem.
\section{Approach}
\subsection{\fontsize{9.9pt}{1em}\selectfont{\textbf{Unsupervised Segmentation for Supervised Training}}}
To harness the bounding boxes annotations, it is desired to estimate segmentation masks from them. This is a widely studied supervised image segmentation problem, and can be addressed by, \eg, GrabCut \cite{rother2004grabcut}. But GrabCut can only generate one or a few samples from one box, which may be insufficient for deep network training.
We propose to generate a set of candidate segments using \emph{unsupervised} region proposal methods (\eg, Selective Search \cite{uijlings2013selective}) due to their nice properties. First, region proposal methods have high recall rates \cite{arbelaez2014multiscale} of having a good candidate in the proposal pool. Second, region proposal methods generate candidates of greater variance, which provide a kind of data augmentation \cite{krizhevsky2012imagenet} for network training. We will show by experiments the improvements of these properties.
The candidate segments are used to update the deep convolutional network. The semantic features learned by the network are then used to pick better candidates. This procedure is iterated.
We formulate this procedure as an objective function as we will describe below.
It is worth noticing that the region proposal is only used for networking training. For inference, the trained FCN is directly applied on the image and produces pixel-wise predictions. So our usage of region proposals does not impact the test-time efficiency.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\linewidth]{fig/epochs}
\caption{Update of segmentation masks during training. Here we show the masks in epoch \#1, epoch \#5, and epoch \#20. Each segmentation mask will be used as the supervision for the next epoch.}
\label{fig:epochs}
\end{figure*}
\subsection{Formulation}
As a pre-processing, we use a region proposal method to generate segmentation masks. We adopt Multiscale Combinatorial Grouping (MCG) \cite{arbelaez2014multiscale} by default, while other methods \cite{uijlings2013selective,krahenbuhl2014geodesic} are also evaluated. The proposal candidate masks are fixed throughout the training procedure. But during training, each candidate mask will be assigned a label which can be a semantic category or background. The labels assigned to the masks will be updated.
With a ground-truth bounding box annotation, we expect it to pick out a candidate mask that overlaps the box as much as possible. Formally, we define an overlapping objective function $\mathcal{E}_o$ as:
\begin{equation}\label{eq:ov}
\mathcal{E}_o = \frac{1}{N}\sum_{S}(1-\text{IoU}(B, S))\delta(l_B, l_S).
\end{equation}
Here $S$ represents a candidate segment mask, and $B$ represents a ground-truth bounding box annotation. $\text{IoU}(B, S)\in[0,1]$ is the intersection-over-union ratio computed from the ground-truth box $B$ and the tight bounding box of the segment $S$. The function $\delta$ is equal to one if the semantic label $l_S$ assigned to segment $S$ is the same as the ground-truth label $l_B$ of the bounding box $B$, and zero otherwise. Minimizing $\mathcal{E}_o$ favors higher IoU scores when the semantic labels are consistent. This objective function is normalized by the number of candidate segments $N$.
With the candidate masks and their estimated semantic labels, we can supervise the deep convolutional network as in Eqn.(\ref{eq:fcn_loss}). Formally, we consider the following regression objective function $\mathcal{E}_r$:
\begin{equation}\label{eq:reg}
\mathcal{E}_r = \sum_{p} e(X_{\theta}(p), l_S(p)).
\end{equation}
Here $l_S$ is the estimated semantic label used as supervision for the network training. This objective function is the same as Eqn.(\ref{eq:fcn_loss}) except that its regression target is the estimated candidate segment.
We minimize an objective function that combines the above two terms:
\begin{equation}\label{eq:obj}
\min_{\theta,\{l_S\}} \sum_{i}(\mathcal{E}_o+\lambda\mathcal{E}_r)
\end{equation}
Here the summation $\sum_{i}$ runs over the training images, and $\lambda=3$ is a fixed weighting parameter. The variables to be optimized are the network parameters $\theta$ and the labeling $\{l_S\}$ of all candidate segments $\{S\}$. If only the term $\mathcal{E}_o$ exists, the optimization problem in Eqn.(\ref{eq:obj}) trivially finds a candidate segment that has the largest IoU score with the box; if only the term $\mathcal{E}_r$ exists, the optimization problem in Eqn.(\ref{eq:obj}) is equivalent to FCN. Our formulation simultaneously considers both cases.
\subsection{Training Algorithm}
\label{sec:alg}
The objective function in Eqn.(\ref{eq:obj}) involves a problem of assigning labels to the candidate segments. Next we propose a greedy iterative solution to find a local optimum.
With the network parameters $\theta$ fixed, we update the semantic labeling $\{l_S\}$ for all candidate segments. In our implementation, we only consider the case in which one ground-truth bounding box can ``activate'' (\ie, assign a non-background label to) one and only one candidate. As such, we can simply update the semantic labeling by selecting a single candidate segment for each ground-truth bounding box, such that its cost $\mathcal{E}_o+\lambda\mathcal{E}_r$ is the smallest among all candidates. The selected segment is assigned the ground-truth semantic label associated with that bounding box. All other pixels are assigned the background label.
The above winner-takes-all selection tends to repeatedly use the same or very similar candidate segments, and the optimization procedure may be trapped in poor local optima. To increase the sample variance for better stochastic training, we further adopt a random sampling method to select the candidate segment for each ground-truth bounding box. Instead of selecting the single segment with the largest cost $\mathcal{E}_o+\lambda\mathcal{E}_r$, we randomly sample a segment from the first $k$ segments with the largest costs. In this paper we use $k=5$. This random sampling strategy improves the accuracy by about 2\% on the validation set.
With the semantic labeling $\{l_S\}$ of all candidate segments fixed, we update the network parameters $\theta$. In this case, the problem becomes the FCN problem \cite{Long2015} as in Eqn.(\ref{eq:fcn_loss}). This problem is minimized by SGD.
We iteratively perform the above two steps, fixing one set of variables and solving for the other set.
For each iteration, we update the network parameters using one training epoch (\ie, all training images are visited once), and after that we update the segment labeling of all images. Fig.\ref{fig:epochs} shows the gradually updated segmentation masks during training.
The network is initialized by the model pre-trained in the ImageNet classification dataset, and our algorithm starts from the step of updating segment labels.
Our method is applicable for the semi-supervised case (the ground-truth annotations are mixtures of segmentation masks and bounding boxes). The labeling $l(p)$ is given by candidate proposals as above if a sample only has ground-truth boxes, and is simply assigned as the true label if a sample has ground-truth masks.
In the SGD training of updating the network, we use a mini-batch size of 20, following \cite{Long2015}. The learning rate is initialized to be 0.001 and divided by 10 after every 15 epochs. The training is terminated after 45 epochs.
\section{Experiments}
In all our experiments, we use the publicly released VGG-16 model\footnote{\url{www.robots.ox.ac.uk/~vgg/research/very_deep/}} \cite{Simonyan2015} that is pre-trained on ImageNet \cite{Russakovsky2014}. The VGG model is also used by all competitors \cite{Long2015,Hariharan2015,Dai2015,Chen2015,mostajabi2014feedforward} compared in this paper.
\subsection{Experiments on PASCAL VOC 2012}
We first evaluate our method on the PASCAL VOC 2012 semantic segmentation benchmark \cite{everingham2010pascal}. This dataset involves 20 semantic categories of objects. We use the ``comp6'' evaluation protocol.
The accuracy is evaluated by mean IoU scores.
The original training data has 1,464 images.
Following \cite{hariharan2011semantic}, the training data with ground-truth segmentation masks are augmented to 10,582 images. The validation and test sets have 1,449 and 1,456 images respectively. When evaluating the validation set or the test set, we only use the training set for training.
A held-out 100 random validation images are used for cross-validation to set hyper-parameters.
\vspace{8pt}
\noindent\textbf{Comparisons of Supervision Strategies}
\setlength{\tabcolsep}{5pt}
\begin{table}[t]
\renewcommand{\arraystretch}{1.1}
\begin{center}
\small
\begin{tabular}{x||x|x|x|x|x}
\hline
data & \multicolumn{3}{ c| }{VOC train} & \multicolumn{2}{ c }{VOC train + COCO}\\
\hline
total \# & \multicolumn{3}{ c| }{10,582} & \multicolumn{2}{ c }{133,869}\\\hline
\tn{supervision} & mask & box & semi & mask & semi\\
\hline
mask \# & 10,582 & - & 1,464 & 133,869 & 10,582\\
box \# & - & 10,582 & 9,118 & - & 123,287\\
\hline
mean IoU & 63.8 & 62.0 & 63.5 & 68.1 & 68.2\\
\hline
\end{tabular}
\end{center}
\caption{Comparisons of supervision in PASCAL VOC 2012 validation.}
\label{tab:voc2012_val_supervision}
\end{table}
Table~\ref{tab:voc2012_val_supervision} compares the results of using different strategies of supervision on the validation set. When all ground-truth masks are used as supervision, the result is our implementation of the baseline DeepLab-CRF \cite{Chen2015}.
Our reproduction has a score of 63.8 (Table~\ref{tab:voc2012_val_supervision}, ``mask only''), which is very close to 63.74 reported in \cite{Chen2015} under the same setting. So we believe that our reproduced baseline is convincing.
When all 10,582 training samples are replaced with bounding box annotations, our method yields a score of 62.0 (Table~\ref{tab:voc2012_val_supervision}, ``box only''). Though the supervision information is substantially weakened, our method shows a graceful degradation (1.8\%) compared with the strongly supervised baseline of 63.8. This indicates that in practice we can avoid the expensive mask labeling effort by using only bounding boxes, with small accuracy loss.
Table~\ref{tab:voc2012_val_supervision} also shows the semi-supervised result of our method. This result uses the ground-truth masks of the original 1,464 training images and the bounding box annotations of the rest 9k images. The score is 63.5 (Table~\ref{tab:voc2012_val_supervision}, ``semi''), on par with the strongly supervised baseline. Such semi-supervision replaces 9/10 of the segmentation mask annotations with bounding box annotations. This means that we can greatly reduce the labeling effort by dominantly using bounding box annotations.
As a proof of concept, we further evaluate using a substantially larger set of boxes. We use the Microsoft COCO dataset \cite{Lin2014} that has 123,287 images with available ground-truth segmentation masks. This dataset has 80 semantic categories, and we only use the 20 categories that also present in PASCAL VOC. For our mask-supervised baseline, the result is a score of 68.1 (Table~\ref{tab:voc2012_val_supervision}). Then we replace the ground-truth segmentation masks in COCO with their tight bounding boxes. Our semi-supervised result is 68.2 (Table~\ref{tab:voc2012_val_supervision}), on par with the strongly supervised baseline. Fig.~\ref{fig:results} shows some visual results in the validation set.
The semi-supervised result (68.2) that uses VOC+COCO is considerably better than the strongly supervised result (63.8) that uses VOC only. The 4.4\% gain is contributed by the extra large-scale bounding boxes in the 123k COCO images. This comparison suggests a promising strategy - we may make use of the larger amount of existing bounding boxes annotations to improve the overall semantic segmentation results, as further analyzed below.
\vspace{8pt}
\noindent\textbf{Error Analysis}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{fig/trimap}
\caption{Error analysis on the validation set. Top: (from left to right) image, ground-truth, \emph{boundary} regions marked as white, \emph{interior} regions marked as white). Bottom: \emph{boundary} and \emph{interior} mean IoU, using VOC masks only (blue) and using extra COCO boxes (red).}
\label{fig:trimap}
\end{figure}
Why can a large set of bounding boxes help improve convolutional networks? The error in semantic segmentation can be roughly thought of as two types: (i) \emph{recognition} error that is due to confusions of recognizing object categories, and (ii) \emph{boundary} error that is due to misalignments of pixel-level labels on object boundaries. Although the bounding box annotations have no information about the object boundaries, they provide extra object instances for recognizing them. We may expect that the large amount of boxes mainly improve the recognition accuracy.
To analyze the error, we separately evaluate the performance on the \emph{boundary} regions and \emph{interior} regions. Following \cite{Kohli2009,Chen2015}, we generate a ``trimap'' near the ground-truth boundaries (Fig.~\ref{fig:trimap}, top). We evaluate mean IoU scores inside/outside the bands, referred to as boundary/interior regions. Fig.~\ref{fig:trimap} (bottom) shows the results of using different band widths for the trimaps.
For the interior region, the accuracy of using the extra COCO boxes (red solid line, Fig.~\ref{fig:trimap}) is considerably higher than that of using VOC masks only (blue solid line). On the contrary, the improvement on the boundary regions is relatively smaller (red dash line \vs blue dash line). Note that correctly recognizing the interior may also help improve the boundaries (\eg, due to the CRF post-processing). So the improvement of the extra boxes on the boundary regions is secondary.
Because the accuracy in the interior region is mainly determined by correctly recognizing objects, this analysis suggests that the large amount of boxes improve the feature quality of a learned BoxSup model for better recognition.
\setlength{\tabcolsep}{8pt}
\renewcommand{\arraystretch}{1.05}
\begin{table}[t]
\small
\begin{center}
\begin{tabular}{x|x}
\hline
masks & mean IoU\\
\hline
\hline
rectangles & 52.3\\
GrabCut & 55.2\\
WSSL \cite{papandreou2015weakly} & 58.5\\
\hline
ours w/o sampling & 59.7\\
ours & \underline{62.0}\\
\hline
\end{tabular}
\end{center}
\caption{Comparisons of estimated masks for supervision in PASCAL VOC 2012 validation. All methods only use 10,582 bounding boxes as annotations, with no ground-truth segmentation mask used.}
\label{tab:voc2012_val_approaches}
\end{table}
\setlength{\tabcolsep}{8pt}
\renewcommand{\arraystretch}{1.1}
\begin{table}[t]
\begin{center}
\begin{tabular}{x|x|x|x}
\hline
& SS & GOP & MCG \\
\hline
mean IoU & 59.5 & 60.4 & \underline{62.0}\\
\hline
\end{tabular}
\end{center}
\caption{Comparisons of the effects of region proposal methods on our method in PASCAL VOC 2012 validation. All methods only use 10,582 bounding boxes as annotations, with no ground-truth segmentation mask used.}
\label{tab:voc2012_val_proposals}
\end{table}
\vspace{8pt}
\noindent\textbf{Comparisons of Estimated Masks for Supervision}
In Table~\ref{tab:voc2012_val_approaches} we evaluate different methods of estimating masks from bounding boxes for supervision. As a na\"{\i}ve baseline, we fill each bounding box with its semantic label, and consider it as a rectangular mask (Fig.~\ref{fig:masks}(c)). Using these rectangular masks as the supervision throughout training, the score is 52.3 on the validation set. We also use GrabCut \cite{rother2004grabcut} to generate segmentation masks from boxes (Fig.~\ref{fig:masks}(d)). With the GrabCut masks as the supervision throughout training, the score is 55.2. In both cases, the masks are not updated by the network feedbacks.
Our method has a score 62.0 (Table~\ref{tab:voc2012_val_approaches}) using the same set of bounding box annotations. This is a considerable gain over the baseline using fixed GrabCut masks. This indicates the importance of the mask quality for supervision. Fig.~\ref{fig:epochs} shows that our method iteratively updates the masks by the network, which in turn improves the network training.
We also evaluate a variant of our method where each time the updated mask is the candidate with the largest cost, instead of randomly sampled from the first $k$ candidates (see Sec.~\ref{sec:alg}). This variant has a lower score of 59.7 (Table~\ref{tab:voc2012_val_approaches}). The random sampling strategy, which is data augmentation and increases sample variances, is beneficial for training.
Table~\ref{tab:voc2012_val_approaches} also shows the result of the concurrent method WSSL \cite{Chen2015} under the same evaluation setting. Its results is 58.5. This result suggests that our method estimates more accurate masks than \cite{Chen2015} for supervision.
\vspace{8pt}
\noindent\textbf{Comparisons of Region Proposals}
Our method resorts to unsupervised region proposals for training.
In Table \ref{tab:voc2012_val_proposals}, we compare the effects of various region proposals on our method: Selective Search (SS) \cite{uijlings2013selective}, Geodesic Object Proposals (GOP) \cite{krahenbuhl2014geodesic}, and MCG \cite{arbelaez2014multiscale}.
Table \ref{tab:voc2012_val_proposals} shows that MCG \cite{arbelaez2014multiscale} has the best accuracy, which is consistent with its segmentation quality evaluated by other metrics in \cite{arbelaez2014multiscale}. Note that at test-time our method does not need region proposals. So the better accuracy of using MCG implies that
our method effectively makes use of the higher quality segmentation masks to train a better network.
\begin{figure*}[t]
\centering
\includegraphics[width=0.85\linewidth]{fig/results_voc2012}
\caption{Example semantic segmentation results on \textbf{PASCAL VOC 2012} validation using our method. (a) Images. (b) Supervised by masks in VOC. (c) Supervised by boxes in VOC. (d) Supervised by masks in VOC and boxes in COCO.}
\label{fig:results}
\end{figure*}
\vspace{8pt}
\noindent\textbf{Comparisons on the Test Set}
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.1}
\begin{table}[t]
\begin{center}
\begin{small}
\begin{tabular}{x|x|r|r|x}
\hline
method & sup. & mask \# & box \# & mIoU\\
\hline
\hline
FCN \cite{Long2015} & mask & V 10k & - & 62.2\\
\tn{DeepLabCRF} \cite{Chen2015} & mask & V 10k & - & 66.4\\
WSSL \cite{papandreou2015weakly} & box & - & V 10k & 60.4\\
\textbf{BoxSup} & box & - & V 10k & 64.6\\
\textbf{BoxSup} & semi & V 1.4k & V 9k & 66.2\\
\hline
WSSL \cite{papandreou2015weakly} & mask & V+C 133k & - & 70.4\\
\textbf{BoxSup} & semi & V 10k & C 123k & 71.0\\
\textbf{BoxSup} & semi & V 10k & V$_{07}$+C 133k & 73.1 \\
\textbf{BoxSup+} & semi & V 10k & V$_{07}$+C 133k & \textbf{75.2}\\
\hline
\end{tabular}
\end{small}
\end{center}
\caption{Results on \textbf{PASCAL VOC 2012 test} set. In the supervision (``sup'') column, ``mask'' means all training samples are with segmentation mask annotations, ``box'' means all training samples are with bounding box annotations, and ``semi'' means mixtures. ``V'' denotes the VOC data, ``C'' denotes the COCO data, and ``V$_{07}$'' denotes the VOC 2007 data which only has bounding boxes available.}
\label{tab:voc2012_test}
\end{table}
Next we compare with the state-of-the-art methods on the PASCAL VOC 2012 {\em test} set. In Table~\ref{tab:voc2012_test}, the methods are based on the same FCN baseline and thus fair comparisons are made to evaluate the impact of mask/box/semi-supervision.
As shown in Table \ref{tab:voc2012_test}, our \emph{box-supervised} result that only uses VOC bounding boxes is 64.6. This compares favorably with the WSSL \cite{papandreou2015weakly} counterpart (60.4) under the same setting. On the other hand, our \emph{box-supervised} result has a graceful degradation (1.8\%) compared with the \emph{mask-supervised} DeepLab-CRF (66.4 \cite{Chen2015}) using the VOC training data.
Moreover, our semi-supervised variant which replaces 9/10 segmentation mask annotations with bounding boxes has a score of 66.2. This is on par with the mask-supervised counterpart of DeepLab-CRF, but the supervision information used by our method is much weaker.
In the WSSL paper \cite{papandreou2015weakly}, by using all segmentation mask annotations in VOC and COCO, the strongly mask-supervised result is 70.4. Our semi-supervised method shows a higher score of \textbf{71.0}. Remarkably, our result uses the bounding box annotations from the 123k COCO images. So our method has a more accurate result but uses much weaker annotations than \cite{papandreou2015weakly}.
On the other hand, compared with the DeepLab-CRF result (66.4), our method has a 4.6\% gain enjoyed from exploiting the \emph{bounding box} annotations of the COCO dataset. This comparison demonstrates the power of our method that exploits large-scale bounding box annotations to improve accuracy.
\vspace{8pt}
\noindent\textbf{Exploiting Boxes in PASCAL VOC 2007}
To further demonstrate the effect of BoxSup, we exploit the bounding boxes in the PASCAL VOC 2007 dataset \cite{everingham2010pascal}. This dataset has no mask annotations. It is a de facto dataset which mask-supervised methods are \emph{not} able to use.
We exploit all 10k images in the VOC 2007 trainval and test sets. We train a BoxSup model using the union set of VOC~2007 boxes, COCO boxes, and the augmented VOC 2012 training set. The score improves from 71.0 to \textbf{73.1} (Table~\ref{tab:voc2012_test}) because of the extra box training data. It is reasonable for us to expect further improvement if more bounding box annotations are available.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\linewidth]{fig/results_voc2010}
\caption{Example results on \textbf{PASCAL-CONTEXT} validation. (a) Images. (b) Results of our baseline (35.7 mean IoU), trained using VOC masks. (c) Results of BoxSup (40.5 mean IoU), trained using VOC masks and COCO boxes.}
\label{fig:results_voc2010}
\end{figure*}
\vspace{8pt}
\noindent\textbf{Baseline Improvement}
Although our focus is mainly on exploiting boxes as supervision, it is worth noticing that our method may also benefit from other improvements on the mask-sup baseline (FCN in our case). Concurrent with our work, there are a series of improvements \cite{zheng2015conditional,Chen2015} made on FCN, which achieve excellent results using strong mask-supervision from VOC and COCO data.
To show the potential of our BoxSup method in parallel with improvements on the baseline, we use a simple test-time augmentation to boost our results. Instead of computing pixel-wise predictions on a single scale, we compute the score maps from two extra scales ($\pm20\%$ of the original image size) and bilinearly re-scale the score maps to the original size. The scores from three scales are averaged. This simple modification boosts our result from 73.1 to \textbf{75.2} (BoxSup+, Table~\ref{tab:voc2012_test}) in the VOC 2012 test set. This result is on par with the latest results using strong mask-supervision from both VOC and COCO, but in our case the COCO dataset only provides bounding boxes.
\setlength{\tabcolsep}{6pt}
\renewcommand{\arraystretch}{1.1}
\begin{table}[t]
\begin{center}
\begin{tabular}{x|x|x|x|x}
\hline
method & sup. & mask \# & box \# & \tn{mean IoU}\\
\hline
\hline
O$_2$P \cite{carreira2012semantic} & mask & V 5k & - & 18.1\\
CFM \cite{Dai2015} & mask & V 5k & - & 34.4\\
FCN \cite{Long2015} & mask & V 5k & - & 35.1\\
\hline
baseline & mask & V 5k & - & 35.7\\
\textbf{BoxSup} & semi & V 5k & C 123k & \textbf{40.5}\\
\hline
\end{tabular}
\end{center}
\caption{Results on \textbf{PASCAL-CONTEXT} \cite{mottaghi2014role} validation. Our baseline is our implementation of FCN+CRF. ``V'' denotes the VOC data, and ``C'' denotes the COCO data.}
\label{tab:voc2010_val}
\end{table}
\subsection{Experiments on PASCAL-CONTEXT}
We further perform experiments on the recently labeled PASCAL-CONTEXT dataset \cite{mottaghi2014role}. This dataset provides ground-truth semantic labels for the whole scene, including object and stuff (\eg, grass, sky, water). Following the protocol in \cite{mottaghi2014role,Dai2015,Long2015}, the semantic segmentation is performed on the most frequent 59 categories (identified by \cite{mottaghi2014role}) plus a background category. The accuracy is measured by mean IoU scores. The training and evaluation are performed on the training and validation sets that have 4,998 and 5,105 images respectively.
To train a BoxSup model for this dataset, we first use the \emph{box annotations} from all 80 object categories in the COCO dataset to train the FCN (using VGG-16). This network ends with an 81-way (with an extra one for background) layer. Then we remove this last layer and add a new 60-way layer for the 59 categories of PASCAL-CONTEXT. We fine-tune this model in the 5k training images of PASCAL-CONTEXT. A CRF for post-processing is also used. We do no use the test-time scale augmentation.
Table \ref{tab:voc2010_val} shows the results in PASCAL-CONTEXT. The methods of CFM \cite{Dai2015} and FCN \cite{Long2015} are both based on the VGG-16 model. Our baseline method, which is our implementation of FCN+CRF,
has a score of 35.7 using masks of the 5k training images. Using our BoxSup model pre-trained using the COCO boxes, the result is improved to \textbf{40.5}. The 4.8\% gain is solely because of the bounding box annotations in COCO that improve our network training. Fig.~\ref{fig:results_voc2010} shows some examples of our results for joint object and stuff segmentation.
\section{Conclusion}
The proposed BoxSup method can effectively harness bounding box annotations to train deep networks for semantic segmentation. Our BoxSup method that uses 133k bounding boxes and 10k masks achieves state-of-the-art results.
Our error analysis suggests that semantic segmentation accuracy is hampered by the failure of recognizing objects, which large-scale data may help with.
{\small
\bibliographystyle{ieee}
|
1410.6148
|
\section{Introduction}
A chord diagram is a circle (called the backbone) with line segments (called chords) attached at their endpoints.
Chord diagrams have been extensively used in knot theory and its applications, as well as in physics and biology.
They are main tools for finite type knot invariants \cite{Dror}, and are also used for describing RNA secondary
structures \cite{APRW}, for example. A chord diagram is usually
depicted as
a circle in the plane with chords inside the circle. The chords may intersect in the circle, but such intersections
are ignored (chords are regarded as pairwise disjoint).
The genus of a chord diagram
is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands
to the inner boundary circle at the ends of each chord, and it has been studied earlier in
the context of
knot theory. In \cite{STV}, for example, it was pointed out that the genus of a chord diagram equals that of a
surface obtained from the Seifert algorithm, a standard construction of orientable surfaces bounded by knots
from diagrams. This fact was used in \cite{STV}
to define the genus of virtual knots as the minimum of such genera
over all virtual knot diagrams. Such genera was used in \cite{APRW} for the study of RNA foldings.
Thickened chord diagrams were used for the study of DNA structures as well \cite{Jonoska2002}.
The genus of a chord diagram is defined by attaching bands at chord endpoints on the inner boundary circle
of the annulus as mentioned above, and different surfaces could be obtained if some bands are allowed to
be attached on the outer boundary circle of the annulus. It is, then, natural to ask which integers arise as
genera of surfaces if such variants are allowed for thickened chord diagrams. Specifically, we consider the
following questions.
\bigskip
\noindent
{\bf Problem.} For a given positive integer $n$,
(1) determine which sets of integers appear as genus ranges of chord
diagrams with $n$ chords, and
(2) characterize chord diagrams with $n$ chords that have a specified genus range.
\bigskip
The genus range of graphs
has
been studied in topological graph theory~\cite{MoharThomassenBook}. Our focus in this paper is on a special
class of trivalent graphs that arise as chord diagrams, and
the
behavior of their genus ranges for a fixed number of chords.
The genus ranges of 4-regular rigid vertex graphs were studied in \cite{BDJSV}, where the embedding of
rigid vertex graphs is
required to preserve
the given cyclic order of edges at every vertex.
The paper is organized as follows. Preliminary material is presented in Section~\ref{sec-prelim}. A method of
computing the genus by the Euler characteristic is given in Section~\ref{sec-comp}, where results of computer
calculations are also presented. In Section~\ref{sec-prop}, various properties of genus ranges are described,
and some sets of integers are realized as genus ranges in Section~\ref{sec-realize}. In Section~\ref{sec-chara},
results from Sections ~\ref{sec-prop}
and
\ref{sec-realize} are combined to summarize our findings on which
sets of integers can and cannot be realized as genus ranges of chord diagrams for a fixed number of chords.
We also list the sets for which realizability as the genus range of a chord diagram has yet to be determined, and end with some short concluding remarks.
\section{Terminology and Preliminaries}\label{sec-prelim}
This section contains the definitions of the concepts, their basic properties, and the
notations used in this paper.
A {\it chord diagram} consists of a finite number of {\it chords}, that are closed arcs, with their endpoints
attached to a circle, called the {\it backbone}. An example of a chord diagram is given in Fig.~\ref{thicken} (A).
For
more details and the background of chord diagrams, see, for example, \cite{Dror,STV}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=4in]{thicken.eps}\\
\caption{(A) An example of a chord diagram 123132, $*$ indicates the base point;
(B) and (C) Two examples of thickened chord diagrams corresponding to the chord diagram in (A);
(D) Schematic representation of the thickened diagram in (C).}
\label{thicken}
\end{center}
\end{figure}
A {\it double-occurrence} word $w$ over an alphabet set is a word which contains each symbol of the alphabet
set exactly $0$ or $2$ times. Double-occurrence words are also called {\it (unsigned) Gauss codes} in knot
theory \cite{Kauff}.
For a given chord diagram, we obtain a double-occurrence word as follows. If it has $n$ chords, assign distinct
labels (e.g., positive integers $\{ 1, \ldots, n\}$) to the chords. The endpoints of the chords lying on the
backbone inherit the labels of the corresponding chords. Pick and fix a base point $*$ on the backbone
of a chord diagram. The sequence of endpoint labels obtained by tracing the backbone in one direction
(say, clockwise) forms a double-occurrence word corresponding to the chord diagram. Conversely, for a given
double-occurrence word, a chord diagram corresponding to the word is obtained by choosing distinct points
on a circle such that each point corresponds to a letter in the word in the order of their appearance, and then
connecting each pair of points of the same letter by a chord. The chord diagram in Fig.~\ref{thicken} (A) has
the corresponding double-occurrence word $123132$. Equivalence relations are defined on chord diagrams
and double-occurrence words in such a way that this correspondence is bijective. Two double-occurrence
words are equivalent if they are related by cyclic permutations, reversal, and/or symbol renaming.
\smallskip
\noindent
{\bf Notation.} Applying the above mentioned correspondence between chord diagrams and double-occurrence
words, in this paper a double-occurrence word $W$ also represents the corresponding chord diagram.
A {\it thickened} chord diagram (or simply a {\it thickened diagram}) is a compact orientable surface obtained
from a given chord diagram by thickening its backbone circle and chords as depicted in Fig.~\ref{thicken} (B), (C).
The backbone is thickened to an annulus. A band corresponding to each chord is attached to one of two
boundary circles of the annulus. In literature (e.g., \cite{APRW,STV}), all bands are attached to the inner
boundary of the thickened circle as in Fig.~\ref{thicken} (B), and in this case we say that chords are {\it all-in},
or that the chord diagram is of {\it all-in}. For a chord diagram $D$ we denote with $F_D$ the all-in thickened
chord diagram corresponding to $D$. In this paper, we consider thickened chord diagrams with band ends
possibly attached to the outer boundary circle of the annulus, as is one of the ends of chord $1$ in (C). Since
each endpoint of a chord has two possibilities of band ends attachments (inner or outer), there are 4 possible
band attachment cases for each chord, in total $4^n$ surfaces obtained from a chord diagram with $n$ chords.
To simplify exposition, we draw an endpoint of a chord attached to the outer side of the backbone as in
Fig.~\ref{thicken} (D) to indicate that the corresponding thickened diagram is obtained by attaching the
corresponding band end to the outer boundary of the annulus. A band whose one end is connected to the
outside curve of the annulus and the other is connected to the inside part of the curve is said to be a
{\it one-in, one-out chord}.
\smallskip
\noindent
{\bf Convention.} We assume that all surfaces are orientable throughout the paper.
\begin{definition}
Let $g(F)$ denote the genus of a compact orientable surface $F$. The {\it genus range} of a chord diagram
$D$ is the set of genera of thickened chord diagrams, and denoted by ${\rm gr}(D)$:
$$ {\rm gr}(D)=\{ \, g(F) \ | \ F \mbox{ is a thickened chord diagram of } D \, \} .$$
\end{definition}
\begin{figure}[h]
\begin{center}
\includegraphics[width=9cm]{endedge}
\caption{End edges traced by (A) single and (B) two boundary curve(s).}
\label{endedge}
\end{center}
\end{figure}
We use the following terminology in the later sections. The closed backbone arc which is a portion of the
backbone between the first and the last endpoints containing the base point is called the {\it end edge}.
Because the backbone and the chords are thickened to bands that constitute a thickened diagram, we
regard that each backbone arc and each chord has two corresponding boundary curve segments, which may
or may not belong to the same connected component of the boundary. In particular, the boundary curves
corresponding to the end edge may belong to one or two boundary components, as depicted in
Fig.~\ref{endedge} (A) and (B), respectively. In each case, we say that the end edge is {\it traced by a
single (resp. double) boundary curve(s)}.
\section{Computing the Genus Range of a chord diagram} \label{sec-comp}
In this section we recall the Euler characteristic formula used to compute the genus ranges by counting
the number of boundary components, and present outputs of computer calculations.
\subsection{Euler characteristic formula}
First we recall the well-known Euler characteristic formula, establishing the relation between the genus and the
number of boundary components. The Euler characteristic $\chi(F)$ of a compact orientable surface $F$ of
genus $g(F)$ and the number of boundary components $b(F)$ of $F$ are related by $\chi(F)=2-2 g(F) - b(F)$.
A thickened chord diagram $F$ is a compact surface with the original chord diagram $D$ as a deformation
retract. If the number of chords is $n >0$, $n \in \Z$, then there are $2n$ vertices in $D$ and $3n$ edges
($n$ chords and $2n$ arcs on the backbone), so that $\chi(F)=\chi(D)=2n - 3n=-n$. Thus we obtain the
following well known formula, which we state as a lemma, as we will use it often in this paper.
\begin{lemma} \label{lem-euler}
Let $F$ be a thickened chord diagram of a chord diagram $D$. Let $g(F)$ be the genus of $F$, $b(F)$ be
the number of boundary components of $F$, and $n$ be the number of chords of $D$. Then we have
$g(F) =(1/2) (n - b(F) + 2 )$.
\end{lemma}
Thus we can compute the genus range from the set of the numbers of boundary components of thickened
chord diagrams, $\{ \, b(F) \mid F \mbox{ is a thickened chord diagram of } D \, \}$. Note that $n$ and
$b(F)$ have the same parity, as genera are integers.
\subsection{Computer calculations}
In \cite{APRW}, the genera of chord diagrams was defined (which is the genus of all-in chord diagrams), and
an algorithm to compute the number of graphs with a given genus and $n$ chords by means of cycle
decompositions of permutations was presented. Our computer calculation is based on a modified version of
their algorithm. The computational results are posted at \url{http://knot.math.usf.edu/data/} under {\it Tables}.
Computer calculations show that the sets of all possible genus ranges of chord diagrams with $n$ letters for
$n=1, \ldots, 7$ are as follows.
\begin{table}[h]
\centering
\begin{tabular}{rl}
& ${\cal GR}_n$\\
$n$ = 1, 2 : & \{0,1\} \\
$n$ = 3, 4 : & \{0,1\}, \{0,1,2\}, \{1,2\} \\
$n$ = 5, 6 : & \{0,1\}, \{0,1,2\}, \{1,2\}, \{0,1,2,3\}, \{1,2,3\} \\
$n$ = 7 : & \{0,1\}, \{0,1,2\}, \{1,2\}, \{0,1,2,3\}, \{1,2,3\}, \{0,1,2,3,4\}, \{1,2,3,4\}, \{2,3,4\}
\end{tabular}
\end{table}
The following conjectures hold for all examples we computed.
\begin{conjecture} \label{conj-range2}
For any $n>0$, if a chord diagram with $n$ chords has genus range consisting of two numbers, then the
genus range is either $\{ 0,1\}$ or $\{1,2\}$.
\end{conjecture}
\begin{conjecture}
For any $n \neq 2$, there is a unique (up to equivalence) double-occurrence word $11 \cdots nn$ that
corresponds to a chord diagram with the genus range $\{0,1\}$.
\end{conjecture}
We note that there are two 2-letter words, $1122$ and $1212$, and both corresponding chord diagrams
have the genus range $\{ 0, 1\}$.
\begin{conjecture}
For any $n \neq 4$, there is a unique (up to equivalence) chord diagram
with genus range $\{1, 2\}$, and it is
$(123123)(44 \cdots nn)$.
\end{conjecture}
There are several more chord diagrams for $n=4$ with genus range $\{1,2\}$.
\section{Properties of Genus Ranges}\label{sec-prop}
The following is standard for cellular embeddings of general graphs \cite{MoharThomassenBook}, and also
known for 4-regular rigid vertex graphs \cite{BDJSV}. Below we state the property for chord diagrams.
\begin{proposition} \label{lem-consec}
The genus range of any chord diagram consists of consecutive integers.
\end{proposition}
By Proposition~\ref{lem-consec} the genus ranges of chord diagrams are integer intervals, therefore in
the rest of the paper we use the notation $[a, b] = \{ k \in \Z \mid a \leq k \leq b\}.$
\begin{lemma} \label{lem-nosingleton}
There does not exist a chord diagram whose genus range consists only of a singleton.
\end{lemma}
\begin{proof}
Since all-in thickened diagram $F_D$ for a chord diagram $D$ has an outside boundary component and
some inside ones, any chord diagram has a thickened chord diagram with more than one boundary component.
Let $n$ be the number of boundary components of $F_D$, one of which is the outside circle.
Let $c$ be a chord in $D$. Then removing the corresponding band (thickened chord) from $F_D$ either
increases or decreases the number of boundary components of a thickened diagram by exactly one.
If $c$ is traced by a single boundary component, then its band removal splits the component in two parts, and
if $c$ is traced by two components, then the removal of its band connects the two components
as
a single one. Suppose that when a band for $c$ is removed from $F_D$, the number of boundary components
increases by one. Let $D'$ be the chord diagram with $c$ removed from $D$, and consider $F'=F_{D'}$, the
all-in thickened diagram of $D'$. Then the number of boundary components of $F'$ is $n+1$. In this case,
adding the band of $c$ back to $F'$ to obtain $F_D$ will connect two inside boundary components of $F'$.
Instead, connecting both ends of the band of $c$ to the outside boundary circle of $F'$ increases the number of
boundary components by one, and gives rise to a thickened diagram of $D$ with $n+2$
boundary components, and with genus $g(F_D)+1$. Hence, ${\rm gr}(D)$ is not a singleton.
We repeat a similar argument for the case when the number of boundary components of $F_D$ decreases by
one when a band of $c$ is removed. Let $D'$ be the chord diagram with $c$ removed from $D$ and $F'=F_{D"}$
be the all-in thickened diagram of $D'$, then the number of boundary components of $F'$ is $n-1$. Adding to
$F'$ a one-in, one-out chord for $c$ connects the original inside boundary with the outside curve and
decreases the number of boundary components by one. This gives rise to a thickened diagram of $D$ with
$n-2$ boundary components with genus $g(F_D)-1$.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{sum.eps}
\caption{Connected sum of two chord diagrams.}
\label{sum}
\end{figure}
The connected sum of two chord diagrams with base points is defined in a manner similar to the connected sum
of knots, see Fig.~\ref{sum}. A band is attached at the base points preserving orientations to obtain a new
chord diagram. In the figure, the left and right chord diagrams, respectively, before taking connected sum are
represented by double-occurrence words $W_1=123123$ and $W_2=123132$, respectively, and after the
connected sum, it is represented by $W=123123456465$, after renaming $W_2$. We use the notation
$W=W_1 W_2$ to represent the word thus obtained, by renaming and concatenation.
\begin{lemma}\label{lem-connectsum}
Let $W_1$ and $W_2$ be chord diagrams such that the genus ranges of corresponding chord diagrams are
$[g_1, g_1']$ and $[g_2, g_2']$, respectively. Let $e_1,e_2$ be the end edges of $W_1$, $W_2$, respectively.
Then the genus range of the chord diagram corresponding to $W=W_1 W_2$ is
$[g_1 + g_2 - \epsilon, g_1' + g_2' - \epsilon ']$
for some $\epsilon, \epsilon ' \in \{ 0, 1\}$, where $\epsilon, \epsilon'$ are determined as follows.
\begin{description}
\item [($E_0$)] $\epsilon=0$ if and only if at least one of the end edges (say, the end edge $e_1$ of $W_1$) has the
following property: any thickened graph of genus $g_1$ traces $e_1$ by two boundary curves.
\item[($E_1$)] $\epsilon=1$ if and only if both end edges $e_1$ and $e_2$ have the following property: there exist
thickened graphs of genus $g_1$ and $g_2$, respectively, that trace both $e_1$ and $e_2$ by a single boundary
curve.
\item[($E'_0$)] $\epsilon '=0$ if and only if at least one of the end edges (say, $e_1$ of $W_1$) has the following
property: there exists a thickened graph of genus $g_1'$ that traces $e_1$ by two boundary curves.
\item[($E'_1$)] $\epsilon '=1$ if and only if both end edges $e_1$ and $e_2$ of $W_1$ and $W_2$ have the
following property: any thickened graphs of genus $g_1'$ and $g_2'$, respectively, trace $e$ and $e'$ by a single
boundary curve.
\end{description}
\begin{figure}[h]
\centering
\includegraphics[width=15cm]{connect.eps}
\caption{Connected sum and boundary curves: (A) both end edges are traced by a single boundary curve;
(B) one end edge is traced by a single boundary curve but the other is traced by two boundary curves;
(C) both end edges are traced by two boundary curves.}
\label{connect}
\end{figure}
\end{lemma}
\begin{proof}
This is proved by a case-by-case analysis of the number of boundary components and by using
Lemma~\ref{lem-euler}. A similar argument is found in \cite{BDJSV,Jonoska2002}. Let $n_1, n_2$ be the
number of chords of chord diagrams corresponding to $W_1$, $W_2$, respectively. The number of chords
of $W=W_1 W_2$ is $n=n_1+ n_2$. Let $b_1$ and $b_2$ be the number of boundary components of
thickened chord diagrams for $W_1$ and $W_2$, respectively. The number of boundary component $b$ of a
thickened diagram $D$ of $W=W_1 W_2$ after taking the connected sum equals $b_1+b_2-\alpha$ where
$\alpha=0$ or $\alpha =2$. If both end edges $e_1$ and $e_2$ are traced by a single component (the
situation as in Fig.~\ref{connect} (A)) then $\alpha=0$. If at least one end edge $e_1$ or $e_2$ is traced
by two components (the situations Fig~\ref{connect} (B) and (C)), then $\alpha = 2$. Then
Lemma~\ref{lem-euler} implies
\begin{align*}
g &= \frac{1}{2} \left( n - b + 2 \right) = \frac{1}{2} \left[ (n_1 + n_2) - (b_1 + b_2 - \alpha) + 2 \right] \\
&= \frac{1}{2} \left(n_1 - b_1 + 2 \right) + \frac{1}{2} \left( n_2 - b_2 + 2 \right) + \frac{\alpha}{2} - 1 \\
&= g_1 + g_2 + \frac{\alpha}{2} - 1,
\end{align*}
where $g_1$, $g_2$, and $g$ are genera of $W_1$, $W_2$, and $W$, respectively.
For statement $(E_1)$, there are thickened diagrams with minimal genus of $W_1$ and $W_2$ for which
$\alpha = 0$ and whose connected sum preserves the number of boundary components (Fig.~\ref{connect} (A)),
hence the statement follows. The other cases are proved by similar arguments.
\end{proof}
Since we often refer to the number of boundary components tracing the end edge, we define the following
notation. Let $e$ be the end edge of a chord diagram corresponding to a double-occurrence word $W$.
Let $c$ be 1 or 2. We say that $W$ satisfies the condition $A({\rm min}, c)$ (resp. $A({\rm max}, c)$)
if any thickened diagram of minimum (resp. maximum) genus $g$ traces $e$ by a single boundary curve for
$c=1$, and by two boundary curves for $c=2$. Similarly, we say that $W$ satisfies the condition
$E({\rm min}, c)$ (resp. $E({\rm max}, c)$) if there exists a thickened diagram of minimum (resp. maximum)
genus $g$ that traces $e$ by a single boundary curve for $c=1$, and by two boundary curves for $c=2$.
We also simply say $W$ is (of) $A({\rm min}, c)$ etc. Then Lemma~\ref{lem-connectsum} is summarized as follows.
\begin{table}[h]
\centering
\begin{tabular}{clc}
\hline
Cases & \quad $W_1$, $W_2$ & $W_1 W_2$ \\
\hline
$(E_0)$ & one $A({\rm min}, 2)$ & $ \epsilon = 0 $ \\
$(E_1)$ & both $E({\rm min}, 1)$ & $ \epsilon = 1 $ \\
$(E_0')$ & one $E({\rm max}, 2)$ & $ \epsilon ' = 0 $ \\
$(E_1')$ & both $A({\rm max}, 1)$ & $ \epsilon ' = 1 $ \\
\hline
\end{tabular}
\end{table}
If a chord diagram $D'$ is obtained from $D$ by removing some chords, then $D'$ is called a {\it sub-chord diagram}
of $D$. The following lemma covers a large family of chord diagrams that support Conjecture~\ref{conj-range2}.
\begin{lemma} \label{lem-range3words}
If a chord diagram has a sub-chord diagram corresponding to the double-occurrence word $123321$, then
its genus range contains more than $2$ integers.
\end{lemma}
\begin{proof}
Consider the surface $F$ obtained by thickening the chord diagram such that the three parallel chords represented
by 1, 2 and 3 are all-in, and the other chords are all-out. Then it has $4$ inside boundary curves and at least one
outside, total at least $5$. Refer to Fig.~\ref{parallel3}, where other chords are not depicted. Move one end of
chord $1$ from inside to outside, keeping the other inside. Then the total number of boundary curves decreases
by $2$. This is seen as follows. Regard this operation in two steps: (1) remove a band corresponding to chord
$1$ from $F$, and (2) add a band corresponding to $1$ with one-in and one-out ends. The step (1) joins the two
inside boundary curves to a single curve, thus reduces the number of boundary curves by 1. In step (2), the new
one-in, one-out band joins the newly formed inside curve with one of the outside curves, reducing the boundary
curve by 1 again. Hence replacing an all-in chord $1$ in $F$ with one-in, one-out chord decreases the number of
boundary curves by 2. Performing the same procedure for the chord labeled $3$, further decreases the number
of components by $2$. Therefore, the genus range consists of at least $3$ numbers.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{parallel3}
\caption{Three thickened surfaces with distinct genera for a chord diagram containing the sub-chord diagram
$123321$. All other chords (not pictured) are all-out.}
\label{parallel3}
\end{figure}
\section{Realizations of Genus Ranges}\label{sec-realize}
We use the following notations for respective double-occurrence words and corresponding chord diagrams:
\begin{align*}
U_n & = 1122\cdots nn \quad (\; =\ {U_1}^n\; ) , \\
R_n & = 12\cdots n 12 \cdots n , \\
G_\ell &= (1212)(3434) \cdots ((2\ell -1)2 \ell (2\ell -1) 2 \ell) \quad (\; =\ {R_2}^{\ell}\; ) .
\end{align*}
\begin{lemma} \label{lem-repeatword}
For the chord diagram corresponding to $R_n=12\cdots n 12 \cdots n$, where $n=2m$ or $n=2m-1$ and
$n>2$, we have ${\rm gr}(R_n)=[1, m]$.
\end{lemma}
\begin{proof}
For an even $n=2m$ ($m\ge 1$), consider the all-in thickened diagram $F_{R_n}$. Then $F_{R_n}$ has
exactly two boundary components: One inside curve, tracing chords in successive order
(see Fig.~\ref{repeat} (A)), and one outside. Hence $F_{R_n}$ achieves the maximum genus $m$. By
adding a one-in, one-out chord, the two curves are joined to a single component, therefore for an odd $n$,
the resulting surface the maximum genus.
Consider a thickened diagram for $R_n$ where every chord is one-in, one-out (see Fig.~\ref{repeat} (B)).
Each boundary curve traces a single side of two chords. Then the resulting surface has $n$ boundary
components, and genus $1$.
Since the chord diagram $D$ for $R_3=123123$ is isomorphic (as a graph) to the bipartite graph $K_{3,3}$,
$D$ is non-planar and the genus range of any chord diagram that has $R_3$ as a sub-chord diagram does
not contain $0$. The result follows from Lemma~\ref{lem-consec}.
\end{proof}
\begin{figure}[h]
\centering
\begin{overpic}[width=7cm]{repeat}
\put(-5,0){(A)}
\put (40,10){$i$}
\put (-3,25){$i$}
\put (34,0){{\small $i+1$}}
\put (-5,35){{\small $i+1$}}
\put(53,0){(B)}
\end{overpic}
\caption{Thickened diagrams of $R_n$ with the maximum genus (A) and the minimum genus (B).}
\label{repeat}
\end{figure}
\begin{lemma} \label{lem-range2}
(1) For any $n>0$, there exists a chord diagram of $n$ chords with genus range $[0,1]$.
(2) For any $n>2$, there exists a chord diagram of $n$ chords with genus range $[1,2]$.
\end{lemma}
\begin{proof}
The chord diagram $U_1=11$ has genus range $[0,1]$ and also has the properties $A({\rm min}, 2)$ and
$A({\rm max}, 1)$. By Lemma~\ref{lem-connectsum} (cases $(E_0)$ and $(E'_1)$), the chord diagram of
$U_2={U_1}^2$ has genus range $[0,1]$, and its end edge retains the conditions $A({\rm min}, 2)$ and
$A({\rm max}, 1)$. Inductively, $U_n$ has genus range $[0,1]$ for any $n$.
Recall that the chord diagram corresponding to $R_3$ is non-planar. The chord diagram of $R_3$ has genus
range $[1,2]$ (Lemma~\ref{lem-repeatword}), and has property $A({\rm max}, 1)$. Then the chord diagram
of $R_3 U_1$ has genus range $[1,2]$ by Lemma~\ref{lem-connectsum} (cases $(E_0)$ and $(E'_1)$),
and retains the condition $A({\rm max}, 1)$. Inductively, $R_3 U_m$ has genus range $[1,2]$ for any
$m\geq 0$, hence for any $n=m+3>2$.
\end{proof}
\begin{lemma}\label{lem-1212}
For any chord diagram $W$ with ${\rm gr}(W)=[g, g']$, we have ${\rm gr}(R_2 W)=[g, g'+1]$.
\end{lemma}
\begin{proof}
The chord diagram $R_2$ has genus range $[0,1]$ and is of $A({\rm min}, 2)$ and $A({\rm max}, 2)$,
so it is $E({\rm max}, 2)$. By Lemma~\ref{lem-connectsum} (cases $(E_0)$ and $(E'_0)$), we obtain
the result.
\end{proof}
\begin{lemma}\label{lem-repeatrepeat}
For $G_m=(1212)(3434)\cdots ( (2m-1)2m(2m-1)2m )$, we have ${\rm gr}(G_m)= [0, m]$ for any $m>0$.
\end{lemma}
\begin{proof}
This follows from Lemma~\ref{lem-1212} by induction.
\end{proof}
\begin{lemma}\label{lem-repeatrepeat11}
For any $k, m >0 $, we have ${\rm gr}(G_m U_k)=[0, m+1]$.
\end{lemma}
\begin{proof}
The chord diagram $G_m$ is of $E({\rm max}, 2)$. By Lemma~\ref{lem-connectsum} $(E_0')$,
we have ${\rm gr}(G_n U_1 )=[0, m+1]$. The chord diagrams $U_1$ and $G_m U_k$ for $k>0$
are of $A({\rm max}, 1)$, hence by Lemma~\ref{lem-connectsum} $(E'_1)$ the statement holds by
induction.
\end{proof}
We use the notation $X=12341342$.
\begin{lemma}\label{lem-highgenus}
For any $k>0$, $k \in \Z$, there exists a chord diagram with $n$ chords, where $n=4k-1$ or $4k$,
having genus range $[k, 2k]$.
\end{lemma}
\begin{proof}
Computer calculation shows that the chord diagram $D$ corresponding to $W=12312345674675$ has
genus range $[2,4]$. The word $W$ is the concatenation of $R_3=123123$ and $X=12341342$.
Computer calculation also shows that ${\rm gr}(X)=[1,2]$. By Lemma~\ref{lem-repeatword}, we also have
${\rm gr}(R_3)=[1,2]$.
\begin{figure}[h]
\centering
\includegraphics[width=3cm]{F123123}
\put(-13,60){$*$}
\caption{Edges traced by a single boundary curve of a genus $1$ surface.}
\label{F123123}
\end{figure}
The diagram of $R_3$ is of $E({\rm min}, 1)$ as depicted in Fig.~\ref{F123123}. This implies that
$X=12341342$ is of $A({\rm min}, 2)$. (Otherwise $R_3 X$ has minimum genus 1 by
Lemma~\ref{lem-connectsum} $(E_0)$.) A connected sum of two chord diagrams of $A({\rm min}, 2)$
is again a diagram of $A({\rm min}, 2)$ (case (C) in Fig.~\ref{connect}). By Lemma~\ref{lem-connectsum}
$(E_0)$ again, inductively, the minimum genus of $X^k$ is $k$, and $X^k$ is of $A({\rm min}, 2)$.
Any thickened diagram of $R_3$ with genus $2$ must have a single boundary component,
and therefore, every edge is singly traced, hence it is $A({\rm max}, 1)$. Since ${\rm gr}(W)=[2,4]$,
Lemma~\ref{lem-connectsum} $(E'_0)$ implies that $X$ is of $E({\rm max}, 2)$. Note that the end edge
$e$ of $X^k$ for any $k$ is of $E({\rm max}, 2)$ (case (C) in Fig. ~\ref{connect}). By using
Lemma~\ref{lem-connectsum} $(E'_0)$ inductively, we obtain that the maximum genus of $X^k$ is $2k$.
Hence the diagram for $X^{k}$ has genus range $[k, 2k]$ and $4k$ chords. The diagram corresponding
to $X^{k-1} R_3 $ has $n=4k-1$ chords, the minimum genus $(k-1)+1=k$, the maximum genus
$2(k-1)+2=2k$, by Lemma~\ref{lem-connectsum} as desired.
\end{proof}
We note here that computer calculation was critical for this proof, since it would otherwise be difficult to
determine the genus range of $R_3 X=12312345674675$.
The proof of Lemma~\ref{lem-highgenus} shows that $X^k$ is of $A({\rm min}, 2)$ and $E({\rm max}, 2)$
for every $k>0$.
\begin{lemma}\label{lem-hk}
For any $h>0$ and $ k\geq 0$, there is a chord diagram $W(h, k)$ of $n=4k+h$ chords with genus range
$[k, 2k+1]$.
\end{lemma}
\begin{proof}
Let $W(h, k)= U_h X^{k}$ which has $n=4k+h $ chords. The diagram $U_1$ is of $A({\rm min}, 2)$, and
inductively, so is $U_h$ for any $h>0$. Lemma~\ref{lem-connectsum} $(E_0)$ implies that $W(h, k)$ has
the minimum genus $k$.
We have that $U_h$ has genus range $[0,1]$ and is of $A({\rm max}, 1)$, and $X^k$ is of $E({\rm max}, 2)$
(Proof of Lemma~\ref{lem-highgenus}). Hence $W(h,k)$ has highest genus $2k+1$. The statement holds
for $k=0$ as well, from the proof of Lemma~\ref{lem-range2} (1).
\end{proof}
\begin{lemma}\label{lem-morehk}
For any $h>0$ and $k, \ell \geq 0$, there is a chord diagram $V(h, k, \ell)$ with $n= 4k + 2\ell +h $ chords
such that ${\rm gr}(V (h,k, \ell) )=[k, 2k + \ell + 1 ]$. In the case $h=0$, for any $k, \ell \geq 0$, there is a
chord diagram $V(0, k, \ell)$ with $n= 4k + 2\ell $ chords such that ${\rm gr}(V (0,k, \ell) )=[k, 2k + \ell ]$.
\end{lemma}
\begin{proof}
We consider $V(h, k, \ell)= W(h,k) G_\ell = U_h X^k G_\ell$. We see that $G_\ell$ is of $A({\rm min}, 2)$
and $E({\rm max}, 2)$ inductively from the proof of Lemma~\ref{lem-1212}. Hence the minimum genus of
$V (h,k,\ell )$ is $k$ for any $h, k, \ell \geq 0$ by Lemma~\ref{lem-connectsum} $(E_0)$.
Recall that $X^k$ is of $E({\rm max}, 2)$ and $G_\ell$ is of $E({\rm max}, 2)$ for any $k, \ell >0$.
Hence $V(0, k, \ell)$ is of $E({\rm max}, 2)$ and ${\rm gr}( V(0, k, \ell)) = {\rm gr}(X^kG_\ell) =[k, 2k+ \ell]$ by
Lemma~\ref{lem-connectsum} $(E'_0)$, proving the second statement of the lemma. Because $V(0, k, \ell)$
is of $E({\rm max}, 2)$ and ${\rm gr}( U_h)=[0,1]$, Lemma~\ref{lem-connectsum} $(E_0')$ implies
$V (h,k, \ell) = [k, 2k + \ell + 1 ]$ for $h>0$, $k, \ell \geq 0$.
\end{proof}
\begin{proposition}
For any $g, g'$ such that $g' \ge 2g $ there is a chord diagram with genus range $[g, g']$.
\end{proposition}
\begin{proof}
For $g' >2g$, we set $\ell=g' - 2g$. Then the chord diagram $V(0,k,\ell)$ in
Lemma~\ref{lem-morehk} has genus range $[g, g']$. If $g'=2g > 0$ then Lemma~\ref{lem-highgenus}
provides a desired chord diagram.
\end{proof}
\section{Towards Characterizing Genus Ranges} \label{sec-chara}
In this section we state and prove the main theorem. Recall from Lemma~\ref{lem-euler} that any chord
diagram of $n$ chords, the genus $g$ of a thickened diagram is at most $\lceil n/2 \rceil$.
\begin{theorem}
There exists a chord diagram with $n$ chords and genus range $[g, g']$ whenever $g, g'$ satisfy one
of the following conditions:
(1) $g'=2g$ and either $g=1$ or $g'=\lceil n/2 \rceil$, or
(2) $0 \leq 2 g < g' \leq \lceil n/2 \rceil$.
\end{theorem}
\begin{proof}
Let $m=\lceil n/2 \rceil $.
Case (1): The case of $[g,g']=[1,2]$ follows from Lemma~\ref{lem-range2} (2). In the case of
$0< g'=2g=m$, setting $k=g$ in Lemma~\ref{lem-highgenus}, we obtain a chord diagram with genus
range $[k, 2k]=[g, g']$ with $n=2m=4k$ or $n=2m-1=4k-1$ as desired.
Case (2): Suppose $0 \leq 2 g < g' \leq m$. First we consider the case $n=2m-1$. Set
$\ell=g'-2g-1 \geq 0$ and $h=2m- 2g' +1 >0$. Then $V(h, g, \ell)$ in Lemma~\ref{lem-morehk}
has genus range $[g, 2g+\ell +1]=[g, g']$ and the number of chords is
$4g + 2 \ell + h= 4g + 2( g' - 2g -1) + (2m-2g' +1 )= 2m-1=n, $ as desired.
Next we consider the case $n=2m$. If $m=g'$, set $\ell= g' - 2g >0$. Then the chord diagram
$V(0, g, \ell)$ in Lemma~\ref{lem-morehk} has genus range $[g, 2g + \ell]=[g, g']$ and the number
of chords $4g + 2 \ell = 2g'=2m=n$, as desired. If $m> g'$, set $\ell = g' - 2g -1$ and
$h'= m - g' = m - 2g - \ell \geq 0$. Then the chord diagram corresponding to $V(2h', g, \ell)$ in
Lemma~\ref{lem-morehk} has genus range $[g, 2g + \ell + 1 ]=[g, g']$ and the number of chords
$4g + 2 \ell + 2h' = 2g' + 2h'=2m=n$, as desired.
\end{proof}
\begin{figure}[h]
\centering
\begin{overpic}[width=6cm]{gr}
\put(-35,89){$m=\lceil n/2 \rceil $}
\put(5,99){$b$}
\put(44,102){$b=2a$}
\put(73,102){$b=a+1$}
\put(100,90){$b=a$}
\put(49,70){\Large ?}
\put(95,5){$a$}
\end{overpic}
\caption{Realizing genus ranges for chord diagrams with $n$ chords.}
\label{gr}
\end{figure}
The situation of the theorem is represented in the graph of Fig.~\ref{gr}. Each lattice point of coordinate
$(a,b)$ represents the genus range $[a,b]$. A black dot represents that there is a chord diagram of the
corresponding genus range. A circle with backslash inside, located on the line $b=a$, represents that there
is no singleton genus range by Lemma~\ref{lem-nosingleton}. White dots between two lines $b=a$ and
$b=2a$, and those on the line $b=2a$, denote the cases for which we do not know whether there are
diagrams of those ranges. Note that only two points are realized on the lines $b=2a$ and $b=a+1$. Other
points on the integer lattice, not indicated in the figure, are excluded from the Euler characteristic formula
(Lemma~\ref{lem-euler}).
\section{Concluding Remarks} \label{sec-concl}
In this paper, we studied sets of genus values, called the genus ranges, for thickened chord diagrams.
Variations of surfaces occur when bands that correspond to chords are attached to outside circle boundary
of the backbone of a chord diagram. Computer calculations and constructive methods were used to prove
the results. For a fixed number of chords, we investigated which ranges can and cannot occur. It may be
of interest to investigate the ranges for which we have not been able to determine whether they can be
realized or not.
\section*{Acknowledgements}
This research was partially supported by National Science Foundation DMS-0900671 and National Institutes
of Health R01GM109459. The content is solely the responsibility of the authors and does not necessarily
represent the official views of the NSF or NIH.
|
1410.6096
|
\section{Introduction}
The performance of many coherent signal processing applications is limited by decoherence \cite{mandel1995a}. Among the most promising materials offering long optical and spin coherence lifetimes are cryogenically-cooled rare-earth-ion-doped crystals \cite{macfarlane1987a,sun2005a,tittel2010a,thiel2011a}. The optically-addressable 4f$^N$-4f$^N$ electronic transitions in these materials are only weakly affected by the crystal environment, allowing slow decoherence and many other properties approaching that of isolated atoms. Due to these unique properties, rare-earth-ion-doped crystals have been employed in a wide variety of photonic signal processing demonstrations that include radio-frequency signal processing (see Ref. \cite{babbitt2014a} and references therein), holographic optical memories \cite{renn2002a,schlottau2004a}, high-resolution ultrasound imaging \cite{li2008a,tay2010a}, laser frequency stabilization \cite{strickland2000a,thorpe2011a,chen2011a,thiel2014a}, optical quantum memories \cite{tittel2010a,lvovsky2009a,bussieres2013a}, and many others.
Recent studies by Thiel et al. \cite{thiel2014b} have found 1\%Tm$^{3+}$:Y$_3$Ga$_5$O$_{12}$ (Tm:YGG) to have superior optical decoherence properties compared to known Tm$^{3+}$-doped and many other rare-earth-ion-doped materials. While Tm:YGG has an absorption strength that is weaker than other Tm$^{3+}$-doped materials and only somewhat larger than Eu$^{3+}$ and Tb$^{3+}$ materials \cite{thiel2011a}, the absorption strength can be enhanced for applications by utilizing impedance-matched cavities if necessary \cite{afzelius2010b,moiseev2010a}. Motivated by the potential to provide a suitable combination of properties needed for some demanding photonic and quantum applications, we performed a detailed study of decoherence of the inhomogeneously broadened lowest $^3$H$_6$ to $^3$H$_4$ transition in Tm:YGG at temperatures as low as 1.2 K.
Decoherence in rare-earth-ion-doped materials can be caused by a number of physical mechanisms, including phonon interactions, fluctuating electromagnetic fields arising from random nuclear and electronic spin-flips in the host crystal, and dynamic disorder modes in the host lattice structure \cite{macfarlane1987a,sun2005a}. To elucidate these processes in Tm:YGG, we employ two- and three-pulse photon echo measurements to measure the temperature and magnetic field dependence of the optical coherence lifetime as well as time-dependent broadening caused by spectral diffusion phenomena. Moreover, spectral hole burning and photon echoes are used to show uniform decoherence properties as a function of wavelength, verifying that all Tm$^{3+}$ ions in the lattice experience the same decoherence dynamics. Our results show that at temperatures less than 1 K, refined crystal growth, and possible optimization of crystal orientations, coherence lifetimes approaching one millisecond may be possible in this system.
\section{Tm:YGG crystal properties}
The crystal Y$_3$Ga$_5$O$_{12}$ (YGG) has cubic symmetry described by space group Ia3d with eight formula units per unit cell \cite{menzer1928a}. The Y$^{3+}$ ions occupy crystallographically equivalent lattice sites with dodecahedral point symmetry (D$_2$ point group) that have six different local site orientations related to each other through the overall cubic symmetry of the crystal \cite{dillon1961a}. Trivalent rare-earth ions substitute for Y$^{3+}$ in the lattice without charge compensation and the stoichiometric concentration of Y$^{3+}$ sites is 1.298$\times$10$^{22}$ ions$/$cm$^3$. The transition between the lowest energy components of the $^3$H$_6$ and $^3$H$_4$ multiplets occurs at a vacuum wavelength of 795.325 nm in this material, and single crystal YGG has a predicted isotropic index of refraction of $n=1.95$ at this wavelength, \cite{wemple1973a} a value that we confirm using an optical reflectometer. While offering unusually long excited-state and coherence lifetimes of 1.3 and 0.49 milliseconds respectively \cite{thiel2014b}, this transition exhibits a weaker oscillator strength in this material than found in other leading Tm$^{3+}$-doped systems \cite{sun2005a,thiel2014b}. The relatively small absorption is an apparent consequence of the transition only being weakly allowed due to small perturbations of the ideal D$_2$ point symmetry \cite{sun2005a}, likely caused by the unusually high density of anti-site defects that occur in the garnets \cite{brandle1974a,dong1991a,lupei1995a}. Spectral diffusion in Tm:YGG is expected to arise from spin flips of the gallium and yttrium nuclear spins in the host lattice. Yttrium has a single spin $1/2$ isotope $^{89}$Y with a free-nucleus gyromagnetic ratio of 2.1 MHz/T and natural abundance of 100\% \cite{lee1967a}. Gallium has two spin 3/2 isotopes $^{69}$Ga and $^{71}$Ga with free-nucleus gyromagnetic ratios of 10.2 MHz/T and 13.0 MHz/T, and natural abundance of 60\% and 40\%, respectively \cite{lee1967a}.
\section{Experimental methods}
The samples are mounted in an Oxford Optistat liquid helium cryostat. For temperatures below 2.17 K, the samples are immersed in superfluid liquid helium and the temperature is determined by measuring the vapor pressure of the liquid. For higher temperatures, the samples are cooled by a constant flow of helium exchange gas with the temperature of the gas measured using a Rh-Fe resistance sensor in the cryostat. Homogeneous magnetic fields of up to $\sim$ 500 G are applied using a water-cooled Helmholtz coil. A higher field of 6.4 kG is generated by mounting the crystal between a pair of N50-grade NdFeB block magnets immersed in the liquid helium. The magnetic field strength is determined from room temperature measurements corrected for the temperature dependence of the NdFeB magnetic field at $\sim$ 2 K. A Coherent 899-21 Ti:Sapphire ring laser is used as the light source with an estimated linewidth of $\sim$ 100 kHz and output power of typically 425 mW. The frequency is monitored using a Burleigh WA-1500 wavemeter with absolute accuracy of better than 1 GHz and a relative precision of better than 100 MHz. The maximum laser power at the sample is typically 100 mW for coherent transient measurements and $<$ 1 mW for hole burning.
A pair of acousto-optic modulators (AOMs) in series gates the laser beam to generate pulses for photon echo and hole burning measurements. Spectral holes are burned and probed by ramping the laser frequency using a third double-passed AOM driven by a voltage-controlled-oscillator. Another AOM is placed before the detector to block excitation pulses and selectively pass emitted echo signals. In this configuration, the measured on/off dynamic range of the generated pulses is $>$80 dB and the extinction of the gating AOM is typically 40 dB.
Optical transmission is detected using a New Focus 1801 amplified silicon photodiode. To detect photon echoes, a Hamamatsu R928 photomultiplier with extensive light baffles is used with a voltage of -1250 V. When measuring weaker echo signals, the photomultiplier termination is chosen to be 1-10 k$\Omega$ to maximize the voltage gain. Echo signals are digitized, the background light level subtracted, and the signal is integrated over time to measure the total power emitted in the echo. The voltage gain of the detection is automatically adjusted at each data point to maximize the sensitivity and effectively provide logarithmic amplification of the signal.
We employ pulse lengths of 200 ns for all pulses in two-pulse and three-pulse echo measurements to provide sufficient echo signal strength while maximizing the pulse bandwidth to reduce the impact of laser frequency noise. Due to persistence, efforts are made to minimize hole burning and accumulated echo distortions of the observed signals whenever a magnetic field is applied to the crystal. This included continuously scanning the laser frequency so that different regions of the inhomogeneous line are sampled on each shot, with typical continuous laser scan ranges during echo measurements of 500 MHz over 625 seconds.
Measurements are carried out on a 10 mm-thick single crystal of Tm:YGG from Scientific Materials Corp. (growth number 4-223). For all measurements, the laser propagates along a $<$110$>$-direction in the crystal, and the magnetic field and linear optical polarization are parallel to the orthogonal $<$111$>$ axis. While polishing the sample, we observed a noticeable tendency for the surfaces to chip, a potential indication of large internal strains in the crystal. To further investigate this, we examined the crystal using a polariscope and observed unusually large inhomogeneous strain-induced birefringence throughout the sample. See Fig. \ref{fig:birefringence} for images of the Tm:YGG crystal taken with unpolarized light (a) and illuminated in a linear polariscope for two different orientations of the crossed polarizers (b) and (c). While an isotropic crystal would normally appear dark, the presence of large strain-induced birefringence produces the patterns of light and dark fringes shown in Fig. \ref{fig:birefringence}. Similar birefringence was found for other samples obtained from the boule, suggesting the strain is a result of the growth process and not the details of the sample preparation. Otherwise, the sample was transparent and colorless with no other apparent defects observed optically and in the Laue x-ray diffraction pattern.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{birefringence.pdf}
\caption{
a) Picture of a 1 mm square grid pattern viewed through the 10 mm thick 1\%Tm:YGG crystal used for all measurements. b) Strain-induced birefringence observed throughout the crystal using a polariscope. c) Birefringence pattern with the crossed polarizers rotated 90 degrees relative to (b).
}
\label{fig:birefringence}
\end{center}
\end{figure}
\section{Results}
\subsection{Frequency dependence}
Previous broadband absorption measurements of the 795 nm transition of Tm:YGG found an estimated full-width at half-maximum linewidth of 55 GHz and peak absorption of 0.53 cm$^{-1}$ after deconvolving the spectrometer resolution from the measured spectrum \cite{sun2005a}. The linewidth is more than a factor of two larger than that of the closely related Tm$^{3+}$:YAG material, an indication of an increase in random nano-scale lattice strains potentially caused by a higher density of Y-Ga anti-site defects in YGG than Y-Al anti-site defects in YAG \cite{dong1991a,stanek2013a}. As an initial step in evaluating whether the increased inhomogeneous bandwidth may be fully employed and to determine if any potential underlying “parasitic” absorption is present due to highly perturbed Tm$^{3+}$ sites in the lattice, we perform spectral hole burning across the entire absorption line. In these measurements, optical pumping by the excitation laser was used to generate complete transparency at each optical frequency across the absorption line. The observed variation in maximum hole depth is recorded as a function of frequency, and is shown in Fig. \ref{fig:lineshape}. The resulting lineshape is fit well by a Lorentzian with FWHM of 56 GHz and peak absorption coefficient of 0.41 cm$^{-1}$, giving reasonable agreement with the broadband absorption measurements reported in \cite{sun2005a}. The center of the absorption line is found to be 376.943 THz, or 795.325 nm in vacuum \cite{thiel2014b}.
To further probe the underlying decoherence dynamics across the entire inhomogeneous line, we measure two-pulse photon-echo-excitation spectra \cite{sun2012a}. For these measurements, two excitation pulses with a fixed relative delay of $t_{12}$ are used to first prepare a coherent superposition of the ground and excited electronic states for the ensemble of resonant Tm$^{3+}$ ions in the crystal, and then to rephase the inhomogneous-broadening-induced dephasing and subsequently produce a coherent burst of radiation, or photon echo, after an additional delay of $t_{12}$. The integrated power of the emitted photon echo is monitored as the frequency of the laser is varied across the absorption line. The echo power $I_{echo}$ is extremely sensitive to the specific decoherence dynamics at each transition frequency across the absorption line, so that variations in the properties of the resonant ions will cause large variations in the shape of the measured excitation spectrum and reveal underlying spectral structure. Specifically, the echo power depends on the material absorption coefficient $\alpha$ and sample length $L$ according to \cite{sun2012a, thiel2012a}
\begin{equation}
I_{echo}\sim [e^{-\alpha L} \textrm{sinh}(\alpha L/2)]^2 e^{\tfrac{-4 t_{12}}{T_2}},
\label{echodep1}
\end{equation}
and for the case of small $\alpha L$, as we have for this crystal, Eq. \ref{echodep1} simplifies to
\begin{equation}
I_{echo}\sim [\alpha(\nu)]^2 e^{\tfrac{-4 t_{12}}{T_{2}(\nu)}}.
\label{echodep2}
\end{equation}
The dependence described by Eq. \eqref{echodep2} provides a powerful tool for interpreting the photon echo excitation spectra. If the inhomogeneous lineshape $\alpha(\nu)$ is already known from direct absorption measurements, this relation allows deviations from the square-root of the observed echo excitation spectrum to be interpreted in terms of variation in optical coherence lifetime $T_2 (\nu)$, identifying subgroups of ions that experience different dynamics. Alternatively, if the coherence lifetime $T_2 (\nu)$ is constant across the inhomogeneous line, i.e. $T_2 (\nu) \equiv T_2$, Eq. \eqref{echodep2} can be used to extract the absorption lineshape from the photon-echo-excitation spectrum.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{lineshapeboth.pdf}
\caption{Spectral hole burning and photon-echo-excitation spectra of the inhomogenous absorption line. The photon echo excitation spectrum (circles) closely matches the lineshape determined using spectral hole burning (squares), indicating that the entire inhomogeneous line has similar decoherence properties. Parameters extracted from fits to echo excitation spectra (solid line) and hole burning (dashed line) are shown.
}
\label{fig:lineshape}
\end{center}
\end{figure}
The measured photon-echo-excitation spectrum is shown in Fig. \ref{fig:lineshape}. We plot the square root of the echo area (i.e. $\sqrt{I_{echo}}$) to provide a direct comparison with the absorption coefficient. The spectrum is smooth and exhibits a Lorentzian lineshape centered at 376.945 THz with a FWHM of 57 GHz, matching the results obtained from spectral hole burning to within experimental accuracy. The decrease in echo excitation signal for the tails of the absorption line is likely a result of the very weak echo signals for frequencies where the absorption coefficient is small but could potentially indicate a slight increase in decoherence for the more strongly perturbed Tm$^{3+}$ ions. Nevertheless, the echo excitation results verify that the decoherence properties are uniform across more than 100 GHz of absorption bandwidth with no “bad” spectral regions due to underlying structure or unresolved defect lines.
\subsection{Magnetic field dependence}
Next, we measure two pulse photon echo decays at the center of the absorption line (795.325 nm) with and without an applied magnetic field. If spectral diffusion occurs over the timescale of the coherence lifetime due to time-dependent perturbations caused by dynamics in the ions’ environments, the progressive acceleration of decoherence can cause the observed photon echo decay shape to become non-exponential \cite{hu1978a}. A typical photon echo decay at 1.9 K with no applied magnetic field is plotted in Fig. \ref{fig:2pe}, revealing a non-exponential shape that indicates the presence of spectral diffusion in Tm:YGG. The decay is fit using the empirical Mims expression \cite{mims1968a} given by
\begin{equation}
I (t_{12}) = I_0 e^{-2(\tfrac{2 t_{12}}{T_2} )^x},
\label{mims}
\end{equation}
where $t_{12}$ is the delay between the two excitation pulses, $I_0$ is the initial integrated echo intensity at $t_{12}=0$, $T_2$ is the effective coherence lifetime related to the effective homogeneous linewidth $\Gamma_h=1/\pi T_2$ \cite{macfarlane1987a}, and $x$ is an empirical parameter that depends on the magnitude, rate, and nature of the spectral diffusion mechanism. For the zero field data, a $1/e$ coherence lifetime of 220 $\mu$s is determined by fitting Eq. \ref{mims} to the measured decay curve, as shown by the solid red line in Fig. \ref{fig:2pe}. The fit yields a relatively small exponent of $x=1.33$ indicating a modest effect of spectral diffusion on the coherence lifetime under these conditions.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{2pe.pdf}
\caption{Example two-pulse photon echo decays for different temperatures and applied magnetic field strengths. Measurements at 1.9 K in zero field (squares) and with a magnetic field of 365 G (circles) reveal an increase in effective coherence lifetime from 220 $\mu$s to 360 $\mu$s. Further increasing the field to 454 G, reducing the temperature to 1.2 K, and attenuating the laser to reduce ISD-induced broadening yields a coherence lifetime of 490 $\mu$s (triangles). Parameters extracted from each fit (solid lines) are shown.
}
\label{fig:2pe}
\end{center}
\end{figure}
For spectral diffusion arising from fluctuating local magnetic fields caused by nuclear or electron spin flips of host lattice constituents, paramagnetic impurities, or other magnetic defects we expect an applied magnetic field to affect the decoherence dynamics by reducing the magnetic entropy of the spin system. Even at very low strengths, a magnetic field can act to detune different spin-flip transitions and inhibit off-diagonal spin-spin coupling, slowing spectral diffusion from spin flip-flops in the environment and reducing the observed homogeneous linewidth. We find that spectral diffusion in Tm:YGG is significantly reduced by the magnetic field, as shown by the data in Fig. \ref{fig:2pe} for an applied field of 365 G, which results in the coherence lifetime increasing to a value of 360 $\mu$s. Increasing the field to 454 G, reducing the temperature to 1.2 K, and reducing the excitation intensity to minimize broadening due to instantaneous spectral diffusion (ISD) \cite{thiel2014c} results in an even longer coherence lifetime of 490 $\mu$s with $x=1.35$ as shown in Fig. \ref{fig:2pe} \cite{thiel2014a}. The decay exhibits a non-exponential shape, indicating that the coherence lifetime is still limited by spectral diffusion in this low-temperature case. Since spectral diffusion depends on temperature, magnetic field strength and orientation, and perhaps even crystal quality, we expect that optimization of the system could further increase the coherence lifetime.
\subsection{Temperature dependence}
Because different decoherence mechanisms each exhibit a characteristic temperature dependence, studying the variation in homogeneous linewidth with temperature can provide insight into the material dynamics that cause optical decoherence. Quantifying the relationship between coherence lifetime and temperature is important for understanding the physical processes limiting the coherence lifetimes and therefore for predicting potential performance under new conditions. For Tm:YGG, we expect elastic Raman scattering of phonons \cite{mccumber1982a} to be the dominant decoherence mechanism at low temperatures since the large crystal field splittings of 70 cm$^{-1}$ and 26 cm$^{-1}$, in the ground- and excited-state multiplets respectively \cite{sun2005a}, minimize other direct phonon interactions for temperatures below 10 K. In some materials, an additional broadening component can be present due to thermally activated low-energy dynamic structural fluctuations, often described using the theory of two-level systems (TLS) developed for amorphous solids \cite{anderson1972a,phillips1972a}; this results in a quasi-linear term in the temperature dependent linewidth \cite{flinn1994a,macfarlane2004a}. It is known that a low density of TLS may appear even in single crystals if large random lattice strains are present \cite{watson1995a,macfarlane2004a}. Because of the large inhomogeneous broadening in Tm:YGG as well as the observation of significant strain birefringence (see Fig. \ref{fig:birefringence}), we include TLS as a potential decoherence mechanism in our analysis of the observed behavior. As a result, we expect the homogeneous linewidth to be described by
\begin{equation}
\Gamma_h (T) = \Gamma_0 + \alpha_{TLS} T^{\beta} + \alpha_R T^7,
\label{temp}
\end{equation}
where $\Gamma_0$ is the linewidth at zero Kelvin. In the analysis of our measurements, we employ the common approximation of setting the TLS exponent $\beta$ equal to one since the TLS contribution to the linewidth is too weak to unambiguously determine the exponent over the measured temperature range.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{temp.pdf}
\caption{Temperature dependence of the homogeneous linewidth in zero magnetic field exhibiting effects of elastic phonon scattering above 4 K and a weak linear broadening below 4 K that may indicate the presence of decoherence due to TLS in the lattice. Parameters extracted from fits with (solid line) and without (dashed line) TLS included in the model are shown.
}
\label{fig:temp}
\end{center}
\end{figure}
We measure the variation in coherence lifetime using two-pulse photon echoes and plot the corresponding homogeneous linewidths on a log-log scale in Fig. \ref{fig:temp}. For temperatures above 4 K, a rapid increase in linewidth which is characteristic of the Raman phonon scattering process is observed. At lower temperatures, the dependence cannot be described by either Raman or any other potential direct phonon mechanisms. This much weaker dependence on temperature could indicate the presence of decoherence due to TLS disorder modes in the lattice. The fit of Eq. \eqref{temp} to the data is shown by the solid line, with good agreement over the entire measured temperature range with values of $\Gamma_0 =$ 540 Hz, $\alpha_R =$ 0.017 Hz/K$^7$, and $\alpha_{TLS}$ = 240 Hz/K. For comparison we also plot the best fit of Eq. \eqref{temp} without the TLS component ($\alpha_{TLS} =$ 0). Another mechanism that could potentially exhibit a linear temperature dependence is decoherence due to spin flips of paramagnetic chemical impurities in the crystal \cite{sabinsky1970a}; however, that mechanism would also exhibit a significant magnetic field dependence that we do not observe when comparing decoherence observed at 454 G and 6.4 kG (see Fig. \ref{fig:SDhighB}), suggesting that TLS are the more probable source of decoherence. Since the thermal broadening below 4 K is relatively weak, further measurements are required to definitively verify the presence of broadening due to TLS in this system.
\subsection{Time-dependent spectral diffusion broadening}
Finally, we study the time evolution of spectral diffusion-induced decoherence using stimulated photon echo measurements. Stimulated photon echoes are produced by creating a population grating in the inhomogeneous line using a pair of excitation pulses separated by $t_{12}$, and then scattering a third excitation pulse from the spectral grating after a time delay of $t_{23}$. Just as the temperature dependence provides insights into the material dynamics, the time dependence of spectral diffusion broadening can be used to identify different decoherence mechanisms. Spectral diffusion due to a single class of dilute point perturbers, such as paramagnetic impurities, produces an increase in linewidth with time that approaches a maximum broadening of $\Gamma_{SD}$ at characteristic rate $R_{SD}$, with the parameter values determined by the details of the mechanisms \cite{mims1968a,bottger2006a,klauder1962a,mims1972a}. In contrast, the theoretical models for spectral diffusion due to a distribution of low-energy TLS modes predict a logarithmic increase in linewidth with $t_{23}$ \cite{black1977a,breinl1984a,littau1992a,silbey1996a}. Incorporating both mechanisms into the model for the time evolution of the linewidth, the spectral diffusion broadening can be described by the relation
\begin{equation}
\Gamma_h (t_{23})= \Gamma_0 + \gamma_{TLS} \textrm{ log}(\frac{t_{23}}{t_0}) + \frac{1}{2} \Gamma_{SD} (1-e^{-R_{SD} t_{23}}),
\label{SD}
\end{equation}
where $\Gamma_0$ is the homogeneous linewidth at zero delay, $\gamma_{TLS}$ is the TLS coupling coefficient, and $t_0$ corresponds to the minimum measurement timescale ($t_0 =$ 1 $\mu$s for our experiment).
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{sd.pdf}
\caption{Time evolution of the effective homogeneous linewidth at 2.0 K and 358 G revealing the presence of spectral diffusion likely due to fluctuating $^{69}$Ga and $^{71}$Ga nuclear spins. Parameters extracted from fits with (solid) and without (dashed) additional spectral diffusion due to TLS are shown.
}
\label{fig:SD}
\end{center}
\end{figure}
We measure the variation in effective homogeneous linewidth using stimulated echoes as a function of $t_{23}$ at 2.0 K and 358 G and plot the results in Fig. \ref{fig:SD}. The dependence exhibits the characteristic shape of spectral diffusion broadening due to a single class of perturbers, such as nuclear spin flips in the host lattice \cite{klauder1962a,mims1972a}. The fit of Eq. \ref{SD} without including TLS ($\gamma_{TLS} = 0$) is shown by the dotted line, indicating 5.0 kHz of spectral diffusion acting with a rate of 180 Hz. Based on the shape and magnitude of the spectral diffusion, it seems likely that the dominant source is nuclear spin flips of neighboring gallium in the lattice. To investigate the possibility that TLS contribute to the spectral diffusion, we fit the full expression in Eq. \ref{SD} to the data, as shown by the solid line in Fig. \ref{fig:SD}. Inclusion of the TLS term only results in a slight improvement in the fit of the model to the data; as a result, more detailed studies over a wider range of parameters would be required to unambiguously determine whether TLS contribute to the spectral diffusion time-dependence over these measurement timescales.
To determine whether any further increase in magnetic field strength reduces the impact of spectral diffusion, stimulated photon echo decay measurements are performed under a higher magnetic field strength of 6.4 kG. Photon echo decays are measured with a fixed waiting time of $t_{23} =$ 30 ms at which the spectral diffusion broadening reached the maximum value in the measurements at low fields. Results from measurements at temperatures of 1.9 K and 1.2 K are plotted in Fig. \ref{fig:SDhighB}. Fitting each curve to Eq. \eqref{mims} gives $x=0$ and coherence lifetimes of $\sim$ 60 $\mu$s corresponding to effective homogeneous linewidths of several kHz. These values are comparable to the measurements at lower field, indicating that the additional order-of-magnitude increase in magnetic field strength does not have a significant effect on the spectral diffusion broadening, a result consistent with decoherence due to nuclear spin flips alone or in combination with potential TLS disorder modes in the lattice.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{sdhighb.pdf}
\caption{Stimulated photon echo decays with a delay of $t_{23}$ = 30 ms and a magnetic field of 6.4 kG revealing a small reduction in effective homogeneous linewidth as the temperature is reduced from 1.9 K (squares) to 1.2 K (circles).
}
\label{fig:SDhighB}
\end{center}
\end{figure}
\section{Projection of Additional Improvements}
Using the data presented above, we may make rough projections of improved decoherence properties that may be achievable. At lower temperatures of 0.5 K, we combine the estimated thermal broadening due to TLS of 240 Hz/K from the analysis shown in Fig. \ref{fig:temp} with the zero excitation linewidth of 580 Hz at 1.2 K determined from an ISD analysis on Tm:YGG discussed in Ref. \cite{thiel2014c}. Together, these results predict that the homogeneous linewidth could be as narrow as 410 Hz at 0.5 K for the current crystal, corresponding to a coherence lifetime of 780 $\mu$s. Furthermore, if an improved crystal were to be grown without the TLS that appear to contribute to the linewidth, then the current results suggest that the homogeneous linewidth could be as narrow as 290 Hz, corresponding to a 1.1 ms coherence lifetime. All of these projections, however, have to be considered in view of the nuclear-spin-induced spectral diffusion that is a limiting factor at timescales greater than milliseconds. Therefore, to exploit such long coherence lifetimes under current conditions, applications must operate in less than the spectral diffusion timescales. For example, to increase the storage times of quantum memories based on Tm:YGG, further spectroscopic studies in conjunction with other methods such as coherent population manipulation or optical stimulation methods should be explored.
\section{Conclusion}
Our measurements show that Tm:YGG offers both (i) uniform coherence properties across the broad 56 GHz-wide inhomogeneous line and (ii) coherence lifetimes that are significantly longer than other known Tm-doped materials. Thus Tm:YGG is a superior candidate for broadband and spectrally multiplexed photonic applications.
We measured the magnetic field and temperature dependence of coherence lifetimes with values increasing up to 490 $\mu$s with a magnetic field of 454 Gauss and a temperature of 1.2 K. Although the spectral-diffusion-induced decoherence does not appear to be reduced by further increases in magnetic field strength, employing lower temperature or different crystal orientations could result in further reduction. It is instructive to compare the present results for Tm:YGG to the measured spectral diffusion in 0.1\%Tm:YAG for similar temperatures and magnetic field strengths. In Tm:YAG, the observed spectral diffusion is caused by $^{27}$Al nuclear spin flips is described by values of $R_{SD}$ = 140 Hz and $\Gamma_{SD}$ = 41 kHz \cite{thiel2013a}. For Tm:YGG, we find that the spectral diffusion has a similar rate but with nearly an order of magnitude weaker broadening, a larger difference than would be expected by only considering the relative magnitude of the aluminum and gallium nuclear magnetic moments. Moreover, there are indications that weak dynamic disorder modes in the crystal may be responsible for some of the decoherence observed at the lowest temperatures. A comparison of material properties from different growths or garnet compositions may suggest improved crystal growth or treatment strategies to reduce or eliminate the effects of dynamic disorder modes.
\section{Acknowledgements}
The authors acknowledge support from Alberta Innovates Technology Futures (AITF), the National Engineering and Research Council of Canada (NSERC), the Defense Advanced Research Projects Agency (DARPA) Quiness program (contract no. W31P4Q-13-l-0004), and the National Science Foundation (NSF) award nos. PHY-1212462 and PHY-1415628. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA or NSF. W.T. is a senior fellow of the Canadian Institute for Advanced Research (CIFAR).
|
1908.08789
|
\section{Introduction}
As three-dimensional analogues of graphene, Dirac and Weyl semimetals have attracted considerable attention in the last years \cite{fisher2019topological,PhysRevLett.108.140405,liu2014discovery,lv2015experimental,xu2015discovery}. Both Dirac and Weyl materials are characterized by linearly dispersing valence and conduction bands that cross at discrete point in momentum space, giving rise to low-energy excitations behaving like Dirac or Weyl fermions.
Recently, a novel class of topological materials, nodal-line materials, has been predicted \cite{burkov2011weyl,burkov2011topological}. In comparison to Dirac and Weyl semimetals, band crossing in nodal-line semimetals occurs along continuous lines. Since 2011, several materials were proposed to be nodal-line semimetals \cite{yu2015topological,chen2015topological,okamoto2016low}, and some of them have been confirmed experimentally using such techniques as angle-resolved photoemission spectroscopy (ARPES) \cite{bian2016topological,schoop2016dirac,topp2016non}, magnetotransport \cite{emmanouilidou2017magnetotransport,matusiak2017thermoelectric,hu2016evidence}, and optical \cite{Shao1168} measurements.
The family of ternary compounds ZrSi$X$ ($X$=S, Se, Te) is a typical example of nodal-line semimetal with well separated Dirac cones \cite{PhysRevB.95.161101,schoop2016dirac}. The presence of topologically nontrivial linear bands in ZrSi$X$ has been observed experimentally by several methods, including ARPES \cite{schoop2016dirac,topp2017surface,chen2017dirac,fu2017observation}, scanning probe techniques \cite{Butler2017Quasi,lodge2017observation}, as well as thermoelectric \cite{matusiak2017thermoelectric} and magnetotransport \cite{pezzini2018unconventional,hu2017nearly,wang2016evidence,singha2017large,ali2016butterfly,pan2018three} measurements of quantum oscillations. Among ZrSi$X$s, ZrSiS is especially prospective material for optoelectronic applications due to its high carrier mobility \cite{matusiak2017thermoelectric,sankar2017crystal}, thermal stability \cite{lam1997new}, and non-toxic nature \cite{neupane2016observation}. A significant attention has been paid to ZrSiS due to its unusual properties observed in experiment.
Particularly, ARPES experiments reveal that ZrSiS hosts two kinds of nodal lines. While in the first kind the degeneracy of Dirac points is protected by non-symmorphic symmetry, in the second kind the degeneracy is lifted by the spin-orbit coupling, inducing a small gap of the order of 10 meV \cite{schoop2016dirac}.
The upper limit of this gap ($\sim$ 30~meV) is observed by recent low-frequency optical measurements \cite{schilling2017flat}. Compared to other known 3D Dirac materials, the energy range of the linearly dispersing bands in ZrSiS reaches 2 eV, making this material a promising candidate for studying Dirac fermions. Apart from Dirac physics, extremely strong Zeeman splitting with a large $g$-factor has been observed by measuring de Haas-van Alphen (dHvA) oscillations \cite{hu2017nearly}.
There is also evidence of an important role of the correlation effects in ZrSiS and related materials.
The unusual mass enhancement of charge carriers in ZrSiS has been recently observed experimentally at low-temperatures \cite{pezzini2018unconventional}, which can be understood in terms of unconventional electron-hole pairing \cite{rudenko2018excitonic,scherer2018}. Last but not least, recent high-pressure electrical transport measurements pointed to the possibility of a topological phase transition in ZrSiS below 0.5 GPa \cite{vangennep2019possible}.
In comparison to conventional metals, Dirac semimetals have raised intense interest both from fundamental and applied perspectives due to their intriguing optical properties \cite{fisher2019topological,kuzmenko2008universal,wang2019anomalous}.
Recently, optical spectra of ZrSiS were measured in a large frequency range, from the near-infrared to the visible \cite{schilling2017flat}. It was found that the absorption spectrum remain almost unchanged for photon energies in the range from 30~meV to 350~meV \cite{schilling2017flat}.
As has been pointed out by B\'acsi and Virosztek \cite{PhysRevB.87.125425}, in a noninteracting electron system with two symmetric
energy bands touching each other at the Fermi level, the real part of the interband optical conductivity $\sigma_{1}(\omega)$ demonstrates a power-law frequency dependence with $\sigma_{1} \propto (\frac{\hbar\omega}{2})^{(d-2)/z}$, where $d$ and $z$ are the dimension of the system and the power law of the band dispersion, respectively \cite{PhysRevB.87.125425}.
The flat optical conductivity is typical for graphene ($d$=2 and $z$=1), being a universal constant for Dirac electrons in two dimensions \cite{kuzmenko2008universal,mak2008measurement}.
In three dimensions, this behavior is not universal. Linear dependence is reported in point-node Dirac or Weyl semimetals as ZrT$_{5}$ \cite{PhysRevB.92.075107}, TaAs \cite{PhysRevB.93.121110}, and Cd$_{3}$As$_{2}$ \cite{PhysRevB.93.121202} ($d$=3 and $z$=1).
The flatness of the optical conductivity in ZrSiS is determined by an appropriate combination of intraband and interband transitions \cite{habe2018dynamical}.
Followed by the flat region, the optical conductivity in ZrSiS exhibits a characteristic U-shape ending at a sharp peak around 1.3 eV \cite{schilling2017flat,ebad2019chemical}. Interestingly, the optical response is strongly anisotropic with the 1.3 eV peak appearing in the in-plane [100] direction only \cite{habe2018dynamical}. Besides, essentially anisotropic magnetoresistance in ZrSiS has been measured experimentally \cite{lv2016extremely,wang2016evidence}. Recent findings on the family of compounds ZrSi$X$ ($X$=S, Se, Te) and ZrGe$X$ ($X$=S, Te) suggest that their optical properties are closely connected to the interlayer bonding, and can be tuned by external pressure \cite{ebad2019chemical}.
Unlike infrared and visible spectral regions, ultraviolet optical response of ZrSiS has not been studied yet.
Besides that, previous works focus on the optical properties of pristine ZrSiS, while effect of strain, has not been addressed in detail.
The ultraviolet region is especially appealing for plasmonic applications, for which ZrSiS appears promising due to its high carrier mobility, closely related to the sustainability of plasmonic modes.
Short propagation length (lifetime) of plasmons in typical plasmonic materials (e.g., noble metals) represents a bottleneck for applications \cite{politano2015influence}.
At the same time, the application domain of ultraviolet plasmonics is highly diverse. It includes biochemical sensing applications \cite{mcmahon2013plasmonics,taguchi2012tailoring}, photodetection \cite{dhanabalan2016present}, nano-imaging \cite{zhang2015ultraviolet}, material characterization \cite{nakashima2004deep}, and absorption of radiation \cite{hedayati2014plasmonic}.
In this paper, we study broadband optical properties of ZrSiS crystals with a special emphasis on the effect of external strain. To this end, we use first-principles calculations in combination with the random phase approximation for the dielectric screening. We find that although the low-energy optical conductivity remains frequency-independent under uniaxial loading of up to 10 GPa, the corresponding spectral region is narrowing with increasing stress.
In the presence of tension, we observe an electronic Lifshitz transition at around 2 GPa. This transition results in a suppressed intraband screening, which reduces the spectral weight in the infrared region.
Apart from the flat optical conductivity at low energies, our calculations show that ZrSiS is characterized by high-energy plasma excitations with frequencies around 20 eV.
Given that the optical response in ZrSiS is highly anisotropic, it permits the existence of low-loss hyperbolic plasmons in the ultraviolet spectral range.
The paper is organized as follows. In Sec.~II, we describe our computational method and calculation details. Optical properties of pristine ZrSiS are presented in Sec.~III, where we specifically focus on the low- and high-energy spectral regions. In Sec.~IV, we study the effect of external strain on the optical conductivity and plasma excitations in ZrSiS. In Sec.~V, we summarize our findings.
\section{Calculation details}
\subsection{Electronic structure}
ZrSiS is a layered crystal with a tetragonal structure and space group P4/$nmm$ (No.129). Its structure is formed by Zr-S layers sandwiched between Si layers, and periodically repeated in the direction normal to the layers, as shown in Figure~\ref{fig:label1}(a). The equilibrium lattice constants obtained from full structural optimization at the DFT level are $a=3.56$~\AA~(in-plane) and $c=8.17$ \AA ~(out-of-plane).
The DFT electronic structure calculations are performed within the pseudopotential plane-wave method as implemented in {\sc quantum espresso} \cite{giannozzi2017advanced} simulation package. We use generalized gradient approximation (GGA) \cite{perdew1996generalized} in combination with norm-conserving pseudopotentials \cite{hamann1979norm}, in which $4s$ and $4d$ electrons of Zr, $3s$ and $3p$ electrons of Si, as well as $3s$ and $3p$ electrons of S were treated as valent. The reciprocal space was sampled by a uniform ($24\times24\times8$) {\bf k}-point mesh. In the calculations, we set the energy cutoff for the plane-wave basis to 80 Ry, and a self-consistency threshold for the total energy to $10^{-12}$ Ry. The atomic structure and lattice parameters were optimized until the residual forces on each atom were less than $10^{-5}$ Ry/Bohr. The effect of spin-orbit coupling is not taken into account in our study as it is only relevant for low temperatures ($<100$ K) and in the low-frequency region ($< 20$~meV) \cite{schilling2017flat}.
All crystal graphics was generated by means of {\sc xcrysden} visualization package \cite{kokalj2003computer}.
\subsection{Dielectric function}
Dielectric function $\epsilon(\mathbf{q},\omega)$ was calculated within the random phase approximation (RPA) using {\sc yambo} \cite{marini2009yambo} package.
Its standard form as function of wave vector $\mathbf{q}$ and frequency of incident photon $\omega$ reads:
\begin{equation}
\epsilon(\mathbf{q},\omega)=1-v(\mathbf{q})\chi^{0}(\mathbf{q},\omega),
\label{dielec}
\end{equation}
where $v(\mathbf{q})=\frac{4\pi e^2}{|\mathbf{q}|^{2}}$ is the bare Coulomb potential,
$\chi^{0}$ is the irreducible response function evaluated within the independent particle approximation \cite{marini2009yambo}:
\begin{multline}
\chi^{0}(\mathbf{q},\omega) = \frac{2}{V} \sum_{{\bf k},nm} \rho^{*}_{nm\mathbf{k}}(\mathbf{q})\rho_{nm\mathbf{k}}(\mathbf{q})
\\
\times \left[\frac{f_{n\mathbf{k}-\mathbf{q}}(1-f_{m\mathbf{k}})}{\omega+\varepsilon_{n\mathbf{k}-\mathbf{q}}-\varepsilon_{m\mathbf{k}}+i\eta}-\frac{f_{n\mathbf{k}-\mathbf{q}}(1-f_{m\mathbf{k}})}{\omega+\varepsilon_{m\mathbf{k}}-\varepsilon_{n\mathbf{k}-\mathbf{q}}-i\eta}\right],
\label{inter}
\end{multline}
where
\begin{equation}
\rho_{nm\mathbf{k}}(\mathbf{q})=\langle n\mathbf{k}|e^{i\mathbf{q}\cdot\mathbf{r} } |m\mathbf{k}-\mathbf{q} \rangle
\label{rho}
\end{equation}
is the dipole transition matrix element,
$f_{n\mathbf{k}}$ is the Fermi occupation factor, for which $T=300$ K was used in all calculations, $|n\mathbf{k}\rangle$ is the Bloch eigenstate corresponding to the band $n$ and wave vector $\mathbf{k}$, and $V$ is the cell volume.
To avoid computationally demanding calculations, we assume the scalar form of $\epsilon({\bf q}, \omega)$ and $\chi^0({\bf q},\omega)$, meaning that only ${\bf G}=0$ and ${\bf G}'=0$ elements of the full matrices are calculated. Physically, this approximation corresponds to the situation, in which the local field effects are neglected, i.e. $\epsilon({\bf r}_1,{\bf r}_2) \simeq \epsilon(|{\bf r}_1-{\bf r}_2|)$. This approximation is well justified for 3D systems with weak inhomogeneities of the charge density \cite{onida2002electronic}.
In Eq.~(\ref{inter}), $\eta$ is the damping parameter playing the role of the electron linewidth, which can be attributed to the imaginary part of the self-energy, $\eta
\sim \mathrm{Im}[\Sigma(\omega,\mathbf{k})]$ \cite{marder2010condensed}. Here, we do not detail the scattering mechanism and consider $\eta$ as a free parameter.
\begin{figure*}[tbp]
\includegraphics[width=18cm]{model_bandpdos_fs.jpg}
\caption{(a) Schematic representation of the ZrSiS crystal structure; (b) Calculated band structure and orbital-resolved density of states in the vicinity of the Fermi energy; (c) Three-dimensional view of the Fermi surface with purple and cyan colors denoting valence and conduction states, respectively. Black lines mark the Brillouin zone boundaries. Dashes blue lines connect the high-symmetry points used in (b). }
\label{fig:label1}
\end{figure*}
To reproduce the quantities measured in optical experiments, one needs to evaluate the long-wavelength limit of the dielectric function,
\begin{equation}
\epsilon_{\alpha \alpha}(\omega)\equiv \lim_{\mathbf{q} \to 0} \epsilon(\mathbf{q},\omega),
\end{equation}
where $\alpha$ is the direction of the incident light, and the limit is taken with ${\bf q}$ parallel to $\alpha$.
Taking this limit numerically is a computationally nontrivial task as it requires high density of {\bf q}-point to be included in the calculations. This can be avoided by expanding the dipole transition matrix elements at $\mathbf{q} \to 0$ using $e^{i\mathbf{q}\cdot\mathbf{r} } \approx 1 +i \mathbf{q}\cdot\mathbf{r}$.
To this end, the matrix elements $\mathbf{r}_{nm\mathbf{k}}=\langle n \mathbf{k}|\mathbf{r}|m\mathbf{k} \rangle$ needs to be computed. Within the periodic boundary conditions using the relation $[\mathbf{r},H]=\mathbf{p}+[\mathbf{r},V_{nl}]$ one arrives at \cite{sangalli2019many}
\begin{equation}
\langle n \mathbf{k}|\mathbf{r}|m\mathbf{k} \rangle = \frac{\langle n \mathbf{k}|\mathbf{p}+[\mathbf{r},V_{nl}]|m\mathbf{k} \rangle}{\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k}}}
\end{equation}
where $V_{nl}$ is the nonlocal part of the pseudopotential.
At ${\bf q}\to 0$, Eq.~(\ref{inter}) does not explicitly takes intraband transitions into account. Since ZrSiS is a semimetal, the intraband transition provide an important contribution to the dielectric response at low energies. To account for this contribution, we calculate the Drude corrections to the dielectric function $\epsilon^{\mathrm{intra}}_{\alpha \alpha}(\omega)=\epsilon^{\mathrm{intra}}_{1,\alpha \alpha}(\omega)+i\epsilon^{\mathrm{intra}}_{2,\alpha \alpha}(\omega)$, which are evaluated from the standard free-electron plasma model \cite{dressel2002electrodynamics}:
\begin{equation}
\begin{aligned}
\epsilon^{\mathrm{intra}}_{1,\alpha \alpha}(\omega)=1-\frac{\omega_{p,\alpha \alpha}^{2}}{\omega^{2}+\delta^{2}}, \\
\epsilon^{\mathrm{intra}}_{2,\alpha \alpha}(\omega)=\frac{\delta\omega_{p,\alpha \alpha}^{2}}{\omega^{3}+\omega\delta^{2}}.
\label{Drude_eps}
\end{aligned}
\end{equation}
Here, $\delta$ has similar physical meaning as $\eta$ in Eq.~(\ref{inter}), and $\omega_{p,\alpha \alpha}$ is the $\alpha$-component of the (unscreened) plasma frequency given by \cite{lee1994first,harl2007ab}:
\begin{equation}
\omega^{2}_{p,\alpha\beta}=-\frac{4\pi e^{2}}{V}\sum_{n,\mathbf{k}}\frac{\partial f_{n\mathbf{k}}}{\partial \varepsilon_{n{\bf k}}} v^{\alpha}_{n{\bf k}} v^{\beta}_{n{\bf k}}
\label{plasma}
\end{equation}
where
$v^{\alpha}_{n{\bf k}}=\hbar^{-1}\partial \varepsilon_{n{\bf k}}/\partial k_{\alpha}$
is the $\alpha$-component of the group velocity of the electrons with wave vector ${\bf k}$ at band $n$. In this work, the plasma frequency is calculated using the {\sc simple} code \cite{prandini2019simple}.
The intraband contribution to the optical conductivity can be calculated accordingly, using the well-known expressions \cite{marder2010condensed}:
\begin{equation}
\begin{aligned}
\sigma^{\mathrm{intra}}_{1,\alpha\alpha}(\omega)=\frac{\omega \epsilon^{\mathrm{intra}}_{2,\alpha\alpha}(\omega)}{4\pi} \\
\sigma^{\mathrm{intra}}_{2,\alpha\alpha}(\omega)=1 - \frac{\omega \epsilon^{\mathrm{intra}}_{1,\alpha\alpha}(\omega)}{4\pi}
\end{aligned}
\end{equation}
\section{Optical properties of pristine \texorpdfstring{Z\MakeLowercase{r}S\MakeLowercase{i}S}{ZrSiS}}
\subsection{Low-energy region}
We first calculate the electronic structure of ZrSiS for its equilibrium crystal structure.
In Figure~\ref{fig:label1}, we show the band structure, density of states projected on $s$-, $p$-, and $d$-orbitals (PDOS), and the corresponding Fermi surface.
The most prominent feature of the band structure is a series of linearly dispersing bands with the Dirac-like crossings in the vicinity of the Fermi energy ($\varepsilon_F$). The linear bands extend over a rather large energy range of up to 2 eV. From Figure~\ref{fig:label1}(b), one can see that DOS exhibits a minimum at $\varepsilon_F$, as expected near the band crossing points. In the range from $-1$ to 0 eV, the valence states are entirely formed by linearly dispersed bands, while the states above $\varepsilon_F$ are mixed with quadratic bands, giving rise to a larger DOS for the conduction band.
As can be seen from PDOS, $d$-orbitals have dominant contribution to the states near $\varepsilon_F$. At $\varepsilon \lesssim1$ eV there is a comparable contribution from $p$-orbitals.
In Figure~\ref{fig:label1}(c), we show the corresponding Fermi surface. It is composed of two distinct parts, corresponding to electron (cyan) and hole (purple) states. Each part is formed by four disconnected pockets. As we will see below, the Fermi surface topology plays an important role in the optical properties of strained ZrSiS.
\begin{figure}[btp]
\includegraphics[width=8.5cm]{cond.jpg}
\caption{(a) Real part of the optical conductivity shown as a function of the photon energy with incidence along in-plane [100] and out-of-plane [001] crystallographic directions; (b) Real part of the in-plane optical conductivity calculated for different damping parameters $\eta$; (c) Imaginary part of the optical conductivity calculated along [100] and [001] directions; (d) Real part of the in-plane dielectric function calculated for different $\eta$.}
\label{fig:label2}
\end{figure}
\begin{figure*}[tbp]
\includegraphics[width=13cm]{pristine_dielectric.jpg}
\caption{Imaginary (upper panels) and real (lower panels) parts of the dielectric function of pristine ZrSiS calculated as a function of the photon energy $\omega$ for a series of wave vectors $\mathbf{q}$ along the in-plane (left panels) and out-of-plane (right panels) directions. Inset shows a zoom-in of the high-energy region where $\epsilon_1({\bf q},\omega)=0$. }
\label{fig:label3}
\end{figure*}
After the ground state electronic structure is obtained, we calculate the dielectric functions, and the corresponding optical conductivities. We start from the ${\bf q}\to 0$ limit and first calculate the unscreened plasma frequencies using Eq.~(\ref{plasma}). We arrive at $\omega_{p,xx}=3.15$ eV and $\omega_{p,zz}=1.08$ eV for the in-plane [100] and out-of plane [001] components, respectively. The value obtained for the [100] directions is in good agreement with the experimental estimate of 2.88 eV \cite{schilling2017flat}.
In Figures~\ref{fig:label2}(a) and \ref{fig:label2}(c), we show the real and imaginary parts of the optical conductivity calculated in the region up to 2 eV for [100] and [001] directions of photon propagation. The spectral weight obtained for the in-plane direction is significantly larger compared to the out-of-plane direction. This indicates a strong anisotropy between the optical response in ZrSiS.
In order to assess sensitivity of the optical conductivity to the effects induced by finite electron linewidth, in Figure~\ref{fig:label2}(b) we show the real part of the low-energy optical conductivity calculated for different parameters $\eta$ at the range from 20 to 60 meV.
From Figure~\ref{fig:label2}(b), one can clearly see the prominent flat conductivity from 0.1 to 0.4 eV. The flat conductivity $\sigma_{\mathrm{flat}}$ is estimated to be $\sim$7000 $\Omega^{-1}$cm$^{-1}$, which is good agreement with the experimental result of 6600 $\Omega^{-1}$cm$^{-1}$ \cite{schilling2017flat}. The flatness is well reproduced for $\eta=30$--$40$ meV, while larger values result in a noticeable smearing of the flat region. For $\eta \lesssim 20$ meV, one can see the emergence of an oscillatory behavior. This behavior is of the numerical origin, and can be associated with insufficient sampling of the Brillouin zone. In what follows, we set $\eta=40$~meV in all low-energy ($0$--$2$ eV) conductivity calculations.
This value is in agreement with the electron linewidth experimentally estimated in ZrSiS as $\sim$30 meV at 300 K \cite{schilling2017flat}. Following the flat region, there appears a U-shaped optical conductivity around $1.3$~eV \cite{ebad2019chemical}. The peak above the U-shaped region at $\sim$1.3~eV is only found for the in-plane direction, while it is absent in the out-of-plane direction. This peak mainly originates from the excitation between the linearly dispersing bands near $\varepsilon_{F}$ and from the transitions between quadratic bands in the direction from Z to R.
In Figure~\ref{fig:label2}(d), we show the real part of the calculated in-plane dielectric function. The condition $\epsilon_1(\omega^{\mathrm{scr}}_p)=0$ allows us to estimate the screened plasma frequency, which is found to be $\omega^{\mathrm{scr}}_{p}$ $\sim 1$ eV. Having determined $\omega^{\mathrm{scr}}_{p}$, we can estimate the effective screening induced by the interband transitions \cite{dressel2002electrodynamics}. The corresponding dielectric constant $\epsilon_{\infty}=(\omega_{p}/\omega^{scr}_{p})^{2}\approx 9$, which is consistent with the experimental value of $\sim7.8$ \cite{schilling2017flat}.
To understand the effect of finite electron linewidth on $\omega^{scr}_{p}$, we also plot $\epsilon_{1}(\omega)$ for different parameters $\eta$ in Figure~\ref{fig:label2}(d).
Compared to the flatness of the optical conductivity, the screened plasma frequency is almost insensitive to $\eta$.
\subsection{High-energy region}
We now turn to the optical response in the high energy region, $\omega>2$ eV. Here, we focus at the plasmonic excitations and consider momentum-resolved dielectric function $\epsilon({\bf q},\omega)$, which is shown in Figure~\ref{fig:label3} as a function of the photon energy for a series of small wave vectors ${\bf q}$ in both in-plane and out-of-plane directions. At $\omega \gtrsim$ 10 eV, $\epsilon({\bf q},\omega)$ is monotonic at small ${\bf q}$, with $\epsilon({\bf q},\omega) \rightarrow 1$ as $\omega \rightarrow \infty$, which is expected from the Drude model [Eq.~(\ref{Drude_eps})]. The most interesting energy region is determined by the condition $\epsilon_1(\mathbf{q},\omega)=0$, which defines the existence of plasma excitations. From Figure~\ref{fig:label3}, one can see that this criterion is fulfilled for two different energy regions: $\omega_p \sim 5$--7 and $\omega_p \sim 19$--20 eV.
To gain more insights in the plasmonic response, we calculate the energy loss function
\begin{equation}
L(\mathbf{q},\omega)=-\mathrm{Im}\left[\frac{1}{\epsilon(\mathbf{q},\omega)}\right],
\end{equation}
which can be associated with the Electron Energy Loss Spectroscopy (EELS) spectra.
Figure~\ref{fig:label4} shows $L({\bf q},\omega)$ calculated along the in-plane and out-of-plane directions of ZrSiS. In both cases, one can see a sharp peak around 20 eV, while there is no indication of the energy loss at lower energies. This means that the plasma oscillations around 5--7 eV are strongly damped. This can be understood from Figure~\ref{fig:label3}, where $\epsilon_2({\bf q},\omega)$ exhibits a peak around $\omega \sim 5$ eV, indicating strong absorption in this region. On the other hand, $\epsilon_2({\bf q},\omega)$ is almost zero around $\omega \sim $ 20 eV, indicating that high-energy plasmons are characterized by low losses, and could be observed experimentally. Recently, similar behavior has been experimentally observed in bulk black phosphorus crystal in the same frequency region \cite{nicotra2018anisotropic}.
The dispersion of bulk plasmons can be fitted with a second-order polynomial:
\begin{equation}
E(\mathbf{q}) = E(0) + A \, \mathbf{q}^{2},
\label{dispersion}
\end{equation}
where $E({0})$ is the plasmon energy at $\mathbf{q} \to {0}$ and $A$ is the dispersion coefficient. From Figure~\ref{fig:label4}, it can be seen that the calculated dispersion can indeed be fitted with Eq.~(\ref{dispersion}). Interestingly, although the plasma frequency is nearly independent of the direction of light propagation, the dispersion of high-energy plasmon modes is strongly anisotropic.
The existence of high-energy plasmons in ZrSiS might be beneficial in the context of ultraviolet optical devices \cite{sang2013comprehensive}. At the same time, strongly anisotropic dispersion of plasmon modes may give rise to unconventional plasma excitations, known as hyperbolic plasmons \cite{shekhar2014hyperbolic}.
\begin{figure*}[ht]
\includegraphics[width=13cm]{pristine_eels.jpg}
\caption{Upper panels: Electron energy loss spectrum $L({\bf q},\omega)$ as a function of the photon energy $\omega$ and momentum $\mathbf{q}$ calculated for the in-plane (left) and out-of-plane (right) directions. Lower panels: Dispersion of the high-energy plasmon $\omega_p({\bf q})$ calculated along the in-plane (left) and out-of-plane (right) directions.}
\label{fig:label4}
\end{figure*}
\begin{figure}[ht]
\includegraphics[width=8.5cm]{hyperbolic_regime.jpg}
\caption{(a) Product of the in-plane and out-plane real dielectric functions shown as a function of energy; (b) Reciprocal-space representation of the constant-energy surfaces of two possible hyperbolic plasmon modes in ZrSiS, denoted as regime I (left) and II (right). The color shows the magnitude of $k_z$.}
\label{fig:hyper}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=9.0cm]{strain_stress.jpg}
\caption{Black curve: The $zz$-component of the stress tensor ($\sigma_{zz}$) as a function of the uniaxial strain $u_{zz}$ in ZrSiS. Orange curve: In-plane strain $u_{xx}$ versus out-of-plane strain $u_{zz}$. $\nu=-\mathrm{d}u_{xx}/\mathrm{d}u_{zz}$ is the corresponding Poisson ratio estimated by linear regression.}
\label{fig:stress}
\end{figure}
\begin{figure*}[ht]
\includegraphics[width=17cm]{strain_band_fs_nodal.jpg}
\caption{(a) Band structures calculated in the vicinity of the Fermi energy for different values of the uniaxial strain $u_{zz}$ in ZrSiS. Positive and negative values correspond to compression and tension, respectively. The related stress is given in parentheses;
(b) The Fermi surfaces, and (c) corresponding nodal-line structure shown for the case of tensile strain, at which electronic Lifshitz transition is taking place. Circles and arrows highlight the location where the transition occurs.}
\label{fig:label5}
\end{figure*}
Hyperbolic plasmons appear in crystals with strong anisotropy, in which effective permittivity changes sign with respect to the electric field direction \cite{gomez2015hyperbolic}.
The dispersion relation of light propagating in homogeneous layered material is determined by the relation:
\begin{equation}
\frac{(k^{2}_{x}+k^{2}_{y})}{\epsilon_{zz}(\omega)}+\frac{k_{z}^{2}}{\epsilon_{xx}(\omega)}=\frac{\omega^{2}}{c^{2}},
\end{equation}
where $\epsilon_{xx}$ and $\epsilon_{zz}$ are the frequency-dependent permittivities along the in-plane and out-of-plane directions, respectively.
For frequencies at which $\epsilon_{xx}(\omega)\cdot\epsilon_{zz}(\omega)<0$, the equation above describes a hyperboloid. This situation is considerably different from the closed spherical or elliptic dispersion typical for conventional materials with $\epsilon_{xx}(\omega)\cdot\epsilon_{zz}(\omega)>0$ \cite{gjerding2017layered,guo2012applications}. Depending on the form of the isofrequency surface, one can distinguish between the two types of hyperbolic materials: Type I if the hyperboloid is two-sheeted ($\epsilon_{zz}<0, \epsilon_{xx}>0$), and type II if the hyperboloid is single-sheeted ($\epsilon_{zz}>0, \epsilon_{xx}<0$).
In Figure~\ref{fig:hyper}(a), we show the corresponding permittivities calculated in ZrSiS as a function of the photon energy. One can see that the condition $\epsilon_{xx}(\omega)\cdot\epsilon_{zz}(\omega)<0$ is fulfilled in a narrow energy region around $\sim$5 eV and $\sim$20 eV, which are the frequencies at which the conventional bulk plasmon modes are found. In both cases, the hyperbolic plasmons may appear in a frequency range of about 0.6 eV.
Both hyperbolic modes demonstrate the dispersion relation of type I, corresponding to a two-sheeted hyperboloid, shown in Figure~\ref{fig:hyper}(b).
Simliar to other natural hyperbolic materials, hyperbolic regimes in ZrSiS appear only above the onset of intraband transition \cite{gjerding2017layered}. Since electronmagnetic waves propagating in hyprobolic materials follow the hyperbolic dispersion, hyprobolic media supports propagation of high-$\mathbf{k}$ waves that are evanescent in conventional media \cite{gomez2015hyperbolic}. Due to the properties of high-$\mathbf{k}$ waves, hyperbolic material have many potential applications, including negative refraction \cite{yao2008optical, hoffman2007negative}, sub-wavelength modes \cite{kapitanova2014photonic} and thermal emission engineering \cite{biehs2012hyperbolic}. We note, however, since the $\sim$5 eV mode is strongly damped, its practical significance is questionable.
\section{Optical properties of uniaxial strained \texorpdfstring{Z\MakeLowercase{r}S\MakeLowercase{i}S}{ZrSiS}}
Earlier studies on the family of compounds ZrSi$X$ ($X$=S, Se, Te) suggest that their physical properties are closely connected with the interlayer bonding. Moreover, the ratio of the out-of-plane and in-plane lattice constants $c/a$ can be considered as a measure for the interlayer bonding strength in these systems \cite{ebad2019chemical,ebad2019infrared,topp2016non}. In this regard, uniaxial strain applied in the out-of-plane direction is a promising way to tune the materials' properties.
Inspired by recent experimental works, which indicate the possibility of a topological phase transition in nodal-line semimetals under external pressure \cite{vangennep2019possible,ebad2019infrared}, here we study how the uniaxial strain would affect the optical properties of ZrSiS.
Before discussing the effect of strain on the electronic structure, we briefly focus on the mechanical properties of ZrSiS. We apply uniaxial strain in the direction perpendicular to the ZrSiS layers by varying the out-of-plane lattice constant $c$, and relaxing the in-plane lattice constant $a$.
The stress is defined as $\sigma_{ij}=\frac{1}{\Omega}\frac{\partial F}{\partial u_{ij}}$, where $\Omega$ is the volume of unit cell, and $u_{ij}$ is the strain tensor. In our case, we focus on uniaxial strain assuming in-plane relaxation ($\sigma_{xx}=\sigma_{yy}=0$) and the absence of shear strain, i.e. $u_{xy}=u_{xz}=u_{yz}=0$. $F$ is the free energy of the crystal, which in the case of tetragonal symmetry (point group $D_{4h}$) is given by \cite{landau1989course}
\begin{multline}
F=\frac{1}{2}\lambda_{xxxx}(u^{2}_{xx}+u^{2}_{yy})+\frac{1}{2}\lambda_{zzzz}u^{2}_{zz}+ \\
\lambda_{xxzz}(u_{xx}u_{zz}+u_{yy}u_{zz})+\lambda_{xxyy}u_{xx}u_{yy},
\end{multline}
where $\lambda$ is the tensor of elastic moduli.
The calculated stress-strain curves are shown in Figure~\ref{fig:stress}. In case of uniaxial compressive strain along the out-of-plane direction ($u_{zz}$), the $\sigma_{zz}$ vs. $u_{zz}$ curve is nearly linear, indicating typical elastic regime and applicability of the Hooke's law. On the other hand, as can be seen from Figure~\ref{fig:stress}, the tensile strain is highly nonlinear already at $2\%$ tension. The observed nonlinearity of the elastic properties indicates a considerable modification of the electronic structure upon tensile strain. In Figure~\ref{fig:stress}, we also show the dependence of the in-plane strain $u_{xx}$ with respect to $u_{zz}$. For $u_{zz}$ in the range from $-$5\% to $+$5\%, we obtain a perfect linear dependence, which allows us to estimate the Poisson's ratio. We obtain $\nu=-\mathrm{d}u_{xx}/\mathrm{d}u_{zz}=0.24$, which is in agreement with the results of previous studies \cite{2017Possion}.
\begin{figure}[tbp]
\includegraphics[width=8.5cm]{strain_constant.jpg}
\caption{Strain-dependent low-energy plasma frequency $\omega_{p}$ (black) estimated using Eq.~(\ref{plasma}) for in-plane and out-of-plane directions, and intraband screening constant $\epsilon_{\infty}=(\omega_{p,xx}/\omega_{p,xx}^{scr})^2$ (red) calculated for the in-plane direction shown as a function of the uniaxial strain. Positive and negative values correspond to compression and tension, respectively.
$P_{1}=-1.3$ GPa and $P_{2}=-3.4$ GPa are the critical stress values around which the electronic Lifshitz transition takes place. }
\label{fig:strain_plasma}
\end{figure}
\begin{figure}[tbp]
\includegraphics[width=8.7cm]{strain_cond.jpg}
\caption{Real part of the optical conductivity $\sigma_{1}$ (upper panels) and dielectric function $\epsilon_{1}$ (lower panels) calculated for the in-plane direction under compressive (a) and tensile (b) strain as a function of the photon energy $\omega$. The inset figures use logarithmic scale. }
\label{fig:strain_cond}
\end{figure}
Let us now discuss the strain-dependent electronic properties of ZrSiS. In Figure~\ref{fig:label5}(a), we show the band structures for the case of compressive and tensile uniaxial strain $u_{zz}$ of $1\%$, $3\%$, and $5\%$.
One can see that the linear dispersion of states near the Fermi energy is unaffected by the uniaxial strain in the range from $-5\%$ to $+5\%$. The position of the Dirac points near the Fermi energy changes slightly, which is not expected to have any noticeable effects on the optical transitions at low energies.
On the other hand, the position of the nonsymmorphic Dirac node at the X and R points is more susceptible to strain.
As has been pointed out by Andreas \textit{et al.}, the location of these points in ZrSi$X$ ($X$=S, Se, Te) correlates strongly with the chemical pressure $c/a$ \cite{topp2016non}.
The most prominent effect of strain on the electronic structure of ZrSiS is the shift of the quadratic electron band along the energy axis. The tensile strain pushes this band toward the Fermi energy, while the compressive strain has the opposite effect. At around 2\% tensile strain, the electron states along the Z--R line cross the Fermi energy.
The optical conductivity has contributions from both free carriers (Drude) and interband transitions in the vicinity of the Dirac points. The quadratic band, which crosses the nodal line at some k-points reduces the transition probability between the linear bands. This behavior is expected to have influence on the optical properties in the low-energy region.
Upon uniaxial compression of ZrSiS, its Fermi surface does not undergo any considerable modification, remaining topologically equivalent to the Fermi surface of pristine ZrSiS shown in Figure~\ref{fig:label1}(c). In contrast, in case of tensile strain the Fermi surfaces changes its topology as a consequence of the emerged conduction states with quadratic dispersion. One can distinguish between two Lifshitz transition occurring in stretched ZrSiS. When tensile stress reaches $P_{1}\sim1.3$ GPa, the previously disconnected hole pockets merge with each other, forming a ring at $k_z=\pi/c$. The corresponding merging region is highlighted in Figure~\ref{fig:label5}(b).
Up to $4\%$ tension, the electron and hole pockets are connected along the Z--R direction. When tensile stress reaches $P_{2} \sim 3.4$ GPa, a gap is being formed between the electron and hole pockets, manifesting itself the second transition in the Fermi surface topology [highlighted in Figure~\ref{fig:label5}(b)].
We also examine the nodal-line structure under tensile strain focusing at {\bf k}-points where Lifshitz transition takes place at the Fermi surface. The corresponding structure is shown in Figure~\ref{fig:label5}(c). Below $-2\%$ strain, the nodal lines form a continuous cage-like structure in the Brillouin zone. When tensile strain is increased, the nodal lines oriented in the $k_z$ direction get disconnected from the nodal loop at $k_{z}=\pm \frac{\pi}{c}$ along the Z--R direction at the cage corners. The corresponding separation between the nodal lines is increasing with strain, and can also be directly attributed to the appearance of the quadratic band along Z--R. Besides, one can see that the curvature of the nodal loop at $k_{z}=\pm \frac{\pi}{c}$ changes its sign when strain increases from $-1\%$ to $-4\%$.
We now examine the effect of strain on the interband screening.
To this end, we first calculate the unscreened plasma frequency shown in Figure~\ref{fig:strain_plasma} for the two crystallographic directions. While out-of-plane plasma frequency $\omega_{p,zz}$ exhibits a pronounced linear dependence as a function of strain, the in-plane plasma frequency, $\omega_{p,xx}$ demonstrates a more sophisticated dependence. Different behavior of $\omega_{p,zz}$ and $\omega_{p,xx}$ can be attributed to the difference in the Fermi velocities along the $x$- and $z$- directions.
The strain-dependent screened plasma frequency $\omega^{scr}_{p}$ can be obtained from $\epsilon_{1}(\omega)$ shown in Figure~\ref{fig:strain_cond}.
For compressive strain and small tensile strain up to $2\%$, $\omega^{scr}_{p}$ remains nearly a constant of around 1.0 eV. The situation for larger tensile strain is different. Due to the electronic Lifshitz transition, the nodal structure of $\epsilon_{1}(\omega)$ changes, leading to an enhancement of $\omega^{scr}_{p}$, which reaches $\sim$1.3 eV at $4\%$ tension.
The related interband screening $\epsilon_{\infty}=(\omega_p/\omega_p^{scr})^2$ is changing accordingly. As it is shown in Figure~\ref{fig:strain_plasma}, the in-plane component of $\epsilon_{\infty}$ is remaining around 9--10 up to 1\% tension, after which it decreases rapidly until the tension reaches 4\%, i.e. after the Fermi surface modification has occured. In this regime, $\epsilon_{\infty} \sim 3$--4,
similar to the experimental values reported for ZrSiTe ($\sim 3.3$) \cite{ebad2019infrared}. This result is in favor of the chemical pressure mechanism proposed to describe the difference between the ZrSi$X$ ($X$=S, Se, Te) family members. Overall, the interband screening in moderately stretched ZrSiS is reduced considerably, which is expected to influence the optical response.
\begin{figure}[t]
\includegraphics[width=9.0cm]{strain_plasmon.jpg}
\caption{(a) Plasma frequency $\omega_p$ as a function of momentum ${\bf q}$ calculated for uniaxially strained ZrSiS along the in-plane (left) and out-of-plane (right) directions; (b) Plasma frequency in the long-wavelength limit (${\bf q}\to 0$) [fitted with Eq.~(\ref{dispersion})] as a function of strain; (c) Plasmon dispersion [fitted with Eq.~(\ref{dispersion})] as a function of strain. }
\label{fig:label6}
\end{figure}
The in-plane conductivity calculated for different values and types of strain is shown in Figure~\ref{fig:strain_cond} as a function of the photon energy.
The frequency-independent conductivity region tends to narrow (broaden) as the compressive (tensile) strain is applied.
Besides, the spectral weight in the low-energy region almost linearly enhances with load, gaining $\sim$50\% at $5\%$ compression.
On the contrary, the tensile strain reduces the spectral weight, yet not monotonously. At $\sim$3\% tension the optical conductivity is dropped, which apparently associated with the reduction of the interband contribution to the dielectric screening discussed earlier.
The observed lowering of the spectral weight in stretched ZrSiS is in line with the smaller flat optical conductivity observed in ZrSiSe with a larger $c/a$ lattice parameter \cite{ebad2019chemical}.
At a larger energy scale, the effect of strain is less pronounced in the optical properties. In the range from 0.5 eV to 1.2 eV the optical conductivity is redshifted upon compression, while at larger frequencies it is blueshifted. The opposite situation is observed for the case of tensile strain. At low energies, the optical conductivity is mainly determined by the transitions between the linear bands in the electronic structure, as well as by the details of the Fermi surface. At energies above 1 eV the transitions between the parabolic bands become important, whose position on the energy axis is largely dependent on strain.
As a consequence, the characteristic U-shape of the optical conductivity around 1 eV almost disappears for more than $4\%$ tensile strain.
Finally, we would like to comment on the effect of strain on the high-energy plasma excitations in ZrSiS. As this energy region is almost unrelated to the Fermi surface properties, the corresponding effect is less significant. In Figure~\ref{fig:label6}, we show the dispersion of the high-energy plasmon mode, as well as the corresponding parameters entering Eq.(\ref{dispersion}). Although the plasma frequency almost linearly changes with strain, the effect does not exceed a few percent for 5\% strain.
In contrast, the dispersion of the plasma excitations can be tuned effectively by the compressive strain. While the dispersion along the out-of-plane direction decreases with strain gaining 30$\%$ at $+5\%$, the opposite effect is observed along the in-plane direction.
\section{Conclusions}
Based on first-principles calculations, we have systematically studied optical properties of nodal-line semimetal ZrSiS in the presence of uniaxial strain.
We find that the characteristic frequency-independent optical conductivity is robust with respect to external uniaxial compression of up to 10 GPa. The compressive strain increases the spectral weight at low energies, but leads to a narrowing the flat conductivity region. The case of tensile strain is found to be more interesting. Upon tensile stress of 2 GPa, the Fermi surface undergoes a Lifshitz transition, resulting in a weakening of the interband dielectric screening. As a result, the spectral weight in the infrared region is reduced. The results obtained for stretched ZrSiS correlate with the properties of ZrSiSe and ZrSiTe, materials with larger lattice constants $c/a$. We, therefore, confirm the chemical pressure mechanism proposed in Ref.~\cite{ebad2019chemical} to describe variability in the electronic and optical properties of the ZrSi$X$ ($X$=S, Se, Te) family of compounds.
On the other hand, the uniaxial tensile stress up to 2 GPa could be applied experimentally by flexure-based four-point mechanical wafer bending setup \cite{suthram2006piezoresistance}.
In the high-energy region, we found one lossy and one lossless plasmon modes at $\sim$5 and $\sim$20 eV, respectively. Although the frequencies of these modes remain almost unchanged in the presence of strain of up to 5\%, their dispersion can be effectively tuned.
Being a layered material, ZrSiS exhibits strongly anisotropic dielectric response between the in-layer and stacking directions. This gives rise to the possibility of existence of hyperbolic plasmons in ZrSiS. Our calculations show that the hyperbolic regime indeed may exist within a frequency range of 0.6 eV around $\sim$5 and $\sim$20 eV.
Overall, our findings provide insights into the mechanism behind the formation of optical properties in nodal-line semimetals ZrSi$X$, and pave the way for further optical studies, particularly in the ultraviolet spectral range.
\\
\newline
\newpage
\begin{acknowledgements}
SY acknowledges financial support from the National Key R$\&$D Program of China (Grant No. 2018FYA0305800) and National Science Foundation of China (Grant No. 11774269). A.N.R. acknowledges travel support from FLAG-ERA JTC2017 Project GRANSPORT. Numerical calculations presented in this paper have been performed on a supercomputing system in the Supercomputing Center of Wuhan University.
\end{acknowledgements}
\bibliographystyle{achemso}
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1907.07770
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\section{Introduction}
A molecule is an example of a $n$-body system formed by the nuclei and the electrons of its constituent atoms. The first step of the Born-Oppenheimer approximation allows to represent the nuclei as points in Euclidean space.
The space defined by the degrees of freedom of the molecular system after elimination of the symmetry group actions is called the \textit{internal configuration or conformational space}. The potential energy surface (PES) describes the energy of a molecule. It is a function on the internal conformational space of the molecule.
The study of the conformational spaces of molecules and their associated PES is of particular interest in Chemistry. It can help to understand the relationship between structure and properties of molecules. For instance, PES can give insight into the structure and dynamics of macromolecules such as proteins. It has been shown that protein native-state structures can arise from considerations of symmetry and geometry associated with the polypeptide chain \cite{ATSFM}. Furthermore, understanding of the binding pose of a drug with its potential target requires knowledge of both their underlying PES, and conformational spaces \cite{Tao2001}.
Conformer generation is regarded as a fundamental factor in drug design \cite{Confgen}. Therefore, several methods exist that produce conformer sets, sampling the conformational space. One of the challenges is obtaining an algorithm that reduces the number of duplicate conformers. This could be achieved if the symmetries of a given molecule are taken into account. Also, it has been claimed that prediction of melting points can be improved by taking molecular symmetry into account \cite{Wei,Pinal}.
In this paper we explore the geometry and topology of the conformational space of molecules and their quotients by symmetry groups. Despite the importance of the group of symmetries and its action on the space of conformers, to our knowledge there are no works on spaces of conformers that include it. Furthermore, conformational spaces are often discussed only in terms of their torsional degrees of freedom, operating under the rigid geometry hypothesis \cite{Gibson1997}. The impact of other degrees of freedom on the conformational spaces themselves is often ignored.
\subsection{Geometric and topological methods in data analysis}
Topology-based data analysis methods have seen continued interest in recent years. Persistent homology and discrete Morse theory are two topological data analysis tools, which are closely interlinked, and which we have applied to the exploration of conformation spaces of molecules.
Persistent homology, which may be more familiar in the chemistry community, is a method of assigning numerical descriptors to data, based on topological notions of shape, which emerges through a process of creating a combinatorial structure, called a simplicial complex, from the data, together with a filter function. These descriptors satisfy robust stability results with respect to the so-called bottleneck distance. The use of this method allowed us to explore the topology of the conformation spaces.
Discrete Morse theory is mathematically very closely linked to persistent homology, but it has different applications. It is used for topological simplification, for distilling the information down to the most relevant. We used it to explore the energy landscapes via their extrema.
In the paper \emph{PHoS: Persistent Homology for Virtual Screening} \cite{PHoS}, Keller et al. use multi-dimensional persistence to investigate molecules in the context of drug discovery. Their idea is using two filter functions, one of which is a scalar field defined around the molecule. This is similar to our ideas, however we focus on the conformational space, rather than individual molecules, and instead of merely calculating the persistence of the scalar function (the energy landscape, in our case), we explore it using discrete Morse theory.
Discrete Morse theory has recently been used to reconstruct hidden graph-like structures from potentially noisy data. This has found application in vastly diverse areas. For example, Sousbie et al. used the simulated density field of dark matter to reconstruct the network of filaments in the large scale distribution of matter in the universe, the so-called cosmic web \cite{Sousbie1}. Given a collection of GPS trajectories, Dey et al. recovered the hidden road network by modelling it as reconstructing a geometric graph embedded in the plane \cite{Wang1}. Another paper by Delgado-Friedrichs et al. defines skeletons and partitions of greyscale digital images by modelling a greyscale image as a cubical complex with a real-valued function defined on its vertices and using discrete Morse theory \cite{cubical}.
A more fundamental application of discrete Morse theory in topological data analysis is topological simplification. Here, the link with persistent homology allows a topology-based denoising of data, as explored in \cite{Edelsbrunner2000} and \cite{Bauer2012}.
This methodology has already been introduced to the chemical setting as well. Gyulassi et al. \cite{porous} used Morse-theoretic approaches to investigate the properties of a simulated porous solid as it is hit by a projectile by generating distance fields containing a minimal number of topological features, and using them to identify features of the material. In \cite{Beketayev2011}, the authors construct the Morse-Smale complex of
Our approach relies on these results, however our focus is somewhat different. We use the connection between Morse theory and persistent homology to construct a combinatorial summary of the conformation space of a given molecule, which takes into account both the topological properties of the conformation space, as well as the energy landscape defined on it.
\subsection{The workflow and outline of the paper}
The general outline of the paper is shown in Figure \ref{fig:workflow}. Starting with the computational details in the second section, we explain the conformer generation procedure used, followed by the calculations of the potential and free energy surfaces. Next, we move on to outline the geometric and topological methods used for the analysis of the conformational space.
Section 3 discusses our mathematical framework with regards to the conformational spaces arising from the molecular graphs and group actions on these spaces, as well as the metrics defined on them. This section contains our original theoretical results.
The fourth section discusses the results of our analyses performed on several benchmark molecules. These are based on the mathematical framework of the previous section, and use the aforementioned mathematical data analysis methods.
Finally, we end with our conclusions in Section 5.
In the Appendix, we explain in more detail the mathematical methods applied in the paper.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{workflow.png}
\caption{The workflow of the paper.}
\label{fig:workflow}
\end{figure}
\section{Computational details}
\subsection{Conformer Generation Procedure}
The task of creating sets of molecular conformations is inherently complex due to the large number of degrees of freedom in a molecule. Furthermore, it is often the case that in reality what is actually desired is a set of low-energy structures, and often the ability of an algorithm or program to create these conformers is used as its quality metric \cite{Ebejer2012}. In general conformer generation procedures can be separated into knowledge-based, grid search, or distance geometry based approaches. Knowledge-based approaches use known low-energy conformers, such as crystal structures, to define rules which can then be used to generate conformers for a new molecule. Grid search approaches simply enumerate combinations of different degrees of freedom. Finally, distance geometry uses upper and lower bounds to create sets of conformations that satisfy these bounds. The reader is directed to \cite{Ebejer2012} for more information regarding these methods.
For this work, we are in general unrestricted by energy, instead using a more general \emph{physicality} criteria. This is because we would like to sample the conformational space as best as we can, rather than simply obtain representative low energy conformers. However, if energy were totally unrestricted, this would lead to conformers that we would consider unphysical, in particular caused by clashes between atoms. This can be rectified by recognising that any commonly used forcefield would give such a configuration an energy orders of magnitude higher than any other, due to the near exponential scaling of any reasonable Pauli repulsion approximation.
RDKit \cite{Landrum2018} has been used in this work to create conformation sets, using the ETKDG method \cite{Riniker2015}. This creates a set of conformers with reasonable distributions in bond lengths and angles, but fairly fixed torsions. To ensure that the conformational space was covered, each conformer had its torsions determined from independent uniform distributions on $(-\pi,\pi]$. Lastly, conformers with energies in excess of $200$kcal/mol were removed, ensuring there were no atomic clashes.
As well as studying conformer sets with variation in all molecular degrees of freedom, we have also generated sets with only torsional variation. Firstly, conformer sets have been created through a grid search in the torsion degrees of freedom. This set also requires energy pruning to remove unphysical molecules. Secondly, a metadynamics \cite{Laio2002} approach was used. This also allowed the creation of free energy landscapes and do not require energy pruning. The drawback from this approach is that we no longer have information as to all degrees of freedom, instead reducing our space to the reaction coordinates.
Lastly, we have studied the conformational space of cyclooctane, using data obtained from \cite{Martin2010}. This is a set of 6040 conformations of cyclooctane, varying in torsion angles. Hydrogen coordinates were found through a constrained geometry optimisation. The reader is referred to \cite{Martin2010,Brown2008} for more information. Table \ref{tab:molecule_betti} contains information as to the size of the generated conformer sets.
\subsection{Potential energy and free energy surfaces}
Operating within the Born-Oppenheimer approximation, we can consider the molecular energy to be a function of the atomic coordinates. There are many different methods for calculating this molecular energy, broadly split into those that are classical and quantum mechanical. Here, we use a classical forcefield to calculate the potential energy of a single conformer. These forcefields contain parameters describing the relative strength of bond bending, bond stretching, and torsional rotations within a molecule, broadly written as:
\begin{equation}
E_{molecule} = \sum_{bonds} k(x-x^\prime)^2 + \sum_{angles} t(\theta - \theta^\prime)^2 + \sum_{torsions} \left( 1+\sum_n V_n \cos(n\omega)\right)
\end{equation}
Where $k$, $t$ and $V$ modulate the relative strength of interactions, and non-bonded terms have been dropped as we are not studying systems with more than one molecule in this work. We use the MMFF94 forcefield as implemented in RDKit \cite{Halgren1996} to calculate the potential energy of a single conformer.
Often, a more useful quantity to study is the free energy. The free energy can be thought of as a 'smoothed out' potential energy, where various degrees of freedom integrated out through an appropriately weighted Boltzmann average. In our work, we use metadynamics \cite{Laio2002} to calculate free energy surfaces over the torsional degrees of freedom for a molecule.
Both the potential and free energy functions can be considered as maps from a conformational space to a subset of the real line. The potential energy would be a map from the full conformational space, whereas the free energy is a map from the torsion-only subspace of the conformational space.
\begin{comment}
\subsection{Conformational Space Representations}
\label{sampledPoints}
\ingrid{We need to rewrite this subsection to make it consistent with the theoretical discussion}
\lee{Do we want to move it there?}
\subsubsection{Vector Representation}
The vector representation of a conformational space utilises the embedding of a conformer in $\mathbb{R}^3$ to generate a vector in $\mathbb{R}^{N\times3}$, where $N$ is the number of atoms in the molecule. The vector representation of the conformational space is therefore the set of vectors in this high-dimensional space, equipped with the Euclidean metric. This representation has been used in other studies of conformational spaces, such as in \cite{Martin2010,Martin2011}.
\subsubsection{Metric Representation}
Alternatively, the metric representation of a conformational space uses a pairwise dissimilarity metric between two conformers to generate an $n\times n$ distance matrix. A common distance metric used in molecular sciences is the root-mean-square deviation (RMSD):
$$
d(C_1,C_2) = \sqrt{\frac{\sum_{i\in\text{atoms}}|\vec{x}_{i,C_1}-\vec{x}_{i,C_2}|^2}{N}}
$$
which is commonly used when aligning chemical structures such as proteins \cite{Duan2014}, and can be shown to be a metric \cite{Steipe2002,Steipe2002a,Sadeghi2013}.
\end{comment}
\subsection{Local PCA and orientability}
Computations of the distance matrix associated to the internal configuration spaces $({C_{\mathcal M}^{int}}, d_P)$ and $({C_{\mathcal M}^{int}}, d_\mathcal O)$ were carried out in Matlab using the in-built function `Procrustes'.
We estimated the local dimension of ${C_{\mathcal M}^{int}}$ using local PCA. The algorithm was implemented in Matlab, and it is shown in Figure \ref{pca}. A more detailed exposition on local PCA is presented in Appendix \ref{a:lpca}. Let $\mathcal S=\{C_i\}_{i=1}^N$ be a data set of $N$ conformers of a molecule $\mathcal M$. Given a conformer $C_i\in\mathcal S$ we computed its $k$-nearest neighbours, where $k \ll N$ using the the Matlab function `knn' together with the results of Procrustes. We gave a matrix representation to each element of the permutation invertion group (GPI) which is defined in Section 3. We used this representation to compute the distance matrix associated to the conformational spaces $(\mathcal C^{int}_\mathcal M,d_P)$ and $(\mathcal C_\mathcal M^{int},d_\mathcal O)$ along with its local dimension.
\begin{algorithm}
\begin{algorithmic}[1]
\caption{Local PCA with group actions }
\label{pca}
\Require Data set $\mathcal S$ of $N$ conformers, $C_1,\ldots,C_N$, of a molecule with $n$ atoms in $\mathbb R^{3n}$, $N> 3n$
\Require The automorphism group $P$ of the molecular graph, as a subgroup of the symmetric group $S_n$
\Require A constant $\gamma \in (0,1)$; higher values of $\gamma$ result in higher predicted dimension
\For {$i \in \{1,\ldots,N\}$}
\For {$j \in \{1,\ldots,N\}$}
\State Let $\tilde p_{j} \in P$ and $\tilde A_j \in SO(3)$ be elements minimising $d_F(C_i,A(C_j \cdot p))$, $p\in P$, $A \in SO(3)$
\State Set $\tilde C_{ij} = d_{\mathcal O}(C_i,C_j)=d_F(C_i,\tilde A_j(C_j \cdot \tilde p_j))=d_P(C_i,C_j \cdot \tilde p_j)$
\EndFor
\State
Let $NN \subseteq \{1,\ldots,n\}$ be such that $\{ \tilde C_{ij} \mid j \in NN \}$ are the $k$ lowest values of $\tilde C_{ij}$ for $1 \leq j\leq N$
\State Compute PCA for $\{ \tilde A_j(C_j \cdot \tilde p_j) \mid j \in NN \}$ and let $\lambda_1,\ldots,\lambda_{3n}$ be the resulting eigenvalues
\State Let $d_i \in \{ 1,\ldots,3n \}$ be the smallest $d$ such that $(\sum_{k=1}^d \lambda_k) / (\sum_{k=1}^{3n} \lambda_k) > \gamma$;
\State \quad $d_i$ is the predicted dimension of $\mathcal{OC}_\mathcal{M}^{int}$ at $C_i$
\EndFor
\State The predicted dimension $ld$ of $\mathcal{OC}_\mathcal M^{int}$ is the median of the $d_i$, $1 \leq i \leq N$
\end{algorithmic}
\end{algorithm}
We tested orientability of the conformational spaces. In \cite{SH} Singer et al.\ developed an algorithm to detect orientability on large data set which are sample from manifolds. In Algorithm \ref{orienta} we present a version of this algorithm that includes the group action of a discrete group.
\begin{algorithm}
\begin{algorithmic}[1]
\caption{Orientability of conformational spaces} \label{orienta}
\Require Data set $\mathcal S$ with $N$ conformers $C_i$ of a molecule with $n$ atoms
\State Perform Algorithm \ref{pca} to obtain the predicted dimension $ld$, and a $3N\times ld$ matrix $O_i$, $i\in\{1,\dots, N\}$, with column vectors form an orthonormal basis that approximates the tangent space $T_{ C_i}\mathcal C_\mathcal M^{int}$.
\State For neighbour conformers $C_i$ and $C_j$ obtain $O_{ij}=\underset{O\in O(ld)}{argmin}\|O-O^T_iO_j\|_F$.
\State Let $Z$ be the $N\times N$ matrix with entries given by $z_{ij}=det O_{ij}$ for nearby points and 0 otherwise.
\State Define the matrix $A=D^{-1}Z$, where $D$ is diagonal and $D_{ii}=\sum_{i=1}^N|z_{ij}|$.
\State Compute the top eigenvector $v_{top}$ of A.
\State Decide the orientability analysing the histogram of the coordinates of $v_{top}$.
\end{algorithmic}
\end{algorithm}
\subsection{Persistent homology}
Persistent homology \cite{Edelsbrunner2000,Zomorodian1} is a method of topological data analysis, which has been used to analyse different types of data sets from different areas in recent years, including chemistry.
It is a method of calculating topological, or more accurately, homological, features at different spatial resolutions. Features that persist for a wider range of the spatial parameter are deemed to be more likely to represent true features of the underlying space the data was sampled from, rather than noise, sampling errors or particular choices of some parameters.
In order to calculate persistent homology of a data set, we need to represent the data as a space with a triangulation, called a simplicial complex. For a set of points $S$, with $\vert S \vert = k$, the most common way to do this is to define the $k$-simplex (or $k$-dimensional polytope) $\Delta^S$ with the points as its vertices.
This work involves the use of persistent homology in two different contexts. First of all, we use it to investigate the homology of the points sampled from the conformation space. With some sensible assumptions on the sampling quality of these points, we can assume that we can deduce from this the homology of the conformation space, and therefore say something about its topological features.
Secondly, we use the connection between persistent homology and discrete Morse theory in our analysis of the energy landscapes defined on the conformation spaces.
\subsection{Multidimensional or parametrised persistence vs. discrete Morse theory}
In order to investigate the energy landscapes on the conformation spaces via persistent homology, we need to filter the conformation space (or its combinatorial approximation) by the real-valued energy function. In essence, we are filtering our simplex $\Delta^S$ by both the pairwise distance function $f$ defined above, and the energy function $E:\Delta^S\to \mathbb{R}$.
However, we are not actually interested in the two-parameter persistence module we get this way.
Instead, we wish to construct a combinatorial structure, called the \emph{Morse-Smale complex} of the energy function, which represents the associated gradient flow and summarises it according to its critical points. This construction is described in more detail in the appendix.
In order to construct the Morse-Smale complex with the correct number of critical points, we need to filter a simplicial complex which has the Euler characteristic of the conformational space.
Given a value of $f$ for which the preimage $f^{-1}$ is a triangulation with the 'correct' topology of our conformational space, we can use the lower-star filtration on this complex using the filter function $E$, and compute one-parameter persistence.
In practice, the construction we use to get the triangulation, are $3D$ alpha complexes, which can substantially reduce the computational complexity from having to construct Rips complexes with a threshold value. In fact, alpha complexes are thresholded Rips complexes, but there are easier implementations available, e.g. in Matlab.
To describe this construction, first let us say a word about Voronoi diagrams. Given a set $S$ of points in Euclidean space $\mathbb{R}^n$, one defines convex polytopes $V_s$, $s\in S$ called Voronoi cells, which consist of all points $x\in\mathbb{R}^n$ such that the distance between $x$ and $s$ is less than the distance between $x$ and $s'$ for any other $s'\in S$. The subsets $V_s$ give a tessellation of $\mathbb{R}^n$.
Given a finite set of points $S\subset \mathbb{R}^n$ and a real number $r\in \mathbb{R}$, one defines the region $R_s(r) = \bar{B}_s(r)\cap V_s$, where $\bar{B}_s(r)$ is the closure of the ball of radius $r$ centred at $s$. Now we can form the $\alpha$-complex (or alpha complex) $K_r$ as follows: a subset $\sigma\subset S$ is called an $\alpha$-simplex if
$$\bigcap_{s\in\sigma} R_s(\sigma)\neq\emptyset.$$
After we have discovered the topology of the conformational space from which we sampled our point set $S$, we can safely choose a triangulation $T$ of the point cloud that reflects this topology (i.e. choose a radius for the Rips complex) and linearly extend the energy function $E:S\to\mathbb{R}$ to the entire simplicial complex, which gives us a piecewise-linear function $\tilde{E}:T\to \mathbb{R}$. This allows us to explore the energy landscape of the given molecule.
Let $M$ be a compact manifold and let $f$ be a Morse function defined on $M$. Then the alternating sum of the number of critical points of index $k$ of $f$ equals the Euler characteristic of the manifold. The same goes for a simplicial complex. Therefore, we have a linear relation between the minima of our energy landscape and the other critical points, which is unique to the conformation space (unique in the sense that it depends on the topology of the conformation space).
The Morse-Smale complex is commonly applied for surface segmentation \cite{Sousbie1,cubical,Wang1}. Each segment of the surface has uniform integral lines. This leads to a reduction of information. Instead of the entire energy landscape, we are left with a cell complex which accounts for the unique features of the energy function. Each cell has uniform flow, meaning that we can read off the Morse-Smale complex directly which energy minimum a given conformer will flow to as it loses energy. It also shows the unstable equilibria, the maxima and saddle points where there is no flow, but the slightest perturbation can lead to a more drastic change of the conformation.
Moreover, this cell complex still has the same homology as the conformation space, combining the characteristics of both the conformation space and the energy landscape in a compact, combinatorial structure, which can be regarded as a unique descriptor of the molecule.
\section{A mathematical framework for conformational spaces of molecules}\label{MF}
\subsection{Molecular graphs and conformational spaces}
Our model of conformation spaces of molecules will be given by embeddings of graphs in $\mathbb{R}^3$. In such graphs, we combinatorially encode atoms and bonds in a molecule as vertices and edges of a graph, respectively. To introduce this, we need to define molecular graphs.
\begin{De}
\label{d:mg}
A \emph{molecular graph} is a tuple $\mathcal{M} = (V,E,c_V,L,\Theta)$ consisting of the following data.
\begin{enumerate}
\item $\Gamma = (V,E)$ is a finite undirected \emph{graph}: that is, $V$ is a finite set, and $E \subset V \times V$ is a subset such that, for any $v,w \in V$, we have $(v,v) \notin E$, and $(v,w) \in E$ if and only if $(w,v) \in E$. We refer to $V$ as the set of \emph{vertices} of $\mathcal{M}$, and to $E$ as the set of \emph{edges} of $\mathcal{M}$.
\item $c_V: V \to \mathbb{N}$ is a \emph{vertex colouring}; for $v \in V$, we will think of $c_V(v)$ as the chemical element the atom corresponding to $v$ represents, and will set $c_V(v)$ to be the atomic number of this element.
\item $L: E \to (0,\infty)$ is a set of \emph{length constraints}; $L(e)$ will be called the \emph{length} of an edge $e \in E$.
\item $\Theta: E_2 \to (0,\pi]$ is a set of \emph{angle constraints}, where
\[
E_2 = \{ (v,w_1,w_2) \in V \times V \times V \mid (v,w_1),(v,w_2) \in E, w_1 \neq w_2 \}
\]
is viewed as the set of \emph{adjacent edges} in $\Gamma$; we will refer to $\Theta(v,w_1,w_2)$ as the \emph{angle} between edges $(v,w_1) \in E$ and $(v,w_2) \in E$.
\end{enumerate}
For consistency, we also require that $L(v,w) = L(w,v)$ for every $(v,w) \in E$ and $\Theta(v,w_1,w_2) = \Theta(v,w_2,w_1)$ for every $(v,w_1,w_2) \in E_2$.
\end{De}
\begin{remark}
In Definition \ref{d:mg} the length and angle constraints have the following chemical meanings. The bond constraint is associated to the average of a bond length between two atoms A and B, whose centres of mass are modelled as points in $\mathbb R^n$. The angle constraint is associated to the angle given by the hybridation type of the atom around which the bond angle is defined. Therefore in this definition we assume that the bond lengths and bond angles are rigid. That is, they are constant for every conformer.
\end{remark}
To define an embedding of a molecular graph $\mathcal{M}$, note that the length constraints can be used to make $\mathcal{M}$ into a geodesic metric space. Indeed, we may define the \emph{geometric realisation} of $\mathcal{M}$ as a metric space defined by gluing an interval $[0,L(e)]$ for each edge $e \in E$ along the vertices $V$ in the obvious way. From now on, slightly abusing notation, we will identify a molecular graph $\mathcal{M}$ with its geometric realisation.
\begin{De} \label{d:emb}
An \emph{configuration} of a molecular graph $\mathcal{M} = (V,E,c_V,L,\Theta)$ is an embedding (injective continuous map) $\varphi: \mathcal{M} \to \mathbb{R}^3$ from the geometric realisation of $\mathcal{M}$ to $\mathbb{R}^3$ such that
\begin{enumerate}
\item \label{i:demb-isom} for any edge $e \subseteq \mathcal{M}$, the restriction $\varphi|_e: e \to \mathbb R^3$ is an isometric embedding; and
\item for any adjacent edges $(v,w_1),(v,w_2) \in E$ (that is, for any $(v,w_1,w_2) \in E_2$), we have $\angle(\varphi(w_1),\varphi(v),\varphi(w_2)) = \Theta(v,w_1,w_2)$, where $\angle(x,y,z)$ is the angle between the line $yx$ and the line $yz$ in $\mathbb{R}^3$ for $x,y,z \in \mathbb{R}^3$.
\end{enumerate}
The \emph{conformational space} of $\mathcal{M}$, denoted $\mathcal C_\mathcal M$, is the set of all embeddings $\varphi$ of $\mathcal{M}$.
\end{De}
From the condition \eqref{i:demb-isom} in Definition \ref{d:emb} and from the fact that geodesics in $\mathbb{R}^3$ are unique, it follows that an embedding $\varphi: \mathcal{M} \to \mathbb{R}^3$ can be recovered uniquely from its values on the finite set $V$. In particular, if $V = \{ v_1,\ldots,v_n \}$ contains $n$ points, then any embedding $\varphi$ can be recovered from the tuple $C_\varphi = (\varphi(v_1),\ldots,\varphi(v_n)) \in \mathbb{R}^{3n}$, called the \emph{vector representation} of $\varphi$. The map $\mathcal C_\mathcal M \to\mathbb R ^{3n}, \varphi \mapsto C_\varphi$ is therefore injective. We use this map to realise $C_{\mathcal M}$ as a subspace of $\mathbb{R}^{3n}$; in particular, as $\mathbb R^{3n}$ is a topological space, this induces a subspace topology on $\mathcal C_\mathcal M$. Thus, from now on, we will regard $\mathcal C_\mathcal M$ as a topological space.
In order to analyse the connectivity properties of $\mathcal C_\mathcal M$, we will use the following.
\begin{De}\label{d:emg}
The \emph{orientation} of an embedding $\varphi$ of the molecular graph $\mathcal{M}$ is the map $O_\varphi: E_3 \to \{-1,0,1\}$, where
\[
E_3 = \{ (v,w_1,w_2,w_3) \in V \times V \times V \times V \mid (v,w_i) \in E, w_i \neq w_j \text{ whenever } i \neq j \},
\]
defined by
\[
O_\varphi(v,w_1,w_2,w_3) = \operatorname{sign}\ (\varphi(w_1)-\varphi(v)) \cdot [(\varphi(w_2)-\varphi(v)) \times (\varphi(w_3)-\varphi(v))],
\]
where $\operatorname{sign} c = \begin{cases} -1 & \text{if } c < 0, \\ 0 & \text{if } c = 0, \\ 1 & \text{if } c > 0, \end{cases}$ for any $c \in $.
\end{De}
Given a molecular graph $\mathcal{M} = (V,E,c_V,L,\Theta)$ and a quadruple $(v,w_1,w_2,w_3) \in E_3$, one can see that the number $|O_\varphi(v,w_1,w_2,w_3)| \in \{0,1\}$ is independent of the embedding $\varphi$ of $\mathcal{M}$. Indeed, it follows from the fact that any two points $x,y \in S^2$ on the sphere $S^2 \subseteq \mathbb R^3$ are joined by a unique geodesic on $S^2$ (unless $y = -x$) that we have
\begin{equation} \label{e:Ozero}
\begin{aligned}
O_\varphi(v,w_1,w_2,w_3) = 0 \qquad \Leftrightarrow \qquad \text{either } &\theta_{12}+\theta_{13}+\theta_{23} = 2\pi \\ \text{or } &\theta_{ij}+\theta_{ik}=\theta_{jk} \text{ for some } \{i,j,k\}=\{1,2,3\},
\end{aligned}
\end{equation}
where $\theta_{ij} = \Theta(v,w_i,w_j)$. We will call a tuple $(v,w_1,w_2,w_3) \in E_3$ \emph{planar} if $O_\varphi(v,w_1,w_2,w_3) = 0$ for some, or any, $\varphi \in \mathcal C_\mathcal M$ (that is, if both sides of \eqref{e:Ozero} are true). We will say that the molecular graph $\mathcal{M}$ is \emph{planar} if all tuples in $E_3$ are planar. Note that, by convention, a disconnected molecular graph is planar if and only if all its connected components are planar.
\begin{remark}
A planar graph $\Gamma$ is a graph for which there exists an embbeding from the geometric realisation of $\Gamma$ into $\mathbb R^2$. Thus our definition of a planar conformation is more restrictive and does not coincide with that of a planar graph.
\end{remark}
On the other hand, the map $O_\varphi$ itself is \emph{not} independent of the embedding $\varphi$, unless all tuples in $E_3$ are planar. Indeed, given an embedding $\varphi$ of $\mathcal{M}$ and a quadruple $(v,w_1,w_2,w_3) \in E_3$, it is easy to check that $O_{\iota \circ \varphi}(v,w_1,w_2,w_3) = -O_\varphi(v,w_1,w_2,w_3)$, where $\iota: \mathbb R^3 \to \mathbb R^3, x \mapsto -x$ is the antipodal map (it is clear from the definition that $\iota \circ \varphi: \mathcal{M} \to \mathbb R^3$ is also an embedding of $\mathcal{M}$). This allows us to show the following result.
\begin{proposition} \label{p:disconn}
If a molecular graph $\mathcal{M}$ is not planar, then the conformational space $\mathcal C_\mathcal M$ is not path connected. More precisely, if $\varphi,\psi \in \mathcal C_\mathcal M$ and $(v,w_1,w_2,w_3) \in E_3$ are such that $O_\varphi(v,w_1,w_2,w_3) = -O_\psi(v,w_1,w_2,w_3) \neq 0$, then $\varphi$ and $\psi$ are in different path components of $\mathcal C_\mathcal M$.
\end{proposition}
\begin{proof}
As $\mathcal{M}$ is not planar, there exists a quadruple $(v,w_1,w_2,w_3) \in E_3$ that is not planar. Let $\varphi \in \mathcal C_\mathcal M$ be any embedding: then clearly $\iota \circ \varphi \in \mathcal C_\mathcal M$, where $\iota: \mathbb{R}^3 \to \mathbb{R}^3$ is the antipodal map. Thus the first part of the Proposition follows from the second part, by taking $\psi = \iota \circ \varphi$.
To prove the second part of the Proposition, suppose for contradiction that $\varphi,\psi \in \mathcal C_\mathcal M$ are in the same path component of $\mathcal C_\mathcal M$. Then there exists a path $\alpha: [0,1] \to \mathcal C_\mathcal M$ such that $\alpha(0) = \varphi$ and $\alpha(1) = \psi$. Consider the map
\begin{align*}
D: [0,1] &\to \mathbb{R}, \\
t &\mapsto (\alpha_t(w_1)-\alpha_t(v)) \cdot [(\alpha_t(w_2)-\alpha_t(v)) \times (\alpha_t(w_3)-\alpha_t(v))]
\end{align*}
where $\alpha_t = \alpha(t): \mathcal{M} \to \mathbb{R}^3$. The map $D$ is clearly continuous, and it follows from the assumption that $D(0) = -D(1) \neq 0$. Therefore, by the Intermediate Value Theorem, there exists $t \in (0,1)$ such that $D(t) = 0$. But this implies that $O_{\alpha(t)}(v,w_1,w_2,w_3) = 0$, contradicting the fact that $(v,w_1,w_2,w_3) \in E_3$ is not planar. Thus $\mathcal C_\mathcal M$ is not path connected, as required.
\end{proof}
\subsection{Group actions on conformational spaces}
It is known that the quantum mechanical description of a molecular systems must be invariant to the several types of transformations \cite{Bunker}, in which the following are included:
\begin{enumerate}[1)]
\item Rotation of the positions of all particles about any axis through the centre of mass.
\item Translation in space.
\item Permutation of the positions of any set of identical nuclei.
\item Simultaneous inversion of the positions of all particles in the centre of mass.
\end{enumerate}
Therefore it is crucial to study the symmetries present in our model of the conformational space $\mathcal C_\mathcal M$ by studying actions of groups on $\mathcal C_\mathcal M$. In what follows, for any $n \in \mathbb{N}$, let $S_n$ be the symmetric group on $n$ elements (the group of all permutations of an $n$-element set), let $D_n$ be the dihedral group of order $2n$ (the group of all symmetries of a regular $n$-gon), and let $\mathbb{Z}_n$ be the group of integers modulo $n$. Let $E(3)$ the group of isometries of $\mathbb R^3$ and let $SE^+(3)$ be the subgroup of $E(3)$ of orientation preserving isometries. It is known that there are group isomorphisms $E(3)\cong O(3)\ltimes \mathbb R^3$ and $SE(3)^+\cong SO(3)\ltimes \mathbb R^3$, where $O(3)$ and $SO(3)$ are the group of real invertible matrices with determinant $\pm1$ and $+1$, respectively.
In \cite{Gui}, Guichardet approches to configuration spaces of molecules defining the following spaces of $n$-points, $n\geq3$:
\begin{equation}\label{gui0}
X_\mathcal M^0=\{(x_1,\dots,x_n)\in\mathbb R^{3n}|x_i\neq x_j,\; \mathrm{if}\; i\neq j\}
\end{equation}
\begin{equation}\label{gui1}
X_\mathcal M^{1}=\{(x_1,\dots,x_n)\in \mathbb R^{3n}|x_i\neq x_j,\; \mathrm{if}\; i\neq j,\;\sum_{i=1}^nm_ix_i=0\}
\end{equation}
The group $\mathbb R^3$ of $E(3)$ acts on $X^0_\mathcal M$ by translation and it is easy to see that $X_\mathcal M ^1=X_\mathcal M^0\times \mathbb R^3$. Adding the condition $span(x_j,\dots,x_n)=\mathbb R^3$ in \eqref{gui1} we obtain another, configuration space, $X_\mathcal M^{P}$. The group $SO(3)$ acts freely on $X_\mathcal M^{P}$ and there is a principal $SO(3)$-bundle:
\begin{equation}
\xymatrix{
SO(3)\ar[r]&X_\mathcal M^{P}\ar[r]^-{\pi_{1}}&X_\mathcal M^{int}}
\end{equation}
where $X_\mathcal M^{int}$ is homeomorphic to the orbit space $X_\mathcal M^{P}/SO(3)$. In our model there are restrictions on the nuclei positions which are imposed by the underlying molecular graph. Therefore it is expected that our $\mathcal C_\mathcal M$ is a subspace of $X_\mathcal M^{P}$.
Let $\mathcal M=(V.E,c_V,L,\Theta)$ be a molecular graph such that $|V|=n$. Given a conformation $C_\varphi$ of a $\mathcal M$, $\mathcal C_\mathcal M=(\varphi(v_1),\dots,\varphi(v_n))\in\mathbb R^{3n}$, the groups $E(3)$ and $O(3)\cong E(3)/\mathbb R^3$ act on it by isometries. We define the following space
\begin{equation}
\mathcal C_\mathcal M^{P}=\{C_\varphi=(\varphi(v_1),\dots,\varphi(v_n)) \in\mathbb R^3|\sum_{i=1}^nc_V(v_i)\varphi (v_i)=0\}
\end{equation}
We can also see that $\mathcal C_\mathcal M\cong C^P_\mathcal M\times\mathbb R^3$. We might assume additionally that $|V|\geq 3$. The group $SO(3)$ acts on $\mathcal M$ by rotation which induces an action $C^P_\mathcal M$. Let $\mathcal \mathcal C_\mathcal M^{int}$ be the orbit space of the action of $SO(3)$ on $\mathcal C_\mathcal M^P$. We give the space $C^{int}_\mathcal M$ the quotient topology. Then we have the following result.
\begin{theorem}\label{t:pb}
Let $\mathcal M$ be a molecular graph. If $\Theta(E_2)\nsubseteq\{\pi\}$ then the quotient map $q:C^P_\mathcal M\to C^{int}_\mathcal M$ defines a principal $SO(3)$-bundle over $\mathcal C_\mathcal M^{int}$. Moreover this principal $SO(3)$-bundle is trivial.
\end{theorem}
\begin{proof}
Let $C_\varphi=(\varphi(v_1),\dots,\varphi(v_n))\in \mathcal C_\mathcal M ^P\subset \mathbb R^{3n-3}$. Let us write $x_i=\varphi(v_i)$ for all $0\leq i\leq n$. The elements of $ SO(3)$ act on $C_\varphi=(x_1,\dots,x_n)\in \mathcal C_\mathcal M^P$ with the canonical action. Let $\pi:\mathcal C_\mathcal M^P\to \mathcal C_\mathcal M^{int}$ be the quotient map. By assumption there exists a triple $(v,W_1,W_2)\in E_2$ such that $\Theta(v,w_1,w_2)\neq\pi$ then this action has no fixed points. It follows that $q^{-1}(C_\varphi)\cong SO(3)$ for all $C_\varphi\in \mathcal C_\mathcal M^P$. Therefore, the quotient map $q:\mathcal C_\mathcal M^P\to \mathcal C_\mathcal M^{int}$ defines a principal $SO(3)$-bundle over $\mathcal C_\mathcal M^{int}$. It is a routine calculation to check that $s$ is a continuous map. Since the map $\pi$ that defines the principal $SO(3)$-bundles has a section, then this bundle is trivial.
Now we show that the bundle $\pi:\mathcal C_\mathcal M^P\to \mathcal C_\mathcal M^{int}$ is trivial, that is $\mathcal C_\mathcal M^P\cong \mathcal C_\mathcal M^{int}\times SO(3)$. It suffices to show that there is a map $s:\mathcal C_\mathcal M^{int}\to \mathcal C_\mathcal M^P$ such that the composition $\pi\circ s$ is the identity map on $\mathcal C_\mathcal M^{int}$. This is equivalent to choosing one point from each $SO(3)$ orbit in $\mathcal C_\mathcal M^P$ in a continuous way. Now since the action of $SO(3)$ there exists a unique $A\in SO(3)$ and $D\in\mathbb R^3$ such that
\begin{align*}
A\varphi(v)+D&= (0,0,0)\\
A\varphi(w_1)+D&=(L(v,w_1),0,0)\\
A\varphi(w_2)+D&= (L(v,w_2)\cos\theta,L(v,w_2)\sin\theta,0)
\end{align*}
where $\theta=\Theta(v,w_1,w_2)$. We define a map $s:\mathcal C_\mathcal M^{int}\to \mathcal C_\mathcal M^P$ by sending $\pi(C_\varphi)$ to $C_{A\varphi}$. By construction the map $\pi\circ s$ is the identity on $\mathcal C_\mathcal M^{int}$.
\end{proof}
We will start analysing the action of discrete groups. In \cite{LH}, Longuet-Higgins introduces a symmetry group of non-rigid molecules, called the \emph{complete nuclear permutation inversion group}, or \emph{CNPI group} for short. Suppose we are given a molecule $M$ consisting of $n = n_1+\cdots+n_k$ atoms where $n_i \geq 1$ is the number of atoms of element $c_i$, for some distinct chemical elements $c_1,\ldots,c_k$. We let the \emph{conformational space} $\mathcal C_\mathcal M$ of $\mathcal M$ be the set of all embeddings $V \to \mathbb{R}^3$, where $V$ is a finite set of cardinality $n$ (corresponding to the $n$ atoms of $M$); we then have an obvious injective map $\mathcal C_\mathcal M \to \mathbb{R}^{3n}$, which makes $\mathcal C_\mathcal M$ into a topological space as before. The $CNPI$ group of $M$ is then $CNPI_\mathcal M = S_{n_1} \times \cdots \times S_{n_k} \times C_2$. This group acts on $\mathcal C_\mathcal M$ as follows:
\begin{enumerate}
\item for $i=1,\ldots,k$, the group $S_{n_i}$ permutes the images of the atoms of element $c_i$; and
\item the non-trivial element of $C_2$ sends $\varphi \in C_\mathcal M$ to $\iota \circ \varphi$, where $\iota: \mathbb{R}^3 \to \mathbb{R}^3$ is the antipodal map.
\end{enumerate}
This construction fits in our setting as follows. Let $\Gamma = (V,\varnothing)$ be the \emph{discrete} graph on $|V|=n=n_1+\cdots+n_k$ vertices -- that is, $\Gamma$ has no edges. Label the vertices of $\Gamma$ as $$V = \{ v_{1,1},\ldots,v_{1,n_1},v_{2,1},\ldots,v_{2,n_2},\ldots,v_{k,1},\ldots,v_{k,n_k} \},$$ and define $c_V: V \to \mathbb{N}$ by $c_V(v_{i,j}) = c_i \in \mathbb{N}$. Note that since $\Gamma$ has no edges we also have $E_2 = \varnothing$, and so the functions $L: \varnothing \to (0,\infty)$ and $\Theta: \varnothing \to (0,\pi]$ in Definition \ref{d:mg} are defined uniquely. Thus, we have a molecular graph $\mathcal{M} = (V,\varnothing,c_V,L,\Theta)$. As a topological space, $\mathcal{M}$ is a discrete space of cardinality $n$, and an embedding $\mathcal{M} \to \mathbb{R}^3$ is just a choice of $n$ distinct points in $\mathbb{R}^3$. Thus, our construction generalises the construction in \cite{Gui}. The graph permutation inversion group, introduced below, generalises the concept of the CNPI group. Loosely speaking, this is the group of automorphisms that preserve the structure of the molecular graph $\mathcal{M}$.
\begin{De} \label{d:gp}
Let $\mathcal{M} = (V,E,c_V,L,\Theta)$ be a molecular graph. A \emph{graph permutation} of $\mathcal{M}$ is a bijection $g: V \to V$ such that
\begin{enumerate}
\item for any $v,w \in V$, we have $(v,w) \in E$ if and only if $(g(v),g(w)) \in E$;
\item $c_V(v) = c_V(g(v))$ for every $v \in V$;
\item $L(v,w) = L(g(v),g(w))$ for every $(v,w) \in E$; and
\item $\Theta(v,w_1,w_2) = \Theta(g(v),g(w_1),g(w_2))$ for every $(v,w_1,w_2) \in E_2$.
\end{enumerate}
Moreover, let $\psi \in \mathcal C_\mathcal M^{int}$, and let $C_\psi \subseteq \mathcal C_\mathcal M$ be the set of embeddings $\varphi \in \mathcal C_\mathcal M$ such that $O_\varphi \equiv O_\psi$; by Proposition \ref{p:disconn}, $C_\psi$ is a union of path components of $\mathcal C_\mathcal M$. Then a graph permutation $g$ is said to be \emph{orientation preserving} with respect to $C_\psi$ if
\begin{enumerate}
\setcounter{enumi}{4}
\item $O_\psi(v,w_1,w_2,w_3) = O_\psi(g(v),g(w_1),g(w_2),g(w_3))$ for every $(v,w_1,w_2,w_3) \in E_3$,
\end{enumerate}
and \emph{orientation reversing} with respect to $C_\psi$ if
\begin{enumerate}[\quad\ (1')]
\setcounter{enumi}{4}
\item $O_\psi(v,w_1,w_2,w_3) = -O_\psi(g(v),g(w_1),g(w_2),g(w_3))$ for every $(v,w_1,w_2,w_3) \in E_3$.
\end{enumerate}
We denote by $\operatorname{Sym}_C^+(\mathcal{M})$ and $\operatorname{Sym}_C^-(\mathcal{M})$ the sets of graph permutations that are orientation preserving and orientation reversing with respect to $C = C_\psi$, respectively, and we let $\operatorname{Sym}_C^{\pm}(\mathcal{M}) = \operatorname{Sym}_C^+(\mathcal{M}) \cup \operatorname{Sym}_C^-(\mathcal{M})$. Clearly, the sets $\operatorname{Sym}_C^+(\mathcal{M})$ and $\operatorname{Sym}_C^\pm(\mathcal{M})$ are groups under composition.
\end{De}
It is clear that a graph permutation $g$ defines a map $g: \mathcal C_\mathcal M \to \mathcal C_\mathcal M$ by permuting the points $\{ \varphi(v) \mid v \in V \}$ for an embedding $\varphi: \mathcal{M} \to \mathbb{R}^3$. Moreover, if $\psi \in C_\psi \subseteq \mathcal C_\mathcal M$ are as in Definition \ref{d:gp}, then for a graph permutation $g$ that is orientation preserving with respect to $C_\psi$, this map restricts to a map $g: C_\psi \to C_\psi$. Similarly, if a graph permutation $g$ is orientation reversing with respect to $C_\psi$, then we have a map $\hat\iota \circ g: C_\psi \to C_\psi$, where $\hat\iota: \mathcal C_\mathcal M \to \mathcal C_\mathcal M$ is defined by $\hat\iota(\varphi) = \iota \circ \varphi$ and $\iota: \mathbb{R}^3 \to \mathbb{R}^3$ is the antipodal map. It is also easy to check given two such maps, each of which is either $g$ for $g$ orientation preserving or $\hat\iota \circ g$ for $g$ orientation reversing with respect to $C_\psi$, the composite of these two maps will also have this form. Thus, we define the \emph{graph permutation inversion group}, or \emph{GPI group}, of $\mathcal{M}$ with respect to $C = C_\psi$ to be
\[
GPI_{\mathcal{M},C} = \left\{ g \mid g \in \operatorname{Sym}_C^+(\mathcal{M}) \right\} \sqcup \left\{ \hat\iota \circ g \mid g \in \operatorname{Sym}_C^-(\mathcal{M}) \right\}.
\]
Note that
\[
GPI_{\mathcal{M},C} \cong \begin{cases} \operatorname{Sym}_C^\pm(\mathcal{M}) \times \mathbb{Z}_2 & \text{if $\mathcal{M}$ is planar}, \\ \operatorname{Sym}_C^\pm(\mathcal{M}) & \text{otherwise}. \end{cases}
\]
We will usually omit $C$ from the notation and will simply talk about the GPI group $GPI_{\mathcal{M}}$ of $\mathcal{M}$.
The following theorem is clear because $GPI$ is a finite group. An introduction to orbifolds is given in Appendix \ref{pgborbi}.
\begin{theorem}\label{t:orbifoldcs}
Let $\mathcal M$ be a molecular graph and $G$ be its $GPI$ group. Suppose that $\mathcal C_\mathcal M^{int}$ is a manifold. Then there is a properly discontinuous action of the group $G$ on $\mathcal C_{\mathcal M}^{int}$. In particular, if $GPI$ is non trivial, the quotient space $\mathcal C_{\mathcal M}^{int}/G$ has the structure of an orbifold.
\end{theorem}
\begin{example}
Let $\mathcal M=(\Gamma,c_V,\Theta)$ be the molecular graph associated to pentane. If we only consider the carbon atoms with a rigid conformation, then the orbifold conformational space is homeomorphic to the quotient of a torus $S^1\times S^1$ by the action of the group Aut$(\Gamma)=C_2$. The groups $C_2$ acts on the conformational space by permuting the ends of the molecular graph $\mathcal M$. This action is equivalent to permute the parameters $t_1$ and $t_2$ associated to two torsion angles. As it is shown in Figure \ref{MB}.
\end{example}
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale=1.5,baseline={(current bounding box.center)}]
\begin{scope}[xshift=-3cm]
\node at (-0.1,2.1) {};
\node at (2.1,-0.1) {};
\draw [thick,<<->] (1,0) -- (0,0) -- (0,1);
\draw [thick,->] (1,0) -- (2,0) -- (2,1);
\draw [thick,->>] (0,1) -- (0,2) -- (1,2);
\draw [thick] (2,1) -- (2,2) -- (1,2);
\draw [dashed] (0,0) -- (2,2);
\draw [<->] (0.5,1) -- (1,0.5) node [above right] {$C_2$};
\end{scope}
\draw [->] (-0.5,1) arc (120:60:1);
\begin{scope}[xshift=0.5cm]
\draw [thick,->] (1,0) -- (2,0) -- (2,1);
\draw [thick,->] (2,1) -- (2,2) -- (0,0) -- (1,0);
\draw [thick,dotted,-open triangle 60] (1,1) -- (1.5,0.5);
\draw [thick,dotted] (1.5,0.5) -- (2,0);
\end{scope}
\node at (3,1) {\huge =};
\begin{scope}[xshift=2.5cm]
\draw [thick,-open triangle 60] (2.5,1.5) -- (2,2) -- (1,1) -- (1.5,0.5);
\draw [thick,-open triangle 60] (1.5,0.5) -- (2,0) -- (3,1) -- (2.5,1.5);
\draw [thick,dotted,->] (2,0) -- (2,1);
\draw [thick,dotted] (2,1) -- (2,2);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{Action of $C_2$ on $S^1\times S^1$. The orbifold conformational space $\mathcal {OC}_\mathcal M^{int}=\mathcal C_\mathcal M^{int}/G$ is homeomorphic to a Moebius strip.} \label{MB}
\end{figure}
\begin{table}
\begin{center}
\begin{tabular}{r cll}
Graph && CNPI group & GPI group \\
\hline\vspace{-1em}\\
$n$-cycle & \begin{tikzpicture}[scale=0.4,baseline={(current bounding box.center)}]
\fill (0,1) circle (3pt);
\fill (0,2.5) circle (3pt);
\fill (1,3.5) circle (3pt);
\fill (2.5,3.5) circle (3pt);
\fill (3.5,2.5) circle (3pt);
\fill (3.5,1) circle (3pt);
\fill (2.5,0) circle (3pt);
\fill (1,0) circle (3pt);
\draw (0,1) -- (0,2.5) -- (1,3.5) -- (2.5,3.5) -- (3.5,2.5) -- (3.5,1) -- (2.5,0) -- (1,0) -- cycle;
\end{tikzpicture} & $S_n \times C_2$ & $D_{2n} \times C_2$ \\
$n$-path & \begin{tikzpicture}[scale=0.4,baseline={(current bounding box.center)}]
\fill (0,0) circle (3pt);
\fill (1,0.5) circle (3pt);
\fill (2,0) circle (3pt);
\fill (3,0.5) circle (3pt);
\fill (4,0) circle (3pt);
\fill (5,0.5) circle (3pt);
\fill (6,0) circle (3pt);
\fill [white] (3,-0.5) circle (3pt);
\fill [white] (3,1) circle (3pt);
\draw (0,0) -- (1,0.5) -- (2,0) -- (3,0.5) -- (4,0) -- (5,0.5) -- (6,0);
\end{tikzpicture} & $S_{n+1} \times C_2$ & $C_2 \times C_2$ \\
pentane & \begin{tikzpicture}[scale=0.4,baseline={(current bounding box.center)}]
\draw (-0.6,-0.3) -- (0,0) -- (1,0.5) -- (2,0) -- (3,0.5) -- (4,0) -- (4.6,-0.3);
\draw (-0.6,0.3) -- (0,0) -- (0,-0.6);
\draw (4.6,0.3) -- (4,0) -- (4,-0.6);
\draw (0.7,1.1) -- (1,0.5) -- (0.9,-0.1);
\draw (3.3,1.1) -- (3,0.5) -- (3.1,-0.1);
\draw (2,0.6) -- (2,-0.6);
\fill (0,0) circle (3pt);
\fill (1,0.5) circle (3pt);
\fill (2,0) circle (3pt);
\fill (3,0.5) circle (3pt);
\fill (4,0) circle (3pt);
\filldraw [fill=white] (-0.6,0.3) circle (3pt);
\filldraw [fill=white] (-0.6,-0.3) circle (3pt);
\filldraw [fill=white] (0,-0.6) circle (3pt);
\filldraw [fill=white] (4.6,0.3) circle (3pt);
\filldraw [fill=white] (4.6,-0.3) circle (3pt);
\filldraw [fill=white] (4,-0.6) circle (3pt);
\filldraw [fill=white] (0.7,1.1) circle (3pt);
\filldraw [fill=white] (0.9,-0.1) circle (3pt);
\filldraw [fill=white] (3.3,1.1) circle (3pt);
\filldraw [fill=white] (3.1,-0.1) circle (3pt);
\filldraw [fill=white] (2,0.6) circle (3pt);
\filldraw [fill=white] (2,-0.6) circle (3pt);
\end{tikzpicture} & $S_5 \times S_{12} \times C_2$ & $(C_3 \wr C_2) \rtimes C_2$ \\
\makecell{tetraethylmethane \\ \hfill (heavy atoms)} & \begin{tikzpicture}[scale=0.4,baseline={(current bounding box.center)}]
\fill (0,0) circle (3pt);
\fill (1,0.5) circle (3pt);
\fill (2,0) circle (3pt);
\fill (3,0.5) circle (3pt);
\fill (4,0) circle (3pt);
\fill (0.5,-1) circle (3pt);
\fill (1.5,-1) circle (3pt);
\fill (3,-0.5) circle (3pt);
\fill (3.5,-1.5) circle (3pt);
\draw (0,0) -- (1,0.5) -- (2,0) -- (3,0.5) -- (4,0);
\draw (0.5,-1) -- (1.5,-1) -- (2,0) -- (3,-0.5) -- (3.5,-1.5);
\end{tikzpicture} & $S_9 \times C_2$ & $S_4$
\end{tabular}
\caption{Graphs and their associated symmetry groups}
\label{t:graphs}
\end{center}
\end{table}
\subsection{Metrics on conformational spaces}
Let $\mathcal M=(V,E,c_V,\Theta)$ be a molecular graph such that $|V|=n$. We can endow a conformational space $\mathcal C_{\mathcal M}^{int}$ with the following metrics:
\begin{enumerate}
\item Given two matrices $X,Y\in\mathrm{Mat}_{3\times n}(\mathbb R)$ representing two conformers in $\mathcal C_\mathcal M^{int}$ the Frobenius distance $d_F(X,Y)$ between them, is defined to be $$d_F(X,Y)=\|X-Y\|_F,$$ where $\|-\|_F$ is a matrix norm of an $3\times n$ matrix $A$ defined as the squared root of the sum of the absolute squares of its entries
\begin{equation}
\|A\|_F=\sqrt{\sum_{i=1}^3\sum _{i=1}^n\mid a_{ij}\mid^2}.
\end{equation}
Observe that given a matrix $A\in \mathrm{Mat}_{p\times n}$ the Frobenius norm $\mathcal C^{int}_\mathcal M$ coincides with the Euclidean metric $\|-\|_2$ in the vector representation of $\mathcal C^{int}_\mathcal M$.
\item
Given two conformers $X,Y\in \mathcal C_\mathcal M^{int}$ and a matrix $R\in SO(3)$, the Procrustes distance function is defined as,
$$d_P(X,Y)=\underset{R\in SO(3)}{\inf}\{\| X-RY\|_F\}$$
\item A common distance metric used in molecular sciences is the root-mean-square deviation (RMSD):
$$d_{RMSD}(X,Y) = \frac{1}{\sqrt{N}}d_P(X,Y)$$
which is commonly used when aligning chemical structures such as proteins \cite{Duan2014}, and can be shown to be a metric \cite{Steipe2002,Steipe2002a,Sadeghi2013}.
\end{enumerate}
Let $\mathcal M$ be a molecular graph and let $ {GPI}$ be the graph permutation inversion group of $\mathcal M$. We can give the orbifold configuration space $O\mathcal C_\mathcal M^{int}$ the following metric
\begin{equation}\label{e:distance2}
d_{\mathcal O}(X,Y)=\underset{g\in{GPI}}{min}\{d_{P}(X,g\cdot Y)\}
\end{equation}
\section{Results and discussion}
Following our discussion in Section \ref{MF}, each conformer $C_{\varphi}$ in the configuration space $\mathcal C_\mathcal M$ is uniquely determined by the positions of the atoms in the molecules. A conformation of a molecule $\mathcal M$ with $n$ atoms is represented by an $n$-by-$3$ matrix with real coefficients. The entries of the $i$-th row vector in this matrix are the spatial coordinates of the $i$-th atom in $C_{\varphi}$. We eliminate the action of the subgroup of translations $T$ by fixing the centre of mass of each conformer at $(0,0,0)\in\mathbb R^3$. From the set of molecular configurations $\mathcal S=\{C_i\}_{i=1}^N$ sampled from the configuration space $\mathcal C_\mathcal M^P$ we generated a data set of the following metric spaces:
\begin{itemize}
\item The metric space $(X,d_F)$. Assuming that the bond angles are almost constant then by Theorem \ref{t:pb}, we have that $\mathcal C_\mathcal M\cong \mathcal C_\mathcal M^{int}\times SO(3)$. We generate a set of $n$-by-3 matrix in $\mathbb{R}^{n\times3}$, where $n$ is the number of atoms in the molecule. Each matrix is associated to an aligned conformer. The euclidean metric has been used in other studies of conformational spaces, such as in \cite{Martin2010,Martin2011}.
\item The metric space $(X,d_P)$ defined by the Procrustes metric. We obtained a distance matrix.
\item The metric space $(X,d_{RMSD})$ defined by the RMSD metric. We obtained a distance matrix.
\item The metric space $(X,d_\mathcal O)$ defined by the orbifold metric. We obtained a distance matrix.
\end{itemize}
\subsection{Local dimension and orientability}
We used geometric and topological data analysis to study the conformational spaces of small molecules: butane, pentane, alanine dipeptide, and cyclooctane. We used principal component analysis, or PCA -- one the most common tools in data analysis.
The data set of conformers associated to a molecule $\mathcal M$, consists of a set $\mathcal S$ of 3-by-$n$ real matrices $\mathcal S=\{A_i\}_{i=1}^N$, with $N$ the number of conformers. We compute the distance matrix to study the geometric and topological properties of the conformational space $\mathcal C_\mathcal M$.
In Table \ref{locpca} we show the result of local dimension and orientability. In our analysis, fluoromethane is the only molecule with no torsional angles. That is, it is the only molecule that has a contractible conformational space.
Local dimension shows that data variability depends, to a great extent, on torsional angles. Indeed, all the non-rigid molecules have at most 2 torsional angles, and this allows to reduce the local dimension of the conformational space from $3n-6$ to 2 or 3 dimensions. In contrast, although fluoromethane is the smallest molecule, the local dimension observed in this case is the highest. For this molecule, the reduction of the local dimension depends on other parameters such as lengths of bonds and angles between them.
One interesting outcome of our local dimensionality study is the detection of singularities. This algorithm approximates the minimal dimension required to span a neighbourhood of a point. If the analysed point is a singular point, its neighbourhood will typically have a higher dimension than the one observed for a point whose neighbourhood can locally be modelled as a subset of Euclidean space. Thus we could detect the singularities in the conformational spaces of cyclooctane and pentane after quotienting out the action of $C_2$. In the former case, the local dimension detected at a singular point is higher that the local dimension at non-singular points. In \cite{Martin2011}, it was shown that the conformational space of cyclooctane is a non-manifold. More specifically, the authors showed that the conformational space can be embedded in 5-dimensional Euclidean space and that it corresponds to the space formed by the Klein bottle and a $2$-sphere intersecting in $2$ circles. The local dimension and the orientability of the clusters was determined. Our results show that the local dimension of the set of singular points is one, whereas the local dimension of the other clusters is 2. Moreover, one of these clusters is orientable and the other is not. Persistent homology of these spaces is shown in Table \ref{locpca}.
\begin{table}[ht]
\centering
\begin{tabular}{c c c c}
\hline
Molecule &Loc. Dim.&dim. singularities &Orientable\\ [0.5ex]
\hline
butane&2&- &Yes\\
pentane&2 &-&Yes\\
alanine dipeptide&2&-&Yes\\
cyclooctane & 2,3 &1&No\\
pentane($C_2$) & 2 & 1 &No\\
Cyclooctane (singular set) & 1 & -&Yes \\
Cyclooctane (sphere)&2&-&Yes \\
Cyclooctane (Klein bottle) & 2&-&No \\[1ex]
\hline
\end{tabular}
\caption{Local dimension and orientability} \label{locpca}
\label{table:nonlin}
\end{table}
\begin{figure}[ht]
\centering
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{locdim_pentane-eps-converted-to.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{locdim_aladip-eps-converted-to.pdf}
\caption{}
\end{subfigure}
\caption{Local dimension of $ \mathcal C_\mathcal M^{int}$ was determined using PCA at each point. Plots (a) and (b) show the local PCA at one point of $\mathcal C_\mathcal M^{int}$ of pentane and alanine dipeptide, respectively, with the euclidean metric. In both cases local PCA suggests that there are two principal components. This implies that the local dimension at a chosen point $C_\varphi\in \mathcal C_\mathcal M^{int}$ of both pentane and alanine dipeptide is 2. }
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{orienta_pentane-eps-converted-to.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{orienta_aladip-eps-converted-to.pdf}
\caption{}
\end{subfigure}
\caption{The results of the detection of orientability of pentane and dipeptide alanine are shown in figures (a) and (b), respectively. There is noise in the data set of pentane, however it is possible to distinguish two well separated region which corresponds to a choice of orientation at each point in both pentane and alanine dipeptide.}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}{0.44\textwidth}
\includegraphics[width=\textwidth]{orienta_cycloo-eps-converted-to.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.50\textwidth}
\includegraphics[width=\textwidth]{orienta_pentaneC2-eps-converted-to.pdf}
\caption{}
\end{subfigure}
\caption{Figure (a) shows the orientability test result for $\mathcal C_\mathcal M^{int}$ of cyclooctane with the euclidean metric whereas figure (b) shows the corresponding result for $\mathcal C_\mathcal M^{int}$ of pentane with the orbifold metric.}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{sphereE_procrustes.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{kleinbE_procrustes.jpg}
\caption{}
\end{subfigure}
\caption{Orientability of clusters of cyclooctane: (a) orientable cluster (b) non-orientable cluster. The clustering method identified two clusters of local dimension 2 which present different orientability properties. }
\end{figure}
\begin{figure}[ht]
\begin{subfigure}{0.50\textwidth}
\includegraphics[width=\textwidth]{ButaneE_procrustes.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}{0.50\textwidth}
\includegraphics[width=\textwidth]{AlaDipE_procrustes.jpg}
\caption{}
\end{subfigure}
\caption{3d-embbeding of $\mathcal C_\mathcal M^{int}$ spanned by heavy atoms of (a) butane and (b) pentane. The scatter plots are coloured by the energy function.}
\end{figure}
\subsection{Persistent homology of conformational spaces}
\begin{table}[ht]
\centering
\begin{tabular}{c c c c c}
\hline
Molecule &$N$& $\beta_0$ & $\beta_1$ & $\beta_2$ \\% [0.5ex] \\% inserts table
\hline
Alanine dipeptide & 9112 & 1 & 2 & 1\\
Pentane & 9108 &1 & 2 & 1\\
Pentane/$C_2$ & 9108& 1 & 1 & 0\\
Cyclooctane (full) & 6040 & 1 & 1 & 2\\
Cyclooctane (sphere) & 2483 & 1 & 0 & 1\\
Cyclooctane (Klein bottle) & 4196 & 1 & 2 & 1\\
Cyclooctane (singularities) &639 &1 & 1&0\\
Cyclooctane (Klein bottle mod 3) & 4196 & 1 & 1 & 0\\
Fluoromethane & 10000 & 1 & 0 & 0\\
\hline
\end{tabular}
\caption{Betti numbers $\beta_k$ for the conformational spaces of the molecules studied in this work, calculated using the RMSD for all molecules and orbifold metric for pentane. The Betti numbers of four subspaces of the conformational space of cyclooctane are shown in the table. }
\label{tab:molecule_betti}
\end{table}
In this section we analyse the topology of the internal configuration spaces $C_{\mathcal M}^{int}$. We investigate whether the choice of metric for the conformational space leads to a significantly different persistent homology.
\subsubsection{Alanine Dipeptide}
The alanine dipeptide molecule can be seen in Figure \ref{fig:aladip_structure}. We note that there are two free torsions in the alanine dipeptide molecule, as the peptide bonds themselves are considered to be inflexible due to its resonance stabilisation. Therefore, ignoring bond stretching and bending, alanine dipeptide would be predicted to have a conformational space of $S^1\times S^1=T^2$.
\begin{figure}[h]
\centering
\includegraphics[width=0.35\linewidth]{alanine_dipeptide_core.png}
\caption{The structure of alanine dipeptide. The alignment core refers to the heavy atoms inside the square box.}
\label{fig:aladip_structure}
\end{figure}
The vector representation of alanine dipeptide was defined by aligning the set of conformers to a minimum energy conformer calculated by density functional theory. Furthermore, we aligned each conformer to a core set of atoms. This flexibility is inherent within the vector representation of the conformational space.
Persistent homology was calculated by using the Rips complex persistence on the Euclidean distance on the vector representation. Persistence was calculated on the vector space using all atoms, and also for heavy (non-hydrogen) atoms. These can be seen in Figure \ref{fig:aladip_vector_representations}.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{aladip_coord_ALL.png}
\caption{All atoms}
\label{subfig:aladip_vector_all}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{aladip_coord_HEAVY.png}
\caption{Heavy atoms only}
\label{subfig:aladip_vector_heavy}
\end{subfigure}
\caption{Persistence of the vector space representation of alanine dipeptide}
\label{fig:aladip_vector_representations}
\end{figure}
Firstly, it is clear that both sub-representations have similar persistent Betti numbers of $(1,2,1)$. This matches those of a torus, as earlier predicted. However, these features appear at different times depending on representation. This can be explained when considering the behavior of the Euclidean metric as the number of dimensions increases. The all atom system is 66-dimensional, whereas the heavy atom system has only 30 dimensions. This leads to a shorter average distance between conformers in the conformational space in the heavy atom system.
To create the RMSD representation for alanine dipeptide, the optimal alignment between every pair of conformers was found by optimising the RMSD between their atoms. This was calculated for both all atom and heavy atom sets. Persistent homology was then calculated using the Rips filtration on the optimum RMSD metric, with the resulting persistence diagram in Figure \ref{fig:aladip_RMSD_representations}.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{aladip_RMSD_all.png}
\caption{All atoms}
\label{subfig:aladip_RMSD_all}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{aladip_RMSD_heavy.png}
\caption{Heavy atoms only}
\label{subfig:aladip_RMSD_heavy}
\end{subfigure}
\caption{Persistence of the RMSD representation of alanine dipeptide}
\label{fig:aladip_RMSD_representations}
\end{figure}
Again, we find similar persistent Betti numbers of (1,2,1). This suggests that the topology of our conformational space is independent of representation - however it will be shown that this is not always the case. Further, there is a much smaller difference in choice of atom subsets in the case of the RMSD representation. This is due to the difference in the behaviour of the RMSD metric itself. For example, features appear slightly earlier in the all atom system. This is because the average displacement of hydrogen atoms tends to be quite small, but the increase in the denominator of the RMSD metric causes a slightly lower metric. In the case of the vector representation, each hydrogen adds an extra 3 dimensions to the Euclidean distance - the RMSD does not suffer from this curse of dimensionality in the same way.
\subsubsection{Pentane}
The structure for pentane can be seen in Figure \ref{fig:pentane_structure}. Similarly to alanine dipeptide, there are two free torsions in pentane. For this section we are ignoring the symmetry of the pentane molecule, and therefore we expect the conformational space to have the topology of the torus $T^2$.
\begin{figure}[h]
\centering
\chemfig{-[:30]-[:-30]-[:30]-[:-30]}
\caption{The structure of pentane}
\label{fig:pentane_structure}
\end{figure}
Similarly to alanine dipeptide, we define our vector representation by aligning each conformer to some reference, in this case a DFT optimised conformation. However, in contrast to alanine dipeptide, we do not align to a core, but instead align to minimise the RMSD between all carbon atoms. Persistence is then calculated analogously to alanine dipeptide.
The RMSD representation for pentane was defined by calculating the optimum RMSD distance between all carbon atoms for each pair of pentane conformers. Persistence was then calculated using this metric. The vector and RMSD representation persistent homology can be found in Figure \ref{fig:pentane_representations}.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{pentane_coords_index.png}
\caption{Vector representation}
\label{subfig:pentane_vector}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{pentane_rmsd_index.png}
\caption{RMSD representation}
\label{subfig:pentane_RMSD}
\end{subfigure}
\caption{Persistence of the different representations of the pentane conformational space. Symmetry is ignored for these representations}
\label{fig:pentane_representations}
\end{figure}
There is a clear difference in the persistent homology for these two representations. Whereas the RMSD representation has persistent Betti numbers of (1,2,1), those of the vector representation are far more unclear. This makes clear one of the main drawbacks of the vector representation of conformational spaces, in that they are dependent on the alignment to a reference. If we had aligned to some core of the three central carbons, we would have seen persistent Betti numbers of (1,2,1), as we did for alanine dipeptide. Furthermore, although aligning each conformer to some reference may make it easier to visualise the entire set at once (as is often done in the study of proteins etc.), it makes pairwise comparisons of different conformers hard to perform. In contrast, the RMSD representation does not suffer from this, and therefore leads to the correct persistent Betti numbers. Furthermore, as we will now see, the RMSD representation allows us to directly take into account molecular symmetry - which would be impossible for the vector representation.
The molecular graph of pentane has an inherent symmetry, as discussed earlier. In particular, the two torsions are equivalent, rather than being distinguishable. It is a standard result that leads to a M\"obius band topology.
We can take this symmetry into account when calculating the RMSD metric between two conformers. This is done by performing two separate RMSD alignments. In the first, we align the two conformers such that each carbon in the first conformer is matched to the same carbon in the second conformer. In the second, we match each carbon in the first conformer to the opposite carbon in the second. We then choose the optimum RMSD to be our metric. The resulting persistent homology can be seen in Figure \ref{fig:rmsd_permute}.
\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{pentane_rmsd_permute.png}
\caption{Persistence of the RMSD representation of pentane conformational space, with symmmetry taken into account}
\label{fig:rmsd_permute}
\end{figure}
There is a significant difference in the persistent homology when compared to the original RMSD representation. In particular, the persistent Betti numbers are now (1,1,0) matching those of the M\"obius band.
\subsection{ Cyclooctane}
\subsubsection{Finding singular points and clustering}
Once singular points were identified, they were removed from the data set. In principle, this separated the data into its manifold components. These can then be found using a clustering algorithm, in our work, HDBScan \cite{Campello2013}. Subsequently, the manifolds were matched, in an attempt to recreate the spherical and Klein bottle components found in the original work. This could then be verified using persistence.
We used the software \emph{Ripser} \cite{ripser1} to compute the persistent homology of our point data sampled from the conformational spaces.
To verify that we had correctly found the spherical component, we calculated the persistent homology of the Rips complex constructed on the RMSD metric between conformers. The resulting persistence diagram can be seen in Figure \ref{fig:CO_S2}. We can see that the persistent Betti numbers are (1,0,1), as expected for a sphere.
\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{cyclooctane_S2.png}
\caption{Persistence of the RMSD representation of the spherical component of the cyclooctane conformational space}
\label{fig:CO_S2}
\end{figure}
Verifying the presence of the Klein bottle component is slightly more involved. Here, we perform persistent homology calculations in the same manner as before. However, we calculate homology over two different fields of coefficients, namely $\mathbb{Z}_2$ and $\mathbb{Z}_3$. The Klein bottle has different (persistent) Betti numbers over these fields, (1,2,1) and (1,1,0) respectively - a torus would not. This allows us to verify with more confidence the presence of a Klein bottle component. The persistence diagrams can be seen in Figure \ref{fig:CO_KB}. The correct persistent Betti numbers are found.
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{cyclooctane_KB_mod2.png}
\caption{$\mathbb{Z}_2$}
\label{subfig:KB_mod2}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\linewidth]{cyclooctane_KB_mod3.png}
\caption{$\mathbb{Z}_3$}
\label{subfig:KB_mod3}
\end{subfigure}
\caption{Persistence diagrams to verify the presence of a Klein bottle component to the cyclooctane conformational space.}
\label{fig:CO_KB}
\end{figure}
\subsection{Energy landscapes}
In order to compute the Morse-Smale complexes of the energy landscapes, we made use of the Topology ToolKit (ttk) \cite{ttk1}, a software platform designed for the topological analysis of scalar data. We also used Matlab's \emph{alphaShape} function for initial processing, which produces an alpha-shape triangulation from a point cloud, of a specified radius. Extracting the boundary of this triangulation gave us the surface triangulation we required. This, together with the energy values at each point was input into the Topology ToolKit software.
We analysed the potential energy landscapes of cyclooctane, alanine dipeptide, pentane and fluoropentane. The conformational spaces of alanine dipeptide, pentane and fluoropentane are all tori, which makes for a side-by-side comparison of different energy landscapes on what is topologically the same conformational space. There is also an analysis of the free energy landscape of alanine dipeptide.
\
The results for cyclooctane are compared with the results from \cite{Martin2011}.
After finding the singular points, and separating out the sphere and Klein bottle components, we did separate analyses of the energy landscapes on these two components. The points sampled from a sphere, an orientable, low-dimensional manifold, can be triangulated so that the resulting simplicial complex has the topology of a sphere. This simplicial complex can then be input into the \emph{ttk} software and filtered by the energy function.
We use the connection between persistent homology and discrete Morse theory, to smooth this energy surface by removing topological features below a certain persistence threshold. In order to do this, we compute the persistence diagram, as well as a statistical summary of it, called the \emph{ttkPersistenceCurve} in the \emph{ttk} software, which plots the distance from the diagonal against the number of persistent points in the persistence diagram. Depicting this curve in the log scale allows us to estimate a sensible level of noise, by observing a change in the gradient of this curve. In Figure \ref{cycloDenoising}, this computation is shown for the spherical part of cyclooctane, while Figure \ref{cyclo} shows the computed Morse-Smale complex.
\begin{figure}
\centering
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\linewidth]{cycloDiag.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\linewidth]{cycloCurve2.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{cycloDiagThresholded.jpeg}
\caption{}
\end{subfigure}
\caption{The noise detection for the spherical part of cyclooctane, computed using the \emph{ttk} software. The first figure (a) shows the zero-dimensional persistence diagram, the second (b) shows the persistence curve with the likely persistence threshold for noise circled in red, and the final (c) image shows the denoised persistence diagram, using the threshold discovered through the persistence curve.}
\label{cycloDenoising}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\linewidth]{cycloEnergy.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\linewidth]{cycloMorseTubes.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{cycloMorse10Tubes.jpeg}
\caption{}
\end{subfigure}
\caption{The Morse-Smale complex for the spherical component of cyclooctane, computed using the \emph{ttk} software. The first figure (a) shows the potential energy function. The second figure (b) shows the Morse-Smale complex without topological simplification and after simplification (c). The red points are the maxima, the white points are saddle points and the blue points are the energy minima.}
\label{cyclo}
\end{figure}
\
For alanine dipeptide, we consider the model of the conformational space where only the two torsional angles are allowed to rotate, giving a torus. The scalar values of the potential energy are then given on this two-dimensional surface, displayed in Figure \ref{alanine}.
\begin{figure}[ht]
\centering
\begin{subfigure}{.24\textwidth}
\includegraphics[width=\linewidth]{alanineDOF1.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\includegraphics[width=\linewidth]{alanineDOF2.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\includegraphics[width=\textwidth]{alanineDOF_6_2Morse.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.24\textwidth}
\includegraphics[width=\textwidth]{alanineDOFMorse.jpeg}
\caption{}
\end{subfigure}
\caption{The Morse-Smale complex for Alanine Dipeptide, computed using the \emph{ttk} software. The figures (a) and (b) show the potential energy function from both sides of the torus. The image (c) shows the Morse-Smale complex, together with the Morse function after very minor topological simplification. The red points are the maxima, the white points are saddle points and the blue points are the energy minima. Finally, image (d) shows the Morse-Smale complex after severe topological simplification, leaving only one minimum.}
\label{alanine}
\end{figure}
We then did the same calculation for the free energy landscape. The results are displayed in Figure \ref{freeEalanine}. The obvious conclusion is that the free energy landscape is significantly less noisy. Another difference is the change in the number of minima. There are far more local maxima and saddle points in the potential energy landscape than in the free energy landscape.
\begin{figure}
\centering
\begin{subfigure}{.245\textwidth}
\includegraphics[width=\linewidth]{alanineFES1.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.245\textwidth}
\includegraphics[width=\linewidth]{alanineFES2.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.245\textwidth}
\includegraphics[width=\linewidth]{alanineFESMorse.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.245\textwidth}
\includegraphics[width=\textwidth]{alanineFES_9.jpeg}
\caption{}
\end{subfigure}
\caption{The analysis of the free energy surface for alanine dipeptide, computed using the \emph{ttk} software. The first two figures (a) and (b)
are showing the energy function from both sides of the surface of the torus, while (c) and (d) are showing the Morse-Smale complex, together with the energy function, before and after topological simplification.}
\label{freeEalanine}
\end{figure}
\
For pentane, the conformational space is a torus as well. The potential energy surface was analysed in the same fashion as for alanine dipeptide. In Figure \ref{pentane}, the results of this analysis are depicted. The symmetry of the molecule can be observed in the energy landscape as well, in contrast to the energy landscape of alanine dipeptide. After thresholding, only one minimum remains.
\begin{figure}
\centering
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\linewidth]{Pentane1.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\linewidth]{Pentane2.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{PentaneMorse3.jpeg}
\caption{}
\end{subfigure}
\caption{The analysis of the free energy surface for pentane, computed using the \emph{ttk} software. The first two figures (a) and (b) are showing the energy function from both sides of the surface of the torus, while (c) is showing the simplified Morse-Smale complex, together with the energy function.}
\label{pentane}
\end{figure}
Fluoropentane is a molecule that is very similar to pentane. It has the exact same underlying graph with a different vertex colouring.
Naturally, this implies that the conformation spaces of the two molecules are identical. This means we can have a direct comparison of the differences in the energy landscapes.
\begin{figure}
\centering
\begin{subfigure}{0.32\textwidth}
\includegraphics[width=\textwidth]{Fluoro1.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{Fluoro2.jpeg}
\caption{}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\includegraphics[width=\textwidth]{FluoroMorse6.jpeg}
\caption{}
\end{subfigure}
\caption{The analysis of the energy landscape for fluoropentane, computed using the \emph{ttk} software. The first two figures (a) and (b) are showing the energy function from both sides of the surface of the torus, while (c) is showing the simplified Morse-Smale complex, together with the energy function.}
\label{pentane}
\end{figure}
Finally, we are able to analyse the energy landscape for the Klein bottle component of cyclooctane. As the Klein bottle is not an orientable manifold, and also cannot be embedded into $\mathbb{R}^3$, we are unable to use \emph{ttk}. However, due to the link between Morse theory and persistent homology, we can use persistent homology to find the values of the extrema.
Firstly, we must create a simplicial complex. It can be seen from the persistence diagram of the Rips filtration of the Klein bottle component that a filtration value of $0.6$ leads to the correct topology. The MMFF94 energy of each simplex is then found, and the persistent homology of the energy function is calculated. This can be seen in Figure \ref{fig:CO_KB_energy}.
The infinite bars correspond to the topology of the space itself (i.e. the Klein bottle). Of interest, however, are the features that have a death value. Those correspond to the local critical values of the energy function. The zero-dimensional points are born at local minima and die at saddle points, while the one-dimensional points are born at saddle points and die at local maxima. We speculate that this methodology, of using persistence to find simplicial complexes and then critical values of complex energy landscapes could prove very fruitful. As an example, we propose a similarity measure in the chemical space making use of the topological properties of the energy landscapes.
\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{CO_energy_height_KB.png}
\caption{Persistence of the sublevel sets of the MMFF94 energy function defined on the Klein bottle component. Calculated with coefficients in $\mathbb{Z}_2$.}
\label{fig:CO_KB_energy}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{CO_energy_height_inverse_KB.png}
\caption{Persistence of the superlevel sets of the MMFF94 energy function defined on the Klein bottle component. Calculated with coefficients in $\mathbb{Z}_2$.}
\label{fig:CO_KB_energy}
\end{figure}
The polynomial associated to a molecular graph $\mathcal M$ is defined as
\begin{equation}
\mathcal P_{\mathcal M}(t):=\sum_{p\in CP} t^{\lambda(p)}
\end{equation}
where $CP$ is the set of critical points of the PES and $\lambda(p)$ is the index of the critical point $p$ (and $c(p)$ is the critical value at $p$. Let $\mathcal P_1$ and $\mathcal P_2$ be two Morse polynomials associated to the molecules $\mathcal M_1$ and $\mathcal M_2$, respectively. We define the potential similarity measure $S:Chem\times Chem \to \mathbb R$ in the space of molecular graphs as follows
\begin{equation}
S(\mathcal M_1,\mathcal M_2):=\int_{0}^1|\mathcal P_{\mathcal M_1}(t)-\mathcal P_{\mathcal M_2}(t)|^2dt
\end{equation}
\section{Conclusions}
We have developed a data-driven approach to understanding molecular conformational spaces and energy landscapes. We have used this method to demonstrate that conformational spaces of linear molecules match chemical intuition, and are namely products of circles caused by torsional flexibility. Further, bond stretching and bending do not change the homology groups of the conformational space as they lead to spaces related via a retraction. By using this method on different commonly used representations of conformational spaces (namely vector and metric spaces), we demonstrated that it is only the metric space representation that can consistently recreate the expected conformational space. Furthermore, we have demonstrated that the conformational space analysis still holds when molecular symmetry is taken into account.
Spaces of conformers of molecules play a fundamental role in Chemistry. Understanding these spaces might lead us to understand the relationship between the structure and the activity of biomolecules, drugs and other important classes of molecules. For many years chemists have modelled molecules using combinatorial objects, such as graphs. Associating both algebraic and geometric objects to molecules seems to provide new ways to study and to understand their chemical properties. In this paper we have shown that there exists a rich variety of algebraic, geometric and topological tools that can be used to model molecules and their conformational spaces. Symmetry groups of molecules are closely related to the topology of the conformational spaces. Methods developed in geometrical and topological data analysis seems to provide a good source of tools to analyse conformational spaces and functions defined on them such as (free) energy landscapes.
|
1902.00107
|
\section{Introduction}
Black-box
optimisation describes a challenging realm of problems where no algebraic model or gradient information is available. The problem is regarded a black box, and knowledge about the problem in hand can only be obtained by evaluating candidate solutions.
General-purpose metaheuristics like evolutionary algorithms, simulated annealing, ant colony optimisers, tabu search, and particle swarm optimisers are well suited for black-box optimisation as they generally work well without any problem-dependent knowledge.
A lot of research has focussed on designing powerful metaheuristics, yet it is often unclear which search paradigm works best for a particular problem class, and whether and how better performance can be obtained by tailoring a search paradigm to the problem class in hand.
Black-box complexity is a powerful tool that describes limits on the efficiency of black-box algorithms.
The black-box complexity of search algorithms captures the difficulty of problem classes in black-box optimisation. It describes the minimum number of function evaluations that \emph{every} black-box algorithm needs to make to optimise a problem from a given class. It provides a rigorous theoretical foundation through capturing limits to the efficiency of all black-box search algorithms, providing a baseline for performance comparisons across all known and future metaheuristics as well as tailored black-box algorithms. Also it prevents algorithm designers from wasting effort on trying to achieve impossible performance.
Many different models of black-box complexities have been developed.
The first black-box complexity model by~\citet{Droste2006BlackBox} makes no restriction on the black-box algorithm. This leads to some
unrealistic results, such as polynomial black-box
complexities of NP-hard problems~\citep{Droste2006BlackBox}.
Subsequent research introduced refined models that restrict the power of black-box algorithms, leading to more realistic results~\citep{Teytaud2006,Doerr2011,Droste2006BlackBox,Doerr2012},
where black-box algorithms can only query for the
relative order of function values of search points
\citep{Teytaud2006,Doerr2011} as well as memory restrictions~\citep{Droste2006BlackBox,Doerr2012} and restrictions on which search points are allowed to be stored~\citep{Doerr2015b,Doerr2018a,Doerr2016}.
\Citet*{Lehre2012} introduced the unbiased black-box model
where black-box algorithms may only use operators without a search bias (see Section~\ref{sec:parallel-black-box-model}). This model initially considered
unary operators (such as mutation) and
was later extended to higher arity operators (such as
crossover) \citep{Doerr2010} and more general search spaces
\citep{Rowe2011}. It also led to the discovery of more efficient EA variants~\citep{Doerr2014}. For further details we refer to the comprehensive survey by~\citet{Doerr2018b}.
A shortcoming of the above models is that they do not capture the
implicit or explicit parallelism at the heart of many common search
algorithms. Evolutionary algorithms (EAs) such as \mlea{}s or
($\mu$,$\lambda$)~EAs generate $\lambda$ offspring in parallel. Using
a large offspring population in many cases can decrease the number of
generations needed to find an optimal solution\footnote{This does not
hold for all problems; \citet*{Jansen2005a}
constructed problems where offspring populations drastically
increase the number of generations.}. However,
the number of function evaluations may increase as evolution can only
act on information from the previous generation. A large offspring
population can lead to inertia that slows down the optimisation
process. Existing black-box models are unable to capture this inertia as they assume all search points being created in sequence.
The same goes for parallel metaheuristics such as island models evolving multiple populations in parallel (see, e.\,g.~\citet{Luque2011}). Parallelisation can decrease the number of generations, or parallel time. But the overall computational effort, the number of function evaluations across all islands, may increase. \Citet{Lassig2013a} used the following notion. Let $T_\lambda$ be the random number of generations an island model with $\lambda$ islands (each creating one offspring) needed to find a global optimum for a given problem. If using $\lambda$ islands can decrease the parallel time by a factor of order $\lambda$, compared to just one island, $\lambda \cdot \E{T_\lambda} = O(\E{T_1})$, this is called a \emph{linear speedup} (with regards to the parallel time, the number of generations). A linear speedups means that the total number of function evaluations, $\lambda \cdot \E{T_\lambda}$, does not increase beyond a constant factor.
Previous work~\citep{Lassig2013a,Lassig2011a,Mambrini2012} considered illustrative problems from pseudo-Boolean optimisation and combinatorial optimisation, showing sufficient conditions for linear speedups. However, the absence of matching lower bounds makes it impossible to determine exactly for which parameters~$\lambda$ linear speedups are achieved.
We provide a parallel black-box model that captures and quantifies the inertia caused by offspring populations of size $\lambda$ and parallel EAs evaluating~$\lambda$ search points in parallel. We present lower bounds on the black-box complexity for the well known \LO problem and for the general class of functions with a unique optimum, revealing how the number of function evaluations increases with the problem size~$n$ and the degree of parallelism,~$\lambda$.
The results complement existing upper bounds~\citep{Lassig2013a}, allowing us to characterise the realm of linear speedups, where parallelisation is effective.
Our lower bound for functions with a unique optimum is asymptotically tight: we show that for the \OM problem, a \lEA with an adaptive mutation rate is an optimal parallel unbiased black-box algorithm. Adaptive mutation rates decrease the expected running time by a factor of $\ln \ln \lambda$, compared to the \lEA with the standard mutation rate~$1/n$ (see~\citet*{Doerr2015}).
The paper extends a previous conference paper~\cite{Badkobeh2014} with parts of the results.
A major novelty in this manuscript is the introduction of black-box complexity results with tail bounds. Existing black-box complexity results only make statements about the \emph{expected} number of evaluations it takes to find a global optimum\footnote{A notable exception is the $p$-Monte Carlo runtime introduced by~\citet{Doerr2015b}, defined as the minimum number of
steps needed in order to find an optimum with probability at least $1-p$.}. However, it is often not clear whether the expectation is a good reflection of the performance observed in practice.
We provide black-box complexity lower bounds that apply with an overwhelming probability.
More precisely, using the notation $\ln^+ x := \max(1, \ln x)$, we show for every target search point~$x^*$ we can choose that \emph{every} $\lambda$\nobreakdash-parallel unary unbiased black-box algorithm needs at least
\begin{equation}
\label{eq:lower-bound}
\max\left\{\frac{c\lambda n}{\ln^+\lambda}, (1-\delta) n \ln n\right\}
= \mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+\lambda} + n \ln n\right)}
\end{equation}
function evaluations to find~$x^*$, with an overwhelming probability\footnote{An overwhelming probability is defined as $1-2^{-\Omega(n^\varepsilon)}$ for some constant~$\varepsilon > 0$.}, where
$c$ is a constant with $c \ge 1/60$.
The leading constant $1-\delta$ in the $n \ln n$ term can be chosen\footnote{The precise result contains a trade-off between the leading constant and the exponent of the overwhelming probability formula, see Theorem~\ref{the:main-result-tail-bounds}.} arbitrarily close to~1. This means that it is practically impossible for any unary unbiased black-box algorithm to find a designated target with less than $\frac{c\lambda n}{\ln^+\lambda}$ or less than $(1-\delta)n \ln n$ evaluations. The latter bound applies to parallel and non-parallel unary unbiased algorithms.
In addition, if the probability of finding a single target~$x^*$ in the stated time is exponentially small, the probability of finding \emph{many} target points is still exponentially small.
This simple union bound argument opens up a range of opportunities for obtaining stronger results that are much more relevant to practice than the state-of-the-art. Our method is powerful and versatile since we can choose any set of target search points, up to an exponential size. This allows for different applications.
\begin{enumerate}
\item Considering global optimisation, our lower bound~\eqref{eq:lower-bound} applies to highly multimodal functions, even allowing for up to exponentially many optima.
Apart from results tailored to specific problem classes~\citep{Doerr2013b}, the only generic black-box complexity results we are aware of apply to functions with one unique global optimum. This innovation is significant as most functions in practice have multiple or many optima.
\item Choosing all local optima as target search points, we also get that for functions with up to exponentially many local optima, every $\lambda$-parallel unary unbiased algorithm needs at least the stated time~\eqref{eq:lower-bound} to find any \emph{local} optimum.
\item Since we can have exponentially many target search points, we can even afford to consider all search points within an almost linear Hamming distance to any local optimum as target. Then our results imply that even the time to get close to any local or global optimum is bounded by~\eqref{eq:lower-bound}.
\end{enumerate}
We demonstrate the applicability and versatility of our main result by deriving the first black-box complexity results for a wide range of illustrative function classes, from synthetic problems (\twomax, \textsc{H-IFF}, \textsc{Jump}$_k$, \textsc{Cliff}) that are very popular in the evolutionary computation literature to classes of benchmark functions~\cite{Jansen2016} and important problems from combinatorial optimisation such as \textsc{Vertex Colouring}, \textsc{MinCut}, \textsc{Partition}, \textsc{Knapsack} and \textsc{MaxSat}.
In addition to providing a solid unifying theoretical foundation for black-box algorithms, we believe that our results are of immediate relevance to practice.
Our black-box complexity with tail bounds gives hard limits on the capabilities of black-box algorithms. These limits can be used to set stopping criteria appropriately, avoiding stopping an algorithm before it has had a chance to come close to local or global optima. They are useful to set parameters such as the offspring population size~$\lambda$: if we have a limited computational budget of $T$ evaluations, \eqref{eq:lower-bound} implies that we must choose $\lambda$ satisfying $\lambda/\ln^+ \lambda \le T/(cn)$ as for larger values $T$ is lower than~\eqref{eq:lower-bound}, meaning that every $\lambda$\nobreakdash-parallel unary unbiased black-box algorithm fails badly with overwhelming probability.
Moreover, our lower bounds can serve as baseline in performance comparisons across various algorithms. And, last but not least, knowing what is \emph{impossible} is vital for guiding the search for the \emph{best possible} algorithm. The feasibility of this approach is demonstrated in this work as we present an optimal $\lambda$-parallel algorithm for \onemax that uses parallelism most effectively.
\section{A Parallel Black-Box Model}
\label{sec:parallel-black-box-model}
Following~\citet{Lehre2012}, we only use unary unbiased variation operators, i.\,e., operators creating a new search point out of one search point. This includes local search, mutation in evolutionary algorithms, but it does not include recombination.
Unbiasedness means that there is no bias towards particular regions of the search space; in brief, for $\{0, 1\}^n$, unbiased operators must treat all bit values $0, 1$ and all bit positions $1, \dots, n$ symmetrically (see~\citet{Lehre2012,Rowe2011} for details). This is the case for many common operators such as standard bit mutation.
Unbiased black-box algorithms query new search points based on the past history of function values, using unbiased variation operators. We define a \emph{$\lambda$-parallel unbiased black-box algorithm} in the same way, with the restriction that in each round $\lambda$ queries are made in parallel (see Algorithm~\ref{alg:parallel-black-box}). We use the abbreviation \uar for \emph{uniformly at random}. These $\lambda$ queries only have access to the history of evaluations from previous rounds; they cannot access information from queries made in the same round.
We refer to these $\lambda$ search points as \emph{offspring} to indicate search points created in the same round.
\begin{algorithm}[htb]
\caption{$\lambda$-parallel unbiased black-box algorithm} \label{alg:parallel-black-box}
\begin{algorithmic}[1]
\STATE Let $t := 0$. Choose $x^1(0), \dots, x^\lambda(0)$ \uar, compute $f(x^1(0)), \dots, f(x^\lambda(0))$, and let $I(0) := \{f(x^1(0)), \dots, f(x^\lambda(0))\}$.
\REPEAT
\FOR{$1 \le i \le \lambda$}
\STATE Choose an index $0 \le j \le t$ according to $I(t)$.
\STATE Choose an unbiased variation operator $p_v(\cdot \mid x(j))$ according to $I(t)$.
\STATE Generate $x^i(t+1)$ according to~$p_v$.
\ENDFOR
\FOR{$1 \le i \le \lambda$}
\STATE Compute $f(x^i(t))$ and let $I(t) := I(t) \cup \{f(x^i(t))\}$.
\ENDFOR
\STATE Let $t := t + 1$.
\UNTIL{termination condition met}
\end{algorithmic}
\end{algorithm}
This black-box model includes offspring populations in evolutionary algorithms, for example \mlea{}s or ($\mu$,$\lambda$)~EAs (modulo minor differences in the initialisation). It can further model parallel evolutionary algorithms such as cellular EAs with $\lambda$ cells, or island models with $\lambda$ islands, each of which generates one offspring in each generation.
The \lEA maintains the current best search point $x$ and creates $\lambda$ offspring by flipping each bit in~$x$ independently with probability~$p$ (with default $p=1/n$). The best offspring replaces its parent if it has fitness at least $f(x)$.
\begin{algorithm}[htb]
\caption{\lEA} \label{alg:lEA}
\begin{algorithmic}[1]
\STATE Choose $x$ \uar.
\REPEAT
\FOR{$1 \le i \le \lambda$}
\STATE Create $y_i$ by copying~$x$ and flipping each bit independently with probability $1/n$.
\ENDFOR
\STATE Choose $z \in P_t$ \uar from $\arg\max\{f(y_1), \dots, f(y_\lambda)\}$.
\STATE \algorithmicif{} {$f(z) \ge f(x)$} \algorithmicthen{} $x = z$
\UNTIL{termination condition met}
\end{algorithmic}
\end{algorithm}
\subsection{Parallel black-box complexity}
The \emph{unbiased black-box complexity (uBBC)} of a function class $\mathcal{F}$ is the minimum worst-case runtime among all unbiased black-box algorithms~\citep{Lehre2012} (equivalent to Algorithm~\ref{alg:parallel-black-box} with $\lambda=1$).
The \emph{unbiased $\lambda$-parallel black-box complexity ($\lambda$-upBBC)} of a function class $\mathcal{F}$ is defined as the minimum worst-case number of function evaluations among all unbiased $\lambda$-parallel algorithms satisfying the framework of Algorithm~\ref{alg:parallel-black-box}.
With increasing $\lambda$ access to previous queries becomes more and more restricted. It is therefore not surprising that the black-box complexity is non-decreasing with growing~$\lambda$.
For every family of function classes $\mathcal{F}_n$ and all $\lambda \in \N$,
\begin{align}
\label{eq:hierarchy-of-lambda-BBC}
\uBBC(\mathcal{F}_n) &\; \le \upBBC(\mathcal{F}_n) \le \lambda \cdot \uBBC(\mathcal{F}_n)
\end{align}
as any unbiased algorithm can be simulated by a $\lambda$-parallel unbiased black-box algorithm using one query in each round.
The following lemma shows that the parallel black-box complexity increases with the degree of parallelism, modulo possible rounding issues.
\begin{lemma}
\label{lem:bbc-grows}
For any $\alpha,\beta \in \mathbb{N}$, if $\alpha \le \beta$ then
\[
\upBBC[\alpha]{}(\mathcal{F}_n) \le \frac{\alpha}{\beta}\left\lceil \frac{\beta}{\alpha}\right\rceil\cdot \upBBC[\beta]{}(\mathcal{F}_n)\]
In particular, if $\frac{\beta}{\alpha} \in \mathbb{N}$ then $\upBBC[\alpha]{} \le \upBBC[\beta]{}$.
\end{lemma}
A proof (in the context of distributed black-box complexity) was given in~\cite[Lemma~4]{Badkobeh2015}.
Lemma~\ref{lem:bbc-grows} implies the following for all function classes $\mathcal{F}_n$ (we omit $\mathcal{F}_n$ for brevity):
First, if $\frac{\beta}{\alpha} \in \mathbb{N}$ then $\upBBC[\alpha]{} \le \upBBC[\beta]{}$. Otherwise, $\upBBC[\alpha]{} \le (1+\frac{\alpha}{\beta})\cdot\upBBC[\beta]{}
\le 2\cdot\upBBC[\beta]{} $ because $\lceil \frac{\beta}{\alpha}\rceil \le 1+\frac{\beta}{\alpha}$ and $1+\frac{\alpha}{\beta} \le 2$. In particular, this implies that for all $\alpha < \beta \in \N$,
\begin{equation}
\label{eq:bbc-does-not-decrease}
\upBBC[\beta]{} = \Omega(\upBBC[\alpha]{}).
\end{equation}
We conclude that the $\lambda$-parallel black-box complexity
does not asymptotically decrease with the degree of parallelism, $\lambda = \lambda(n)$. This implies that there is a \emph{cut-off point}
such that for all $\lambda = \Oh{\lambda^*}$ the $\lambda$-parallel unbiased black-box complexity of $\mathcal{F}_n$ is asymptotically equal to the regular unbiased black-box complexity.\footnote{Strictly speaking, we should be writing $\lambda(n) = \Oh{\lambda^*(n)}$ as the degree of parallelism may depend on~$n$. We omit this parameter for ease of presentation. Asymptotic statements always refer to~$n$.}
\begin{definition}
\label{def:cutoff}
A value $\lambda^*$ is a cut-off point if
\begin{itemize}
\item for all $\lambda = \Oh{\lambda^*}$, $\upBBC{} = \Oh{\uBBC}$ and
\item for all $\lambda = \omega(\lambda^*)$, $ \upBBC{} = \omega(\uBBC)$.
\end{itemize}
\end{definition}
Such a cut-off point always exists because due to~\eqref{eq:bbc-does-not-decrease} the parallel black-box complexity cannot decrease asymptotically, and values of $\Oh{\uBBC}$ can always be attained for suitable~$\lambda^*$, e.\,g.\ for $\lambda^* := 1$. Furthermore, the $\lambda$-parallel black-box eventually diverges for very large~$\lambda$ (e.\,g. $\lambda = \omega(\uBBC)$) as trivially $\upBBC{} \ge \lambda$.
Note that cut-off points are not unique: if $\lambda^*$ is a cut-off point, then every $\lambda' = \Theta(\lambda^*)$ is also a cut-off point.
A cut-off point determines the realm of linear speedups~\citep{Lassig2013a}, where parallelisation is most effective. Below the cut-off, for an optimal parallel black-box algorithm the number of function evaluations does not increase (beyond constant factors), but the number of rounds decreases by a factor of~$\Theta(\lambda)$. The number of rounds corresponds to the parallel time if all $\lambda$ evaluations are performed on parallel processors. Hence, below the cut-off it is possible to reduce the parallel time proportionally to the number of processors, without increasing the total computational effort (by more than a constant factor).
\section{Parallel Black-Box Complexity of LeadingOnes}
\label{sec:lo}
We consider the function $\LO(x) := \sum_{i=1}^n \prod_{j=1}^i x_j$, counting the number of leading ones in~$x$.
It is an example of a unimodal function where a specific bit needs to be flipped.
Similarly, $\LZ(x)$ counts the number of leading zeros in~$x$.
We first provide a tool for estimating the progress made by $\lambda$ trials, which may or may not be independent. It is based on moment-generating functions (mgf).
\begin{lemma}\label{lemma:mgf-max-bound}
Given $X_1,\ldots,
X_\lambda\in\mathbb{N}$, where $X_i$s are random variables, not necessarily
independent. Define $X_{(\lambda)}:=\max_{i\in[\lambda]} X_i$, if there exists $\eta,D\geq 0$, such that for all
$i\in[\lambda]$, it holds $\E{e^{\eta X_i}}\leq D$, then
$\E{X_{(\lambda)}}\leq (\ln(D\lambda)+1)/\eta$.
\end{lemma}
\begin{proof}
Note first that for any $i\in[\lambda]$ and $j\in\mathbb{N}$,
it follows from Markov's inequality that
$ \Pr(X_i\geq j)
= \Pr(e^{\eta X_i}\geq e^{\eta j})
\leq e^{-\eta j}\E{e^{\eta X_i}}
\leq e^{-\eta j}D.
$
Now, let $k:=\ln(D\lambda)/\eta$. It then follows by a union bound that
\begin{align*}
\E{X_{(\lambda)}}
=\;& \sum_{i=1}^\infty \Pr(X_{(\lambda)}\geq i)
\leq k + \sum_{i=1}^\infty \Pr(X_{(\lambda)}\geq k+i)\\
\leq\;& k + \sum_{i=1}^\infty\sum_{j=1}^\lambda \Pr(X_{j}\geq k+i)
\leq k + \sum_{i=1}^\infty \lambda e^{-\eta (k+i)}D \\
=\;& k + e^{-\eta k}\frac{D\lambda }{e^\eta-1}
\leq k + e^{-\eta k} D\lambda/\eta
= (\ln(D\lambda)+1)/\eta. \qedhere
\end{align*}
\end{proof}
For the \LO function, the $\lambda$-parallel black-box complexity is
as follows.
\begin{theorem}
\label{the:LO}
Let $\ln^+ x := \max(1, \ln x)$.
The $\lambda$-parallel unbiased black-box complexity of \LO is
\[
\mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+(\lambda/n)} + n^2\right)} \text{\quad and \quad} \Oh{\lambda n + n^2}.
\]
The cut-off point is
$
\lambda^*_{\LO} = n.
$
The corresponding parallel time for an optimal algorithm is $\mathord{\Omega}\mathord{\left(\frac{n}{\ln^+(\lambda/n)} + \frac{n^2}{\lambda}\right)}$ and $\Oh{n + \frac{n^2}{\lambda}}$.
\end{theorem}
This result solves an open problem from~\citet{Lassig2013a}, confirming that the analysis of the realm of linear speedups for \LO from~\citet{Lassig2013a} is tight.
\begin{proof}[Proof of Theorem~\ref{the:LO}]
The upper bound follows from~\citet[Theorem~1]{Lassig2011a} for a \lEA, as within the context of this bound the \lEA is equivalent to an island model with complete communication topology.
A lower bound $\Omega(n^2)$ follows from~\citet{Lehre2012}, hence the
statement holds for the case $\lambda = O(n)$.
Thus we only need to consider the case $\lambda = \omega(n)$ and to prove a lower bound of $\mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+(\lambda/n)}\right)} = \mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln(\lambda/n)}\right)}$ for this case.
We proceed by drift analysis. Let the
``potential'' of a search point $x$ be
\[
\max_{0\leq j\leq
t,1\le i \le \lambda}\{\LO(x^i(j)), \LZ(x^i(j)), n/2\}
\]
and define the potential of the algorithm, $P_t$ at time
$t$ to be the largest potential among all search points produced until
time~$t$.
Assume that the potential in generation $t$ is $P_t=k$. In any
generation $t$, let $X_i$ for $i\in[\lambda]$ be the indicator
variable for the event that all of the first $k+1$ bit-positions in
individual $i$ are $1$-bits (or $0$-bits). Furthermore, let $Y_i$ be
the number of consecutive 1-bits (respectively 0-bits) from position
$k+2$ and onwards, ie., the number of ``free riders''.
Following the same arguments as in~\citet{Lehre2012}, the probability
that $X_i=1$ is no more than ${1/(k+1)}=O(1/n)$. Defining
$M:=\sum_{i=1}^\lambda X_i$, we therefore have $\E{M}=O(\lambda/n)$. Each
random variable $Y_i$, $i\in[\lambda]$, is stochastically dominated by
a geometric random variable $Z_i$ with parameter $1/2$. The expected
progress in potential is therefore
\begin{align*}
\E{\Delta_{(\lambda)}}
= \E{\max_{i\in[\lambda]} X_iY_i}
\leq \E{\max_{i\in[M]} Z_i}.
\end{align*}
The mgf of the geometric random variable $Z_i$ is
$M_{Z_i}(\eta)=1/(2-e^\eta)$. The tower property of the expectation and Lemma~\ref{lemma:mgf-max-bound} with $\eta:=\ln(3/2)$ and $D:=2$
give
\begin{align*}
\E{\Delta_{(\lambda)}}
\leq\;& \E{\E{\max_{i\in[M]} Z_i\mid M}}\\
\leq\;& \E{(\log(DM)+1)/\eta}\\
\leq\;& (\log(\E{DM})+1)/\eta
= O(\log(\lambda/n)),
\end{align*}
where the last inequality follows from Jensen's inequality and the
last equality follows from $\log(\lambda/n) = \Omega(1)$. With
overwhelmingly high probability, the initial potential is at least
$n/2$. Hence, by classical additive drift theorems~\citep{He2004}, the
expected number of rounds to reach the optimum is $\Omega(n/\log(\lambda/n))$. Multiplying by $\lambda$ gives the number of function evaluations.
\end{proof}
\section{Parallel Black-Box Complexity of Functions with One Unique Optimum}
\label{sec:unique-optimum}
\Citet*{Jansen2005a} considered the \lEA
and established a cut-off point for~$\lambda$ where the running time
increases from $\Theta(n \log n)$ to $\omega(n \log n)$:
\begin{equation}
\label{eq:cutoff-for-1+lambda-on-OneMax}
\lambda^*_{\text{\lEA on \OM}} = \Theta((\ln n)(\ln \ln n)/(\ln \ln \ln n))
\end{equation}
\Citet*{Doerr2015} presented the following tight bounds for bounded~$\lambda$:
\begin{theorem}[Adapted from~\citet*{Doerr2015}]
\label{the:He-Chen-Yao}
The expected optimisation time of the \lEA on \ONEMAX is
\[
\mathord{\Theta}\mathord{\left(n \cdot \frac{\lambda \log \log \lambda}{\log \lambda} + n \log n\right)}
\]
where the upper bound holds for $\lambda = O(n^{1-\varepsilon})$ and the lower bound holds for $\lambda = O(n)$.
\end{theorem}
We show that the parallel black-box complexity is lower than the bound
from Theorem~\ref{the:He-Chen-Yao} for large~$\lambda$ by a factor of
order $\log \log \lambda$.
\begin{theorem}
\label{the:black-box-complexity-onemax}
For any $\lambda \le e^{\sqrt{n}}$ the $\lambda$-parallel unbiased unary black-box complexity for any function with a unique optimum is at least
\[
\mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+ \lambda} + n \log n\right)}.
\]
This bound is tight for \OM, where the cut-off point is
\[
\lambda^*_{\OM} = \Theta(\log(n) \cdot \log \log n).
\]
The corresponding parallel time for an optimal algorithm is $\mathord{\Omega}\mathord{\left(\frac{n}{\ln^+ \lambda} + \frac{n \log n}{\lambda}\right)}$.
\end{theorem}
Note that the cut-off point is higher than the cut-off point for the
\lEA with the standard mutation rate $p=1/n$
from~\eqref{eq:cutoff-for-1+lambda-on-OneMax} and~\citet{Jansen2005a}.
For the proof we consider the progress made during a round of
$\lambda$ variations in terms of a potential function defined in the following. The following definitions and arguments, including several lemmas shown in the following, will also be used in Section~\ref{sec:tail-bounds} to prove lower bounds that hold with overwhelming probability.
Without loss of generality, we assume that the search point $1^n$ is
the optimum. Following \citet{Lehre2012}, we assume a ``mirrored''
sampling process, where every time a bit string~$x$ is queried (including
in the initial generation), the algorithm queries the complement
bit string $\overline{x}$ for ``free''. This makes sense as the complement of any bit string can be generated by flipping all bits. Thus we have to consider the progress towards the global optimum as well as the progress towards its complement.
\begin{definition}
\label{def:progress-measures}
Let $s_0^t$ be the minimum number of zeros in all search points queried in all steps up to time~$t$.
For all $s_0^t \le m \le n-s_0^t$ and $r \in \{0, \dots, n\}$ we define the random variable $\Delta_0(s_0^t, m, r) := \max\{0, s_0^t - |y|_0\}$ where $y$ is a random search point obtained by applying unbiased variation with radius~$r$ to a search point with $m$ zeros.
Define $s_1^t$ and $\Delta_1$ symmetrically with respect to the number of ones.
Due to mirrored sampling, we always have $s_0^t = s_1^t$, hence we simply write $s^t$ or just $s$ if we refer to the current point in time.
Then we define the progress in terms of the potential as $\Delta(s, m, r) = \max\{\Delta_0(s, m, r), \Delta_1(s, m, r)\}$.
\end{definition}
Note in particular that for all $z \in \mathbb{N}$ we have
\begin{align}
\label{eq:bound-on-Delta}
\Prob{\Delta(s, m, r) \ge z}
&\le \Prob{\Delta_0(s, m, r) \ge z} + \Prob{\Delta_1(s, m, r) \ge z}
\end{align}
Also note that by symmetry of zeros and ones $\Delta_0(s, m, r)$ has the same distribution as $\Delta_1(s, n-m, r)$, hence it suffices to study the distribution of $\Delta_0$.
We also have for all $s, s \le m \le n-s, r$
\begin{equation}
\label{eq:Delta-symmetry}
\Delta_0(s, m, r) = \Delta_0(s, n-m, n-r)
\end{equation}
as flipping all bits (in the transition from $m$ to $n-m$) and then flipping all but $r$ bits in the variation has the same effect as flipping $r$ bits in the first place.
Hence it suffices to consider $\Delta_0(s, m, r)$ for $s \le m \le n/2$.
Now consider the progress $\Delta_0(s, m, r)$. Let $Z$ be the number of 0\nobreakdash-bits that flipped to~1, then
there are $r-Z$ new 0-bits that were originally~1. Therefore, the
number of 0\nobreakdash-bits in the new generated search point is $m-Z+(r-Z)$
where $Z$ can be described by the hypergeometric distribution with
parameters $n, m$ and $r$. We only make progress if the number of
0-bits in the new search point is less than $s$. Hence the progress
(decrease in $0$-potential) is
\begin{align*}
\Delta_0(s,m,r)=\;& \max\{ Z-(r-Z)+(s-m),0\}\\
=\;& \max\{ 2Z-r+s-m,0\}.
\end{align*}
We show a tail inequality for hypergeometric variables and use this to derive a progress bound for the 0-potential.
\begin{lemma}
\label{lem:hypergeometric-tail}
Let $Z$ be a hypergeometrically distributed random variable with parameters $n$ (number of balls), $m$ (number of red balls), and $r$ (number of balls drawn). For all ${z \in \N_0}$,
\[
\Prob{Z = z} \le \binom{r}{z} \cdot \frac{m^z}{n^z}
\le \left(\frac{4m}{n}\right)^z
\]
where the second inequality holds for $z \ge r/2$.
\end{lemma}
\begin{proof}
We assume $z \le m$ and $z \le r$ as otherwise $\Prob{Z = z} = 0$.
We further assume $z \ge 1$ as for $z=0$ the probability bound is~1 and the statement is trivial.
Now,
\begin{align}
\Prob{Z = z}
=\;& \binom{m}{z}\binom{n-m}{r-z}/\binom{n}{r}\notag\\
=\;& \frac{m!(n-m)!r!(n-r)!}{z!(m-z)!(r-z)!(n-m-r+z)!n!}\notag\\
=\;& \binom{r}{z} \cdot \frac{m!(n-m)!(n-r)!}{(m-z)!(n-m-r+z)!n!}\label{eq:progress-factorials}.
\end{align}
The fraction can be written as
\allowdisplaybreaks[0]
\begin{align*}
& \frac{m(m-1)\cdot \ldots \cdot (m-z+1)}{n(n-1)\cdot \ldots \cdot(n-z+1)} \cdot
\frac{(n-m)(n-m-1)\cdot \ldots \cdot(n-m-r+z+1)}{(n-z)(n-z-1) \cdot \ldots \cdot(n-r+1)
\end{align*}
\allowdisplaybreaks[4]
Since $z \le m$, the second fraction above is at most~1. The first fraction is at most $m^z/n^z$
as $(m-i)/(n-i) \le m/n$ for all $i \in \N$ and $m \le n$.
Plugging this into~\eqref{eq:progress-factorials} and using $\binom{r}{z} \le 2^r \le 2^{2z} = 4^z$ for $z \ge r/2$ yields
\[
\Prob{Z = z} \le \binom{r}{z} \cdot \frac{m^z}{n^z}
\le \left(\frac{4m}{n}\right)^z. \qedhere
\]
\end{proof}
The following lemma shows that for any radius~$r$ the probability of having a progress of~$z$ decreases exponentially with~$z$.
\begin{lemma}\label{lemma:improve-prob}
Let $s$ denote the current 0-potential.
If $s\leq m\leq n/8$, then for all $z\in\mathbb{N}$ and $r\in \{1, \dots, n\}$,
\[
\Prob{\Delta_0(s, m, r) = z} \le \left(\frac{1}{2}\right)^{z/2}.
\]
\end{lemma}
\begin{proof}
Applying Lemma~\ref{lem:hypergeometric-tail} to a hypergeometric random variable $Z$ with parameters $m$ and $r$ we have, for all $z \in \N_0$,
\begin{align*}
\Prob{\Delta_0(s, m, r) = z}
=\;& \Prob{Z =\frac{z + r + m - s}{2}}\\
\le\;& \left(\frac{4m}{n}\right)^{(z+r+m-s)/2} \le \left(\frac{1}{2}\right)^{z/2}. \qedhere
\end{align*}
\end{proof}
The following lemma gives another tail bound that will be used to exclude steps where a search point of potential $m \gg s$ is chosen for variation. The probability of having a positive progress decreases rapidly with growing $m-s$.
\begin{lemma}
\label{lem:applying-Chvatal}
For every $s \le m \le n/2$ and every $r \in \{1, \dots, n\}$
\[
\Prob{\Delta_0(s, m, r) > 0} \le \exp\left(-\frac{(m-s)^2}{2r}\right).
\]
\end{lemma}
\begin{proof}
We use Chv\'{a}tal's tail bound~\citep{Chvatal1979}:
$\Prob{Z\geq \E{Z}+r\delta} \leq
\exp(-2\delta^2r)$, where $\E{Z}=\frac{rm}{n}$.
\begin{align*}
& \Prob{\Delta_0(s,m,r)>0}\\
= \;& \Prob{Z > \frac{r+m-s}{2}}\\
= \;& \Prob{Z > \frac{r m}{n} +r \cdot \left(\frac{r+m-s}{2r}-\frac{m}{n}\right)}\\
\leq\;& \Prob{Z \geq \frac{rm}{n} + r \cdot \left(\frac{m-s}{2r}\right)}\\
\leq\;& \exp\left(-2r \left(\frac{m-s}{2r}\right)^2\right)
= \exp\left(-\frac{(m-s)^2}{2r}\right).\qedhere
\end{align*}
\end{proof}
Putting all lemmas together shows that the expected progress is at most logarithmic in $\lambda$.
\begin{lemma}
\label{lem:progress-m}
Let $\Delta_0^{(\lambda)} = \Delta_0^{(\lambda)}(s, m_i, r_i)$ be the maximum of $\lambda$ random variables $\Delta_0(s, m_i, r_i)$ for arbitrary $s \le m_i \le n/2$ and $r_i$, $1\le i\le\lambda$.
For $s \le n/16$ we have
$\E{\Delta_0^{(\lambda)}} = \Oh {\log(\lambda)}$.
\end{lemma}
\begin{proof}
If $n/8 < m_i \le n/2$ then by Lemma~\ref{lem:applying-Chvatal}
\begin{align*}
& \Prob{\Delta_0(s,m_i,r_i)>0} \le e^{-n^2/(512r_i)} \le e^{-\Omega(n)}.
\end{align*}
This means that the probability of making any progress is
exponentially small, for any~$r_i$. Thus $\E{\Delta_0^{(\lambda)}}$ is maximised if we assume that $m_i\le n/8$ for all~$i$.
Under this assumption, applying Lemma~\ref{lemma:improve-prob}, for all $z \in \N_0$,
\begin{align*}
\Prob{\Delta_0(s, m_i, r_i) = z} \le\;& \left(\frac{1}{2}\right)^{z/2}
\end{align*}
hence $\E{e^{\eta \Delta_0(s, m_i, r_i)}} \le D$
for $\eta := \ln(4/3)$ and $D := 9+6\sqrt{2}$. Applying Lemma~\ref{lemma:mgf-max-bound} proves $ \E{\Delta_0^{(\lambda)}} = \Oh{\log \lambda}$.
\end{proof}
Now we are in a position to prove Theorem~\ref{the:black-box-complexity-onemax}.
\begin{proof}[Proof of Theorem~\ref{the:black-box-complexity-onemax}]
The upper bound for \OM will be shown later in
Theorem~\ref{the:upper-bound-lea-adaptive-mutation}. The lower
bound $\Omega(n \log n)$ follows from unbiased unary black-box
complexity~\citep{Lehre2012}. Hence, it suffices to prove
the lower bound $\Omega(\lambda n/\ln^+ \lambda)$ for $\lambda \ge 3$, where $\ln^+ \lambda$ can be replaced by $\ln \lambda$.
Consider any $\lambda$-parallel unary unbiased black-box algorithm.
We grant the algorithm an advantage by revealing all search points with Hamming distance at least $n/16$ to both $0^n$ and $1^n$ at no cost. Hence the potential is always $s \le n/16$.
Let $\Delta_0^{(\lambda)}$ be the progress due to reduction of the
$0$\nobreakdash-potential in one step, and $\Delta_1^{(\lambda)}$ be the progress due to reduction of
the $1$\nobreakdash-potential.
By virtue of the symmetry of $\Delta_0$ and $\Delta_1$, Lemma~\ref{lem:progress-m} also applies to $\Delta_1^{(\lambda)}$.
Hence the expected change in
potential per round is no more than
\[
\E{\Delta_0^{(\lambda)}} + \E{\Delta_1^{(\lambda)}} = O(\log \lambda).
\]
Hence, by the additive drift theorem~\citep{He2004}, the expected number of rounds
until one of the search points $0^n$ or $1^n$ is obtained is
$\Omega(n/\log \lambda)$. Multiplying by~$\lambda$ proves the claim.
\end{proof}
\section{An Optimal Parallel Black-Box Algorithm for OneMax}
\label{sec:optimal-algorithm-onemax}
The following theorem shows that the lower bound on the black-box complexity from Theorem~\ref{the:black-box-complexity-onemax} is tight.
We show that the \lEA has a better optimisation time if the mutation rate is chosen adaptively, according to the current best fitness. This is similar to common ideas from artificial immune systems, particularly the clonal selection algorithm. Adaptive mutation rates for \OM have been studied by~\citet{Zarges2008}, however the standard parameters for the clonal selection algorithm were too drastic to even obtain polynomial running times. Better results were obtained when using a population-based adaptation~\citep{Zarges2009}.
The following result reveals an optimal choice for the mutation rate of the \lEA, depending on~$n$ and $\lambda$.
\begin{theorem}
\label{the:upper-bound-lea-adaptive-mutation}
On OneMax, the expected number of function evaluations of the \lEA with an adaptive mutation rate $p = \max\{\ln(\lambda)/(n\ln(en/i)), \ 1/n\}$, where $i$ is the number of zeros in the current search point, for any $\lambda \le e^{\sqrt{n}}$, is at most
\[
\Oh{\frac{\lambda n}{\ln \lambda} + n \log n}.
\]
The parallel time (number of generations) is $\Oh{\frac{n}{\ln \lambda} + \frac{n \log n}{\lambda}}$.
\end{theorem}
\begin{proof}
Let $i$ be the current number of zeros and $p$ be the mutation rate. The probability of decreasing the number of zeros by any~$k \in \N$ with $k \le i$ is at least
\begin{align*}
\Prob{\Delta \ge k} \ge\;& \binom{i}{k} \cdot p^{k} \cdot (1-p)^{n-k}\\
\ge\;& \frac{i^{k}}{k^k} \cdot p^{k} \cdot (1-p)^{n-k}
= (1-p)^{n-k} \cdot \left(\frac{ip}{k}\right)^{k}.
\end{align*}
Then the probability that one of $\lambda$ offspring will decrease the number of zeros by at least $k$ is at least, using $1-(1-p)^\lambda \ge 1-e^{-p\lambda} \ge 1 - 1/(1+p\lambda) = p\lambda/(1+p\lambda)$,
\begin{align*}
\Prob{\Delta_{(\lambda)} \ge k} \ge 1-(1-\Prob{\Delta \ge k})^\lambda
\ge\;& \frac{\lambda (1-p)^{n-k} \cdot (ip/k)^{k}}{1 + \lambda (1-p)^{n-k} \cdot (ip/k)^{k}}.
\end{align*}
Hence for any $k \le i$ the expected drift is at least
\begin{align*}
\E{\Delta_{(\lambda)}} \ge\;& k \cdot \frac{\lambda (1-p)^{n-k} \cdot (ip/k)^{k}}{1 + \lambda (1-p)^{n-k} \cdot (ip/k)^{k}}.
\end{align*}
For $i > en/\ln \lambda$, which implies $pn > 1$, we set $k := pn = \ln(\lambda)/\ln(en/i)$. We have $k \le i$ since $k \le \ln(\lambda) \le \sqrt{n} \le en/\ln \lambda$.
We use $k := 1$ for $i \le en/\ln \lambda$, the realm where $p=1/n$.
This results in the following drift function~$h$:
\[
h(i) := \begin{cases}
\frac{\lambda (1-1/n)^{n-1} \cdot i/n}{1 + \lambda (1-1/n)^{n-1} \cdot i/n} & \text{if $i \le en/\ln \lambda$}\\
pn \cdot \frac{\lambda (1-p)^{n-pn} \cdot (i/n)^{pn}}{1 + \lambda (1-p)^{n-pn} \cdot (i/n)^{pn}} & \text{otherwise}
\end{cases}
\]
We estimate the number of function evaluations by multiplying the number of generations by~$\lambda$. The number of generations is estimated using Johannsen's variable drift theorem~\citep{Johannsen2010} in the variant from~\citet{Rowe2013}, with the above function~$h$. This gives an upper bound of
\begin{align*}
\frac{\lambda}{h(1)} + \int_1^n \frac{\lambda}{h(i)} \ \mathrm{d}i
=\;& \frac{1 + \lambda (1-1/n)^{n-1} \cdot 1/n}{(1-1/n)^{n-1} \cdot 1/n} + \lambda \int_1^n \frac{1}{h(i)} \ \mathrm{d}i\\
\le\;& \lambda + en + \lambda \int_1^{en/\ln \lambda} \frac{1}{h(i)} \ \mathrm{d}i + \lambda \int_{en/\ln \lambda}^n \frac{1}{h(i)} \ \mathrm{d}i.
\end{align*}
The first terms are at most
\begin{align*}
\;& \lambda + en + \lambda \int_1^{en/\ln\lambda} \frac{1 + \lambda (1-1/n)^{n-1} \cdot i/n}{\lambda (1-1/n)^{n-1} \cdot i/n} \ \mathrm{d}i\\
\le \;& \frac{\lambda en}{\ln \lambda} + en \left(1 + \int_1^{en/\ln\lambda} \frac{1}{i}\ \mathrm{d}i\right)
\le \frac{\lambda en}{\ln \lambda} + en \cdot (2+\ln n).
\end{align*}
The second integral is bounded as
\begin{align*}
\;& \int_{en/\ln\lambda}^n \frac{1 + \lambda (1-p)^{n-pn} \cdot (i/n)^{pn}}{pn \cdot (1-p)^{n-pn} \cdot (i/n)^{pn}} \ \mathrm{d}i\\
\le\;& \int_{0}^n \frac{\lambda \ln(en/i)}{\ln \lambda} \ \mathrm{d}i + \frac{1}{\ln \lambda} \int_{en/\ln\lambda}^n \frac{\ln(en/i)}{e^{-pn} \cdot (i/n)^{pn}} \ \mathrm{d}i\\
=\;& \frac{2\lambda n}{\ln \lambda} + \frac{1}{\ln \lambda} \int_{en/\ln\lambda}^n \ln(en/i) \cdot (en/i)^{pn} \ \mathrm{d}i\\
=\;& \frac{2\lambda n}{\ln \lambda} + \frac{1}{\ln \lambda} \int_{en/\ln\lambda}^n \ln(en/i) \cdot \lambda \ \mathrm{d}i
\le \frac{3\lambda n}{\ln \lambda}.
\end{align*}
Together, we get an upper bound of
$(3+e)\lambda n/\ln(\lambda) + en \cdot (2+\ln n)$.
\end{proof}
Note that the optimal mutation rate $p = \max\{\ln(\lambda)/(n\ln(en/i)), \ 1/n\}$, in particular the functional relationship between the mutation rate and the current fitness~$i$, is quite hard to guess through experimentation and was only revealed through the present theoretical analysis.
After the result from Theorem~\ref{the:upper-bound-lea-adaptive-mutation} was first published~\citep{Badkobeh2014}, \citet*{Doerr2018} presented a self-adjusting scheme for choosing the mutation rate in the \lEA and showed that it is able to match the upper bound from Theorem~\ref{the:upper-bound-lea-adaptive-mutation} without knowing the functional relationship between the mutation rate and the current fitness.
\section{Tail Bounds}
\label{sec:tail-bounds}
In this section we now show that the lower bound for all $\lambda$\nobreakdash-parallel unbiased unary black-box algorithms from Theorem~\ref{the:black-box-complexity-onemax} holds with high probability. In particular, it also applies to (non-parallel) unbiased unary black-box algorithms, for which only lower bounds on the expectation were known before~\citep{Lehre2012}. Our main result is as follows.
\begin{theorem}
\label{the:main-result-tail-bounds}
For every unary unbiased $\lambda$-parallel black-box algorithm $\mathcal{A}$ and every constant $0 < \delta < 1$, with probability $1-\exp(-\Omega(n^{\delta}/\log n))$ $\mathcal{A}$ does not find any target set of at most $\exp(o(n^{\delta}/\log n))$ search points within time
\[
\max\left\{\frac{\lambda n}{60\ln^+\lambda}, (1-\delta) n \ln n\right\}
= \mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+\lambda} + n \ln n\right)}.
\]
The expected time also satisfies the asymptotic bound.
\end{theorem}
Theorem~\ref{the:main-result-tail-bounds} establishes very general limits to the performance of large classes of algorithms, including mutation-only evolutionary algorithms with standard mutation operator, local search, simulated annealing.
In particular, putting $\delta := 0.01$ (say), Theorem~\ref{the:main-result-tail-bounds} shows that every unary unbiased search algorithm needs to be run for at least $n \ln n$ evaluations as the probability of finding one of few global optima within $0.99n \ln n$ evaluations is overwhelmingly small. The same holds for $\lambda$-parallel unary unbiased algorithms like mutation-only evolutionary algorithms with offspring populations of size~$\lambda$. Here stopping a run before $\lambda n/(60 \ln^+ \lambda)$ evaluations is futile as with overwhelming probability no optimum will have been found yet.
In addition, Theorem~\ref{the:main-result-tail-bounds} makes a statement about a target set of up to exponential size. This means that the lower bounds also apply to functions with many global optima, with respect to the optimisation time, but it can also be used to bound the time to find local optima or any set of high-fitness individuals of size at most $\exp(o(n^{\delta}/\log n))$. Illustrative applications to a broad range of well-known problems will be given in Section~\ref{sec:applications-of-tail-bounds}.
Theorem~\ref{the:main-result-tail-bounds} will be shown by separately showing lower bounds of $\Omega(\lambda n/\log \lambda)$ and $\Omega(n \log n)$ that both hold with overwhelming probability. Throughout this section we again assume ``mirrored'' sampling, i.\,e.\ every queried search point~$x$ also evaluates $\overline{x}$ for free.
\subsection{Lower Bound \boldmath$\Omega(\lambda n/\log \lambda)$ with overwhelming probability}
We start with a bound of $\Omega(\lambda n/\log \lambda)$. Recall from
Definition~\ref{def:progress-measures} that due to mirrored sampling,
we can define the potential as the minimum number zeros, or equivalently
number of ones, in all search points up to time $t$. In order to use
Theorem~\ref{theorem:general-drift-theorem} for a tail bound on the runtime, we
need to study the mgf. of the progress
\begin{align*}
\Delta^{(\lambda)}(s) :=
\max \left\{\Delta^{(\lambda)}_0(s,m,r),\Delta^{(\lambda)}_1(s,m,r)\right\},
\end{align*}
where $\Delta^{(\lambda)}_0(s,m,r)$ is the maximal progress in the
0-potential, and $\Delta^{(\lambda)}_1(s,m,r)$ is the maximal progress
in the 1-potential, given current potential $s$, where the selected
search point has $m$ 0-bits, respectively 1-bits, and $r$ bits are
flipped.
\begin{lemma}\label{lemma:mgf-bound}
Let $s$ denote the current potential.
If $s\leq \frac{n}{8}$ and
$\gamma:=\ln\left(\frac{3}{4}\sqrt{2}\right)$, then
$\expect{e^{\gamma\Delta^{(\lambda)}(s)}} \leq 8\lambda$.
\end{lemma}
\begin{proof}
As noted in Definition~\ref{def:progress-measures} and
(\ref{eq:Delta-symmetry})
\begin{align*}
\Delta_1(s,m,r) = \Delta_0(s,n-m,r) = \Delta_0(s,m,n-r).
\end{align*}
Hence, by a union bound
\begin{multline*}
\prob{\Delta(s,m,r)=z}
\leq \prob{\Delta_0(s,m,r)=z} +
\prob{\Delta_1(s,m,r)=z}\\
= \prob{\Delta_0(s,m,r)=z} +
\prob{\Delta_0(s,m,n-r)=z}
\leq 2^{1-z/2}
\end{multline*}
where the last inequality follows by Lemma \ref{lemma:improve-prob}.
We now have
\begin{align*}
\expect{e^{\gamma\Delta^{(\lambda)}(s, m, r)}}
& = \sum_{z=0}^\infty \prob{\Delta^{(\lambda)}=z}e^{\gamma z},\\
\intertext{by a union bound over $\lambda$ parallel runs}
& \leq \sum_{z=0}^\infty \lambda\max_{r\in[n],s\leq m}\prob{\Delta(s,m,r)=z}e^{\gamma z}\\
\intertext{the definition of $\gamma$ give}
& \leq \lambda \sum_{z=0}^\infty 2\left(\frac{1}{2}\right)^{z/2}\left(\frac{3}{4}\sqrt{2}\right)^z\\
& = \lambda \sum_{z=0}^\infty 2\left(\frac{3}{4}\right)^z = 8\lambda.\qedhere
\end{align*}
\end{proof}
\begin{theorem}
\label{the:tail-bound-lambda-term}
If $\lambda\geq 1$, then $\Prob{T<\frac{\lambda n}{60\ln^+\lambda}}=e^{-\Omega(n)}$.
\end{theorem}
\begin{proof}
Following the proof of
Theorem~\ref{the:black-box-complexity-onemax}, we assume without
loss of generality that the search point $1^n$ is the optimum, and
let $(X_t)_{t\in\mathbb{N}}$ be the potential as defined before.
We apply the last part of Theorem~\ref{theorem:general-drift-theorem} (iv),
with the parameters $g(x):= x$, $\xmin:=1$, $\xmax:=n$, $a:=0,$
$S:=\{0\}\cup[\xmin,\xmax]$, and $\beta_l(t):=8\lambda$, for all
$t\in\mathbb{N}$. We consider the number of \emph{parallel runs}
$T'$ until the process reaches potential $a=0$.
Define $c:=\frac{3}{10}\gamma$ where
$\gamma:=\ln\left(\frac{3}{4}\sqrt{2}\right)$.
By Lemma~\ref{lemma:mgf-bound}
\begin{align*}
\expect{e^{\gamma (g(X_t)-g(X_{t+1}))}\filtcond{ X_t > a}}
\leq\;& \expect{e^{\gamma\Delta^{(\lambda)}(s)}}
\leq 8\lambda = \beta_\ell(t)
\end{align*}
Furthermore, by the definition of the process, the set $S\cap\{x\mid
x\leq a\}=\{0\}$ is absorbing, thus for $t:=\frac{cn}{\ln^+\lambda}$,
\begin{align*}
\prob{T'<t\mid X_0>0}
&\leq \left(\prod_{i=0}^{t-1} \beta_{\mathrm{\ell}}(i)\right)\cdot e^{-\gamma (g(X_0)-g(a))}\\
&< (8\lambda)^t\cdot e^{-\gamma n}\\
& = (8\lambda)^{\frac{c n}{\ln^+\lambda}}\cdot e^{-\gamma n}\\
& = e^{\left(\frac{c n}{\ln^+\lambda}\right)\ln(8\lambda)-\gamma n}
\intertext{using that $\ln(8\lambda)=\ln(\lambda)+3\ln(2)\leq 3\ln^+\lambda$ gives}
& \leq e^{(3c-\gamma)n}\\
& = e^{-\gamma n/10}.
\end{align*}
The result follows by taking into account that the algorithm makes
$\lambda$ fitness evaluations per iteration, \ie, $T=\lambda T',$
and that $c>1/60.$
\end{proof}
\subsection{Lower Bound \boldmath$\Omega(n \log n)$ with overwhelming probability}
Now we show a lower bound of $\Omega(n \log n)$ with overwhelming probability. Note that this result is independent of $\lambda$ and thus unrelated to parallel black-box complexity; it gives limitations for general (parallel or non-parallel) unary unbiased black-box algorithms. Recall that every $\lambda$-parallel unary unbiased algorithm is also a unary unbiased algorithm, hence the result applies to a strictly larger class of algorithms. Previously only lower bounds on the expectation were known: \citet{Lehre2012} showed an asymptotic bound of $\Omega(n \log n)$ and~\citet{Doerr2016a} presented a more precise lower bound of $n \ln n - O(n)$.
\begin{theorem}
\label{the:n-log-n-bound}
For every unary unbiased black-box algorithm $\mathcal{A}$ and every constant $0 < \delta \le 1$, the probability that $\mathcal{A}$ finds any fixed target search point $x^*$ within $(1-\delta)n \ln n$ steps is $\exp(-\Omega(n^{\delta}/\log n))$.
\end{theorem}
Before presenting the proof of Theorem~\ref{the:n-log-n-bound}, we present the main idea behind the proof, and the challenges to overcome.
The proof will be based on the following well-known ``coupon collector'' argument that we discuss first for a simple algorithm such as Randomised Local Search (RLS) or the (1+1)~EA. For these algorithms, we can argue that with high probability there will be $cn$ bits in the initial search point that differ from the optimum, for an appropriate constant $0 < c < 1/2$.
Each such bit has a probability of $1/n$ of being flipped in each step of the algorithm. For a time period of $T:= (1-\delta) (n-1) \ln n$ steps, the probability that any fixed bit is never being flipped is at least
\[
\left(1-\frac{1}{n}\right)^T \ge \left(1-\frac{1}{n}\right)^{(1-\delta)(n-1) \ln n} \ge
n^{-(1-\delta)}
\]
using $(1-1/n)^{n-1} \ge 1/e$. Now the probability that there is a bit among the $cn$ incorrect bits that is never being flipped is at least
\[
\left(1 - n^{-(1-\delta)}\right)^{cn} \le \exp(-cn^{\delta}).
\]
This implies that with the above probability the optimum has not been found in $T = \Omega(n \log n)$ steps.
This argument works for RLS and the (1+1)~EA for the following reasons:
\begin{enumerate}
\item The algorithms evolve a single lineage from the initial search point, which allows us to argue with ``incorrect'' bits that need to be flipped at least once.
\item The same variation operator is applied at all times, which establishes the formula $(1-1/n)^T$.
\item All bits are treated independently, which is implicitly used in the derivation of the term $(1-n^{-(1-\delta)})^{cn}$.
\end{enumerate}
In order to prove Theorem~\ref{the:n-log-n-bound}, we have to consider \emph{all} unary unbiased black-box algorithms, for which the above properties do not hold. In particular, algorithms may easily generate several lineages. This makes it unclear how ``incorrect'' bits can be defined.
Also note that an algorithm might flip many ``incorrect'' bits in one step simply by choosing a very large radius. So the simple argument that we need to flip all incorrect bits at least once breaks down.
Algorithms may choose different variation operators at different times, possibly depending on fitness values generated so far. This makes it difficult to argue that no variation flips a bit over a period of time. Finally, mutations with a fixed radius $r \ge 2$ may introduce dependencies between bits, which needs to be addressed.
We tackle these challenges as follows.
Assume w.\,l.\,o.\,g.\ that $x^* = 1^n$.
We give away knowledge of all search points $x$ that have Hamming distance at least $n^* := n/(2^{13}\ln n)$ to both $0^n$ and $1^n$. Hence we start with a potential of $s = n^*$. Moreover, whenever the algorithm decreases the potential from $s$ to $s' < s$, we grant the algorithm knowledge of all solutions with Hamming distance at least $s'$ from both $0^n$ and $1^n$. This assumption implies that the current knowledge of the algorithm can be fully described by the current potential, and the progress of the algorithm can be bounded by considering the transitions of the potential.
Note that all solutions with the same potential are isomorphic to the algorithm. Pick a set of $n^*$ bit positions, w.\,l.\,o.\,g.\ the first $n^*$ ones. We define these bits as ``incorrect'' bits that need to be set to 1 in order to reach the optimum. Since the behaviour of the algorithm is fully determined by the current potential, and the bit positions are irrelevant for transitions between potential values, we may assume w.\,l.\,o.\,g.\ that the algorithm, whenever performing a variation of a search point $x_t$ with $\ones{x_t}$ ones, it always picks $x_t$ from the set of all search points with $\ones{x_t}$ ones such that $\min\{n-\ones{x_t}, n^*\}$ ``incorrect'' bit positions have value~0. Such a search point always exists as otherwise the potential would be less than $\ones{x_t}$ at time~$t$, which is a contradiction.
Now variations that decrease the potential by decreasing the number of zeros will fix some of the incorrect bits accordingly. Variations that do not decrease the potential only create search points that are already known and thus can be ignored as they have no effect. Hence we require that these incorrect bits are flipped \emph{in variations that decrease the potential}.
Having laid the foundation for arguing with ``incorrect'' bits being fixed, we now show that with overwhelming probability, $\mathcal{A}$ does not find $1^n$ within $T := (1-\delta)(n-1) \ln n$ steps.
Note that $\mathcal{A}$ can choose the radius in each step. We distinguish between single-bit variations where $r=1$ (or, symmetrically, $r=n-1$) and multi-bit variations where $2 \le r \le n-2$. We first show that in at most~$T$ steps with multi-bit variations, not too many incorrect bits are being fixed. Then we show later that at most $T$ single-bit variations are not enough to fix all incorrect bits that are not being fixed by multi-bit variations. Note that the algorithm can interleave single-bit variations and multi-bit variations arbitrarily. Our arguments work for arbitrary sequences of single-bit and multi-bit variations; they even hold if the algorithm is allowed to make $T$ single-bit variations \emph{and} $T$ multi-bit variations at the cost of $T$ queries.
The following lemma considers multi-bit variations and bounds transition probabilities of the potential.
\begin{lemma}
\label{lem:bounding-probability-of-progress}
Let $s \le n^*$ for $n^* := n/(2^{13}\ln n)$, then for every $m \in [s, 2n^*] \cup [n-2n^*, n-s]$, every radius $2 \le r \le n-2$ and every $1 \le z \le n$ we have
\[
\Prob{\Delta_0(s, m, r) = z} \le \left(\frac{16n^*}{n}\right)^2 \cdot 2^{-z}.
\]
If $2n^* < m < n-2n^*$ we have
\[
\Prob{\Delta_0(s, m, r) = z} \le e^{-\Omega({n^*}^2/n)}.
\]
\end{lemma}
\begin{proof}
Recall that by~\eqref{eq:Delta-symmetry} it
suffices to consider the case $m \le n/2$.
If $2n^* \le m \le n/2$ then by Lemma~\ref{lem:applying-Chvatal}
\[
\Prob{\Delta_0(s,m,r) > 0}
\leq \exp\left(-\frac{(m-s)^2}{2r}\right) = e^{-\Omega({n^*}^2/n)}.
\]
Now assume $s \le m \le 2n^*$.
As shown in the proof of Lemma~\ref{lem:progress-m},
\[
\Prob{\Delta_0(s, m, r) = z} \le \left(\frac{4m}{n}\right)^{(z+r+m-s)/2}
\le \left(\frac{8n^*}{n}\right)^{(z+r)/2}
\]
We claim that the above is bounded by $\left(\frac{4n^*}{n}\right)^2 \cdot 2^{-z}$ for all $z \ge 1$ and $r \ge 2$.
Note that for $z=1$ and $r=2$ we have $\Prob{\Delta_0(s, m, r) = z} = 0$ as the progress must be an even number.
For $z=1$ and $r \ge 3$ we get
\[
\left(\frac{8n^*}{n}\right)^{(z+r)/2} \!\! = \left(\frac{8n^*}{n}\right)^2 \cdot \left(\frac{8n^*}{n}\right)^{(r-3)/2}
\le \left(\frac{16n^*}{n}\right)^2 \cdot 2^{-1}.
\]
For $z=2$ we get
\[
\left(\frac{8n^*}{n}\right)^{(z+r)/2} \!\! = \left(\frac{8n^*}{n}\right)^2 \cdot \left(\frac{8n^*}{n}\right)^{(r-2)/2}
\le \left(\frac{16n^*}{n}\right)^2 \cdot 2^{-2}.
\]
For $z \ge 3$ we have, using $(8n^*/n)^{1/2} \le 1/2$,
\[
\left(\frac{8n^*}{n}\right)^{(z+r)/2} \!\! \le \left(\frac{8n^*}{n}\right)^2 \cdot \left(\frac{8n^*}{n}\right)^{z/2}
\le \left(\frac{8n^*}{n}\right)^2 \cdot 2^{-z}. \qedhere
\]
\end{proof}
Using Lemma~\ref{lem:bounding-probability-of-progress} now allows us to express the progress of any algorithm using stochastic domination and a combination of two simple random variables:
\begin{lemma}
\label{lem:dominating-distribution}
Let $s \le n^*$ for $n^* := n/(2^{13}\ln n)$, then for every $s \le m \le n-s$ and every radius $2 \le r \le n-2$ the progress $\Delta(s, m, r)$ is stochastically dominated by
\[
2X_t Y_t
\]
where $X_t \in \{0, 1\}$ is a Bernoulli variable with $\Prob{X_t = 1} = \left(\frac{16n^*}{n}\right)^2$ and $Y_t$ is a geometric random variable with parameter $1/2$, $X_t$ and $Y_t$ being independent of each other and independent of other time steps $t' \neq t$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:bounding-probability-of-progress} and the definition of $X_t, Y_t$,
\[
\Prob{\Delta_0(s, m, r) = z} \le \left(\frac{16n^*}{n}\right)^2 \cdot 2^{-z} = \Prob{X_t Y_t = z}
\]
for every~$z \ge 1$ and all $m \in [s, 2n^*] \cup [n-2n^*, n-s]$. The same clearly also holds in case $2n^* < m < n-2n^*$ by the second statement of Lemma~\ref{lem:bounding-probability-of-progress}.
The probability bounds for $\Delta_0$ also apply to $\Delta_1$ by symmetry of zeros and ones, and thus by the union bound $\Prob{\Delta(s, m, r) \ge z} \le \Prob{\Delta_0(s, m, r) \ge z} + \Prob{\Delta_1(s, m, r) \ge z}$ we get
$
\Prob{\Delta(s, m, r) \ge z} \le 2 \cdot \Prob{X_t Y_t \ge z}
$.
\end{proof}
We use Lemma~\ref{lem:dominating-distribution} to show tail bounds for the progress made in multi-bit variations. The following lemma shows that at most half of the incorrect bits are being fixed by multi-bit variation steps, even when considering a time span of $n \ln n$ steps instead of $(1-\delta)n \ln n$.
\begin{lemma}
\label{lem:tail-bound-for-multi-bit-variations}
Let $n^* := n/(2^{13}\ln n)$. Within $T := n \ln n$ multi-bit variation steps at most $n^*/2$ incorrect bits are being fixed, with probability
$1-2^{-\Omega(n/\log n)}$.
\end{lemma}
\begin{proof}
We give a tail bound for the sum of variables $X_t Y_t$ defined in Lemma~\ref{lem:dominating-distribution}; by stochastic domination, the tail bound then also holds for the real progress. Recall that $X_t$ as well as $Y_t$ are both sequences of iid variables and that all variables are mutually independent.
By Chernoff bounds, with overwhelming probability the number of $X_t$ variables attaining value~1 is bounded by at most twice its expectation:
\begin{align*}
\Prob{\sum_{t=1}^T X_t \ge 2T \left(\frac{16n^*}{n}\right)^2} \le\;& \exp\left(-\frac{T}{3} \left(\frac{16n^*}{n}\right)^2\right)
= e^{-\Omega(n/\log n)}.
\end{align*}
If $\sum_{t=1}^T X_t \le \left\lfloor 2T \left(\frac{16n^*}{n}\right)^2 \right\rfloor =: k$ then there are at most $k$ variables $Y_t$ that contribute to $\sum_{t=1}^T X_t Y_t$. For ease of notation, we assume that these are variables $Y_1, \dots, Y_k$.
We apply Chernoff bounds for sums of geometric random variables~\cite[Theorem~3]{Doerr2011d} to bound the contribution of $k$ variables $Y_1, \dots, Y_k$. Note that ${\E{\sum_{t=1}^k Y_t} = 2k}$.
\begin{align*}
\Prob{\sum_{t=1}^k Y_t \ge 4k} \le\;& \exp\left(-\frac{k-1}{4}\right)
= e^{-\Omega(n/\log n)}.
\end{align*}
Hence if both ``typical'' events occur,
\begin{align*}
\sum_{t=1}^T 2X_t Y_t \le\; 8k
\le\;& 16T \cdot \frac{16^2 n^*}{n^2} \cdot n^*
= 16n \ln(n) \cdot \frac{2^{-5}}{n \ln n} \cdot n^*
=\; n^*/2.
\end{align*}
Taking the union bound for the two probabilities $2^{-\Omega(n/\log n)}$ that the typical events do not happen completes the proof.
\end{proof}
Now we are ready to give a proof for Theorem~\ref{the:n-log-n-bound}.
\begin{proof}[Proof of Theorem~\ref{the:n-log-n-bound}]
As explained earlier, it suffices to consider $n^*$ incorrect bits and to show that with the claimed probability not all of these bits will be fixed within $T$ unbiased variations.
Lemma~\ref{lem:tail-bound-for-multi-bit-variations} implies that with overwhelming probability there exist $n^*/2$ incorrect bits that are not being fixed by up to~$T$ multi-bit variations. We now use coupon collector argument (similar to those sketched earlier) to show that, in up to $T$ single-bit variations, with overwhelming probability these $n^*/2$ incorrect bits will not all be fixed.
The probability that any fixed bit~$i$ will not be flipped in a single-bit variation amongst the first $T$ steps is at least, using $(1-1/x)^{x-1} \ge 1/e$ for $x > 1$,
\[
\left(1 - \frac{1}{n}\right)^T
= \left(1 - \frac{1}{n}\right)^{(1-\delta)(n-1) \ln n}
\ge n^{-(1-\delta)}.
\]
Hence the probability that a fixed bit~$i$ will be flipped in at to $T$ single-bit variations is at least $1-n^{-(1-\delta)}$. Hence the probability that all of the $n^*/2$ incorrect bits are being flipped in $T$ steps is at most
\[
(1-n^{-(1-\delta)})^{n^*/2} \le \exp(-\Omega(n^{\delta}/\log n)). \qedhere
\]
\end{proof}
Theorems~\ref{the:tail-bound-lambda-term} and~\ref{the:n-log-n-bound} imply our main result, Theorem~\ref{the:main-result-tail-bounds}.
\begin{proof}[Proof of Theorem~\ref{the:main-result-tail-bounds}]
Fix a target search point $x^*$ from the target set.
By Theorem~\ref{the:tail-bound-lambda-term} the probability of finding $x^*$ within $\frac{\lambda n}{60\ln^+\lambda}$ steps is $\exp(-\Omega(n))$.
Applying Theorem~\ref{the:n-log-n-bound} with parameter $\delta$ yields that the probability of finding $x^*$ within $(1-\delta) n \ln n$ steps is $\exp(-\Omega(n^{\delta}/\log n))$.
By the union bound, the probability that one of these lower bounds does not apply is $\exp(-\Omega(n)) + \exp(-\Omega(n^{\delta}/\log n)) \le 2\exp(-\Omega(n^{\delta}/\log n))$. Repeating the above arguments for all target search points and using a union bound over
at most $\exp(o(n^{\delta}/\log n))$ search points yields an overall probability bound of
\begin{align*}
& \exp(o(n^{\delta}/\log n)) \cdot 2\exp(-\Omega(n^{\delta}/\log n)) \\
=\;& \exp(-\Omega(n^{\delta}/\log n)+ o(n^{\delta}/\log n) + \ln 2)\\
=\;& \exp(-\Omega(n^{\delta}/\log n)).
\end{align*}
Finally, the claimed equality
\[
\max\left\{\frac{\lambda n}{60\ln^+\lambda}, (1-\delta) n \ln n\right\}
= \mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+\lambda} + n \ln n\right)}
\]
follows from $\max\{x, y\} \ge (x+y)/2$ and $1-\delta = \Omega(1)$.
\end{proof}
\section{Black-Box Complexity Results for Illustrative Function Classes}
\label{sec:applications-of-tail-bounds}
In this section we give a number of examples of how to exploit the fact that our lower bounds apply to the time for finding an arbitrary target set of up to exponentially many search points. This leads to novel results for functions with many global optima, but can also be used to bound the time for reaching local optima or search points within a certain distance from any local or global optimum.
\subsection{Black-Box Complexity Lower Bounds for Functions with Many Optima}
Previous black-box complexity results like Theorem~\ref{the:black-box-complexity-onemax} or results on (non-parallel) unbiased black-box complexity~\cite{Lehre2012} were limited to functions with a unique optimum. These results apply to popular test functions like \onemax and \LO and function classes like linear functions or monotone functions~\cite{monotone-journal}. However, they do not apply when considering functions with more than one optimum.
Apart from tailored analyses for specific problems classes (e.\,g.\ problems from combinatorial optimisation~\citep{Doerr2013b}), we are not aware of any generic black-box complexity results that apply to functions with multiple optima.
Theorem~\ref{the:main-result-tail-bounds} overcomes this limitation, yielding novel black-box complexity results for the unary unbiased black-box complexity and its $\lambda$-parallel variant across a range of problems with several global optima, including some widely studied problem classes. These black-box complexity results give general limitations that can serve as baselines for performance comparisons and guide the search for the most efficient algorithms, including those using parallelism most effectively (as demonstrated successfully for \onemax in Section~\ref{sec:optimal-algorithm-onemax}).
There are many examples of relevant problem classes to which Theorem~\ref{the:main-result-tail-bounds} applies. The most obvious class is that of all functions with $\exp(o(n^{\delta}/\log n))$ optima. Note that when choosing, say, $\delta := 0.995$ then $\exp(n^{0.99}) \le \exp(o(n^{\delta}/\log n))$; the reader may choose to think of the latter expression as $\exp(n^{0.99})$ as this may be easier to digest.
Following~\citet{Witt2006}, the mentioned function class includes problems where all optima have at most $n^\delta/\log^3 n$ ones or at most $n^\delta/\log^3 n$ zeros.
This is because the number of such search points is bounded by
\begin{equation}
\label{eq:area-of-hypercube}
2\sum_{i=0}^{n^\delta/\log^3 n} \binom{n}{i} = \mathord{O}\mathord{\left(n^{n^\delta/\log^3 n}\right)}
= \exp(o(n^{\delta}/\log n)),
\end{equation}
where the last step used $n^{n^{\delta/\log^3 n}} = \exp(\Theta(n^{\delta/\log^2 n})) = \exp(o(n^{\delta/\log n}))$.
In the following we survey a number of illustrative problems that have been studied previously and for which we give the first black-box complexity results.
In terms of combinatorial problems, there are a lot of well-studied problems with a property called \emph{bit-flip symmetry}: flipping all bits gives a solution of the same fitness. This means that there are always at least two global optima. Such problems have been popular as search algorithms need to break the symmetry between good solutions~\citep{Goldberg2002}.
Well-known examples include the function $\twomax := \max\{\sum_{i=1}^n x_i, {\sum_{i=1}^n (1-x_i)}\}$ \citep{Goldberg2002}, which has been used as a challenging test bed in theoretical studies of diversity-preserving mechanisms~\citep{Oliveto2018,Covantes2018a,CovantesOsuna2018}.
The function \textsc{H-Iff} (Hierarchical If and only If)~\citep{Watson1998} consists of hierarchical building blocks that need to attain equal values in order to contribute to the fitness. It was studied theoretically~\citep{Dietzfelbinger2003,Goldman2016} and is frequently used in empirical studies, see, e.\,g.~\citep{Thierens2013,Goldman2015}.
In terms of classical combinatorial problems, the \textsc{Vertex Colouring} problem asks for an assignment of colours to vertices such that no two adjacent vertices share the same colour. For two colours, a natural setting is to use a binary encoding for the colours of all vertices and to maximise the number of bichromatic edges (edges with differently coloured end points). A closely related setting is that of simple Ising models, where the goal is to \emph{minimise} the number of bichromatic edges. For bipartite (that is, 2-colourable) graphs, this is identical to maximising the number of bichromatic edges as inverting one set of the bipartition turns all monochromatic edges into bichromatic ones and vice versa.
Previous theoretical work includes evolutionary algorithms on ring/cycle graphs~\citep{Fischer2005}, the Metropolis algorithm on toroids~\citep{Fischer2004} and evolutionary algorithms on binary trees~\citep{Sudholt2005}.
Other combinatorial problems with bit-flip symmetry include cutting and selection problems. Given an undirected graph, the problems \textsc{MaxCut} and \textsc{MinCut} seek to partition the graph into two non-empty sets such as to maximise or minimise the number of edges running between those two sets, respectively. Using a straightforward binary encoding for all vertices, this results in bit-flip symmetry and multiple optima. Theoretical studies of evolutionary algorithms on cutting problems include~\citet{Neumann2011} and~\citet{Sudholt2010}; the latter paper considers a simple instance of two equal-sized cliques that leads to two complementary optima.
Concerning selection problems, the well-known NP hard \textsc{Partition} problem asks whether it is possible to schedule a set of $n$ jobs on two identical machines such that both machines will have identical loads. An optimisation problem is obtained by trying to minimise the load of the fuller machine, also called the \emph{makespan}. A straightforward encoding is used: every bit indicates which machine the corresponding job should be assigned to.
\Citet{Witt2005} analysed the performance of the \EA for this problem, including random instance models where job sizes are drawn randomly from a real range, according to a uniform or an exponential distribution, respectively. In both cases such instances will almost surely have two complementary optima\footnote{More than two optima only exist if there are different combinations of job sizes (beyond symmetries) that add up to the same value. Since the weight of each job size is drawn from a continuous range and the number of values that could lead to equal values is finite, this almost surely never happens.}.
\Citet*{Wegener2005c} considered monotone polynomials: a sum of monomials (products of variables, e.\,g.\ $x_1 x_3 x_4$) with positive weights. Here $1^n$ is always a global optimum, but more optima can exist if there are variables that do not appear in any monomial: each such variable doubles the number of optima as it is not relevant for the fitness. Hence if there are $o(n^\delta/\log n)$ such variables then there are at most $2^{o(n^\delta/\log n)} \le \exp(o(n^\delta/\log n))$ optima.
\Citet*{Jansen2016} presented instance classes called \emph{nearest peak functions} and \emph{weighted nearest peak functions}. Both are defined with respect to an arbitrary number of peaks: search points with an associated height and slope. For nearest peak functions the fitness of a search point is determined by its closest peak: for the peak itself the fitness is equal to the height of the peak and for other search points the fitness decreases gradually with the distance from the peak, according to the slope of the peak. Weighted nearest peak functions are defined similarly, but all peaks are considered and higher peaks can dominate shallower peaks. This function class was introduced as a test bed allowing to create an arbitrary number of optima. It is shown in~\citet{Jansen2016} that the set of local optima is a subset of all peaks. Hence the number of peaks is an upper bound on the number of global (and local) optima. The two function classes were named Jansen-Zarges function classes in~\citet{Covantes2018a}, where they were used as benchmarks for the \emph{clearing} diversity mechanism.
Finally we consider random planted \textsc{Max-3-Sat} instances as a popular benchmark model in both experimental~\cite{Goldman2014} and theoretical studies~\cite{Sutton2014,Doerr2015a,Buzdalov2017}. The fitness function is the number of satisfied clauses and each clause contains exactly 3 literals
(negated or non-negated variables from the set $\{x_1, \dots, x_n\}$).
In this model, we fix a planted optimum $x^*$ and generate clauses independently such that they are satisfied by~$x^*$. This means that at least one literal needs to evaluate to \texttt{true} in~$x^*$. The variables for each clause are chosen uniformly at random (with or without replacement) from $\{x_1, \dots, x_n\}$. We may assume that instances are generated by first deciding which of the 3 literals will match~$x^*$ and which won't. In a second step, the indices of variables will be picked. We further assume that there is at least a constant probability $c_1$ of a clause having one matching literal and at least a constant probability $c_3$ of a clause having three matching variables\footnote{This is the case in~\cite{Sutton2014,Doerr2015a,Buzdalov2017} where implicitly $c_1 = 3/7$ and $c_3 = 1/7$ and in~\cite{Goldman2014} where $c_1=4/6$ and $c_3=1/6$. The latter probabilities favour clauses with only one matching literal in order not to give an obvious bias towards the values of~$x^*$. Note that we do not care about the value of $c_2$ (two matching literals).}.
In this setup, $x^*$ is a global optimum, but there may be more global optima. We argue that the number of optima is bounded if the number of clauses, $m$, is chosen large enough.
Consider a solution $x$ with Hamming distance $H := H(x, x^*)$ to~$x^*$.
We argue that for any clause, the probability that the clause will be satisfied under~$x$ is $\Omega(H/n)$. If $H \le n/2$ then with probability $c_1$ we will choose one matching literal and the probability that only the variable of this literal will be chosen among the $H$ ones that differ in $x$ and $x^*$ is $\Omega(H(n-H)^2/n^3) = \Omega(H/n)$. Likewise, if $H > n/2$ then with probability $c_3$ we will choose three matching literals and the probability that they are all different in $x$ and $x^*$ is $\Omega(H^3/n^3) = \Omega(H/n)$.
Now since all clauses are generated independently, the probability that all $m$ clauses are satisfied under~$x$ is $(1-\Omega(H/n))^m \le \exp(-\Omega(Hm/n))$.
Hence for all search points~$x$ with $H \ge n^\delta/\log^3 n$ the probability that $x$ is a global optimum is at most $\exp(-\Omega(n^{\delta}/(\log^3 n) \cdot m/n)) = \exp(-\Omega(n \log n))$ if the number of clauses is $m = \Omega(n^{2-\delta}\log^4 n)$. In this case, the probability that any such search point will be a global optimum is at most $2^n \cdot \exp(-\Omega(n \log n)) = \exp(-\Omega(n \log n))$, a failure probability so small that it can be absorbed in the failure probabilities for our tail bounds.
Now, with overwhelming probability the number of global optima is bounded by the number of search points with Hamming distance less than $n^\delta/\log^3 n$ from~$x^*$. By~\eqref{eq:area-of-hypercube}, this number is $\exp(o(n^{\delta}/\log n))$.
The following theorem summarises all the above.
\begin{theorem}
\label{the:main-result-tail-bounds-applications}
Every unary unbiased $\lambda$-parallel black-box algorithm $\mathcal{A}$
needs more than
\[
\max\left\{\frac{\lambda n}{60\ln^+\lambda}, (1-\delta) n \ln n\right\}
= \mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+\lambda} + n \ln n\right)}
\]
evaluations, with probability $1-\exp(-\Omega(n^{\delta}/\log n))$, to find a global optimum for all of the following settings.
\begin{enumerate}
\item All functions with $\exp(o(n^{\delta}/\log n))$ optima.
\item All functions where all optima have at most $n^\delta/\log^3 n$ ones or at most $n^\delta/\log^3 n$ zeros.
\item $\twomax := \max\{\sum_{i=1}^n x_i, \sum_{i=1}^n (1-x_i)\}$.
\item \textsc{H-Iff} (Hierarchical If and only If).
\item Vertex colouring/Ising model problems: maximising or minimising the number of bichromatic edges when trying to colour a connected bipartite graph with 2 colours.
\item \textsc{MinCut} instances with two equal-sized cliques.
\item \textsc{Partition} instances having two symmetric optimal solutions (which almost surely applies to random instances)
\item Monotone polynomials with positive weights where all but $o(n^\delta/\log n)$ variables appear in at least one monomial.
\item Jansen-Zarges nearest peak functions and weighted nearest peak functions with $\exp(o(n^\delta/\log n))$ peaks.
\item Random planted \textsc{Max-3-Sat} instances as described above with at least $m=\Omega(n^{2-\delta}\log^4 n)$ clauses.
\end{enumerate}
The expected time also satisfies the asymptotic bound.
\end{theorem}
\subsection{Lower Bounds on the Time to Reach Local Optima}
For many multimodal problems evolutionary algorithms are likely to need a much larger time than indicated by the lower bounds from Theorem~\ref{the:main-result-tail-bounds-applications}. When put in perspective, our bounds may appear to be quite loose for some of the harder problems considered. However, our lower bounds can also be applied to bound the time until any unary unbiased black-box algorithm has found a local optimum, or any search point of reasonably high fitness, if the number of such points is bounded.
Example applications include functions with $\exp(o(n^{\delta}/\log n))$ \emph{local} optima, including those where all local optima have at most $n^\delta/\log^3 n$ ones or at most $n^\delta/\log^3 n$ zeros. The latter function class includes the well-known $\textsc{Jump}_k$ functions~\citep{Droste2002,Dang2017}, where a gap of Hamming distance~$k$ has to be
``jumped'' to reach a global optimum,
with parameter $k \le n^\delta/\log^3 n$: here all search points with $k$ zeros are local optima, in addition to the global optimum~$1^n$. A similar function class $\textsc{Cliff}_d$ was used in~\cite{Jagerskupper2007a,Paixao2016,Corus2017}, where the same holds for $d$ in lieu of~$k$; the difference between these two functions is that in the region ``between'' local and global optima $\textsc{Jump}_k$ has a gradient pointing back towards the local optima whereas $\textsc{Cliff}_d$ points towards the global optimum~$1^n$.
Functions with difficult local optima include a modified version of $\twomax$ used in~\citep{Friedrich2009}: in $\twomax' := \max\{\sum_{i=1}^n x_i, \sum_{i=1}^n (1-x_i)\} + \prod_{i=1}^n x_i$ the point $1^n$ is the only global optimum and $0^n$ is a local optimum it is very hard to escape from.
A combinatorial example of a \textsc{MaxSat} instance with difficult local optima was studied in the context of evolutionary algorithms in~\citet{Droste2002a}, with variables $x_1, \dots, x_n$ and clauses
\begin{equation}
\label{eq:maxsat-instance}
\{(x_i \vee \overline{x_j} \vee \overline{x_k}) \mid i \neq j \neq k \neq i\} \cup \{(x_i) \mid 1 \le i \le n\}.
\end{equation}
Here the optimum is again $1^n$, and all $n$ search points with a single 1-bit are local optima. Likewise, the \textsc{MinCut} instance from Theorem~\ref{the:main-result-tail-bounds-applications} has $O(n)$ local optima as well: all search points with exactly one 1-bit or one 0-bit are locally optimal. \Citet{Sudholt2010} further presents a hard \textsc{Knapsack} instance with $(n+1)/2$ ``small'' objects of weight and value $n$ and $(n-1)/2$ ``big'' objects of weight and value $n+1$. The weight limit is set to $(n+1)/2 \cdot n$, such that including all small objects yields a global optimum, but selecting all but one big object gives a local optimum. Similar as above, the number of local optima is $O(n)$.
Finally, the arguments for Jansen-Zarges function classes also hold with respect to the number of local optima.
The following theorem summarises all the above.
\begin{theorem}
\label{the:main-local-optima-applications}
Every unary unbiased $\lambda$-parallel black-box algorithm $\mathcal{A}$
needs more than
\[
\max\left\{\frac{\lambda n}{60\ln^+\lambda}, (1-\delta) n \ln n\right\}
= \mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+\lambda} + n \ln n\right)}
\]
evaluations, with probability $1-\exp(-\Omega(n^{\delta}/\log n))$, to find a \textbf{local} or global optimum for all of the following settings.
\begin{enumerate}
\item All functions with $\exp(o(n^{\delta}/\log n))$ local optima.
\item All functions where all local optima have at most $n^\delta/\log^3 n$ ones or at most $n^\delta/\log^3 n$ zeros.
\item $\textsc{Jump}_k$ functions with $k \le n^\delta/\log^3 n$.
\item $\textsc{Cliff}_d$ functions with $d \le n^\delta/\log^3 n$.
\item $\twomax := \max\{\sum_{i=1}^n x_i, \sum_{i=1}^n (1-x_i)\}$ as well as the modified \twomax function $\twomax' := \max\{\sum_{i=1}^n x_i, \sum_{i=1}^n (1-x_i)\} + \prod_{i=1}^n x_i$
\item \textsc{MinCut} instances with two equal-sized cliques.
\item The hard \textsc{MaxSat} instance from~\eqref{eq:maxsat-instance}.
\item The hard \textsc{Knapsack} instance mentioned above.
\item Jansen-Zarges nearest peak functions and weighted nearest peak functions with $\exp(o(n^\delta/\log n))$ peaks.
\end{enumerate}
The expected time also satisfies the asymptotic bound.
\end{theorem}
We can even push our applications a bit further. Again using~\eqref{eq:area-of-hypercube}, there are at most $\exp(o(n^{\delta}/\log n))$ search points within a Hamming ball of radius $n^\delta/\log^3 n$ around any search point. If there are $\exp(o(n^{\delta}/\log n))$ global or local optima then the number of all search points within the union of Hamming balls around all these points is still
$\exp(o(n^{\delta}/\log n)) \cdot \exp(o(n^{\delta}/\log n)) = \exp(o(n^{\delta}/\log n))$. Hence our main result from Theorem~\ref{the:main-result-tail-bounds} still applies when considering the time to get to within Hamming distance $n^\delta/\log^3 n$ of any global or local optimum.
\begin{theorem}
Theorem~\ref{the:main-result-tail-bounds-applications} and Theorem~\ref{the:main-local-optima-applications} still apply when replacing ``to find a global optimum'' with ``to find any search point within Hamming distance $n^\delta/\log^3 n$ to any global optimum'' in
Theorem~\ref{the:main-result-tail-bounds-applications} and
replacing ``to find a local or global optimum'' with
``to find any search point within Hamming distance $n^\delta/\log^3 n$ to any local or global optimum'' in Theorem~\ref{the:main-local-optima-applications}.
\end{theorem}
In particular, this implies that with overwhelming probability no unary unbiased black-box algorithm can find a search point of fitness at least $n-n^\delta/\log^3 n$ for \onemax, \LO and \twomax within the stated time.
In other words, the expected fitness after the stated time is $n-n^\delta/\log^3 n + o(1)$ (where the $o(1)$ term accounts for an exponentially small failure probability, in case of which the fitness could be as large as~$n$). Such results are known as \emph{fixed-budget results}~\citep{Jansen2014,Doerr2013c}. This shows that our $\lambda$\nobreakdash-parallel black-box complexity results with tail bounds can be applied in a large variety of settings.
\section{Conclusions and Future Work}
We have introduced the parallel unbiased black-box complexity to quantify the limits on the performance of parallel search heuristics, including offspring populations, island models and multi-start methods. We proved that \emph{every} $\lambda$-parallel unbiased black-box algorithm needs at least $\Omega(\lambda n/\log^+(\lambda) + n \log n)$ function evaluations on every function with unique optimum, and at least $\Omega(\lambda n/(\log^+(\lambda/n)) + n^2)$ function evaluations on \LO. Corresponding parallel times are by a factor of~$\lambda$ smaller.
For \LO and \OM we identified the cut-off point for $\lambda$, above which the asymptotic number of function evaluations increases, compared to non-parallel algorithms ($\lambda=1$). All smaller $\lambda$ allow for linear speedups with regard to the parallel time. For \OM this cut-off point is higher than that for the standard \lEA; optimal performance for all~$\lambda$ is achieved by a \lEA with an adaptive mutation rate.
In a novel and more detailed analysis we have established tail bounds showing that the lower bound $\mathord{\Omega}\mathord{\left(\frac{\lambda n}{\ln^+\lambda} + n \ln n\right)}$ holds with overwhelming probability, for parallel and non-parallel algorithms (where $\lambda=1$) and for finding any target set of search points we can choose. This makes it a very general, powerful and versatile statement: we obtain lower bounds on the optimisation time on functions with many optima, the time to find a local optimum, and the time to even get close to any local or global optimum. We demonstrated the usefulness of this approach by deriving the first black-box complexity lower bounds for a range of popular and illustrative problems, from
synthetic problems (\twomax, \textsc{H-IFF}, \textsc{Jump}$_k$, \textsc{Cliff}) to classes of multimodal benchmark functions~\cite{Jansen2016} and important problems from combinatorial optimisation such as \textsc{Vertex Colouring}, \textsc{MinCut}, \textsc{Partition}, \textsc{Knapsack} and \textsc{MaxSat}.
A major open problem for future work is to derive lower bounds for the $\lambda$-parallel unbiased black-box complexity when allowing binary operators like crossover, or operators combining many search points as in EDAs or swarm intelligence algorithms. Currently even in the non-parallel case no non-trivial lower bounds on the binary unbiased black-box complexity are known.
\bibliographystyle{apalike}
|
2208.01686
|
\section{Introduction}
Rigidity and deformability problems of a given isometric immersion are fundamental
problems of the theory of isometric immersions. Of particular interest is the
classification of all noncongruent minimal surfaces in a space form, that are isometric to a given one.
This problem was raised by Lawson in \cite{Law} and partial answers were provided by several authors.
For instance, see \cite{Cal2, J, L, Law,M1, N, S, S1, V9,V08}.
The aforementioned problem has drown even more attention for minimal
surfaces in spheres. That is mainly due to the difficulty that arises from the fact that the Gauss map is
merely harmonic, in contrast to minimal surfaces in the Euclidean space where the Gauss map
is holomorphic. The classification problem of minimal surfaces in spheres that are
isometric to minimal surfaces in the sphere $\mathbb{S}^3$ was raised by Lawson
in \cite{L}, where he stated a conjecture that is still open. This conjecture has been
only confirmed for certain classes of minimal surfaces in spheres (see \cite{N, S, S1,
V9,V08}). It is worth noticing that a surface is locally isometric to a minimal surface
in $\mathbb{S}^3$ if its Gaussian curvature $K$ satisfies the spherical Ricci condition
$$
\Delta\log(1-K)=4K,
$$
away from totally geodesic points, where $\Delta$ is the Laplacian operator of the surface
with respect to its
induced metric.
In this paper, we turn our interest to a distinguished class of minimal surfaces in
spheres, the so-called {\textit{pseudoholomorphic curves}} in the nearly K{\"a}hler
sphere $\mathbb{S}^6.$ This class of surfaces was introduced by Bryant
\cite{Br} and has been widely studied (cf. \cite{BVW, H, EschVl}). The
pseudoholomorphic curves in $\mathbb{S}^6$ are nonconstant smooth maps from
a Riemann surface into the nearly K{\"a}hler sphere $\mathbb{S}^6,$ whose
differential is complex linear with respect to the almost complex structure of
$\mathbb{S}^6$ that is induced from the multiplication of the Cayley numbers.
In analogy with Calabi's work \cite{Cal2}, in the present paper we focus on the
following problem:
\begin{quotation}
\textit{Classify noncongruent minimal surfaces in spheres that are isometric to a given
pseudoholomorphic curve in the nearly K{\"a}hler sphere $\mathbb{S}^6.$}
\end{quotation}
One of the aims in this paper is to investigate the moduli space of all noncongruent substantial minimal
surfaces $f\colon M\to\mathbb{S}^n$ that are isometric to a given pseudoholomorphic curve
$g\colon M\to\mathbb{S}^6.$ By substantial, we mean that $f(M)$ is not contained in any totally geodesic submanifold of $\mathbb{S}^n.$
It is known \cite{Br, EschVl} that any pseudoholomorphic curve $g\colon M\to
\mathbb{S}^6$ is $1$-isotropic (for the notion of $s$-isotropic surface see Section 2). The nontotally geodesic pseudoholomorphic curves
in $\mathbb{S}^6$ are either substantial in a totally geodesic $\mathbb{S}^5\subset
\mathbb{S}^6$ or substantial in $\mathbb{S}^6$ (see \cite{BVW}). In the latter case, the curve is either
null torsion (studied by Bryant \cite{Br}) or nonisotropic. It turns out that null torsion curves are isotropic.
In order to study the above problem we have to deal
separately with these three classes of pseudoholomorphic curves. It is worth noticing that a characterization of Riemannian metrics
that arise as induced metrics on each class of these pseudoholomorphic curves was given in \cite{EschVl, V16} (for details see Section 5).
Flat minimal
surfaces in odd dimensional spheres (see \cite{K11, Br2}) are obviously isometric to any flat pseudoholomorphic curve in $\mathbb{S}^5.$ In \cite{VT} we provided a method to produce nonflat minimal surfaces in odd dimensional
spheres that are isometric to pseudoholomorphic curves in $\mathbb{S}^5.$ More precisely, let
$g_{\theta}, 0\leq \theta<\pi$, be the associated family of a simply connected
pseudoholomorphic curve $g\colon M\to\mathbb{S}^{5}.$ We consider the surface
$G\colon M\to\mathbb{S}^{6m-1}$ defined by
\begin{equation}\label{gth}
G=a_{1}g_{\theta _{1}}\oplus \cdots\oplus a_{m}g_{\theta _{m}},
\end{equation}
where $a_{1},\dots\,,a_{m}$ are any real numbers with $\sum_{j=1}^{m}a_{j}^
{2}=1,$ $0\leq \theta _{1}<\cdots<\theta_{m}<\pi,$ and $\oplus $ denotes the
orthogonal sum with respect to an orthogonal decomposition of the Euclidean space
$\mathbb{R}^{6m}.$ It is easy to see that $G$ is minimal and isometric to $g.$
It was verified in \cite{VT} that minimal surfaces given by \eqref{gth} belong to the class of
exceptional surfaces that was studied in \cite{V08, V16}. These are minimal surfaces
whose all Hopf differentials are holomorphic, or equivalently all curvature ellipses of
any order have constant eccentricity up to the last but one (see Sections 2 and 3 for details). In addition, in \cite{VT} it was proved
that minimal surfaces in spheres that are isometric to a given pseudoholomorphic curve in $\mathbb{S}^5$ are exceptional
under appropriate global assumptions. In fact, we proved that besides flat minimal
surfaces in odd dimensional spheres, the only simply connected exceptional surfaces
that are isometric to a pseudoholomorphic curve in $\mathbb{S}^5$ are of the type \eqref{gth}.
Describing the moduli space of noncongruent minimal surfaces in spheres that are isometric
to a given pseudoholomorphic curve in the nearly K{\"a}hler $\mathbb{S}^6$ in full generality, turns out to be
a hard problem. To begin with, we investigate this moduli space in the class
of exceptional substantial surfaces in $\mathbb{S}^n.$ We denote by
$\mathcal{M}_n^{\mathrm{e}}(g)$ the moduli space of all noncongruent exceptional surfaces
$f\colon M\to\mathbb{S}^n$ that are isometric to a given pseudoholomorphic curve $g\colon M\to\mathbb{S}^6.$
At first we deal with nonflat pseudoholomorphic curves in
a totally geodesic $\mathbb{S}^5\subset\mathbb{S}^6$ in the case where $n$ is odd.
Given such a pseudoholomorphic curve $g,$ we are able to show that
the moduli space $\mathcal{M}_n^{\mathrm{e}}(g)$ is empty unless $n\equiv 5\;\mathrm{mod}\; 6,$
in which case $\mathcal{M}_n^{\mathrm{e}}(g)$ splits as
$$
\mathcal{M}_n^{\mathrm{e}}(g)=\mathbb{S}_\ast^{m-1}\times\Gamma_0,
$$
where $m=(n+1)/6,$
$$
\mathbb{S}_\ast^{m-1}=\Big\{(a_1,\dots,a_m)\in\mathbb{S}^{m-1}
\subset\mathbb{R}^m \colon \prod\limits_{j=1}^{m}a_j\neq0\Big\}
$$
and $\Gamma_0$ is a subset of
$$
\Gamma^m=\big\{(\theta_1,\dots,\theta_m)\in\mathbb{R}^m\colon 0\leq \theta _1<\cdots<\theta_m<\pi
\big\}.
$$
The case where $M$
is simply connected was studied in \cite[Theorem 3]{VT}, where it was proved that $\Gamma_0=\Gamma^m$.
In this paper, we prove that if $\Gamma_0$ is a proper subset of $\Gamma^m$ then it is
locally a disjoint finite
union of $d$-dimensional real analytic subvarieties where $d=0,\dots,m-1$.
If $M$ is compact and not homeomorphic to the torus, then it is shown that $\Gamma_0$ is
a proper subset of $\Gamma^m$ (see Theorems \ref{lineorfinite} and \ref{compactps}). As a
result, we are able to prove the following theorem, which provides an answer to
the aforementioned problem for minimal surfaces in spheres with low codimension.
\begin{theorem}\label{mikradim}
Let $g\colon M\to\mathbb{S}^5$ be a compact pseudoholomorphic curve. If $M$ is not
homeomorphic to the torus, then the moduli space of all noncongruent substantial
minimal surfaces in $\mathbb{S}^n,
\, 4\le n\le 7,$ that are isometric to $g$ is empty, unless $n=5$ in which case the moduli space
is a finite set.
\end{theorem}
The necessity of the assumption
that the surface is not homeomorphic to the torus is justified by the class of flat
tori in $\mathbb{S}^5$ (see Remark \ref{flattori}).
Given a pseudoholomorphic curve
$g\colon M\to\mathbb{S}^6,$
we are able to give the following description of the moduli space
(for the definition of the normal curvatures we refer the reader to Section 2).
\begin{theorem}\label{finiteorcircle}
Let $g\colon M\to \mathbb{S}^6$ be a pseudoholomorphic curve. The moduli space of all noncongruent minimal surfaces
$f\colon M\to \mathbb{S}^6$ that are isometric to $g$ and have the same normal curvatures with $g,$
is either a circle or a finite set.
\end{theorem}
For nonisotropic pseudoholomorphic curves, under
a global assumption on the Euler-Poincar\'{e} number of the second normal bundle (see Sections 2 and 3 for details),
we are able to prove the following result that provides a partial answer to our problem.
\begin{theorem}\label{compnon}
Let $g\colon M\to \mathbb{S}^6$ be a compact substantial pseudoholomorphic curve that is nonisotropic. If the Euler-Poincar\'{e} number of the second normal bundle of $g$ is nonzero,
then there are at most finitely many minimal surfaces in $\mathbb{S}^6$ isometric to
$g$ having the same normal curvatures with $g$.
\end{theorem}
The necessity of the assumption on the codimension and the global assumptions in the above theorem is justified by the fact that the
direct sums of the associated family of a simply connected nonisotropic pseudoholomorphic
curve $g\colon M\to\mathbb{S}^6$ are isometric to $g$ (see Remark \ref{remarknoniso}).
In addition, we prove the following theorem that may be viewed as analogous to the classical
result of Schur (see \cite[p. 36]{Chnew}) in the realm
of minimal surfaces in spheres.
\begin{theorem}\label{teleutaioPhi}
Let $g\colon M\to\mathbb{S}^6$ be a compact, nonisotropic and substantial pseudoholomorphic curve and $\hat{g}\colon M\to\mathbb{S}^n$ be a
substantial minimal surface that is isometric to $g.$
If $\hat{g}$ is not $2$-isotropic and the second normal curvatures $K_2^\perp, \hat{K}_2^\perp$ of the surfaces $g$ and $\hat{g}$
respectively satisfy the inequality $\hat{K}_2^\perp\le K_2^\perp,$ then $n=6.$
Moreover, the moduli space of all such noncongruent minimal surfaces $\hat{g}\colon M\to\mathbb{S}^6$
that are isometric to $g,$
is either a circle or a finite set.
\end{theorem}
Finally, we deal with the third class of pseudoholomorphic curves in $\mathbb{S}^6,$
namely the isotropic ones. It turns out that these surfaces
are rigid. For compact minimal surfaces our result is stated as follows.
\begin{theorem}\label{forintro}
Let $f\colon M\to\mathbb{S}^n$ be a compact substantial minimal surface. If $f$ is isometric to an isotropic pseudoholomorphic curve $g\colon M \to\mathbb{S}^6,$ then $n=6$ and $f$ is
congruent to $g.$
\end{theorem}
The same result holds if instead
of the compactness of the surface we assume that the surface is exceptional.
The paper is organized as follows: In Section 2, we fix the notation and give some
preliminaries. In Section 3, we recall the notion of Hopf differentials and some known results
about exceptional surfaces. In Section 4, we give some basic facts about absolute value type
functions, a notion that was introduced in \cite{EGT,ET} and will be exploited throughout the paper.
In Section 5, we recall some properties of
pseudoholomorphic curves in the nearly K{\"a}hler sphere $\mathbb{S}^{6}.$
In Section 6, we investigate properties of the moduli space of noncongruent minimal surfaces, substantial in odd
dimensional spheres, that are isometric to a given
pseudoholomorphic
curve in $\mathbb{S}^{5}$ and give the proof of Theorem \ref{mikradim}. Section 7 is devoted to the study of nonisotropic
pseudoholomorphic curves in $\mathbb{S}^{6}$ and we give the proofs of Theorems \ref{finiteorcircle}, \ref{compnon} and \ref{teleutaioPhi}.
In the last section, we deal with the case of isotropic pseudoholomorphic curves in $\mathbb{S}^{6}$ and give the proof of Theorem \ref{forintro}.
\section{Preliminaries}
In this section, we collect several facts and definitions about minimal surfaces in spheres.
For more details we refer to \cite{DF} and \cite{DV2k15}.
\vspace{1,5ex}
Let $f\colon M\to\mathbb{S}^n$ be an isometric immersion of a $2$-dimensional
Riemannian manifold. The $k^{th}$\emph{-normal space} of $f$ at $p\in M$ for
$k\geq 1$ is defined as
$$
N^f_k(p)={\rm span}\left\{\alpha^f_{k+1}(X_1,\ldots,X_{k+1}):X_1,\ldots,X_{k+1}\in
T_pM\right\},
$$
where the symmetric tensor
$$
\alpha^f_s\colon TM\times\cdots\times TM\to N_fM,\;\; s\geq 3,
$$
given
inductively by
$$
\alpha^f_s(X_1,\ldots,X_s)=\left(\nabla^\perp_{X_s}\cdots\nabla^\perp_{X_3}
\alpha^f(X_2,X_1)\right)^\perp,
$$
is called the $s^{th}$\emph{-fundamental form} and $\alpha^f\colon TM\times TM\to N_fM$ stands for the standard second fundamental
form of $f$ with values in the normal bundle. Here, $\nabla^{\perp}$ denotes the induced
connection in the normal bundle $N_fM$ of $f$ and $(\,\cdot\,)^\perp$ stands for the
projection onto the orthogonal complement of $N^f_1\oplus\cdots\oplus N^f_{s-2}$ in
$N_fM.$ If $f$ is minimal, then ${\rm dim}N^f_k(p)\le2$ for all $k\ge1$ and any $p\in M$ (cf. \cite{DF}).
A surface $f\colon M\to\mathbb{S}^n$ is called \emph{regular} if for each $k$ the
subspaces $N^f_k$ have constant dimension and thus form normal subbundles. Notice
that regularity is always verified along connected components of an open dense subset of
$M.$
Assume that an immersion $f\colon M\to\mathbb{S}^n$ is minimal and substantial.
By the latter, we mean that $f(M)$ is not contained in any totally geodesic submanifold of $\mathbb{S}^n.$ In this case, the normal bundle
of $f$ splits along an open dense subset of $M$ as
$$
N_fM=N_1^f\oplus N_2^f\oplus\dots\oplus N_m^f,\;\;\; m=[(n-1)/2],
$$
since all higher normal bundles have rank two except possible the last one that has rank
one if $n$ is odd; see \cite{Ch} or \cite{DF}. Moreover, if $M$ is oriented, then an
orientation is induced on each plane subbundle $N_s^f$ given by the ordered basis
$$
\alpha^f_{s+1}(X,\ldots,X),\;\;\;\alpha^f_{s+1}(JX,\ldots,X),
$$
where $0\neq X\in TM,$ and $J$ is the complex structure determined by the orientation and the metric.
If $f\colon M\to\mathbb{S}^n$ is a minimal surface, then at any point $p\in M$ and for each
$N_r^f$, $1\leq r\leq m$, the \emph{$r^{th}$-order curvature ellipse}
$\mathcal{E}^f_r(p)\subset N^f_r(p)$ is defined by
$$
\mathcal{E}^f_r(p) = \left\{\alpha^f_{r+1}(Z^{\varphi},\ldots,Z^{\varphi})\colon\,
Z^{\varphi}=\cos\varphi Z+\sin\varphi JZ\;\mbox{and}\;\varphi\in[0,2\pi)\right\},
$$
where $Z\in T_xM$ is any vector of unit length.
A substantial regular surface $f\colon M\to\mathbb{S}^{n}$ is called
\emph{$s$-isotropic} if it is minimal and at any point $p\in M$ the curvature ellipses
$\mathcal{E}^f_r(p)$ contained in all two-dimensional $N^f_r$$\,{}^{\prime}$s
are circles for any $1\le r\le s.$ It is called \emph{isotropic} if it is $s$-isotropic for any $s.$
The $r$-th \textit{normal curvature} $K_{r}^{\perp}$ of $f$ is defined by
\begin{equation*}
K_{r}^{\perp}={\frac{2}{\pi}}{\hbox {Area}}(\mathcal{E}^f\sb r).
\end{equation*}
If $\kappa _{r}\geq \mu_{r}\geq 0$ denote the length of the semi-axes of the curvature ellipse
$\mathcal{E}^f\sb r,$ then
\begin{equation}\label{elipsi}
K_{r}^{\perp}=2\kappa _{r}\mu _{r}.
\end{equation}
The \textit{eccentricity} $\varepsilon\sb r$ of the curvature ellipse $\mathcal{E}^f\sb r$ is given by
\begin{equation*}
\varepsilon\sb r=\frac{\left(\kappa^{2}_{r}-\mu^{2}_{r}\right)^{1/2}}{\kappa_{r}},
\end{equation*}
where $\left(\kappa^{2}_{r}-\mu^{2}_{r}\right)^{1/2}$ is the distance from the center to a focus,
and can be thought of as a measure of how far $\mathcal{E}^f\sb r$ deviates from being a
circle.
The $a$-\textit{invariants} (see \cite{V16}) are the functions
\begin{equation*}
a^{\pm}_{r}= \kappa _{r}{\pm} \mu _{r}= \left(2^{-r} \Vert \alpha^f_{r+1}
\Vert^{2} \pm K_{r}^{\perp}\right)^{1/2}.
\end{equation*}
These functions determine the geometry of the $r$-th curvature ellipse.
Denote by $\tau^o_f$ the index of the last plane bundle, in the orthogonal decomposition
of the normal bundle. Let $\{e_1,e_2\}$ be a local tangent orthonormal frame and
$\{e_\alpha\}$ be a local orthonormal frame of the normal bundle such that
$\{e_{2r+1},e_{2r+2}\}$ span $N_r^f$ for any $1\le r\le\tau^o_f$ and $e_{2m+1}$
spans the line bundle $N^f_{m+1}$ if $n=2m+1.$ For any $\alpha=2r+1$ or $\alpha=2r+2,$ we set
$$
h_1^{\alpha}=\langle
\alpha^f_{r+1}(e_1,\dots,e_1),e_{\alpha}\rangle,{\ }h_2^{\alpha}=\langle
\alpha^f_{r+1}(e_1,\dots,e_1,e_2),e_{\alpha}\rangle,
$$
where $\langle\cdot,\cdot\rangle$ is the standard metric of $\mathbb{S}^n.$ Introducing the complex valued functions
$$
H_{\alpha}=h_1^{\alpha}+ih_2^{\alpha}\;\;\text{for any}\;\;\alpha=2r+1\;\;\text{or}\;\;\alpha=2r+2,
$$
it is not hard to verify that the $r$-th normal
curvature is given by
\begin{equation}\label{prwthsxeshden}
K_r^{\perp}=i\left(H_{2r+1}{\overline{H}_{2r+2}}-{\overline{H}_{2r+1}}
H_{2r+2}\right).
\end{equation}
The length of the $(r+1)$-th fundamental form $\alpha^f_{r+1}$ is given by
\begin{equation}\label{deuterhsxeshden}
\Vert \alpha^f_{r+1}\Vert ^2=2^r\big(|{H_{2r+1}}|^2+|{H_{2r+2}}|
^2\big),
\end{equation}
or equivalently (cf. \cite{A})
\begin{equation}\label{si}
\Vert \alpha^f_{r+1}\Vert ^2=2^r(\kappa_r^2+\mu_r^2).
\end{equation}
\smallskip
Each plane
subbundle $N_r^f$ inherits a Riemannian connection from that of the normal bundle. Its
\textit{intrinsic curvature} $K^*_r$ is given by the following proposition (cf. \cite{A}).
\begin{proposition}\label{5}
\textit{The intrinsic curvature $K_r^{\ast}$ of each plane subbundle $N_{r}^f$
of a minimal surface} $f\colon M\to \mathbb{S}^{n}$ \textit{is given by}
\begin{equation*}
K_{1}^{\ast}=K_1^{\perp}-{\frac{\Vert \alpha^f_3\Vert ^2}{2K_1^{\perp}}}
\;\; \text{and}\;\; K_r^{\ast}={\frac{K_r^{\perp}}{(K_{r-1}^{\perp})^2}}{\frac{
\Vert \alpha^f_{r}\Vert ^{2}}{2^{r-2}}}-{\frac{\Vert \alpha^f_{r+2}\Vert ^2}{
2^rK_r^{\perp}}}\;\;\text{for}\;\;2\leq r\leq \tau_f^o.
\end{equation*}
\end{proposition}
Let $f\colon M\to\mathbb{S}^n$ be a minimal isometric immersion. If $M$ is
simply connected, there exists a one-parameter \emph{associated family} of minimal
isometric immersions $f_\theta\colon M\to\mathbb{S}^n,$ where $\theta\in\mathbb{S}^1=[0,\pi).$ To see this, for each
$\theta\in\mathbb{S}^1$ consider the orthogonal parallel tensor field
$$
J_{\theta}=\cos\theta I+\sin\theta J,
$$
where $I$ is the identity endomorphism of the tangent bundle and $J$ is the complex
structure of $M$ induced by
the metric and the orientation. Then, the symmetric section $\alpha^f(J_\theta\cdot,
\cdot)$ of the bundle $\text{Hom}(TM\times TM,N_f M)$ satisfies the Gauss, Codazzi and
Ricci equations, with respect to the same normal connection; see \cite{DG2} for details.
Therefore, there exists a minimal isometric immersion $f_{\theta}\colon M\to
\mathbb{S}^n$ whose second fundamental form is given by
\begin{equation*}
\alpha^{f_{\theta}}(X,Y)=T_\theta\alpha^f(J_{\theta}X,Y),
\end{equation*}
where $T_\theta\colon N_fM\to N_{f_{\theta}}M$ is a parallel
vector bundle isometry that identifies the normal subspaces
$N_s^f$ with $N_s^{f_\theta}$, $s\geq 1.$
\section{Hopf differentials and Exceptional surfaces}
Let $f\colon M\to\mathbb{S}^n$ be a minimal surface. The complexified tangent bundle
$TM\otimes \mathbb{C}$ is decomposed into the eigenspaces $T^{\prime}M$ and $T^{\prime \prime}M$ of the complex structure
$J$, corresponding to the
eigenvalues $i$ and $-i.$ The $(r+1)$-th fundamental form $\alpha^f_{r+1}$, which takes values in
the normal subbundle $N_{r}^f$, can be complex linearly extended to $TM\otimes
\mathbb{C}$ with values in the complexified vector bundle $N_{r}^f\otimes \mathbb{C}$
and then decomposed into its $(p,q)$-components, $p+q=r+1,$ which are tensor
products of $p$ differential 1-forms vanishing on $T^{\prime \prime}M$ and $q$ differential 1-forms vanishing
on $T^{\prime}M.$ The minimality of $f$ is equivalent to the vanishing of the
$(1,1)$-part of the second fundamental form. Hence, the $(p,q)$-components of $\alpha^f_{r+1}$
vanish unless $p=r+1$ or $p=0,$ and consequently for a local complex coordinate $z$
on $M$, we have the following decomposition
\begin{equation*}
\alpha^f_{r+1}=\alpha_{r+1}^{(r+1,0)}dz^{r+1}+\alpha_{r+1}^{(0,r+1)}d\bar{z}^
{r+1},
\end{equation*}
where
\begin{equation*}
\alpha_{r+1}^{(r+1,0)}=\alpha^f_{r+1}(\partial,\dots,\partial),\;\;\alpha_{r+1}^{(0,r+1)}=\overline{\alpha_{r+1}^{(r+1,0)}}\;\;\;\text{and}\;\;\;\partial ={\frac{1}{2}}\big({\frac{\partial}{\partial x}}-i{\frac{\partial}{\partial y}}\big).
\end{equation*}
The \textit{Hopf differentials} are the differential forms (see \cite{V})
\begin{equation*}
\Phi _{r}=\langle \alpha_{r+1}^{(r+1,0)},\alpha_{r+1}^{(r+1,0)}\rangle dz^{2r+2}
\end{equation*}
of type $(2r+2,0),r=1,\dots,[(n-1)/2],$ where $\langle \cdot,\cdot\rangle$ denotes the extension of the usual Riemannian metric
of $\mathbb{S}^n$ to a complex bilinear form. These forms are
defined on the open subset where the minimal surface is regular and are independent of the choice of coordinates, while
$\Phi _{1}$ is globally well defined.
Let $\{e_1,e_2\}$ be a local orthonormal frame in the tangent bundle.
It will be convenient to use complex vectors, and we put
\begin{equation*}
\text{ }E=e_1-ie_2\;\;
\text{and}\;\; \phi =\omega _{1}+i\omega _2,
\end{equation*}
where $\{\omega_1,\omega_2\}$ is the dual frame. We choose a local complex
coordinate $z=x+iy$ such that $\phi =Fdz.$
From the definition of Hopf differentials, we easily obtain
\begin{equation*}
\Phi _{r}={\frac{1}{4}}\left({\overline{H}_{2r+1}^2}+{\overline{H}_{2r+2}^2}
\right) \phi^{2r+2}.
\end{equation*}
Moreover, using (\ref{prwthsxeshden}) and (\ref{deuterhsxeshden}), we find that
\begin{equation}\label{what}
\left\vert \langle \alpha_{r+1}^{(r+1,0)},\alpha_{r+1}^{(r+1,0)}\rangle \right\vert
^2=\frac{F^{2r+2}}{2^{2r+4}}\left(\Vert \alpha^f_{r+1}\Vert
^4-4^r(K_r^{\perp})^2\right).
\end{equation}
Thus, the zeros of $\Phi _r$ are precisely the points where the $r$-th curvature ellipse $
\mathcal{E}^f\sb r$ is a circle. Being $s$-isotropic is equivalent to $\Phi_r=0$ for any
$1\le r\le s.$
The Codazzi equation implies that $\Phi _{1}$ is always holomorphic (cf. \cite{Ch,ChW}).
Besides $\Phi_1$, the rest Hopf differentials are not always holomorphic. The following
characterization of the holomorphicity of Hopf differentials was given in \cite{V08}, in
terms of the eccentricity of curvature ellipses of higher order.
\begin{theorem}\label{ena}
Let $f\colon M\to \mathbb{S}^n$ be a minimal surface. Its Hopf differentials
$\Phi _{2},\dots,\Phi_{r+1}$ are holomorphic if and only if the higher curvature ellipses
have constant eccentricity up to order $r.$
\end{theorem}
A minimal surface in $\mathbb{S}^n$ is called \textit{$r$-exceptional} if all Hopf
differentials up to order $r+1$ are holomorphic, or equivalently if all higher curvature
ellipses up to order $r$ have constant eccentricity. A minimal surface in $\mathbb{S}^n$
is called \textit{exceptional} if it is $r$-exceptional for $r=[(n-1)/2-1].$
This class of minimal surfaces may be viewed as the next simplest to superconformal ones.
In fact, superconformal minimal surfaces are indeed exceptional, characterized by the fact that all
Hopf differentials vanish up to the
last but one, which is equivalent to the fact that all higher curvature ellipses are circles up
to the last but one. As a matter of fact, there is an abundance of exceptional surfaces.
We recall some results for exceptional surfaces proved in \cite{V08}, that will be used in
the proofs of our main results.
\begin{proposition}\label{3i}
Let $f\colon M\to\mathbb{S}^n$ be an $(r-1)$-exceptional surface. At regular
points the following hold:
(i) For any $1\leq s\leq r-1,$ we have
\begin{equation*}
\Delta \log \left\Vert \alpha_{s+1}\right\Vert ^2=2\big((s+1)K-K_s^{\ast}\big),
\end{equation*}
where $\Delta $ is the Laplacian operator with respect to the induced metric $ds^{2}.$
(ii) If $\Phi _{r}\neq 0$\textit{, then}
\begin{equation*}
\Delta \log \left(\left\Vert \alpha_{r+1}\right\Vert^2+2^rK_r^{\perp}\right) =2\big((r+1)K-K_r^{\ast}\big)
\end{equation*}
and
\begin{equation*}
\Delta \log \left(\left\Vert \alpha_{r+1}\right\Vert^2-2^rK_r^{\perp}\right) =2\big((r+1)K+K_r^{\ast}\big).
\end{equation*}
(iii) If $\Phi _{r}=0$\textit{, then}
\begin{equation*}
\Delta \log \left\Vert \alpha_{r+1}\right\Vert^2=2\big((r+1)K-K_r^{\ast}\big).
\end{equation*}
(iv) The intrinsic curvature of the $s$-th
normal bundle $N_s^f$\textit{\ is} $K_{s}^{\ast}=0$
if $1\leq s\leq r-1$ and $\Phi _s\neq 0.$
\end{proposition}
A remarkable property of exceptional surfaces is that singularities of the higher normal bundles
are of holomorphic type and can be smoothly extended to vector bundles. This fact was proved in
\cite[Proposition 4]{V08}.
\begin{proposition}\label{neoksanaafththfora}
Let $f\colon M\to \mathbb{S}^n$ be an $r$-exceptional surface. Then the set $L_0$,
where $f$ fails to be regular, consists of isolated points and all $N_s^f$'s and the Hopf
differentials $\Phi_s$'s extend smoothly to $L_0$ for any $1\leq s\leq r.$
\end{proposition}
\section{Absolute value type functions}
For the proof of our results, we shall use the notion of absolute value type functions
introduced in \cite{EGT,ET}. A smooth complex valued function $p$ defined on a
Riemann surface is called of \textit{holomorphic type} if locally $p=p_0p_1,$ where $p_0$
is holomorphic and $p_1$ is smooth without zeros. A function $u\colon M\to\lbrack 0,+
\infty)$ defined on a Riemann surface $M$ is called of \textit{absolute value type} if there
is a function $p$ of holomorphic type on $M$ such that $u=|p|.$
The zero set of such a function on a connected compact oriented surface $M$ is either
isolated or the whole of $M$, and outside its zeros the function is smooth. If $u$ is a
nonzero absolute value type function, i.e., locally $u=|t_0|u_1$, with $t_0$ holomorphic,
the order $k\ge1$ of any point $p\in M$ with $u(p)=0$ is the order of $t_0$ at $p.$ Let
$N(u)$ be the sum of the orders for all zeros of $u.$ Then $\Delta\log u$ is bounded on
$M\smallsetminus\left\{u=0\right\}$ and its integral is computed in the following lemma
that was proved in \cite{EGT,ET}.
\begin{lemma}\label{forglobal}
Let $(M,ds^2)$ be a compact oriented two-dimensional Riemannian manifold with area
element $dA.$
(i) If $u$ is an absolute value type function on $M,$ then
\begin{equation*}
\int_{M}\Delta \log udA=-2\pi N(u).
\end{equation*}
(ii) If $\Phi $ is a holomorphic symmetric $(r,0)$-form on $M,$ then either $\Phi =0$ or
$N(\Phi)=-r\chi (M),$ where $\chi (M)$ is the Euler-Poincar\'{e} characteristic of $M.$
\end{lemma}
The following lemma, that was proved in \cite{N}, provides a sufficient condition for a
function to be of absolute value type.
\begin{lemma}\label{dena}
Let $D$ be a plane domain containing the origin with coordinate $z$ and $u$ be a real
analytic nonnegative function on $D$ such that $u(0)=0.$ If $u$ is not identically zero
and $\log u$ is harmonic away from the points where $u=0$, then $u$ is of absolute
value type and the order of the zero of $u$ at the origin is even.
\end{lemma}
\section{Pseudoholomorphic curves in $\mathbb{S}^6$}
In this section we summarize some well known facts about pseudoholomorphic
curves in the nearly K{\"a}hler sphere $\mathbb{S}^6$. It is known that the
multiplicative structure on the Cayley numbers $\mathbb{O}$ can be
used to define an almost complex structure on the sphere $\mathbb{S}^6$ in
$\mathbb{R}^7.$ This almost complex structure is not integrable but it is nearly
K{\"a}hler. A \textit{pseudoholomorphic curve}, which was introduced by Bryant \cite{Br}, is a nonconstant smooth map
$g\colon M\to\mathbb{S}^6$ from a Riemann surface $M$ into the nearly K{\"a}hler
sphere $\mathbb{S}^6,$ whose differential is complex linear.
It is known \cite{Br, EschVl} that any pseudoholomorphic curve $g\colon M\to\mathbb{S}
^6$ is $1$-isotropic. The nontotally geodesic pseudoholomorphic curves in $\mathbb{S}^6$
are are either substantial in a totally geodesic $\mathbb{S}^5\subset
\mathbb{S}^6$ or substantial in $\mathbb{S}^6$ (see \cite{BVW}). In the latter case, the curve is either
null torsion (studied by Bryant \cite{Br}) or nonisotropic. It turns out that null torsion curves are isotropic.
The following theorem \cite{EschVl} provides a characterization of Riemannian metrics
that arise as induced metrics on pseudoholomorphic curves in $\mathbb{S}^5.$
\begin{theorem}\label{eschvlach}
Let $(M,ds^2)$ be a simply connected Riemann surface, with Gaussian curvature $K\leq 1$ and Laplacian operator
$\Delta$. Suppose that the function $1-K$ is of absolute value type. Then there exists an isometric pseudoholomorphic curve $g\colon M\to\mathbb{S}^5$ if and only if
\[
\Delta\log(1-K)=6K.\tag{$\ast$}
\]
In fact, up to translations with elements of $G_2$, that is the set $Aut(\mathbb{O})\subset SO(7),$ there is precisely one associated family of such maps.
\end{theorem}
The above result shows that a minimal surface in a sphere is locally isometric to a
pseudoholomorphic curve in $\mathbb{S}^5$ if its Gaussian curvature satisfies the
condition $(\ast)$ at points where $K<1$ or equivalently if the metric
$d\hat{s}^2=(1-K)^{1/3}ds^2$ is flat.
Let $g\colon M\to\mathbb{S}^5$ be a pseudoholomorphic curve and let
$\xi\in N_fM$ be a smooth unit vector field that spans the extended line bundle $N_2^g$ over the isolated set of points where $f$ fails to be regular (see Proposition \ref{neoksanaafththfora}). The
surface $g^*\colon M\to\mathbb{S}^5$ defined by $g^*=\xi$ is called the \textit{polar surface} of $g.$ It has been proved in \cite[Corollary 3]{V16} that the surfaces $g$ and $g^*$ are
congruent.
The following theorem \cite{V16} provides a characterization of Riemannian metrics
that arise as induced metrics on nonisotropic substantial pseudoholomorphic curves in $\mathbb{S}^6.$
\begin{theorem}
Let $(M,ds^2)$ be a simply connected Riemann surface, with Gaussian curvature $K\leq 1$ and Laplacian operator
$\Delta$. Suppose that the function $1-K$ is of absolute value type. Then there exists a nonisotropic pseudoholomorphic curve $g\colon M\to\mathbb{S}^6,$ unique up to translations with elements of $G_2,$ if and only if
\[
\Delta\log\left((1-K)^2\left(1-6K+\Delta\log\left(1-K\right)\right)\right)=12K.
\]
Moreover the following holds:
\begin{equation}\label{trik}
6K-1<\Delta\log(1-K)<6K.
\end{equation}
\end{theorem}
We recall the following theorem \cite{EschVl}, which provides a characterization of Riemannian metrics
that arise as induced metrics on isotropic substantial pseudoholomorphic curves in $\mathbb{S}^6.$
\begin{theorem}
Let $(M,ds^2)$ be a simply connected Riemann surface, with Gaussian curvature $K\leq 1$ and Laplacian operator
$\Delta$. Suppose that the function $1-K$ is of absolute value type. Then there exists an isotropic pseudoholomorphic curve $g\colon M\to\mathbb{S}^6,$ unique up to translations with elements of $G_2,$ if and only if
\[
\Delta\log(1-K)=6K-1.\tag{$\ast\ast$}
\]
\end{theorem}
\section{Isometric deformations of pseudoholomorphic curves in $\mathbb{S}^5$}
We are interested in nontrivial isometric deformations of pseudoholomorphic
curves in $\mathbb{S}^5.$ Given a pseudoholomorphic curve
$g\colon M\to\mathbb{S}^5,$ we would like to describe the moduli space of all
noncongruent substantial minimal surfaces $f\colon M\to\mathbb{S}^n$ that are locally
isometric to the curve $g.$ For the class of the exceptional surfaces we denote the
above mentioned space by $\mathcal{M}_n^{\mathrm{e}}(g).$ Hereafter we assume
that $n$ is odd and $M$ is nonflat.
If $M$ is simply connected, it has been proved in \cite[Theorem 3]{VT} that $n\equiv 5\;
\mathrm{mod}\; 6,$ and
$$
\mathcal{M}_n^{\mathrm{e}}(g)=\mathbb{S}_\ast^{m-1}\times\Gamma^m,
$$
where $m=(n+1)/6,$
$$
\mathbb{S}_\ast^{m-1}=\Big\{\bold{a}=(a_1,\dots,a_m)\in\mathbb{S}^{m-1}
\subset\mathbb{R}^m \colon \prod\limits_{j=1}^{m}a_j\neq0\Big\}
$$
and
$$
\Gamma^m=\big\{{\pmb{\theta}}=(\theta_1,\dots,\theta_m)\in[0,\pi)
\times\cdots\times[0,\pi)\colon 0\leq \theta _1<\cdots<\theta_m<\pi
\big\}.
$$
Our aim in this section is to study the moduli space of noncongruent isometric deformations of a
nonsimply connected pseudoholomorphic curve $g\colon M\to\mathbb{S}^5.$ We
consider the covering map $\Pi\colon\tilde{M}\to M,$
$\tilde{M}$ being the universal cover of $M$ with the metric and orientation that
make $\Pi$ an orientation preserving local isometry. Corresponding objects on
$\tilde{M}$ are denoted with tilde. Then the map $\tilde{g}\colon \tilde{M}\to
\mathbb{S}^5$ with $\tilde{g}=g\circ\Pi$ is a pseudoholomorphic curve. Obviously,
since $\tilde{g}$ is simply connected, we know from \cite[Theorem 3]{VT} that
$$
\mathcal{M}_n^{\mathrm{e}}(\tilde{g})=\mathbb{S}_\ast^{m-1}\times
\Gamma^m.
$$
For any $(\bold{a},\pmb{\theta})\in\mathbb{S}_\ast^{m-1}\times
\bar{\Gamma}^m,$ where $\bar{\Gamma}^{m}$ is the closure of ${\Gamma}^m,$ we consider
the minimal surface
$\tilde{g}_{\bold{a},\pmb{\theta}}\colon \tilde{M}\to\mathbb{S}^
{6m-1}\subset\mathbb{R}^{6m}$ defined by
\[
\tilde{g}_{\bold{a},{\pmb{\theta}}}=a_1\tilde{g}_{\theta_{1}}\oplus\cdots
\oplus a_m\tilde{g}_{\theta_m},
\]
where $\oplus$
denotes the orthogonal sum with respect to an orthogonal decomposition of
$\mathbb{R}^{6m}.$ Each surface $\tilde{g}_{\theta_{j}}\colon \tilde{M}\to\mathbb{S}
^5, \,j=1,\dots,m,$ is a member of the associated family of $\tilde{g}.$
Clearly, given an exceptional surface $f\colon M\to\mathbb{S}^n
$ in the moduli space of the curve $g,$ the minimal surface $\tilde
{f}\colon \tilde{M}\to\mathbb{S}^n$ with $\tilde{f}=f\circ\Pi$ belongs to the moduli
space $\mathcal{M}_n^{\mathrm{e}}(\tilde{g})$ of the curve
$\tilde{g}.$ Therefore, the moduli space $\mathcal{M}_n^{\mathrm{e}}(g)$ can
be described as the subset of all $(\bold{a},\pmb{\theta})$ in $\mathcal{M}^{\mathrm
{e}}_n(\tilde{g})$ such that $\tilde{g}_{\bold{a},\pmb{\theta}}$ factors as
$F\circ\Pi$ for some exceptional surface $F\colon M \to\mathbb{S}^n.$
We follow this notation throughout this section.
The group $\mathcal{D}$ of \textit{deck transformations} of
the covering map $\Pi\colon \tilde{M}\to M$ consists of all diffeomorphisms
$\sigma\colon\tilde{M}\to\tilde{M}$ such that $\Pi\circ\sigma=\Pi.$
We need the following lemmas.
\begin{lemma}\label{deckbig}
For each $\sigma\in\mathcal{D}$ the surfaces $\tilde{g}_{\bold{a},\pmb{\theta}}$
and $\tilde{g}_{\bold{a},\pmb{\theta}}\circ\sigma$ are congruent for every $(\bold
{a},\pmb{\theta})\in\mathbb{S}_\ast^{m-1}\times
\bar{\Gamma}^m,$ that is there exists $\Phi_{\pmb{\theta}}(\sigma)\in\mathrm{O}
(n+1)$ such that
\begin{equation*}
\tilde{g}_
{\bold{a},{\pmb{\theta}}}\circ\sigma=\Phi_{\pmb{\theta}}(\sigma)\circ\tilde{g}_
{\bold{a},{\pmb{\theta}}}.
\end{equation*}
\end{lemma}
\begin{proof}
It follows from \cite[Proposition 9]{DV} that the surfaces $\tilde{g}_{\theta}$ and $\tilde{g}_
{\theta}\circ\sigma$ are congruent for all $\theta\in[0,\pi).$ Therefore, there exists
$\Psi_\theta(\sigma)\in\mathrm{O}(7)$ such that
\begin{equation} \label{decksmall}
\tilde{g}_\theta\circ\sigma=
\Psi_\theta(\sigma)\circ\tilde{g}_\theta
\end{equation}
for every $\theta\in[0,\pi).$
We define the isometry $\Phi_{\pmb{\theta}}(\sigma)\in\mathrm{O}(n+1)$ given by
\[
\Phi_{\pmb{\theta}}(\sigma)=\Psi_{\theta_1}(\sigma)\oplus\cdots\oplus\Psi_
{\theta_m}(\sigma),
\]
with respect to an orthogonal decomposition $\mathbb{R}^{6m}=\mathbb{R}^{6}
\oplus\cdots\oplus\mathbb{R}^{6}.$
That
\begin{equation*}
\tilde{g}_
{\bold{a},{\pmb{\theta}}}\circ\sigma=\Phi_{\pmb{\theta}}(\sigma)\circ\tilde{g}_
{\bold{a},{\pmb{\theta}}}
\end{equation*}
holds, follows directly from \eqref{decksmall}.
\end{proof}
\begin{remark}\label{remark1}
The isometry $\Phi_{\pmb{\theta}}(\sigma)$ is real analytic with respect to
${\pmb{\theta}}$ (cf. \cite{EQ}).
\end{remark}
\begin{lemma}\label{lemmaphitheta}
If $(\bold{a},\pmb{\theta})$ belongs to $\mathcal{M}_n^{\mathrm{e}}
(\tilde{g}),$ then $(\bold{a},\pmb {\theta})$ belongs to $\mathcal{M}^{\mathrm
{e}}_n(g)$ if and only if
\begin{equation}\label{phitheta}
\Phi_{\pmb{\theta}}(\mathcal{D})=\left\{\mathrm{Id}\right\}.
\end{equation}
\end{lemma}
\begin{proof}
Let $(\bold{a},\pmb{\theta})\in\mathcal{M}^{\mathrm{e}}_
n(g).$ There exists an exceptional surface $F\colon M\to\mathbb
{S}^{n}$ such that
\begin{equation*}
F\circ\pi=\tilde{g}_{\bold{a},{\pmb{\theta}}}.
\end{equation*}
Composing with an arbitrary $\sigma\in\mathcal{D}$ and using
Lemma \ref{deckbig}, we obtain
\begin{equation*}
\tilde{g}_{\bold{a},{\pmb{\theta}}}=\Phi_{\pmb{\theta}}(\sigma)\circ\tilde{g}_
{\bold{a},{\pmb{\theta}}}.
\end{equation*}
The fact that $\tilde{g}_{\bold{a},{\pmb{\theta}}}$ has substantial
codimension yields \eqref{phitheta}.
Conversely, assume that \eqref{phitheta} holds. We will prove that $\tilde{g}_
{\bold{a},{\pmb{\theta}}}$ factors as $F\circ\Pi$ where $F\colon M\to\mathbb{S}
^{n}$ is an exceptional surface. At first we claim that $\tilde{g}_{\bold{a},{\pmb
{\theta}}}$ remains constant on each fiber of the covering map $\Pi.$ Indeed, let
$\tilde{p}_1, \tilde{p}_2$ belong to the fiber $\Pi^{-1}(p)$ for some $p\in M.$ Then there exists
a deck transformation $\sigma$ such that $\sigma(\tilde{p}_1)=\tilde{p}_2.$ Using
Lemma \ref{deckbig} and \eqref{phitheta}, we obtain
\begin{eqnarray}
\tilde{g}_{\bold{a},{\pmb{\theta}}}(\tilde{p}_2)
&=&\tilde{g}_{\bold{a},{\pmb{\theta}}}\circ\sigma(\tilde{p}_1) \nonumber\\
&=&\Phi_{\pmb{\theta}}(\sigma)\circ\tilde{g}_{\bold{a},{\pmb{\theta}}}(\tilde{p}
_1) \nonumber\\
&=&\tilde{g}_{\bold{a},{\pmb{\theta}}}(\tilde{p}_1).\nonumber
\end{eqnarray}
Then $\tilde{g}_
{\bold{a},{\pmb{\theta}}}$ factors as $F\circ\Pi,$ where $F\colon M\to\mathbb{S}
^{n}$ is a minimal surface. It remains to prove that $F\in\mathcal{M}^{\mathrm{e}}_n(g).$ Since $\Pi$ is an
orientation preserving local isometry, it is obvious that $F$ is an exceptional surface.
\end{proof}
The following theorem provides properties of exceptional surfaces that are locally isometric to a pseudoholomorphic curve in
$\mathbb{S}^5.$
\begin{theorem}\label{lineorfinite}
If $g$ is a nonflat pseudoholomorphic curve in $\mathbb{S}^5,$ and $n$ is odd, then
the moduli space $\mathcal{M}_n^{\mathrm{e}}(g)$ splits as $\mathbb{S}^{m-1}_*\times\Gamma_0,$ where $\Gamma_0$
is a subset of $\Gamma^m.$ If $\Gamma_0$ is a proper subset of $\Gamma^m,$ then
it is locally a disjoint finite
union of $d$-dimensional real analytic subvarieties where $d=0,\dots,m-1$. Moreover,
the subset $\Gamma_0$ has the property that for each point
${\pmb{\theta}}\in\Gamma_0,$ every straight line through ${\pmb{\theta}}$ that
is parallel to every coordinate axis of $\mathbb{R}^m$ either intersects $\Gamma_0$ at finitely
many points, or at a line segment.
\end{theorem}
\begin{proof}
Lemma \ref{lemmaphitheta} implies that $\mathbb{S}_*^{m-1}\times
\left\{{\pmb{\theta}}\right\}$ is contained in $\mathcal{M}_n^{\mathrm{e}}(g)$ for
each $(\bold{a},{\pmb{\theta}})\in\mathcal{M}_n^{\mathrm{e}}(g).$ Therefore,
the moduli space splits as
$$
\mathcal{M}_n^{\mathrm{e}}(g)=\mathbb{S}_*^{m-1}\times\Gamma_0,
$$
where $\Gamma_0$ is a subset of $\Gamma^m.$ Additionally, Lemma \ref{lemmaphitheta}
implies that ${\pmb{\theta}}\in\Gamma_0$ if and only if $\Phi_{\pmb{\theta}}(\mathcal{D})
=\left\{\mathrm{Id}\right\}.$ Fix $\sigma\in\mathcal{D}$. Then $\Phi_{\pmb{\theta}}(\sigma)
=\mathrm{Id}$ and $\Gamma_0$ is a real analytic set (see Remark \ref{remark1}). If $\Gamma_0$ is a proper subset of
$\Gamma^m$, according to Lojasiewicz's structure theorem \cite[Theorem
6.3.3]{KP}) the set $\Gamma_0$ locally decomposes as
\[
\Gamma_0=\mathcal{V}^0\cup\mathcal{V}^1\cup\cdots\cup\mathcal{V}^{m-1},
\]
where each $\mathcal{V}^d,\ 0\le d\le m-1,$ is either empty or a disjoint finite union
of $d$-dimensional real analytic subvarieties.
Let ${\pmb{\theta}}=(\theta_1,\dots,\theta_l,\dots,\theta_m)\in\Gamma_0.$ Suppose that the
straight line through ${\pmb{\theta}}$ that is parallel to the $l$-th coordinate axis of $\mathbb{R}^m$
is not a finite set. Thus, this line contains a sequence ${\pmb{\theta}}^{(i)}=(\theta_1,\dots,
\theta_l^{(i)},\dots,\theta_m), i\in\mathbb{N}.$ By passing if necessary to a subsequence, we
may assume that this sequence converges to ${\pmb{\theta}}^{\infty}=(\theta_1,\dots,
\theta_l^{\infty},\dots,\theta_m),$ where $\theta_l^{\infty}=\lim\theta_l^{(i)}.$
Clearly ${\theta}_{l-1}\le{\theta}_l^{\infty}\le{\theta}_{l+1}.$
At first we suppose that ${\theta}_{l-1}<{\theta}_l^{\infty}<{\theta}_{l+1},$ that is ${\pmb{\theta}}^{\infty}\in\Gamma_0.$ Fix $\sigma\in
\mathcal{D}.$ Lemma \ref{lemmaphitheta} implies that
$\Phi_{{\pmb{\theta}}^{(i)}}(\sigma)=\mathrm{Id}$ and consequently $\Phi_{{\pmb{\theta}}^{\infty}}(\sigma)=\mathrm{Id}$.
We define the function
$$
h(\theta)=\left(\Phi_{({\theta}_1,\dots,
{\theta}_{l-1},\theta,{\theta}_{l+1},\dots{\theta}_m)}(\sigma)\right)_{ij}, \, \theta\in[\theta_{l-1},\theta_{l+1}),
$$
where $\big(\Phi_{\pmb{
\theta}}(\sigma)\big)_{ij}$ denotes the $(i,j)$-element of the matrix of $\Phi_{\pmb{
\theta}}(\sigma)$ with respect to the standard basis of $\mathbb{R}^{n+1}$.
From the mean value theorem we have that there exists $\xi_1^{(i)}$ between $\theta_l^{(i)}$ and
${\theta}_l^{\infty}$ such that $(dh/d\theta)(\xi_1^{(i)})=0$ and hence $(dh/d\theta)(\theta_l^{\infty})
=0.$ Applying
again the mean value theorem, we obtain
that there exists $\xi_2^{(i)}$ between $\xi_1^{(i)}$ and $\theta_l^{\infty}$ such that
$(d^2h/d\theta^2)(\xi_2^{(i)})=0.$ Inductively, we have
that the $k$-th derivative satisfies $(d^kh/d\theta^k)(\theta_l^{\infty})
=0$ for any
$k.$ The analyticity of $h$ (see Remark \ref{remark1})
yields that $h=\delta_{ij}$ on $[\theta_{l-1},\theta_{l+1}),$ where $\delta_{ij}$ is the Kr\"onecker delta.
Now without loss of generality, assume that ${\theta}_{l-1}={\theta}_l^{\infty}<{\theta}_{l+1}.$
Clearly ${\pmb{\theta}}^{\infty}\notin\Gamma_0.$ We fix $\sigma\in\mathcal{D}$
and extend $\Phi_{\pmb{\theta}}$ in the obvious way. Then $\Phi_{{\pmb{\theta}}^{(i)}}
(\sigma)=\mathrm{Id}$ and consequently $\Phi_{{\pmb{\theta}}^{\infty}}(\sigma)=\mathrm{Id}$
and the claim follows as before.
\end{proof}
We now provide a result for compact pseudoholomorphic curves in $\mathbb{S}^5.$
\begin{theorem}\label{compactps}
If $g$ is a compact pseudoholomorphic curve in $\mathbb{S}^5$ that is not
homeomorphic to the torus, then the moduli space $\mathcal{M}_n^{\mathrm{e}}
(g),$ with $n$ odd, is given by $\mathcal{M}_n^{\mathrm{e}}(g)=\mathbb{S}^{m-1}_*\times
\Gamma_0,$ where $\Gamma_0$ is a proper subset of $\Gamma^m$ that is locally a disjoint finite
union of $d$-dimensional real analytic subvarieties where $d=0,\dots,m-1.$ Moreover,
every straight line through each point ${\pmb{\theta}}\in\Gamma_0$ that
is parallel to every coordinate axis of $\mathbb{R}^m$ intersects $\Gamma_0$ at finitely
many points.
\end{theorem}
\begin{proof}
Suppose to the contrary that the intersection of $\Gamma_0$ with the straight line through $\pmb
{\theta}$ that is parallel to the first coordinate axis is an infinite set. For a fixed $\bold{a}\in
\mathbb{S}_*^{m-1},$ we choose $\pmb{\theta}_1,\dots, \pmb
{\theta}_N\in\Gamma_0$ that belong to this straight line. Hence $(\bold{a},\pmb
{\theta}_j)\in\mathcal{M}^{\mathrm{e}}_n(g)$ for all $\pmb{\theta}_j=(
\theta_{j1},\dots,\theta_{jm}),\ j=1,\dots,N.$ Consequently there exist exceptional
surfaces $F_j\colon M\to\mathbb{S}^n$ such that $F_j\circ\pi=\tilde{g}_{\bold{a},
\pmb{\theta}_j}.$
We claim that the set of all coordinate functions associated to vectors $\bold{v}=(v_1,0,\dots,0)$ in $\mathbb{R}^{6m}$
of all surfaces $F_j$'s are linearly independent. It is
sufficient to prove that if
\begin{equation}\label{linear}
\sum\limits_{j=1}^N\langle F_j,\bold{v}\rangle=0,
\end{equation}
then $\bold{v}=0.$ From \eqref{linear} we obtain
$$
\sum\limits_{j=1}^N\langle F_j\circ\pi,\bold{v}\rangle=0,
$$
or equivalently
$$
a_1\sum\limits_{j=1}^N\langle \tilde{g}_{\theta_{j1}},v_1\rangle=0.
$$
In analogy with the argument in the proof of \cite[Theorem 2]{DV}, we finally conclude
that $v_1=0$ and the claim is proved.
The contradiction follows easily since the coordinate functions of the surfaces $F_j$'s are eigenfunctions of
the Laplacian operator with corresponding eigenvalue 2 and the vector space of the eigenfunctions
has finite dimension. Hence $\Gamma_0\neq\Gamma^m$ and the proof follows from Theorem \ref{lineorfinite}.
\end{proof}
\begin{remark}\label{flattori}
The assumption in Theorem \ref{compactps} that the pseudoholomorphic curve
$g$ is not homeomorphic to the torus
is essential and can not be dropped. According to results due to Kenmotsu \cite{K2, K11}
the moduli space of all minimal surfaces in odd dimensional spheres that are isometric to
a flat pseudoholomorphic torus in $\mathbb{S}^5$ is not a finite set.
\end{remark}
\begin{proof}[Proof of Theorem \ref{mikradim}]
It follows from \cite[Theorem 5]{VT} and \cite[Corollary 1]{VT} that any minimal surface $f\colon M\to\mathbb{S}^n$ that is
isometric to $g$ is exceptional and $n=5.$ Then Theorem \ref{compactps} above completes the proof.
\end{proof}
\section{Isometric deformations of nonisotropic pseudoholomorphic curves in $\mathbb{S}^6$}
In this section, we mostly deal with noncongruent isometric deformations of
pseudoholomorphic curves in $\mathbb{S}^6$ that are always 1-isotropic (see \cite{V16}) but
in general not 2-isotropic.
For a given nonisotropic pseudoholomorphic curve
$g\colon M\to\mathbb{S}^6,$
our aim is to describe the moduli space $\mathcal{M}
^K_n(g)$
of all noncongruent minimal surfaces $f\colon M\to\mathbb{S}^n$ that are locally
isometric to the curve $g$, having the same normal curvatures up to order 2 with
the curve $g.$
From \cite[Corollary 5.4(ii)]{V} we know that two
locally isometric 1-isotropic surfaces in $\mathbb{S}^6$ with the same normal curvatures, belong locally
to the same associated family. In particular, if $g$ is simply connected then $\mathcal{M}^K_6
(g)=[0,\pi).$
Hereafter we are interested in the case where the pseudoholomorphic curve $g$ is nonsimply connected. We consider the covering map $\Pi\colon\tilde{M}\to M,$ $\tilde{M}$ being the universal cover of $M$ equipped with the metric and
orientation that make $\Pi$ an orientation preserving local isometry. Corresponding objects on
$\tilde{M}$ are denoted with tilde. Then the map $\tilde{g}\colon \tilde{M}\to\mathbb{S}^6$
with $\tilde{g}=g\circ\Pi$ is up to congruence a pseudoholomorphic curve. Hence, the moduli space $\mathcal{M}^K_6(g)$ of
the curve $g$ can be described as the set of all $\theta\in\mathcal{M}^K_6
(\tilde{g})=[0,\pi)$ such that $\tilde{g}_{\theta}$ factors as $\tilde{g}_{\theta}=g_\theta\circ\Pi$
for a minimal surface $g_\theta\colon M\to\mathbb{S}^6$
and $\tilde{g}_{\theta}$ is a member in the associated family of $\tilde{g}.$
We follow this notation throughout this section.
\begin{lemma}\label{lemmaPhitheta}
(i) For each $\sigma\in\mathcal{D},$ the surfaces $\tilde{g}_{\theta}$
and $\tilde{g}_{\theta}\circ\sigma$ are congruent for every $\theta\in[0,\pi],$ that is there exists $\Psi_{\theta}(\sigma)\in\mathrm{O}
(7)$ such that
\begin{equation} \label{deckagain}
\tilde{g}_
{\theta}\circ\sigma=\Psi_{\theta}(\sigma)\circ\tilde{g}_
{\theta}.
\end{equation}
(ii) If $\theta$ belongs to $\mathcal{M}^K_6(\tilde{g}),$ then $\theta$ belongs to $\mathcal{M}^K_6(g)$ if
and only if
\begin{equation}\label{phithetaIII}
\Psi_{\theta}(\mathcal{D})=\left\{\mathrm{Id}\right\},
\end{equation}
where $\Psi_{\theta}\in\mathrm{O}(7).$
\end{lemma}
\begin{proof}
(i) From \cite[Proposition 9]{DV} we have that for any $\sigma$ in the group
$\mathcal{D},$ the surfaces $\tilde{g}_\theta\colon\tilde{M}\to\mathbb{S}^6$ and $\tilde{g}_\theta\circ\sigma\colon \tilde{M}\to\mathbb{S}^6$ are congruent for any $\theta\in\mathcal{M}^K_6(g).$
Therefore, there exists
$\Psi_\theta(\sigma)\in\mathrm{O}(7)$ such that \eqref{deckagain} holds
for every $\theta\in\mathcal{M}^K_6(g).$
(ii) Take $\theta\in\mathcal{M}^K_6(g).$ Then, there exists a minimal surface $g_{\theta}\colon M\to\mathbb
{S}^{6}$ such that $g_{\theta}\circ\pi=\tilde{g}_{\theta}.$
Composing with an arbitrary $\sigma\in\mathcal{D}$ and using
\eqref{deckagain} we obtain
\begin{equation*}
\tilde{g}_{\theta}=\Psi_{\theta}(\sigma)\circ\tilde{g}_{\theta}.
\end{equation*}
Since $\tilde{g}_{\theta}$ has substantial
codimension \eqref{phithetaIII} yields.
Conversely assume that \eqref{phithetaIII} holds. We will prove that
$\tilde{g}_{\theta}$ factors as $\tilde{g}_{\theta}=g_{\theta}\circ\Pi,$ where $g_{\theta}\colon M\to\mathbb{S}^6$ is
a minimal surface. At first we claim that $\tilde{g}_{\theta}$ remains constant on each fiber of
the covering map $\Pi.$ Indeed, let
$\tilde{p}_1, \tilde{p}_2$ belong to $\Pi^{-1}(p)$ for some $p\in M.$ Then there exists
a deck transformation $\sigma$ such that $\sigma(\tilde{p}_1)=\tilde{p}_2.$ Using
\eqref{deckagain}, we obtain
\begin{eqnarray}
\tilde{g}_{\theta}(\tilde{p}_2)
&=&\tilde{g}_{\theta}\circ\sigma(\tilde{p}_1) \nonumber\\
&=&\Psi_{\theta}(\sigma)\circ\tilde{g}_{\theta}(\tilde{p}
_1) \nonumber\\
&=&\tilde{g}_{\theta}(\tilde{p}_1).\nonumber
\end{eqnarray}
Then $\tilde{g}_{\theta}$ factors as $\tilde{g}_{\theta}=g_{\theta}\circ\Pi,$ where $F\colon M\to\mathbb{S}
^{n}$ is a minimal surface. It remains to prove that $g_{\theta}\in\mathcal{M}^K_6(g).$ Since $\Pi$ is an
orientation preserving local isometry, it is obvious that $F$ is a minimal surface.
\end{proof}
Now we are able to prove Theorem \ref{finiteorcircle}.
\begin{proof}[Proof of Theorem \ref{finiteorcircle}]
If $g$ is substantial in a totally geodesic $\mathbb{S}^5,$ then from \cite[Theorem 1]{DV},
the moduli space of $g$ is either a circle or a finite set.
If $g$ is isotropic and substantial in $\mathbb{S}^6,$ then Theorem \ref{congruent2iso}
implies that the moduli space of $g$ consists of a single point.
Suppose now that $g$ is substantial in $\mathbb{S}^6$ and nonisotropic. Assume that $\mathcal{M}^K_6(g)$ is not finite. Thus, there exists a sequence $\theta^{(i)}, i\in
\mathbb{N},$ that belongs to $\mathcal{M}^K_6(g).$ By passing if necessary to a subsequence, we
assume that this sequence converges to $\theta^{\infty}\in[0,\pi].$
From Lemma \ref{lemmaPhitheta}(ii), we derive that $\Psi_{\theta^{(i)}}(\mathcal{D})=\left\{\mathrm{Id}\right\}$ for
every $i\in\mathbb{N}$ and $\Psi_{\theta^{\infty}}(\mathcal{D})=\left\{\mathrm{Id}\right\}.$
Fix a $\sigma\in\mathcal{D}.$
We define the function
$$
h(\theta)=\left(\Psi_{\theta}(\sigma)\right)_{ij}, \theta\in[0,\pi],
$$
where $\left(\Psi_{\theta}(\sigma)\right)_{ij}$ denotes the $(i,j)$-element of the matrix of $\Psi_
{\theta}(\sigma)$ with respect to the standard basis of $\mathbb{R}^{7}$.
By the mean value theorem, there exists $\xi_1^{(i)}$ between $\theta^{(i)}$ and
${\theta}^{\infty}$ such that $(dh/d\theta)(\xi_1^{(i)})=0$ and hence $(dh/d\theta)(\theta^{\infty})
=0.$ Applying
repeatedly the mean value theorem, we obtain inductively
that the $k$-th derrivative satisfies $(d^kh/d\theta^k)(\theta^{\infty})
=0$ for any $k.$ The analyticity of $h$ (cf. \cite{EQ}) implies that $h=\delta_{ij},$ where $\delta_{ij}$ is the Kr\"onecker delta.
\end{proof}
We now turn our attention to the study of isometric deformations of compact nonisotropic pseudoholomorphic curves in $\mathbb{S}^6.$ We will need the following lemmas:
\begin{lemma}\label{antistoixo}
Let $g\colon M\to\mathbb{S}^6$ be a nonisotropic pseudoholomorphic curve.
For each $g_\theta,\theta\in\mathcal{M}^K_6(g),$ there exists a parallel and orthogonal
bundle isomorphism $T_\theta\colon N_gM\to N_{g_\theta}M$ such that the second
fundamental forms of $g$ and $g_\theta$ are related by
\[
\alpha^{g_\theta}(X,Y)=T_\theta\circ\alpha^g(J_\theta X,Y),\ \ X,Y\in TM.
\]
\end{lemma}
\begin{proof}
Since $g$ and $g_\theta$ have the same normal curvatures, it follows from \cite[Corollary 5.4(ii)]{V} that for any simply connected
subset $U$ of $M$ there exists a parallel and orthogonal bundle isomorphism $T_\theta^U\colon
N_g U\to N_{g_\theta}U$ such that the second fundamental forms of the surfaces $g|_U$ and
$g_{\theta}|_U$ are related by
\[
\alpha^{g_\theta|_U}(X,Y)=T_\theta^U\circ\alpha^{g|_U}(J_\theta X,Y),\ \ X,Y\in TM.
\]
Let $U,V$ be simply connected subsets of $M$ with $U\cap V\neq\varnothing.$
Then on $U\cap V$ we have
\[
T_\theta^U\circ\alpha^{g|_U}(J_\theta X,Y)=T_\theta^V\circ\alpha^{g|_V}(J_\theta X,Y),
\]
for every $X,Y\in TM.$ Equivalently we obtain
\[
\left(T_\theta^U-T_\theta^V\right)\circ\alpha^{g|_{U\cap V}}(X,Y)=0
\]
and obviously $\left(T_\theta^U-T_\theta^V\right)(N^{g|_{U\cap V}}_1)=0.$
Differentiating we obtain $\left(T_\theta^U-T_\theta^V\right)(N^{g|_{U\cap V}}_2)=0,$ which yields
that $T_\theta^U=T_\theta^V$ on $U\cap V.$ Thus,
$T_\theta^U$ is globally well defined.
\end{proof}
For each orthonormal frame along any minimal surface, one has the connection forms (cf. \cite{V}).
\begin{lemma}\label{ksanauseful}
Let $g\colon M\to\mathbb{S}^6$ be a substantial nonisotropic pseudoholomorphic curve and
let $M_1$ be the zero set of the second Hopf differential $\Phi_2.$ Around each point of $M\smallsetminus M_1,$ there exist a local
complex coordinate $(U,z),$ $U\subset M\smallsetminus M_1$ and orthonormal
frames $\{e_1,e_2\}$ in $TU,$ $\{e_3,e_4\}$ in $N^g_1U$ and $\{e_5,e_6\}$ in $N^g_2U$ which agree with the given orientations such that:
(i) $e_5$ and $e_6$ give respectively the directions of the major and the minor axes of the second curvature ellipse, and
(ii) $H_5=\kappa_2,$ $H_6=i\mu_2$ and $\kappa_2$ and $\mu_2$ are smooth real functions. Moreover, the connection and the normal connection forms, with respect to this frame, are given respectively, by
\begin{equation}\label{connectionforms}
\omega_{12}=-\frac{1}{6}*d\log(\kappa_2^2-\mu_2^2),\,\,\omega_{34}=2\omega_{12}+*d\log\kappa_1, \,\, \omega_{56}=\frac{\kappa_2\mu_2}{\kappa_2^2-\mu_2^2}*d\log\frac{\mu_2}{\kappa_2},
\end{equation}
where $*$ stands for the Hodge operator.
\end{lemma}
\begin{proof}
(i) Take an arbitrary orthonormal frame $\{E_1,E_2\}$ in $TU.$ Arguing
pointwise in $U$ we have that
$$\max\limits_{\theta\in[0,2\pi)}\Vert\alpha^g_3(X_\theta,X_\theta,X_\theta)\Vert=
\kappa_2 \text{\,\, and\,\,}
\min\limits_{\theta\in[0,2\pi)}\Vert\alpha^g_3(X_\theta,X_\theta,X_\theta)\Vert=\mu_2,
$$
where $X_\theta=\cos\theta E_1+\sin\theta E_2.$
Assume that the function $f(\theta)=\Vert\alpha^g_3(X_\theta,X_\theta,X_\theta)\Vert^2$ attains
its maximum at $\theta_0\in[0,2\pi).$ Since $f'(\theta_0)=0,$ we find that
\[
\langle \alpha^g_3(X_{\theta_0},X_{\theta_0},X_{\theta_0}),\alpha^g_3(X_{\theta_0},X_{\theta_0},X_{\theta_0})\rangle=0,
\]
or equivalently
\[
2\langle \alpha^g_3(E_1,E_1,E_1),\alpha^g_3(E_1,E_1,E_2) \rangle\cos 6\theta =
\left( \Vert\alpha^g_3(E_1,E_1,E_1)\Vert^2-\Vert\alpha^g_3(E_1,E_1,E_2)\Vert^2\right)\sin 6\theta.
\]
Since the second curvature ellipse is not a circle, we choose a smooth function
$\sigma$ such that
\[
\tan\sigma=
\frac{2\langle \alpha^g_3(E_1,E_1,E_1),\alpha^g_3(E_1,E_1,E_2) \rangle}
{\Vert\alpha^g_3(E_1,E_1,E_1)\Vert^2-\Vert\alpha^g_3(E_1,E_1,E_2)\Vert^2},
\]
or
\[
\cot\sigma=
\frac{\Vert\alpha^g_3(E_1,E_1,E_1)\Vert^2-\Vert\alpha^g_3(E_1,E_1,E_2)\Vert^2}
{2\langle \alpha^g_3(E_1,E_1,E_1),\alpha^g_3(E_1,E_1,E_2) \rangle}.
\]
We now consider the orthonormal frame $\{e_1,e_2\}$ in $TU$ with
$$ e_1=\cos\sigma E_1+\sin\sigma
E_2 \text{\,\,and\,\,} e_2=-\sin\sigma E_1+\cos\sigma E_2.$$
We may also consider the orthonormal frame $\{e_3, e_4\}$
in $N^g_1U$ given by
$$ e_3=\frac{1}{\kappa_1 }\alpha^g(e_1,e_1)\text{\,\,and\,\,}
e_4=\frac{1}{\kappa_1}\alpha^g(e_1,e_2)$$
and the orthonormal frame $\{e_5, e_6\}$ in $N^g_2U$ such
that
$$ e_5= \frac{1}{\kappa_2}\alpha^g_3(e_1,e_1,e_1)\text{\,\,and\,\,} e_6=\frac{1}{\mu_2}\alpha^g_3(e_1,e_1,e_2).$$
Let $\{\tilde{e}_5,
\tilde{e}_6\}$ be an orthonormal frame in $N^g_2U$ as in \cite[Lemma 5]{V16}. Then
the complex valued functions $\tilde{H}_5,\tilde{H}_6$ associated to the
frame $\{\tilde{e}_5, \tilde{e}_6\}$ satisfy
\begin{equation}\label{tildeH}
\tilde{H}_6
=i(\kappa_1-\tilde{H}_5).
\end{equation}
We easily find that
\begin{equation}\label{H5}
\tilde{H}_5=\cos\varphi H_5+\sin\varphi H_6
\end{equation}
and
\begin{equation}\label{H6}
\tilde{H}_6=-\sin\varphi H_5+\cos\varphi H_6,
\end{equation}
where $\varphi$ is the angle between $e_5$ and $\tilde{e}_5.$
Since $H_5=\kappa_2$ and $H_6=i\mu_2,$ equations
\eqref{tildeH}, \eqref{H5} and \eqref{H6} yield $\varphi=0$ and consequently the orthonormal
frames $\{e_5,e_6\}$ and $\{\tilde{e}_5,\tilde{e}_6\}$ coincide.
(ii) It follows directly from \cite[Lemma 6]{V08} that the connection forms $\omega_{34}$ and $\omega_{56}$
are given by \eqref{connectionforms}.
From $\alpha_3(e_1,e_1,e_1)=\kappa_2e_5,$ we obtain
\[
\omega_{35}(e_1)=-\omega_{45}(e_2)=\frac{\kappa_2}{\kappa_1}
\text{\,\,and\,\,\,}
\omega_{36}(e_1)=\omega_{46}(e_2)=0.
\]
Similarly, $\alpha_3(e_1,e_1,e_2)=\mu_2e_6$ implies that
\[
\omega_{46}(e_1)=\omega_{36}(e_2)=\frac{\mu_2}{\kappa_1}
\text{\,\,and\,\,\,}
\omega_{45}(e_1)=\omega_{35}(e_2)=0.
\]
Therefore,
\[
\omega_{35}=\frac{\kappa_2}{\kappa_1}\omega_1,\ \omega_{45}=-\frac{\kappa_2}{\kappa_1}
\omega_2,
\ \omega_{36}=\frac{\mu_2}{\kappa_1}\omega_2 \text{\,\, and\,\,}\omega_{46}=\frac{\mu_2}{\kappa_1}\omega_1.
\]
Using the above, the Ricci equations
\[
\langle R^\perp(e_1,e_2)e_3,e_5\rangle=0 \text{\,\ and \,} \langle R^\perp
(e_1,e_2)e_4,e_6\rangle=0,
\]
where $R^\perp$ is the curvature tensor of the normal bundle, are written equivalently as
\[
3\omega_{12}(e_1)=\frac{\mu_2}{\kappa_2}\omega_{56}(e_1)+e_2\big(\log\frac
{\kappa_2}{\kappa_1}\big)-*d\log\kappa_1(e_1)
\]
and
\[
3\omega_{12}(e_1)=\frac{\kappa_2}{\mu_2}\omega_{56}(e_1)+e_2\big(\log\frac
{\mu_2}{\kappa_1}\big)-*d\log\kappa_1(e_1)
\]
respectively. From these and from the fact that the normal connection form $\omega_{56}$ is given by \eqref{connectionforms}, one can easily obtain
\begin{eqnarray}\label{omegapros}
\omega_{12}(e_1)=-\frac{1}{6}*d\log(\kappa_2^2-\mu_2^2)(e_1).
\end{eqnarray}
Arguing similarly for the Ricci equations
\[
\langle R^\perp(e_1,e_2)e_3,e_6\rangle=0 \text{\,\ and \,} \langle R^\perp
(e_1,e_2)e_4,e_5\rangle=0
\]
we have that
\begin{eqnarray*}
\omega_{12}(e_2)=-\frac{1}{6}*d\log(\kappa_2^2-\mu_2^2)(e_2), \nonumber
\end{eqnarray*}
which combined with \eqref{omegapros} yields the connection form $\omega_{12}$ of \eqref{connectionforms}.
\end{proof}
Let $g\colon M\to\mathbb{S}^6$ be a substantial pseudoholomorphic curve. Assume hereafter
that $g$ is nonisotropic. For each point $p\in M\smallsetminus M_1,$ we consider
$\{e_1,e_2,e_3,e_4,e_5,e_6\}$ being an orthonormal frame on a neighborhood $U
\subset M\smallsetminus M_1$ of $p$ as in Lemma
\ref{ksanauseful}.
We note that the connection form $\omega_{56}$ cannot vanish on any open subset of
$M\smallsetminus M_1.$
Suppose to the contrary that $\omega_{56}=0.$ Then \eqref{connectionforms} implies that
$\mu_2=\lambda \kappa_2$ for some $\lambda\in\mathbb{R}^+$ and from \cite[Theorem 5(iii)]{V16} we obtain
\[
\kappa_2=\frac{\kappa_1}{\lambda+1} \text{\,\, and\,}\ \mu_2=\frac{\lambda\kappa_1}{\lambda+1}.
\]
From \eqref{connectionforms} it follows that the connection form is given by
\[
\omega_{12}=-\frac{1}{3}*d\log\kappa_1,
\]
which implies
\[
6K=\Delta\log{\kappa_1^2}=\Delta\log(1-K).
\]
According to Theorem \ref{eschvlach}, this would imply a reduction of codimension, which is a contradiction.
For any $\theta\in\mathcal{M}^K_6(g),$ let $\{e_1,e_2,T_\theta e_3,T_\theta e_4,T_\theta
e_5,T_\theta e_6\}$ be an orthonormal frame along $g_\theta,$ where $T_\theta$ is the bundle
isomorphism of Lemma \ref{antistoixo}. The complex valued functions $H_3, H_4, H_5, H_6$ of $g,$ associated
to the orthonormal frame $\{e_1,e_2,e_3,e_4,e_5,e_6\}$ and the corresponding
functions $H_3^\theta, H_4^\theta, H_5^\theta, H_6^\theta$ of $g_\theta$, associated to the orthonormal
frame
$\{e_1,e_2,T_\theta e_3,T_\theta e_4,T_\theta e_5,T_\theta e_6\}$ satisfy
\begin{equation}\label{xrhsimoweing}
H_3^\theta=e^{-i\theta}H_3,\,\,H_4^\theta=e^{-i\theta}H_4,\,\,H_5^\theta=e^{-i\theta}H_5\,\text{ and }H_6^\theta=e^{-i\theta}H_6.
\end{equation}
Using \eqref{xrhsimoweing} and the Weingarten formula for $g_\theta,$ we obtain
\begin{equation}\label{weine3}
\tilde{\nabla}_ET_{\theta}e_3=\omega_{34}(E)T_{\theta}e_4+\frac{\kappa_2}{\kappa_1}T_{\theta}e_5-\frac{i\mu_2}{\kappa_1}T_{\theta}e_6-\kappa_1 e^{i\theta}dg_{\theta}(\overline{E}),
\end{equation}
\begin{equation}\label{weine4}
\tilde{\nabla}_ET_{\theta}e_4=-\omega_{34}(E)T_{\theta}e_3+\frac{i\kappa_2}{\kappa_1}T_{\theta}e_5+\frac{\mu_2}{\kappa_1}T_{\theta}e_6+i\kappa_1 e^{i\theta}dg_{\theta}(\overline{E}),
\end{equation}
\begin{equation}\label{weine5}
\tilde{\nabla}_ET_{\theta}e_5=\omega_{56}(E)T_{\theta}e_6-\frac{\kappa_2}{\kappa_1}\left(T_{\theta}e_3+iT_{\theta}e_4\right),
\end{equation}
\begin{equation}\label{weine6}
\tilde{\nabla}_ET_{\theta}e_6=-\omega_{56}(E)T_{\theta}e_5+\frac{i\mu_2}{\kappa_1}\left(T_{\theta}e_3+iT_{\theta}e_4\right),
\end{equation}
where $E=e_1-ie_2$ and $\tilde{\nabla}$ stands for the usual connection in the induced bundle $(i_1\circ f)^*(T\mathbb{R}^7),$ with $i_1\colon \mathbb{S}^6\to\mathbb{R}^7$ being the inclusion map.
\begin{lemma}
Suppose that for $\theta_j\in\mathcal{M}^K_6(g), j=1,\dots,m,$ there exist vectors $v_j\in\mathbb{R}^7,$ such that
\begin{equation*}
\sum\limits_{j=1}^m\langle g_{\theta_j},v_j\rangle=0 \,\,\text{ on }\,\, U.
\end{equation*}
Then the following hold:
\begin{equation}\label{seconduseful}
\sum\limits_{j=1}^me^{i\theta_j}\left(\kappa_2\langle T_{\theta_j}e_5,v_j\rangle-i\mu_2\langle T_{\theta_j}e_6,v_j\rangle\right)=0,
\end{equation}
away from the zeros of $\omega_{56},$ and
\begin{equation}\label{thirduseful}
\overline{E}\Big(\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_6,v_j\rangle\Big)=-\omega_{56}(\overline{E})\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_5,v_j\rangle.
\end{equation}
\end{lemma}
\begin{proof}
Our assumption implies that
\begin{equation*}
\sum\limits_{j=1}^m\langle dg_{\theta_j},v_j\rangle=0.
\end{equation*}
Differentiating and using the Gauss formula we obtain
\begin{equation}\label{auseful}
\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_3-iT_{\theta_j}e_4,v_j\rangle=0.
\end{equation}
Differentiating \eqref{auseful} with respect to $E$ and using \eqref{xrhsimoweing}, \eqref{weine3} and \eqref{weine4}, it follows that
$$
\sum\limits_{j=1}^me^{i\theta_j}\left(\overline{H}_5\langle T_{\theta_j}e_5,v_j\rangle+\overline{H}_6\langle T_{\theta_j}e_6,v_j\rangle\right)=0.
$$
Using that $H_5=\kappa_2$ and $H_6=i\mu_2$ (see Lemma \ref{ksanauseful}(ii)), the above yields \eqref{seconduseful}.
From \eqref{weine6}, we compute that
\[
\overline{E}\Big(\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_6,v_j\rangle\Big)=-\omega_{56}(\overline{E})\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_5,v_j\rangle -\frac{i\mu_2}{\kappa_1}\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_3-iT_{\theta}e_4,v_j\rangle,
\]
which in view of \eqref{auseful} yields \eqref{thirduseful}.
\end{proof}
We recall the following result \cite{V16}.
\begin{lemma}\label{apoesch}
Let $f\colon M\to\mathbb{S}^n$ be a compact exceptional surface. The Euler-Poincar\'{e}
number $\chi(N_r^fM)$ of the $r$-th normal bundle and the Euler-Poincar\'{e}
characteristic $\chi(M)$ of $M$ satisfy the following:
(i) If $\Phi_r\neq0$ for some $1\le r<m,$ where $m=[(n-1)/2],$ then
\[
\chi(N_r^fM)=0 \text{\,\, and \,\,} (r+1)\chi(M)=-N(a_r^+)=-N(a_r^-).
\]
(ii) If $\Phi_r=0$ for some $1\le r\le m,$ then
\[
(r+1)\chi(M)-\chi(N_r^fM)=-N(a_r^+).
\]
(iii)If $\Phi_m\neq0,$ then
\[
(m+1)\chi(M)\mp\chi(N_m^fM)=-N(a_m^\pm).
\]
\end{lemma}
Now we are able to prove Theorem \ref{compnon}.
\begin{proof}[Proof of Theorem \ref{compnon}]
According to Theorem \ref{finiteorcircle}, the space $\mathcal{M}^K_6(g)$ of the
isometric deformations that are isometric to $g$ is either $[0,\pi)$ or a finite subset of $[0,\pi).$
Suppose to the contrary that $\mathcal{M}^K_6(g)=[0,\pi).$ We claim that the
coordinate functions of the minimal surfaces $g_\theta, \theta\in[0,\pi),$ are linearly independent. Since these functions are eigenfunctions of the Laplace operator of $M$ with corresponding eigenvalue 2, this contradicts the fact that the eigenspaces of the Laplace operator are finite dimensional. To prove that the coordinate functions are linearly independent, it is enough to prove that if
\begin{equation}\label{firstusefulagain}
\sum\limits_{j=1}^m\langle g_{\theta_j},v_j\rangle=0,
\end{equation}
for $0<\theta_1<\cdots<\theta_m<\pi,$ then $v_j=0$ for all $1\le j\le m.$
Assume to the contrary that $v_j\neq0$ for all $1\le j\le m.$ Let $M_1=\{p_1,\dots,p_k\}$ be the zero set of $\Phi_2.$ Around each point $p\in M\smallsetminus M_1,$ we choose local complex coordinate $(U,z)$ and an orthonormal frame $\{e_1,e_2,e_3,e_4,e_5,e_6\}$ on $U\subset M\smallsetminus M_1$ as in Lemma \ref{ksanauseful}. We consider the complex valued function
\[
\psi:=\Big(\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_6,v_j\rangle\Big)^2,
\]
where $T_{\theta_j}\colon N_gM\to N_{g_{\theta_j}}M$ is the bundle
isomorphism of Lemma \ref{antistoixo}. Obviously $\psi$ is well defined on $M\smallsetminus M_1.$ Equations \eqref{connectionforms} imply that
\begin{equation*}
E(\kappa_2)=i\mu_2\omega_{56}(E)-3i\kappa_2\omega_{12}(E),
\end{equation*}
and
\begin{equation*}
E(\mu_2)=i\kappa_2\omega_{56}(E)-3i\mu_2\omega_{12}(E).
\end{equation*}
These yield
\[
\omega_{56}(\overline{E})=\frac{i}{\kappa_2^2-\mu_2^2}\left(\kappa_2\overline{E}(\mu_2)-\mu_2\overline{E}(\kappa_2) \right).
\]
Then \eqref{seconduseful} and \eqref{thirduseful} imply that $\overline{E}\big(\psi\left(1-\mu_2^2/\kappa_2^2\right)\big)=0,$ and hence the function $\Psi:=\psi\left(1-\mu_2^2/\kappa_2^2\right)\colon M\smallsetminus M_1\to\mathbb{C}$ is holomorphic. Since $\Psi$ is bounded, its isolated singularities are removable and consequently there exists a constant $c$ such that
\begin{equation}\label{limit}
\psi(\kappa_2^2-\mu_2^2)=c\kappa_2^2\,\,\text{ on\,}\, M\smallsetminus M_1.
\end{equation}
We claim that $c=0.$ Indeed, if $\kappa_2(p_l)=\mu_2(p_l)>0$ for some $1\le l\le k,$ then taking the limit in \eqref{limit} along a sequence of points in $M\smallsetminus M_1$ that converges to $p_l,$ we deduce that $c=0.$
Suppose now that $\kappa_2(p_l)=\mu_2(p_l)=0$ for all $1\le l\le k.$ Let $(V,z)$ be a local complex coordinate around $p_l$ with $z(p_l)=0.$
From the proof of \cite[Proposition 4]{V08} for $s=2$
we obtain
\[
d\overline{H}_5-3i\overline{H}_5\omega_{12}-\overline{H}_6\omega_{56}\equiv 0\;\mathrm{mod}\; \phi,
\]
and
\[
d\overline{H}_6-3i\overline{H}_6\omega_{12}+\overline{H}_5\omega_{56}\equiv 0\;\mathrm{mod}\; \phi.
\]
Writing $\phi=Fdz,$ we deduce that
\[
\frac{\partial\overline{H}_5}{\partial\overline{z}}=3i\overline{H}_5\omega_{12}(\overline{\partial})+\overline{H}_6\omega_{56}(\overline{\partial})
\]
and
\[
\frac{\partial\overline{H}_6}{\partial\overline{z}}=3i\overline{H}_6\omega_{12}(\overline{\partial})-\overline{H}_5\omega_{56}(\overline{\partial}).
\]
Using a theorem due to Chern \cite[p. 32]{Ch}, we may write
\[
\overline{H}_5=z^{m_l}\hat{H}_5\text{\,\,\,\, and \,\,\,}\overline{H}_6=z^{m_l}\hat{H}_6,\]
where $m_l$ is a positive integer and $\hat{H}_5,\hat{H}_6$ are nonzero smooth complex functions.
Since
$$
\alpha_3(E,E,E)=4(\overline{H}_5e_5+\overline{H}_6e_6),
$$
we obtain
\begin{equation}\label{a3tilde}
\alpha_{3}^{(3,0)}=z^{m_l}\hat{\alpha}_{3}^{(3,0)}\,\,\text{ on }\, V,
\end{equation}
where $\hat{\alpha}_{3}^{(3,0)}$ is a tensor field of type $(3,0)$ with $\hat{\alpha}_{3}^{(3,0)}|_{p_l}\neq0.$
We now define the $N_2^g$-valued tensor field $\hat{\alpha}_3:=\hat{\alpha}_{3}
^{(3,0)}+\overline{\hat{\alpha}_{3}^{(3,0)}}.$ It is clear that $\hat{\alpha}_3$
maps the unit circle on each tangent plane into an ellipse, whose length of the semi-axes are denoted by $\hat{\kappa}_2\ge
\hat{\mu}_2\ge0.$ We furthermore consider the differential form of type $(6,0)$
\[
\hat{\Phi}_2:=\langle \hat{\alpha}_{3}^{(3,0)},\hat{\alpha}_{3}^{(3,0)}\rangle dz^6,
\]
which in view of \eqref{a3tilde}, is related to the Hopf differential of $g$ by $\Phi_2=z^{2m_l}\hat{\Phi_2}.$
We split $\Phi_2$ and $\hat{\Phi_2},$ with respect to an arbitrary orthonormal frame $\{\xi_1,\dots\xi_6\},$ where $\{\xi_1,\xi_2\}$
and $\{\xi_5,\xi_6\}$ are arbitrary orthonormal frames of $TV$ and $N^g_2V$ respectively as
\[
\Phi_2=\frac{1}{4}\big(\overline{H}_5^2+\overline{H}_6^2\big)\phi^6=\frac{1}{4}k^{+}_{2}k^{-}_{2}\phi^6,
\]
\[
\hat{\Phi}_2=\frac{1}{4}\big(\overline{\hat{H}}_5^2+\overline{\hat{H}}_6^2
\big)\phi^6=\frac{1}{4}\hat{k}^{+}_{2}\hat{k}^{-}_{2}\phi^6,
\]
where $k^{\pm}_{2}=\overline{H}_5\pm i\overline{H}_6, \hat{k}^{\pm}_{2}=\overline{\hat{H}}_5\pm i\overline{\hat{H}}_6,$
\[
\hat{H}_5=\langle
\hat{\alpha}_{3}(e_1,e_1,e_1),e_5\rangle+i\langle
\hat{\alpha}_{3}(e_1,e_1,e_2),e_5\rangle
\]
and
\[
\hat{H}_6=\langle
\hat{\alpha}_{3}(e_1,e_1,e_1),e_6\rangle+i\langle
\hat{\alpha}_{3}(e_1,e_1,e_2),e_6\rangle.
\]
From \eqref{a3tilde}, we obtain $\overline{H}_{a}=z^{m_l}\overline{\hat{H}}_{a}$ for $a=5,6,$ or equivalently,
$k_2^{\pm}=z^{m_l}\hat{k}_2^{\pm}.$ Observe that $a_2^{\pm}=|k_2^{\pm}|.$ Hence
\begin{equation}\label{kappamu}
\kappa_2=|z|^{m_l}\hat{\kappa}_2,\,\, \mu_2=|z|^{m_l}\hat{\mu}_2.
\end{equation}
Now \eqref{limit} yields
\begin{equation}\label{limitnew}
\psi(\hat{\kappa}_2^2-\hat{\mu}_2^2)=c\hat{\kappa}_2^2\,\,\text{ on\,}\, V\smallsetminus \{p_l\}.
\end{equation}
If $\hat{\kappa}_2(p_l)>\hat{\mu}_2(p_l)$ for all $1\le l\le k,$ then \eqref{kappamu} implies that
$$N(a^+_2)=\sum\limits_{l=1}^{k}m_l=N(a^-_2).$$
Hence, Lemma \ref{apoesch} yields $\mathcal{\chi}(N_2^f)=0,$ which contradicts our assumption. Thus, $\hat{\kappa}_2(p_l)=\hat{\mu}_2(p_l)$ for some $1\le l\le k.$ Taking the limit in \eqref{limitnew}, along a sequence of points in $V\smallsetminus \{p_l\}$ which converges to $p_l,$ we obtain $c\hat{\kappa}_2^2(p_l)=0.$ Since $\hat{\alpha}_3|_{p_l}\neq0,$ we derive that $c=0.$
In view of \eqref{limit}, we conclude that $\psi=0$ on $M\smallsetminus M_1.$ This implies that
\[
\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_6,v_j\rangle=0,
\]
which due to \eqref{seconduseful} gives that
\begin{equation*}
\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_5,v_j\rangle=0.
\end{equation*}
Differentiating this with respect to $E,$ and using \eqref{weine5} and the above, we obtain
\begin{equation*}
\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_3+iT_{\theta_j}e_4,v_j\rangle=0
\end{equation*}
which combined with \eqref{auseful} yields
\begin{equation}\label{more}
\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_3,v_j\rangle=0=
\sum\limits_{j=1}^me^{i\theta_j}\langle T_{\theta_j}e_4,v_j\rangle.
\end{equation}
Differentiating \eqref{more} with respect to $E$ we find that
\begin{equation*}
\sum\limits_{j=1}^me^{2i\theta_j}\langle dg_{\theta_j}(\overline{E}),v_j\rangle=0.
\end{equation*}
Differentiating once more with respect to $E$ and using the minimality of each $g_{\theta_j}$we obtain
\begin{equation*}
\sum\limits_{j=1}^me^{2i\theta_j}\langle g_{\theta_j},v_j\rangle=0.
\end{equation*}
Combining this with \eqref{firstusefulagain}, we obtain
\begin{equation*}
\sum\limits_{j=2}^m\langle g_{\theta_j},w_j\rangle=0,
\end{equation*}
where $w_j:=\lambda_jv_j\neq0, j=2,\dots,m$ and $\lambda_j=\cos2\theta_m-\cos2\theta_1$ or $\lambda_j=\sin2\theta_m-\sin2\theta_1.$ By induction, we finally conclude that $\langle g_{\theta_m},w\rangle=0,$ for some nonzero vector $w.$ Therefore, $g_{\theta_m}$ lies in a totally geodesic $\mathbb{S}^5,$ which is a contradiction and the theorem is proved.
\end{proof}
\begin{remark}\label{remarknoniso}
The global assumptions and the assumption on the codimension in Theorem \ref{compnon} are essential and can not
be dropped. In fact, locally we can produce
minimal surfaces in spheres that are isometric to a nonisotropic pseudoholomorphic curve $g$
in $\mathbb{S}^6.$ More precisely, let
$g_{\theta}, 0\leq \theta<\pi$, be the associated family of a simply connected nonisotropic
pseudoholomorphic curve $g\colon M\to\mathbb{S}^{6}.$ We consider the surface
$G\colon M\to\mathbb{S}^{7m-1}$ defined by
\begin{equation*}
G=a_{1}g_{\theta _{1}}\oplus \cdots\oplus a_{m}g_{\theta _{m}},
\end{equation*}
where $a_{1},\dots\,,a_{m}$ are any real numbers with $\sum_{j=1}^{m}a_{j}^
{2}=1,$ $0\leq \theta _{1}<\cdots<\theta_{m}<\pi,$ and $\oplus $ denotes the
orthogonal sum with respect to an orthogonal decomposition of the Euclidean space
$\mathbb{R}^{7m}.$ Arguing as in \cite{VT}, it is easy to see that the surface $G$ is minimal and isometric to $g.$
\end{remark}
\begin{proposition}\label{1iso}
Let $g\colon M\to\mathbb{S}^6$ be a compact nonisotropic and substantial pseudoholomorphic curve.
If $\hat{g}\colon M\to\mathbb{S}^n$
is a minimal surface that is isometric to $g,$ then $\hat{g}$ is 1-isotropic.
\end{proposition}
\begin{proof}
According to \cite[Theorem 2]{V16}, the function $1-K$ is of absolute value type.
If the zero set of the function $1-K$ is empty, then from the condition \eqref{trik} it follows that $M$
is homeomorphic to the sphere. From \cite{Calabi} we have that $\hat{g}$ is full isotropic and from
\cite{V} it follows that $n=6$ and $\hat{g}$ is congruent to $g.$
Now suppose that the zero set of the function $1-K$ is the nonempty set
$M_0=\left\{ p_1,\dots,p_m\right\} $ with corresponding order $\mathrm{ord}_{p_j}(1-K)=2k_j.$
For each point $p_j, j=1,\dots,m,$ we choose a
local complex coordinate $z$ such that $p_j$ corresponds to $z=0$\ and the induced
metric is written as $ds^2=F|dz|^2.$ Around\ $p_j,$ we have that
\begin{equation}\label{onemoreavt}
1-K=|z|^{2k_j}u_0,
\end{equation}
where $u_0$ is a smooth positive function.
We know that the first Hopf differential $\hat{\Phi}_1=\hat{f}_1dz^4$ of $\hat{g}$ is globally defined
and holomorphic. We claim that $\hat{\Phi}_1$ is identically zero. We assume to
the contrary that it is not identically zero. Hence its zeros are isolated.
Obviously, $\hat{\Phi}_1$ vanishes at each $p_j.$
Thus we may write $\hat{f}_1=z^{l_{1j}}\psi _1$\ around\ $p_j,$ where
$l_{1j}$\ is the order of $\hat{\Phi}_1$\ at $p_j,$ and $\psi _1$ is a nonzero
holomorphic function. Bearing in mind (\ref{what}),\ we obtain
\begin{equation*}
\frac{1}{4}\left\Vert \hat{\alpha}_2\right\Vert^4-(\hat{K_1}^{\perp})^2=2^4F^{-4}|\psi _1|^2|z|^{2l_{1j}}
\end{equation*}
around $p_j.$ In view of (\ref{onemoreavt}) and the fact that $\left\Vert
\hat{\alpha}_2\right\Vert^2=2(1-K),$ we find that the function $u_1\colon M
\smallsetminus M_0\to \mathbb{R}$ defined by
$$
u_1=\frac{(1-K)^2-(\hat{K_1}^{\perp})^2}{(1-K)^2},
$$
is written as
\[
u_1=2^{4}F^{-4}u_0^{-2}|\psi _1|^2|z|^{2(l_{1j}-2k_j)}.
\]
Since $u_1\leq 1,$ from the above we deduce that $l_{1j}\geq 2k_j$ and we
can extend $u_1$ to a smooth function on $M.$ Applying Proposition \ref{3i}(i) for $g$
and Proposition \ref{3i}(ii) for $\hat{g}$ we have
that $\Delta\log u_1=4K_1^*$ away from the isolated zeros of $u_1,$ where
$K_1^*$ is the intrinsic curvature of the first normal bundle $N_1^g.$
Moreover Proposition \ref{3i}(iii) in combination with \eqref{trik} provides
\[
K_1^*+K>0.
\]
Hence, we obtain $\Delta\log u_1+4K>0$ and consequently using Lemma \ref{forglobal} and the Gauss Bonnet theorem
we have that
\begin{equation*}
N(u_1)\le4\chi (M)\le0 ,
\end{equation*}
where $\chi (M)$ is the Euler-Poincar\'{e} characteristic of $M.$
This implies that $N(u_1)=0.$ Equivalently,
$\hat{\Phi}_1$ does not have any zeros, which is a contradiction.
\end{proof}
In view of Proposition \ref{1iso}, the surface $\hat{g}$ in Theorem \ref{teleutaioPhi} is $1$-isotropic and consequently the Hopf differential $\hat{\Phi}_2$ of $\hat{g}$ is not identically zero.
The following lemma will be used for the proof of Theorem \ref{teleutaioPhi}.
\begin{lemma}\label{lemma9}
Under the assumptions of Theorem \ref{teleutaioPhi}, the following assertions hold:
(i) The $a$-invariants of $g$ and $\hat{g}$ satisfy the inequality
\[
a_2^- \hat{a}_2^+\le a_2^+\hat{a}_2^-.
\]
(ii) The eccentricities $\varepsilon_2, \hat{\varepsilon}_2$ of the second curvature
ellipses of $g$ and $\hat{g}$ respectively satisfy the inequality $\varepsilon_2\le\hat{\varepsilon}_2.$
(iii) There exists a constant $c\ge1$ such that the lengths $\kappa_2,\mu_2$ and
$\hat{\kappa}_2,\hat{\mu}_2$ of the semi-axes of the second curvature ellipses of
the surfaces $g$ and $\hat{g}$ respectively satisfy
\begin{equation}\label{peris}
\kappa_2^2-\mu_2^2=c(\hat{\kappa}_2^2-\hat{\mu}_2^2).
\end{equation}
(iv) At a point $p\in M,$ we have that $a_2^+(p)=0$
if and only if $\hat{a}_2^+(p)=0.$
(v) If $\hat{a}_2
^+(p)>0$ at a point $p\in M,$ then $\hat{a}_2^-(p)=0$ if and only if $a_2^-(p)=0.$
\end{lemma}
\begin{proof}
(i) It follows from Proposition \ref{1iso}, Propositions \ref{5} and \ref{3i} and the Gauss
equation that
$\left\Vert \hat{\alpha}_{3}\right\Vert=\left\Vert\alpha_{3}\right\Vert.$ This means
that
\begin{equation}\label{meiwsh}
\hat{\kappa}_2^2+\hat{\mu}_2^2=\kappa_2^2+\mu_2^2.
\end{equation}
Combining the above with our assumption $\hat{\kappa}_2\hat{\mu}_2\le\kappa_2\mu_2,$ we have that
$$\hat{\kappa}_2+\hat{\mu}_2\le\kappa_2+\mu_2\text{\,\,\ and\,\,}\kappa_2-\mu_2\le\hat{\kappa}_2-\hat{\mu}_2.$$
The proof of part (i) follows easily.
(ii) Since $\hat{K}_2^\perp\le K_2^\perp,$ equation \eqref{meiwsh} implies that
\[
\frac{\hat{\kappa}_2\hat{\mu}_2}{\hat{\kappa}_2^2+\hat{\mu}_2^2}\le
\frac{\kappa_2\mu_2}{\kappa_2^2+\mu_2^2}.
\]
We set $\hat{t}_2:=\hat{\mu}_2/\hat{\kappa}_2$ and $t_2:=\mu_2/\kappa_2.$ Obviously,
$0\le\hat{t}_2,t_2\le1$ and
\[
\frac{\hat{t}_2}{1+\hat{t}_2^2}\le\frac{t_2}{1+t_2^2}.
\]
This immediately implies that $\varepsilon_2\le\hat{\varepsilon}_2.$
(iii) From Proposition \ref{3i}(ii) we have that
\begin{equation}\label{prwti}
\Delta\log(\kappa_2+\mu_2)=3K-K_2^*,
\,\,\,\
\Delta\log(\kappa_2-\mu_2)=3K+K_2^*,
\end{equation}
and
\begin{equation}\label{tetarti}
\Delta\log(\hat{\kappa}_2+\hat{\mu}_2)=3K-\hat{K}_2^*,
\,\,\,\
\Delta\log(\hat{\kappa}_2-\hat{\mu}_2)=3K+\hat{K}_2^*.
\end{equation}
Equations \eqref{prwti} and \eqref{tetarti} yield
\begin{equation*}
\Delta\log\left(\left\Vert \alpha_{3}\right\Vert^4-4^2(K_2^{\perp})^2\right)=12K
\text{\,\,and\, \,}
\Delta\log\left(\left\Vert \hat{\alpha}_{3}\right\Vert^4-4^2(\hat{K}_2^{\perp})^2\right)=12K.
\end{equation*}
Inequality $\hat{K}_2^\perp\le K_2^\perp$ yields
\begin{equation}\label{flast}
|f_2|^2\le|\hat{f}_2|^2,
\end{equation}
where $\Phi_2=f_2dz^6$ and $\hat{\Phi}_2=\hat{f}_2dz^6.$ For each point $p_j\in M_0
=\{p_1,\dots,p_m\}, j=1,\dots,m,$ where $M_0$ is the union of the zero sets of $\Phi_2$ and $\hat\Phi_2,$
we choose a local complex coordinate $z$ such that
$p_j$ corresponds to $z=0$ and the induced metric is written as $ds^2=F|dz|^2.$
Suppose that $\hat{\Phi}_2(p_j)=0$ for some $j=1,\dots,m.$ Then Lemma
\ref{lemma9}(ii) implies that $\Phi_2(p_j)=0.$
Thus we may write $f_2=z^{m(p_j)}u$ and $\hat{f}_2=z^{\hat{m}(p_j)}\hat{u}$
around $p_j,$ where $m(p_j)$ and $\hat{m}(p_j)$ are the orders of $\Phi_2$ and $\hat{\Phi}_2$
respectively at $p_j$ and $u$ and $\hat{u}$ are nonzero holomorphic functions. From
\eqref{flast} we have that $\hat{m}\le m,$ and therefore the function $u_2=|f_2|^2/|\hat{f}_2|^2
\colon M\smallsetminus M_0\to \mathbb{R}$ can be extended to a smooth
function on $M.$
Suppose now that $\hat{\Phi}_2(p_j)\neq0$ for some $j=1,\dots,m.$ We have that the function $u_2=|z|^{2m(p_j)}u,$
with $u$ a positive smooth function, can be extended to a smooth
function on $M.$
In both cases we have that the function $u_2$ is subharmonic and the maximum principle
yields \eqref{peris}. Obviously \eqref{peris} gives that the zeros of the second Hopf differential $\Phi_2$ of the curve $g$ coincide with
the zeros of the second Hopf differential $\hat{\Phi}_2$ of the surface $\hat{g}.$
(iv) If $a_2^+(p)=0$ at a point $p\in M,$ we obtain $\kappa_2(p)=
\mu_2(p)=0.$ It follows from \eqref{meiwsh} that $\hat{\kappa}_2(p)=\hat{\mu}_2
(p)=0,$ which is $\hat{a}_2^+(p)=0.$
(v) Part (v) follows immediately from (i) and \eqref{peris} which is equivalently written as
$a_2^+a_2^-=c \hat{a}_2^+\hat{a}_2^-.$
\end{proof}
Now we prove Theorem \ref{teleutaioPhi}.
\begin{proof}[Proof of Theorem \ref{teleutaioPhi}]
Equations \eqref{prwti} and \eqref{tetarti} yield
\begin{equation}\label{lastnoleast}
\Delta\log\frac{a_2^-\hat{a}_2^+}{a_2^+\hat{a}_2^-}
=2(K^*_2-\hat{K}_2^*),
\end{equation}
on $M\smallsetminus M_0,$ where $M_0=\{p_1,\dots,p_m\}$ is the
union of the zero sets of $\Phi_2$ and $\hat\Phi_2.$
For each point $p_j\in M_0=\{p_1,\dots,p_m\}, j=1,\dots,m,$ we choose a local
complex coordinate $z$ such that $p_j$ corresponds to $z=0$ and the induced
metric is written as $ds^2=F|dz|^2.$
We now claim that the function $u=(a_2^-\hat{a}_2^+)/(a_2^+\hat{
a}_2^-)\colon M\smallsetminus M_0\to \mathbb{R}$ can be extended
to a smooth function on $M.$ To this aim we distinguish the following cases:
\textit {Case I:} Suppose that $\hat{a}_2^+(p_j)=0$ for some $j=1,\dots,m.$ Then Lemma
\ref{lemma9}(iv) implies that $a_2^+(p_j)=0.$ Hence $\hat{a}
_2^-(p_j)=a_2^-(p_j)=0.$ The $a$-invariants are absolute value type functions,
thus we may write $a_2^+=|z|^{2m_+}u_+,$ $a_2^-=|z|^{2m_-}u_-,$ $\hat{a}
_2^+=|z|^{2\hat{m}_+}\hat{u}_+$ and $\hat{a}_2^-=|z|^{2\hat{m}_-}\hat{u}_-$
around $p_j,$ where $m_+,m_-,\hat{m}_+$ and $\hat{m}_-$ are the orders of $a_2^+,$
$a_2^-,$ $\hat{a}_2^+$ and $\hat{a}_2^-$ respectively at $p_j$
and $u_+, u_-, \hat{u}_+$ and $\hat{u}_-$ are nonvanishing smooth functions.
From Lemma \ref{lemma9}(i) it follows that
\[
m_-(p_j)+\hat{m}_+(p_j)\ge m_+(p_j)+\hat{m}_-(p_j).
\]
Therefore the function $u=(a_2^-\hat{a}_2^+)/(a_2^+\hat{
a}_2^-)$ can be extended
to a smooth function around $p_j.$
\textit{Case II:} Suppose that $\hat{a}_2^+(p_j)>0$ for some $j=1,\dots,m.$
Lemma \ref{lemma9}(v) implies that either $\hat{a}_2^-(p_j)a_2^-(p_j)>0$ or $\hat{a}_2^-
(p_j)=a_2^-(p_j)=0.$ In the former case, by Lemma \ref{lemma9}(i) we have that $a_2^+(p_j)>0.$
Thus $u$ is well defined at $p_j$.
Now assume that $\hat{a}_2^-
(p_j)=a_2^-(p_j)=0.$ Clearly \eqref{meiwsh} implies that $a_2^+(p_j)>0.$ Since the $a$-invariants are absolute value type functions, we may write $a_2^-=|z|^{2m_-}u_-$ and
$\hat{a}_2^-=|z|^{2\hat{m}_-}\hat{u}_-$ around $p_j,$ where $m_-$ and
$\hat{m}_-$ are the orders of $a_2^-,$ and $\hat{a}_2^-$ respectively at $p_j$
and $u_-$ and $\hat{u}_-$ are nonvanishing smooth functions.
Lemma \ref{lemma9}(i) yields
\[
m_-(p_j)\ge\hat{m}_-(p_j),
\]
therefore the function $u=(a_2^-\hat{a}_2^+)/(a_2^+\hat{
a}_2^-)\colon M\smallsetminus M_0\to \mathbb{R}$ can be extended
to a smooth function around $p_j.$
It follows from Proposition \ref{5}
and \eqref{lastnoleast} that
\begin{equation}\label{least}
\Delta\log u=\frac{2\left\Vert \alpha_{2}\right\Vert^2}{(K_1^\perp)^2}
(K_2^{\perp}-\hat{K}_2^{\perp})+\frac{2\left\Vert \hat{\alpha}_{4}\right\Vert^2}
{4\hat{K}_2^{\perp}}
\end{equation}
away from the isolated zeros of $u.$ Hence $\Delta\log u\ge0$ on $M\smallsetminus M_0.$
By continuity, the function $u$ is subharmonic on $M$
and from the maximum principle we have that $u$ is constant. Then from
\eqref{least} it follows that $\hat{K}_2^\perp= K_2^\perp,$ and $\alpha_{4}=0.$
Hence $f(M)$ is contained in a totally geodesic sphere $\mathbb{S}^6$ in
$\mathbb{S}^n$.
The fact that the set of all noncongruent minimal surfaces $\hat{g},$ as in the statement
of the theorem, that are isometric to $g$ is either a circle or a finite set, follows directly
from Theorem \ref{finiteorcircle}.
\end{proof}
\begin{corollary}
Let $g\colon M\to\mathbb{S}^6$ be a compact nonisotropic and substantial pseudoholomorphic
curve with second normal curvature $K_2^\perp.$ Any
substantial minimal surface $\hat{g}$ in $\mathbb{S}^n, n>6,$ whose second normal curvature
$\hat{K}_2^\perp$ satisfies the inequality $\hat{K}_2^\perp\le K_2^\perp,$
cannot be isometric to $g.$
\end{corollary}
\begin{proof}
Assume that $\hat{g}$ is isometric to $g$. Proposition \ref{1iso} implies that $\hat{g}$ is 1-isotropic.
Suppose that $n>6.$ Then Theorem \ref{teleutaioPhi} implies that $\hat{g}$ is 2-isotropic.
Hence $\hat{\kappa}_2=
\hat{\mu}_2.$ The inequality $\hat{K}_2^\perp\le K_2^\perp,$ in combination
with \eqref{meiwsh} implies that $\kappa_2=\mu_2,$ which is a contradiction.
\end{proof}
\section{Rigidity of isotropic pseudoholomorphic curves in $\mathbb{S}^6$}
In this section, we study the rigidity of isotropic pseudoholomorphic curves in $\mathbb{S}
^6$ among minimal surfaces in spheres. We prove Theorem \ref{forintro}.
\begin{proof}[Proof of Theorem \ref{forintro}]
According to \cite[Theorem 2]{V16}, the function $1-K$ is of absolute value type.
If the zero set of the function $1-K$ is empty, then from condition ($\ast\ast$) it follows that $f$
is homeomorphic to the sphere. From \cite{Calabi} we have that $f$ is full isotropic and from
\cite{V} it follows that $n=6$ and $f$ is congruent to $g.$
Now suppose that the zero set of the function $1-K$
is the set $M_0=\left\{ p_1,\dots,p_m\right\}$ with corresponding order $\mathrm{ord}_{p_j}(1-K)=2k_j.$
For each point $p_j, j=1,\dots,m,$ we choose a
local complex coordinate $z$ such that $p_j$ corresponds to $z=0$\ and the induced
metric is written as $ds^2=F|dz|^2.$ On a neighbourhood of \ $p_j,$ we have that
\begin{equation}\label{againavt}
1-K=|z|^{2k_j}u_0,
\end{equation}
where $u_0$ is a smooth positive function.
We claim that $f$ is $1$-isotropic. The first
Hopf differential $\Phi _1=f_1dz^4$ is globally defined and holomorphic. Hence either $\Phi _1$ is identically zero, or its zeros are isolated. Assume now
that $\Phi _1$ is not identically zero. Since $\Phi _1$ vanishes at each $p_j,$
we may write $f_1=z^{l_{1j}}\psi _1$\ around\ $p_j,$ where
$l_{1j}$\ is the order of $\Phi_1$\ at $p_j,$ and $\psi _1$ is a nonzero
holomorphic function. Bearing in mind \eqref{what}, we obtain
\begin{equation*}
\frac{1}{4}\left\Vert \alpha_2\right\Vert^4-(K_1^{\perp})^2=2^4F^{-4}|\psi _1|^2|z|^{2l_{1j}}
\end{equation*}
around $p_j.$ Using (\ref{againavt}) and the fact that $\left\Vert
\alpha_2\right\Vert^2=2(1-K),$ we have that the function $u_1\colon M
\smallsetminus M_0\to \mathbb{R}$ defined by
$$
u_1=\frac{\left((1-K)^2-(K_1^{\perp})^2\right)^3}{(1-K)^4},
$$
around $p_j,$ is written as
\begin{equation}\label{u_1around}
u_1=2^{12}F^{-12}u_0^{-4}|\psi _1|^6|z|^{6l_{1j}-8k_j}.
\end{equation}
Since $u_1\leq (1-K)^2,$ from \eqref{u_1around} we deduce that $l_{1j}\geq 2k_j$ and we
can extend $u_1$ to a smooth function on $M.$ It follows from Proposition \ref{3i}(ii)
and the condition ($\ast\ast$) that $\Delta\log u_1=4$ away from the isolated
zeros of $u_1$. Thus, by continuity we have that $\Delta u_1\ge 4u_1,$ which holds only for $u_1\equiv0.$
This is a contradiction and hence our claim is proved.
Proposition \ref{3i}(i) and condition ($\ast\ast$) yield
\[
K_1^{\ast}=\frac{1}{2}-K.
\]
From the fact that $K_1^\perp=1-K$ and Proposition \ref{5}, we deduce that $\left\Vert \alpha_{3}\right\Vert^2=1-K.$
We now claim that $f$ is also $2$-isotropic. From \cite[Proposition 4]{V08}, we know that $\Phi _2=f_2dz^{6}$
is globally defined and holomorphic. Hence either $\Phi _2$ is identically zero or its zeros are isolated.
In the former case, $f$ is $2$-isotropic. Assume now that $\Phi _2$ is not identically zero. Obviously, $\Phi _2$ vanishes at each
$p_j.$ Hence we may write $f_2=z^{l_{2j}}\psi _2$\ around\ $p_j,$
where $l_{2j}$\ is the order of $\Phi _2$\ at $p_j,$ and $\psi _2$ is a
nonzero holomorphic function. Bearing in mind (\ref{what}),\ we obtain
\begin{equation*}
\left\Vert \alpha_{3}\right\Vert^4-16(K_2^{\perp})^2=2^{8}F^{-6}|\psi _2|^2|z|^{2l_{2j}}
\end{equation*}
around $p_j.$ In view of (\ref{againavt}), we find that
\begin{equation}\label{u2}
u_2=2^{8}F^{-6}u_0^{-2}|\psi
_2|^2|z|^{2l_{2j}-4k_j},
\end{equation}
where $u_{2}\colon M\smallsetminus M_0\to \mathbb{R}$ is the smooth function
(see Proposition \ref{neoksanaafththfora})
given by
$$
u_2=\frac{(1-K)^2-16(K_2^{\perp})^2}{(1-K)^{2}}.
$$
Since $u_2\leq 1,$ from (\ref{u2}) we deduce that $l_{2j}\geq 2k_j$ and we
can extend $u_2$ to a smooth function on $M.$ It follows from Proposition \ref{3i}(ii)
and condition ($\ast\ast$) that $\Delta\log u_2=2$ away from the isolated
zeros of $u_2$. Thus $\Delta u_2\ge 2u_2,$ which holds only for $u_2\equiv0.$
This is a contradiction and hence our claim is proved. Proposition \ref{3i}(i) and condition ($\ast\ast$)
yield that
$K_2^{\ast}=1/2.$
Since $f$ is 2-isotropic, from \eqref{elipsi} and \eqref{si} we obtain $K_2^\perp=(1-K)/4,$ which in combination with
Proposition \ref{5} implies that $\alpha_{4}=0.$ Therefore $n=6$ and the surface $f$ is congruent to $g$ (cf. \cite[Theorem A]{V}).
\end{proof}
We now prove the following local result for exceptional surfaces.
\begin{theorem} \label{congruent2iso}
Let $f\colon M\to\mathbb{S}^n$ be a substantial exceptional surface that is isometric to an isotropic pseudoholomorphic curve $g\colon M \to\mathbb{S}^6.$ Then $n=6$ and $f$ is congruent to $g.$
\end{theorem}
\begin{proof}
We set $\rho _s:=2^sK_s^{\perp}/\left\Vert \alpha_{s+1}\right\Vert^2.$
Since $f$ is exceptional, the function $\rho _s$ is constant for any $1\leq s\leq r,$ where $r=[(n-1)/2-1].$
The Gauss equation implies $\left\Vert\alpha_2\right\Vert^2=2(1-K).$ Then from Proposition \ref{3i}(i) for $s=1$ and condition ($\ast\ast$), we find $K_1^{\ast}=1/2-K.$
Moreover, we have that $K_1^{\perp}=\rho_1(1-K).$
We claim that $\rho_1=1.$ Assume to the contrary that
$\rho_1\neq1.$ Then $\Phi_1\neq0$ and Proposition \ref{3i}(ii) combined with
condition ($\ast\ast$) yield $K=1/2$, which is a contradiction. Hence
$\rho_1=1$ and from Proposition \ref{5} it follows that $\left\Vert \alpha_{3}\right\Vert^2=1-K.$
By Theorem \ref{ena}, the Hopf differential
$\Phi _{2}$ is holomorphic, hence either it is identically zero or its zeros are isolated. Moreover, we have that $K_2^\perp=2^{-2}\rho_2\left\Vert \alpha_{3}\right\Vert^2.$ We claim that $\rho _{2}=1.$
Assume to the contrary that
$\rho_2\neq1.$ Then $\Phi_2\neq0$ and taking into account
condition ($\ast\ast$), Proposition \ref{3i}(ii) provides a
contradiction. Thus $\Phi _{2}$ is identically zero, or equivalently $\rho _{2}=1.$
From Proposition \ref{3i}(iii) and condition ($\ast\ast$), we obtain $K_{2}^
{\ast}=1/2.$ Proposition \ref{5} yields
$ \alpha_{4}=0,$ which completes our proof.
\end{proof}
|
1910.04099
|
\section{Conclusions and Future Work}
\label{sec:conclusion}
In this paper, we provide a thorough analysis of the three state-of-the-art methods for 3D Manhattan layout reconstruction from a single RGB indoor panoramic image, namely, LayoutNet, DuLa-Net, and HorizonNet. We further propose the improved version called {LayoutNet v2} and {DuLa-Net v2}, which incorporate certain advantageous components from HorizonNet.
LayoutNet v2 performs the best on PanoContext dataset and offers more robustness to occlusion from foreground objects. DuLa-Net v2 outperforms in two of three metrics for Stanford 2D-3D. To evaluate the performance on reconstructing general Manhattan layout shapes, we extend the Matterport3D dataset with general Manhattan layout annotations and introduce the {MatterportLayout} dataset. The annotations contain panoramas of both simple (\textit{e}.\textit{g}.,~ cuboid) and complex room shapes. We introduce two depth based evaluation metrics for evaluating the quality of reconstruction.
Future work can be in three directions: (1) Relax Manhattan constraints to general layout. In real cases indoor layouts are more complex and could have non-Manhattan property like arch. One research direction is to study approaches that could generalize across Manhattan layouts and non-Manhattan ones with curve ceilings or walls. (2) Use additional depth and normal information. Our approach is based on a single RGB image only, and we can acquire rich geometric information from either predicted depth map from a single image, or captured depth maps from sensors. Incorporating depth features to both network predictions and the post-processing step could help for more accurate 3D layout reconstruction; (3) Extend to multi-view based 3D layout reconstruction. Reconstruction from a single image is difficult due to occlusions either from other layout boundaries or foreground objects. We can extend our approach for layout reconstruction from multiple images. Using multiple images can recover a more complete floor plan and scene layout, which has various applications such as virtual 3D room walk through for real estate.
\section{Problem Definition and Datasets}
\section{Datasets}
\label{sec:dataset}
Datasets with detailed ground truth 3D room layouts play a crucial role for both network training and performance validation. In this work, we use three public datasets for the evaluation, which are {\em PanoContext}~\cite{zhang2014panocontext}, {\em Stanford 2D-3D}~\cite{stfd2d3d}, and {\em Matterport3D}~\cite{chang2017matterport3d}. All three datasets are composed of RGB(D) panoramic images of various indoor scene types and differ from each other in the following intrinsic properties: (1) the complexity of room layout; (2) the diversity of scene types; and (3) the scale of dataset. For those datasets lack of ground truth 3D layouts, we further extend their annotations with detailed 3D layouts using an interactive annotator, PanoAnnotator~\cite{yang2018panoannotator}. A few sample panoramic images from the chosen datasets are shown in \figref{fig:dataset}. We will briefly describe each dataset and discuss differences as follows.
\subsection{PanoContext Dataset}
\input{fig_dataset.tex}
PanoContext~\cite{zhang2014panocontext} dataset contains $514$ RGB panoramic images of two indoor environments, \textit{i}.\textit{e}.,~ bedrooms and living rooms, and all the images are annotated as cuboid layouts. For the evaluation, we follow the official train-test split and further carefully split 10\% validation images from the training samples such that similar rooms do not appear in the training split.
\subsection{Stanford 2D-3D Dataset}
Stanford 2D-3D~\cite{stfd2d3d} dataset contains $552$ RGB panoramic images collected from $6$ large-scale indoor environments, including offices, classrooms, and other open spaces like corridors. Since the original dataset does not provide ground truth layout annotations, we manually labeled the cuboid layouts using the PanoAnnotator. The Stanford 2D-3D dataset is more challenging than PanoContext as the images have smaller vertical FOV and more occlusions on the wall-floor boundaries. We follow the official train-val-test split for evaluation.
\subsection{Our Labeled MatterportLayout Dataset}
We carefully selected $2295$ RGBD panoramic images from Matterport3D~\cite{chang2017matterport3d} dataset and extended the annotations with ground truth 3D layouts. We call our collected and relabeled subset the {\em MatterportLayout} dataset.
Matterport3D~\cite{chang2017matterport3d} dataset is a large-scale RGB-D dataset containing over ten thousand RGB-D panoramic images collected from 90 building-scale scenes. Matterport3D has the following advantages over the other datasets
\begin{enumerate}
\item covers a larger variety of room layouts (\textit{e}.\textit{g}.,~ cuboid, ``L''-shape, ``T''-shape rooms, etc) and over 12 indoor environments (\textit{e}.\textit{g}.,~ bedroom, office, bathroom and hallway, etc);
\item has aligned ground truth depth for each image, allowing quantitative evaluations for layout depth estimation; and
\item is three times larger in scale than PanoContext and Stanford 2D-3D, providing rich data for training and evaluating our approaches.
\end{enumerate}
Note that there also exists the Realtor360 dataset introduced in~\cite{yang2019dula}, which contains over $2500$ indoor panoramas and annotated 3D room layouts. However, Realtor360 currently could not be made publicly available due to legal privacy issues.
\input{fig_dataset_barchart}
Moreover, as shown in~\figref{fig:dataset_corner_compare}, our MatterportLayout dataset covers a more diverse range of layout types than Realtor360 .
The detailed annotation procedure and dataset statistics are elaborated as follows.
\section{Introduction}
\label{sec:intro}
Estimating the 3D room layout of indoor environment is an important step toward a holistic scene understanding and would benefit many applications such as robotics and virtual/augmented reality. The room layout specifies the positions, orientations, and heights of the walls, relative to the camera center. The layout can be represented as a set of projected corner positions or boundaries or as a 3D mesh. Existing works apply to special cases of the problem such as predicting cuboid-shaped layouts from perspective images or from panoramic images.
Recently, various approaches~\cite{zou2018layoutnet,yang2019dula,sun2019horizonnet} for 3D room layout reconstruction from a single panoramic image have been proposed, which all produce excellent results. These methods are not only able to reconstruct cuboid room shapes, but also estimate non-cuboid general Manhattan layouts as shown in~\figref{fig:illustration}. Different from previous work~\cite{zhang2014panocontext} that estimates 3D layouts by decomposing a panorama into perspective images, these approaches operate directly on the panoramic image in equirectangular view, which effectively reduces the inference time. These methods all follow a common framework: (1) a pre-processing edge-based alignment step, ensuring that wall-wall boundaries are vertical lines and substantially reducing prediction error; (2) a deep neural network that predicts the layout elements, such as layout boundaries and corner positions~(LayoutNet~\cite{zou2018layoutnet} and HorizonNet~\cite{sun2019horizonnet}), or a semantic 2D floor plan in the ceiling view~(DuLa-Net~\cite{yang2019dula}); and (3) a post-processing step that fits the~(Manhattan) 3D layout to the predicted elements
However, until now, it has been difficult to compare these methods due to multiple different design decisions. For example, LayoutNet uses SegNet as encoder while DuLa-Net and HorizonNet use ResNet; HorizonNet applies random stretching data augmentation~\cite{sun2019horizonnet} in training, while LayoutNet and DuLa-Net do not. Direct comparison of the three methods may conflate impact of contributions and design decisions. We therefore want to isolate the effects of the contributions by comparing performance with the same encoding architectures and other settings.
Moreover,
given the same network prediction, we want to compare the performance by using different post-processing steps~(under equirectangular view or ceiling view).
Therefore, in this paper, the authors of LayoutNet and DuLa-Net work together to better describe the common framework, the variants, and the impact of the design decisions for 3D layout estimation from a single panoramic image. For a detailed comparison, we evaluate performance using a unified encoder (\textit{i}.\textit{e}.,~ ResNet~\cite{he2016deep}) and consistent training details such as random stretching data augmentation, and discuss effects using different post-processing steps.
Based on the modifications to LayoutNet and DuLa-Net listed above, we propose the improved version called {LayoutNet v2}\footnote{Code is available at: \url{https://github.com/zouchuhang/LayoutNetv2}} and {DuLa-Net v2}\footnote{Code is available at: \url{https://github.com/SunDaDenny/DuLa-Net}}, which achieve the state-of-the-art for cuboid layout reconstruction.
To compare performance for reconstructing different types of 3D room shape, we extend the annotations of Matterport3D dataset with ground truth 3D layouts\footnote{Our annotation is available at: \url{ https://github.com/ericsujw/Matterport3DLayoutAnnotation}}. Unlike existing public datasets, such as the PanoContext dataset~\cite{zhang2014panocontext}, which provides mostly cuboid layouts of only two scene types,~\textit{i}.\textit{e}.,~ bedroom and living room, the Matterport3D dataset contains 2,295 real indoor RGB-D panoramas of more than 12 scene types,~\textit{e}.\textit{g}.,~ kitchen, office, and corridor, and 10 general 3D layouts,~\textit{e}.\textit{g}.,~ ``L''-shape and ``T''-shape.
Moreover, we leverage the depth channel of dataset images and introduce two depth-based evaluation metrics for comparing general Manhattan layout reconstruction performance. Depth-based metrics are more general than 3D IoU or 2D pixel labeling error, as they pertain to the geometric error of arbitrary-shaped scenes. Additionally, depth-based metrics could be used to compare layout algorithms in the context of multi-view depth and scene reconstruction. We show in experiments that when the performance gap between the competing methods is small in 2D and 3D IoU metrics, the performance gap in the depth-based metrics is still more distinguishable, which helps to provide fruitful comparisons for quantitative analysis.
The experimental results demonstrate that: (1) LayoutNet's decoder can better capture global room shape context, performing the best for cuboid layout reconstruction and being robust to foreground occlusions.
(2) For non-cuboid layout estimation, DuLa-Net and HorizonNet's decoder can better capture detailed layout shapes like pipes, showing better generalization to various complex layout types. Their simplified network output representation also take much less time for the post-processing step. (3) At the component level, a pre-trained and denser ResNet encoder and random stretching data augmentation can help boost performance for all methods.
For LayoutNet, the post-processing method that works under the equirectangular view performs better. For DuLa-Net and HorizonNet, the post-processing step under ceiling view is more suitable.
We hope our analysis and discoveries can inspire researchers to build up more robust and efficient 3D layout reconstruction methods from a single panoramic image in the future.
Our contributions are:
\begin{itemize}
\item We introduce two frameworks, {LayoutNet v2} and {DuLa-Net v2}, which extend the corresponding state-of-the-art approaches of 3D Manhattan layout reconstruction from an RGB panoramic image. Our approaches compare well in terms of speed and accuracy and achieve the best results for cuboid layout reconstruction.
\item We conduct extensive experiments for {LayoutNet v2}, {DuLa-Net v2} and another state-of-the-art approach, HorizonNet. We discuss the effects of encoders, post-processing steps, performance for different room shapes, and time consumption. Our investigations can help inspire researchers to build up more robust and efficient approaches for single panoramic image 3D layout reconstruction.
\item We extend the Matterport3D dataset with general Manhattan layout annotations. The annotations contain panoramas depicting room shapes of various complexity. The dataset will be made publicly available. In addition, two depth-based evaluation metrics are introduced for measuring the performance of general Manhattan layout reconstruction.
\end{itemize}
\section{Annotated General Manhattan Layout Dataset with Ground Truth Depth}
\subsubsection{Annotation Process of~MatterportLayout}
We first exclude images in the Matterport3D dataset with (1) open 3D space like corridors and stairs, (2) non-flat floors and non-Manhattan 3D layouts, (3) outdoors scenes and (4) artifacts resulting from stitching perspective views
We use PanoAnnotator~\cite{yang2018panoannotator} to annotate ground truth 3D layouts. The interface of the annotation tool is shown in \figref{fig:panoannotator}. The annotation pipeline involves four steps: (1) Select one of the panoramas in the dataset; (2) load an initial layout (this step is optional);
(3) add or remove the 3D wall pieces to match the number of walls that the given layout has; (4) push or pull all the walls to match the panorama; (5) push the ceiling and floor to match the panorama. Finally, we obtain a dataset of $2295$ RGB-D panoramas with detailed layout annotations.
For all the layouts in the dataset, we started with initial layouts from~\cite{fuenmatterport}, which is also annotated with PanoAnnotator.
We refined each layout to achieve accurate room corners, ceiling height, and walls position. More than 25\% of initial layouts were adjusted significantly with at least 10\% difference in 3D IoU.
To evaluate the quality of estimated 3D layouts using the depth measurements, we further process the aligned ground truth depth maps to remove pixels that belong to foreground objects (\textit{e}.\textit{g}.,~ furniture). Specifically, we align the ground truth depth map to the rendered depth map of the annotated 3D layout and mask out inconsistent pixels between two depth maps. For the alignment, we scale the rendered depth by normalizing camera height to 1.6m. We then mask out pixels in the ground truth depth map that are more than 0.15m away from their counterparts in the rendered depth map. In our experiment, we use the unmasked pixels for evaluation. See~\figref{fig:matterport3d} for some examples.
\input{tbl_matterport3D.tex}
\subsubsection{MatterportLayout~Dataset Statistics}
We use approximately 70\%, 10\%, and 20\% of the data for training, validation, and testing. Images from the same room do not appear in different sets. Moreover, we ensure that the proportions of rooms with the same 3D shape in each set are similar. We show in Table~\ref{tab:matterport3d} the total numbers of images annotated for different 3D room shapes. The 3D room shape is classified according to the numbers of corners in ceiling view: cuboid shape has four corners,``L''-shape room has six corners, ``T''-shape has eight corners, etc. The {MatterportLayout} dataset covers a large variety of 3D room shapes, with approximately 52\% cuboid rooms, 22\% ``L''-shape rooms, 13\% ``T''-shape rooms, and 13\% more complex room shapes. The train, validation, and test sets have similar distributions of different room shapes, making our experiments reliable for both training and testing.
\section{Methods Overview}
\label{sec:network}
In this section, we introduce the common framework, the variants, and the impact of the design decisions of recently proposed approaches for 3D Manhattan layout reconstruction from a single panoramic image. \tabref{tab:taxonomy_1} summarizes the key design choices that LayoutNet~(\figref{fig:layoutnet-overview}), DuLa-Net~(\figref{fig:dulanet-overview}) and HorizonNet~(\figref{fig:horizonnet-overview}) originally proposed in their papers respectively. Though all three methods follow the same general framework, they differ in the details. We unify some of the designs and training details and propose our modified LayoutNet and DuLa-Net methods as follows, which show better performance compared with the original ones.
\subsection{General framework}\label{subsec:method}
The general framework can be decomposed into three parts. First, we discuss in~\secref{subsec:preprocess} the input and pre-processing step. Second, we introduce the network design of encoder in~\secref{subsec:encoder}, the decoder of layout pixel predictions in~\secref{subsec:decoder}, and the training loss for each method in~\secref{subsec:loss}. Finally, we discuss the structured layout fitting in~\secref{subsec:opt}.
\subsection{Input and Pre-processing}
\label{subsec:preprocess}
Given the input as a panorama that covers a $360^\circ$ horizontal field of view, the first step of all methods is to align the image to have a horizontal floor plane. The alignment, first proposed by LayoutNet and then inherited by DuLa-Net and HorizonNet, ensures that wall-wall boundaries are vertical lines and substantially reduces error. The alignment goes as follows. First, we estimate the orientation of floor plan under spherical projection using Zhang~\textit{et~al.}'s approach~\cite{zhang2014panocontext} (i.e., selecting long line segments using the Line Segment Detector~(LSD)~\cite{von2008lsd} in each overlapping perspective view), then we vote for three mutually orthogonal vanishing directions using the Hough Transform. Afterward, we rotate the scene and re-project it to the 2D equirectangular projection.
The aligned panoramic image is used for all three methods as input. For better predicting layout elements, LayoutNet and DuLa-Net utilize additional inputs as follows.
LayoutNet additionally concatenates a $512\times1024$ Manhattan line feature map lying on three orthogonal vanishing directions using the alignment method as described in the previous paragraph. The Manhattan line feature map provides additional input features that were shown to have improved the performance quantitatively~\cite{zou2018layoutnet}.
To feed its two-branch network, DuLa-Net creates another ceiling-view perspective image projected from the input panorama image using an E2P module described in the following.
For every pixel in the perspective image (assumed to be square with dimension $w$ squared) at position $(p_x,p_y)$, the position of the corresponding pixel in the equirectangular panorama, $(p\textprime_x,p\textprime_y)$, $-1 \le p\textprime_x \le 1, -1 \le p\textprime_y \le 1$, is derived as follows. First, the field of view of the pinhole camera of the perspective image is defined as $FoV$. Then, the focal length can be derived as:
$$
f = 0.5 * w * \cot(0.5 * \mathrm{FoV})~,
$$
$(p_x, p_y, f)$ is the 3D position of the pixel in the perspective image in the camera space. It is then rotated by 90$^\circ$ or -90$^\circ$ along the x-axis~(counter-clockwise) if the camera is looking upward~(\textit{e}.\textit{g}.,~ looking at the ceiling) or downward~(\textit{e}.\textit{g}.,~ looking at the floor), respectively. Next, the rotated 3D position is projected to the equirectangular space. To do so, the rotated 3D position is first projected onto a unit sphere by vector normalization, the resulting 3D position on the unit sphere is denoted as $(s_x, s_y, s_z)$, and then the following formula is applied to project $(s_x, s_y, s_z)$ back to $(p\textprime_x,p\textprime_y)$, which is the corresponding 2D position in the equirectangular panorama:
\begin{align}
(p\textprime_{x}, p\textprime_{y}) &= (\frac{arctan_2 (\frac{s_{x}}{s_{z}})}{\pi}~, \frac{arcsin (s_{y})}{0.5\pi}),
\end{align}
Finally, $(p\textprime_x,p\textprime_y)$ is used to interpolate a pixel value from the panorama. Note that this process is differentiable so it can be used in conjunction with back-propagation. DuLa-Net peforms E2P with a $FoV$ of $160^\circ$ and produces a perspective image of $512\times512$.
The need for ceiling view branch in DuLaNet and E2P module can be explained by two reasons: (1) DuLa-Net is able to extract more features from ceiling view and (2) referring to the Table 2 in the original paper~\cite{yang2019dula}, the best performance comes from two branches with E2P module.
\input{fig_layoutnet_overview.tex}
\input{fig_dulanet_overview.tex}
\subsection{Encoder}
\label{subsec:encoder}
Regarding the network designs, the original LayoutNet uses SegNet as the encoder while DuLa-Net and HorizonNet use ResNet. For our modified version, we use ResNet uniformly because we find that it shows better performance in capturing layout features than SegNet in experiments~(\secref{text:abla}, \secref{tab:cuboid_abla}). Details are as follows.
The ResNet encoder receives a $512\times1024$ RGB panoramic image under equirectangular view as input. For the three methods, there are differences in the last encoding layers based on their different designs of decoders for different output spaces: for LayoutNet, the last fully connected layer and the average pooling layer of the ResNet encoder are removed. For DuLa-Net, it uses a separate ResNet encoder for both {panorama-branch} and {ceiling-branch}. The {panorama-branch} $B_P$ has an output dimension of $16 \times 32 \times 512$. For the {ceiling-branch} $B_C$, the output dimension is $16 \times 16 \times 512$. Finally, for HorizonNet,
it performs a separate convolution for each of the feature maps produced by each block of the ResNet encoder. The convolution down samples each map by 8 in height and 16 in width, with feature size up-sampled to 256. The feature maps are then reshaped to 256 x 1 x 256 and concatenated based on the first dimension, producing the final bottleneck feature.
\subsection{Layout Prediction}\label{subsec:decoder}
Next, we discuss the decoders. In general, the decoders output predictions of layout pixels in the form of either corner and boundary positions under equirectangular view, or semantic floor plan map under ceiling view. We describe each type of prediction as follows.
\subsubsection{Equirectangular-view prediction}
Decoders of LayoutNet and HorizonNet output layout predictions in equirectangular view solely, while DuLa-Net's two-branch network design means that decoders output predictions in both equirectangular view and ceiling view.
Both LayoutNet and HorizonNet predict layout corners and boundaries under equirectangular projections. For LayoutNet, the decoder consists of two branches. The top branch, the layout boundary map~($\boldsymbol{m_E}$) predictor, decodes the bottleneck feature into a 2D feature map with the same resolution as the input. $\boldsymbol{m_E}$ is a 3-channel probability prediction of wall-wall, ceiling-wall and wall-floor boundary on the panorama, for both visible and occluded boundaries. The boundary predictor contains $7$ layers of nearest neighbor up-sampling operation, each followed by a convolution layer with kernel size of $3\times 3$, and the feature size is halved through layers from $2048$. The final layer is a Sigmoid operation. Skip connections are added to each convolution layer following the spirit of the U-Net structure~\cite{ronneberger2015u}, in order to prevent shifting of prediction results from the up-sampling step. The lower branch, the 2D layout corner map~($\boldsymbol{m_C}$) predictor, follows the same structure as the boundary map predictor and additionally receives skip connections from the top branch for each convolution layer. This stems from the intuition that layout boundaries imply corner positions, especially for the case when a corner is occluded. It's shown in~\cite{zou2018layoutnet} that the joint prediction helps improve the accuracy of the both maps, leading to a better 3D reconstruction result.
We exclude the 3D regressor proposed in~\cite{zou2018layoutnet} as the regressor is shown to be ineffective in the original paper.
The output space of LayoutNet is $O(HW)$, where $H$ and $W$ are the height and width of the input image. This dense prediction limits the network to use deeper encoders such as ResNet-50 or complex decoders to improve performance further. HorizonNet simplifies LayoutNet's prediction by predicting three 1-D vectors with 1024 dimensions instead of two 512x1024 probability maps. The three vectors represent the ceiling-wall and the floor-wall boundary position, and the existence of wall-wall boundary~(or corner) of each image column. HorizonNet further applies an RNN block to refine the vector predictions, which considerably help boost performance as reported in~\cite{sun2019horizonnet}.
For DuLa-Net, its {panorama-branch} $B_P$ predicts floor-ceiling probability map $M_{FC}$ under equirectangular view. $M_{FC}$ has the same resolution as the input. A pixels in $M_{FC}$ with higher value means a higher probability to be ceiling or floor. The decoder of $B_P$ consists of 6 layers. Each of the first 5 layers contains the nearest neighbor up-sampling operation followed by a $3\times3$ convolution layer and ReLU activation function, the channel number is halved from 512~(if using ResNet18 as an encoder). The final layer of the decoder replaces the ReLU by Sigmoid to ensure the data range is in $[0, 1]$. The second branch of DuLa-Net predicts 2D probability map under ceiling view which will be introduced in the next paragraph.
\subsubsection{Ceiling-view prediction}
Ceiling-view prediction is exclusive to DuLa-Net. Contrary to its {panorama-branch} decoder, its {ceiling-branch} decoder, $B_C$, outputs a $512\times512$ probability map in the ceiling view. A pixel with higher value indicates a higher probability to be part of the ceiling. Decoders in both branches have the same architecture.
DuLa-Net then fuses the feature map from the {panorama-branch} to the {ceiling-branch} through the E2P projection module as described in~\secref{subsec:encoder}.
Applying fusion techniques
increases the prediction accuracy. It is conjectured that, in a ceiling-view image, the areas near the image boundary~(where some useful visual clues such as shadows and furniture arrangements exist) are more distorted, which can have a detrimental effect for the {ceiling-branch} to infer room structures. By fusing features from the {panorama-branch}~(in which distortion is less severe), performance of the {ceiling-branch} can be improved.
DuLa-Net applies fusions before each of the first five layers of the decoders. For each fusion connection, an E2P conversion with the $FoV$ set to 160$^\circ$ is taken to project the features under the equirectangular view to the perspective ceiling view. Each fusion works as follows:
\begin{align}
\label{equ:fusion}
f_{B_C}^* = f_{B_C} + \frac{\alpha}{\beta^i} \times f_{B_P},~i \in \{0, 1, 2, 3, 4\},
\end{align}
where $f_{B_C}$ is the feature from {ceiling-branch} and $f_{B_P}$ is the feature from {panorama-branch} after applying the {E2P} conversion. $\alpha$ and $\beta$ are the decay coefficients. $i$ is the index of the layer. After each fusion, the merged feature, $f_{B_C}^*$, is sent into the next layer of ceiling-view decoder.
Note that DuLa-Net's 2D floor plan prediction cannot predict 3D layout height, which is an important parameter for 3D layout reconstruction. To infer the layout height, three fully connected layers are added to the middlemost feature of {panorama-branch}. The dimensions of the three layers are 256, 64, and 1. To make the regression of the layout height more robust, dropout layers are added after the first two layers. To take the middlemost feature as input, DuLa-Net first applies average along channel dimensions, which produces a 1-D feature with 512 dimensions, and take it as the input of the fully connected layers.
\subsection{Loss Function}
\label{subsec:loss}
We discuss the loss functions for each of the original methods.
\subsubsection{LayoutNet}
The overall loss function is:
\begin{align}\label{equ:loss_layoutnet}
L(\boldsymbol{m_E},\boldsymbol{m_C},\boldsymbol{d}) &= -\alpha \frac{1}{n}\sum_{p\in \boldsymbol{m_E}}\big(p^{*}\log p +(1-p^{*})\log (1-p) \big)\nonumber\\
&-\beta \frac{1}{n}\sum_{q\in \boldsymbol{m_C}}\big(q^{*}\log q +(1-q^{*})\log (1-q) \big)
\end{align}
Here $\boldsymbol{m_E}$ is the probability that each image pixel is on the boundary between two walls; $\boldsymbol{m_C}$ is the probability that each image pixel is on a corner; $p$ and $q$ are pixel probabilities of edge and corner with ground truth values of $\hat{p}$ and $\hat{q}$, respectively. The loss is the summation over the binary cross entropy error of the predicted pixel probability in $\boldsymbol{m_E}$ and $\boldsymbol{m_C}$ compared with ground truth
\subsubsection{DuLa-Net}
The overall loss function is:
\begin{align}
\label{equ:loss_dulanet}
&L = E_b(M_{FC}, M_{FC}^*) + E_b(M_{FP}, M_{FP}^*) + \gamma E_{L1}(H, H^*),
\end{align}
Here for $M_{FC}$ and $M_{FP}$, we apply binary cross entropy loss:
\begin{align}
E_b(x, x^*) = -\sum_{i} {x}_i^*\log({x}_i) + (1-{x}_i^*)\log(1-{x}_i).
\end{align}
For $H$ (layout height), we use L1-loss:
\begin{align}
E_{L1}(x, x^*) = \sum_{i} |x_i - x^*_i|.
\end{align}
where $M_{FC}^*$, $M_{FP}^*$ and $H^*$ are the ground truth of $M_{FC}$, $M_{FP}$, and $H$.
\subsubsection{HorizonNet}
For the three channel 1-D prediction, HorizonNet applies L1-Loss for regressing the ceiling-wall boundary and floor-wall boundary position, and uses binary cross entropy loss for the wall-wall corner existence prediction.
\subsection{Structured Layout Fitting}
\label{subsec:opt}
Given the 2D predictions (i.e., corners, boundaries and ceiling-view floor plans), the camera position and 3D layout can be directly recovered, up to a scale and translation, by assuming that bottom corners are on the same ground plane and that the top corners are directly above the bottom ones. The layout shape is constrained to be Manhattan, so that intersecting walls are perpendicular, \textit{e}.\textit{g}.,~ like a cuboid or ``L"-shape in a ceiling view. The final output is a sparse and compact planar 3D Manhattan layout. The optimization can be performed under the equirectangular view or the ceiling view. The former approach is taken by LayoutNet while the latter is taken by DuLa-Net and HorizonNet. In the following, we explain our modified version of the LayoutNet method and the original DuLa-Net and HorizonNet methods in details.
\subsubsection{Equirectangular-view fitting}
Since LayoutNet's network outputs (i.e., 2D corner and boundary probability maps) are under the equirectangular view, the 3D layout parameters are optimized to fit the predicted 2D maps. The initial 2D corner predictions are obtained from the corner probability map (the output of the network) as follows. First, the responses are summed across rows, to get a summed response for each column. Then, local maxima are found in the column responses, with distance between local maxima of at least 20 pixels. Finally, the two largest peaks are found along the selected columns. These 2D corners might not satisfy Manhattan constraints, so we perform optimization to refine the estimates.
The ceiling level is initialized as the average~(mean) of 3D upper-corner heights, and then optimize for a better fitting room layout, relying on both corner and boundary information to evaluate 3D layout candidate $L$:
\begin{align}
Score(L)&= w_{junc}\frac{1}{|C|}\sum_{l_c\in C} P_{\text{corner}}(l_c)\nonumber\\
&+ w_{ceil}\frac{1}{|L_e|}\sum_{l_e\in L_e} P_{\text{ceil}}(l_e)\nonumber\\
&+ w_{floor}\frac{1}{|L_f|}\sum_{l_f\in L_f} P_{\text{floor}}(l_f)
\label{equ:layotnetopt}
\end{align}
where $C$ denotes the 2D projected corner positions of $L$. Cardinality of $L$ is \#walls$\times$ 2. The nearby corners are connected on the image to obtain $L_e$ which is the set of projected wall-ceiling boundaries, and $L_f$ which is the set of projected wall-floor boundaries~(each with cardinality of \#walls). $P_{\text{corner}}(\cdot)$ denotes the pixel-wise probability value on the predicted $\boldsymbol{m_C}$. $P_{\text{ceil}}(\cdot)$ and $P_{\text{floor}}(\cdot)$ denote the probability on $\boldsymbol{m_E}$. LayoutNet finds that adding wall-wall boundaries in the scoring function helps less, since the vertical pairs of predicted corners already reveals the wall-wall boundaries information.
Note that the cost function in~\eqnref{equ:layotnetopt} is slightly different from the cost function originally proposed in LayoutNet - we revise the cost function to compute the average response across layout lines instead of the maximum response. In this way, we are able to produce a relatively smoothed space for the gradient ascent based optimization as introduced below.
The originally proposed LayoutNet uses sampling to find the best ranked layout based on the cost function, which is time consuming and is constrained to the pre-defined sampling space. We instead use stochastic gradient ascent~\cite{robbins1951stochastic} to search for local optimum of the cost function~\footnote{We revised the SGD based optimization implemented by Sun~(with different loss term weights): https://github.com/sunset1995/pytorch-layoutnet}. We demonstrate the performance boost by using gradient ascent in experiments~(\secref{text:abla})
Finally, we made a few extensions. As LayoutNet's network prediction might miss occluded corners, which are important for the post-processing step that relies on Manhattan assumption, we adopt HorizonNet's post-processing step to find occluded corners for initialization before performing the fitting refinement in the equirectangular view.
\input{fig_floorplan_fitting.tex}
\subsubsection{Ceiling-view fitting}
DuLa-Net's network outputs 2D floor plan predictions under ceiling view. Given the probability maps ($M_{FC}$ and $M_{FP}$) and the layout height ($H$) predicted by the network, DuLa-Net reconstructs the final 3D layout in the following two steps:
\begin{enumerate
\item Estimating a 2D Manhattan floor plan shape using the probability maps.
\item Extruding the floor plan shape along its normal according to the layout height.
\end{enumerate}
For step 1, two intermediate maps, denoted as $\fcmap^{C}$ and $\fcmap^{F}$, are derived from ceiling pixels and floor pixels of the {floor-ceiling probability map} using the {E2P} conversion.
DuLa-Net further uses a scaling factor, $1.6/(H - 1.6)$, to register the $\fcmap^{F}$ with $\fcmap^{C}$, where the constant $1.6$ is the distance between the camera and the ceiling.
Finally, a {fused floor plan probability map} is computed as follows:
\begin{align}
\displaystyle
M_{FP}^{fuse} = 0.5*M_{FP} + 0.25*\fcmap^{C} + 0.25*\fcmap^{F}.
\end{align}
\figref{fig:floorplan_fitting} (a) illustrates the above process.
The probability map $M_{FP}^{fuse}$ is binarized using a threshold of $0.5$. A bounding rectangle of the largest connected component is computed for later use.
Next, the binary image is converted to a densely sampled piece-wise linear closed loop and simplify it using the Douglas-Peucker algorithm (see~\figref{fig:floorplan_fitting} (b)).
A regression analysis is run on the edges. The edges are clustered into sets of axis-aligned horizontal and vertical lines.
These lines divide the bounding rectangle into several disjoint grid cells (see~\figref{fig:floorplan_fitting} (c)).
The shape of the 2D floor plan is defined as the union of grid cells where the ratio of floor plan area is greater than $0.5$ (see~\figref{fig:floorplan_fitting} (d)). Note that this post-processing step does not have an implicit constraints on layout shapes~(cuboid or non-cuboid). To evaluate on cuboid room layout, we directly use the bounding rectangle of the largest connected component as the predicted 2D floor plan for DuLa-Net.
For HorizonNet, although the prediction is done under an equirectangular view, the post-processing step is done under a ceiling view. We observe that computing on the ceiling view helps enforcing the constraints that neighboring walls are orthogonal to each other, and to recover occluded wall corners that cannot be detected from equirectangular view.
First, the layout height is estimated by averaging over the predicted floor and ceiling positions in each column. Second, the scaled ceiling boundary and floor boundary are projected to the ceiling view, same as Dula-Net. Following LayoutNet's approach, HorizonNet then initializes the corner positions by finding the most prominent wall-wall corner points and project them to ceiling view. The orientations of walls are retrieved by computing the first PCA component along the projected lines between two nearby corners.
The projected ceiling boundary is represented by multiple groups of 2D pixel points separated by the wall-wall boundary. It then gives a higher score to the PCA vector line with more 2D pixel points within 0.16 meters and selects the vector that obtains the highest score as the wall in every group.
Finally, the 3D layout is reconstructed.
\subsubsection{Handling occlusions from layout boundaries}
For non-cuboid Manhattan layouts, some of the walls can be occluded from the camera position. LayoutNet finds the best-fit layout shape based on the 2D predictions, which might not be able to recover the occluded layout corners and boundaries. DuLa-Net fits a polygon to the predicted 2D floor plan, which explicitly enforces the neighboring walls to be orthogonal to each other. HorizonNet detects occlusions from layout boundaries by checking the orientation of the first PCA component for nearby layout walls. If two neighboring walls are parallel to each other, HorizonNet will hallucinate the occluded walls. We conjecture that the difference in handling occlusions from layout boundaries is the main reason why LayoutNet performs better than DuLa-Net and HorizonNet for cuboid layouts (no occlusions from layout boundaries) while performing slightly worse for non-cuboid layouts.
\subsection{Implementation Details}
\label{subsec:implementation}
We implement LayoutNet and DuLa-Net using PyTorch. For HorizonNet, we directly use their PyTorch source code available online for comparison. For implementation details, we summarize the data augmentation methods in~\secref{subsec:augmentation} and the training scheme and hyper-parameters in~\secref{exp:implementation}.
\subsection{Data augmentation}
\label{subsec:augmentation}
We show in~\tabref{tab:taxonomy_2} the summary of the different data augmentations originally proposed in each method. All three methods use horizontal rotation, left-right flipping and luminance change
to augment the training samples.
We unify the data augmentation by adding random stretching~(introduced below) to our modified LayoutNet and DuLa-Net methods.
\subsubsection{Random Stretching}
Random stretching is introduced by HorizonNet. The augmentation utilizes the $360^{\circ}$ property of panoramic images, projects the pixels into 3D space, stretches pixels along 3D axes, re-projects and interpolates pixels to the equirectangular image to augment training data.
The effectiveness of this approach has been demonstrated in~\cite{sun2019horizonnet}.
\subsubsection{Ground Truth Smoothing}
For LayoutNet, the target 2D boundary and corner maps are both binary maps that consist of thin curves (boundary map) or points (corner map) on the images, respectively. This makes training more difficult. For example, if the network predicts the corner position slightly off the ground truth, a huge penalty will be incurred. Instead, LayoutNet dilates the ground truth boundary and corner map with a factor of 3 and then smooth the image with a Gaussian kernel of $\sigma = 20$.
Note that even after smoothing, the target image still contains $\sim 95\%$ zero values, so the back propagated gradients of the background pixels is re-weighted by multiplying with $0.2$.
This strategy is also taken by HorizonNet for its wall-wall corner existence prediction. It is not taken by DuLa-Net since it predicts the complete floor plan map with clear boundaries.
\input{tbl_taxonomy_2.tex}
\subsection{Training Scheme and Parameters}
\label{exp:implementation}
For our modified LayoutNet and DuLa-Net methods, we use pre-trained weights on ImageNet to initialize the ResNet encoders. We perform random stretching with stretching factors $k_x=1$ and $k_z=2$. For each method, we use the same hyper-parameters for evaluating on the different datasets.
\input{tbl_cuboid_pano.tex}
\input{tbl_cuboid_stdn.tex}
\subsubsection{LayoutNet training}
LayoutNet uses the ADAM~\cite{kingma2014adam} optimizer with $\beta_1=0.9$ and $\beta_2=0.999$ to update network parameters. The network learning rate is $1e^{-4}$.
To train the network, we first train the layout boundary prediction branch, then fix the weights of boundary branch and train the corner prediction branch, and finally we train the whole network end-to-end.
To avoid the unstable learning of the batch normalization layer in ResNet encoder due to smaller batch size, we freeze the parameters of the batch normalization~(bn) layer when training end-to-end. The batch size for ResNet-18 and ResNet-34 encoder is 4, while the batch size for ResNet-50 is 2~(Which is too small to have a stable training of the bn layer, leading performance drops comparing with LayoutNet using ResNet-18 or ResNet-34 encoder as shown in~\tabref{tab:cuboid_pano} and~\tabref{tab:cuboid_stdn} in experiments). We set the term weights in~\eqnref{equ:loss_layoutnet} as $\alpha = \beta = 1$.
\subsubsection{DuLa-Net training}
DuLa-Net uses the ADAM optimizer with $\beta_1=0.9$ and $\beta_2=0.999$. The learning rate is $0.0001$ and batch size is $8$. The training loss converges after about $120$ epochs. For feature fusion, the $\alpha$ and $\beta$ in~\eqnref{equ:fusion} is set to be $0.6$ and $3$. The $\gamma$ in~\eqnref{equ:loss_dulanet} is set to be $0.5$.
\subsection{Summarization of Modifications}
As introduced in~\secref{subsec:method} and~\secref{subsec:implementation}, we unify some of the designs and training details and propose the modified LayoutNet and DuLa-Net methods. For clarity, we summarize our modifications to LayoutNet~(denoted as \textbf{``{LayoutNet v2}"}) and DuLa-Net~(denoted as \textbf{``{DuLa-Net v2}"}) as follows.
\subsubsection{LayoutNet v2}
For LayoutNet v2, we use pre-trained ResNet encoder instead of SegNet encoder trained from scratch. We add random stretching data augmentation. We perform 3D layout fitting using gradient ascent optimization instead of sampling based searching scheme. We extend the equirectangular view optimization for general Manhattan layout.
\subsubsection{DuLa-Net v2}
For DuLa-Net v2, we choose to use deeper ResNet encoders instead of the ResNet-18 one and add random stretching data augmentation.
\section{Experiments and Discussions}
In this section, we evaluate the performance of {LayoutNet v2}, {DuLa-Net v2} and HorizonNet introduced in~\secref{sec:network}. We describe the evaluation metrics in~\secref{exp:metric} and compare the methods on PanoContext dataset and Stanford 2D-3D dataset for cuboid layout reconstruction in~\secref{exp:cuboid}. We evaluate performance on {MatterportLayout} for general Manhattan layout estimation in~\secref{exp:matterport}. Finally, based on the experiment results, we discuss the advantages and disadvantages of each method in~\secref{exp:discuss}.
\input{fig_visual_result_pano.tex}
\subsection{Evaluation Setup}
\label{exp:metric}
We use the following five standard evaluation metrics:
\begin{itemize}
\item \textbf{Corner error}, which is the $L2$ distance between the predicted layout corners and the ground truth under equirectangular view. The error is normalized by the image diagonal length and averaged across all images.
\item \textbf{Pixel error}, which is the pixel-wise semantic layout prediction (wall, ceiling, and floor) accuracy compared to the ground truth. The error is averaged across all images.
\item \textbf{3D IoU}, defined as the volumetric intersection over union between the predicted 3D layout and the ground truth.
The result is averaged over all the images.
\item \textbf{2D IoU}, defined as the pixel-wise intersection over union between predicted layout under ceiling view and the ground truth.
The result is averaged over all the images.
\item \textbf{rmse}, defined as the root mean squared error between predicted layout depth $\hat{d}$ and the ground truth $d$: \\
$\sqrt{\frac1{|d|} \sum_{p\in d}{(d_p - \hat{d}_p)^2}}$ where p represents every pixel in the depth. We use the true camera height, which is 1.6 for each image, to generate the predicted depth map
The result is averaged over all the images.
\item $\boldsymbol{\delta_i}$, defined as the percentage of pixels where the ratio~(or its reciprocal) between the prediction and the label is within a threshold of 1.25
$\frac1{|d|} \sum_{p\in d} \mathbf{1}[\max{(\frac{d_p}{\hat{d_p}},\frac{\hat{d_p}}{d_p})} < 1.25]$.
\end{itemize}
We use corner error, pixel error, and 3D IoU to evaluate performance of cuboid layout reconstruction.
For general Manhattan layout reconstruction, since the predicted layout shape can be different from the ground truth shape, we use 3D IoU, 2D IoU and depth measurements (\textit{i}.\textit{e}.,~ rmse and $\delta_1$) for evaluation.
\label{sec:experiment}
\subsection{Performance on PanoContext and Stanford 2D-3D}
\label{exp:cuboid}
In this experiment, we evaluate the performance of {LayoutNet v2}, {DuLa-Net v2}, and HorizonNet on the PanoContext dataset and Stanford 2D-3D dataset, which is comprised of cuboid layouts. For all three methods, we used a unified (ResNet) encoder and analyzed the performance of using different post-processing steps.
\paragraph{Dataset setting.} For the evaluation on PanoContext dataset, we use both the training split of PanoContext dataset and the whole Stanford 2D-3D dataset for training and vice versa for the evaluation on Stanford 2D-3D dataset. The split for validation and testing of each dataset is reported in~\secref{sec:dataset}. We use the same dataset setting for all three methods.
\paragraph{Qualitative results.} We show in~\figref{fig:pano} the qualitative results of the experiments on PanoContext dataset and Stanford 2D-3D dataset.
All methods offer similar accuracy. LayoutNet v2 slightly outperforms on PanoContext and offers more robustness to occlusion from foreground objects, while DuLa-Net v2 outperforms in two of three metrics for Stanford 2D-3D as shown in Table~\ref{tab:cuboid_pano} and Table~\ref{tab:cuboid_stdn}.
\subsubsection{Evaluation on Unified Encoder}
\label{text:exp_encoder}
\tabref{tab:cuboid_pano} and~\tabref{tab:cuboid_stdn} show the performance for {LayoutNet v2}, {DuLa-Net v2} and HorizonNet on PanoContext dataset and Stanford 2D-3D dataset, respectively. In each row, we report performance by using ResNet-18, ResNet-34, and ResNet-50 encoders respectively. For both {DuLa-Net v2} and HorizonNet, using ResNet-50 obtains the best performance, indicating that deeper encoder can better capture layout features. For {LayoutNet v2}, we spot a performance drop with ResNet-50, this is mainly due to the smaller number of batch size (we use 2 in experiment, which is the maximum available number to run on a single GPU of 12GB) that leads to unstable training of the batch normalization layer in ResNet encoder. We expect an better performance of {LayoutNet v2} with ResNet-50 by training on a GPU with a larger memory, but we consider it as an unfair comparison with the other two methods since the hardware setup is different. In general, {LayoutNet v2} with ResNet-34 outperforms all other methods on PanoContext dataset and obtains lowest pixel error on Stanford 2D-3D dataset. {DuLa-Net v2}, on the other hand, shows the best 3D IoU and corner error on Stanford 2D-3D dataset.
Note that the reported number for HorizonNet with ResNet-50 is slightly lower than that reported in the original paper. This is attributed to the difference in the training dataset, \textit{i}.\textit{e}.,~ the authors used both the training split of PanoContext dataset and Stanford 2D-3D dataset for training. We thus retrain the HorizonNet using our training dataset setting for a fair comparison.
\subsubsection{Ablation Study}
\label{text:abla}
We show in~\tabref{tab:cuboid_abla} the ablation study of different design choices of {LayoutNet v2} on the best performing PanoContext dataset. The first row shows the performance reported in~\cite{zou2018layoutnet}. The proposed {LayoutNet v2} with ResNet encoder, modified data augmentation and post-processing step boosts the overall performance by a large margin ($\sim 10\%$ in 3D IoU).
A large performance drop is observed when training the model from scratch (w/o ImageNet pre-training).
Using gradient ascent for post-processing contributes the most to the performance boost (w/o gradient ascent), while adding random stretching data augmentation contributes less (w/o random stretching).
Freezing batch normalization layout when training end-to-end can avoid unstable training of this layer when the batch size is small (w/o freeze bn layer).
Including all modifications together achieves the best performance.
We show in~\tabref{tab:cuboid_abla_dula} the ablation study for {DuLa-Net v2} on the Stanford 2D-3D dataset.
We obtain a performance boost of $5\%$ in 3D IoU when comparing with the original model~\cite{yang2019dula} by using a deeper ResNet encoder (ResNet-50 vs. ResNet-18). Similar to {LayoutNet v2}, using the random stretching data augmentation (w/o random stretching) improves the performance only marginally.
\input{tbl_cuboid_abla.tex}
\input{tbl_cuboid_abla_dula.tex}
\input{tbl_time.tex}
\paragraph{Comparison with different post-processing steps.}
In this experiment, we compare the performance of {LayoutNet v2} while using the post-processing steps of {DuLa-Net v2} and HorizonNet, and combining its own optimization step with additional semantic loss, respectively.
The post-processing step of HorizonNet utilizes predicted layout boundaries and corner positions in each image column, which can be easily converted from the output of {LayoutNet v2}. To adapt {DuLa-Net v2}'s post-processing step, we train {LayoutNet v2} to predict the semantic segmentation (\textit{i}.\textit{e}.,~ wall probability map) under equirectangular view as an additional channel in the boundary prediction branch. Then, we use the predicted floor-ceiling probability map as input to the post-processing step of {DuLa-Net v2}. Alternatively, we can also incorporate the predicted wall probability map into the layout optimization of {LayoutNet v2}. We add an additional loss term to~\eqnref{equ:layotnetopt} for the average per-pixel value enclosed in the wall region of the predicted probability map with a threshold of 0.5. We set the semantic term weights to 0.3 for grid search in the validation set.
As reported in~\tabref{tab:cuboid_abla} (row~6-8), together with {LayoutNet v2}'s neural network, a post-processing under equirectangular view performs better than the one under ceiling view. We found that the additional semantic optimization did not improve the post-processing step under equirectangular view. This is because the jointly predicted semantic segmentation is not that accurate, achieving only 2.59\% pixel error compared with the 1.79\% pixel error by our proposed LayoutNet v2.
Another interesting study is to see whether the performance of {DuLa-Net v2} and HorizonNet will be affected by using the post-processing step that works on the equirectangular view. However, it is not clear how to convert from their output probability maps to layout boundaries and corner positions, which are the required input for {LayoutNet v2}'s post-processing step.
\input{fig_visual_result_matterport.tex}
\subsubsection{Timing Statistics}
We show in~\tabref{tab:time} the timing performance of {LayoutNet v2} with ResNet-34 encoder, {DuLa-Net v2} with ResNet-50 encoder, and HorizonNet with ResNet-50 encoder. We report the computation time of HorizonNet with RNN refinement branch. Note that HorizonNet without RNN only costs 8ms for network prediction but produces less accurate result compared with other approaches. We report average time consumption for a single forward pass of the network and the post-processing step.
\input{fig_eval_confuse_mat.tex}
\subsection{Performance on {MatterportLayout}}
\label{exp:matterport}
In this experiment, we compare the performance of three methods on estimating the general Manhattan layouts using the {MatterportLayout} dataset.
For a detailed evaluation, we report the performance for layouts of different complexity. We categorize each layout shape according to the number of floor plan corners in the ceiling view, e.g. a cuboid has 4 corners, an ``L''-shape has 6 corners, and a ``T''-shape has 8 corner.
The dataset split used for training/validation/testing is reported in~\secref{sec:dataset}.
\paragraph{Qualitative results.}
\label{exp:matterport_quali}
\figref{fig:matterport_eval} shows the qualitative comparisons of the three methods.
All three methods have similar performance when the room shape is simpler, such as cuboid and `L''-shape rooms. For more complex room shapes, HorizonNet is capable of estimating thin structures like the walls as shown in~\figref{fig:matterport_eval} (6th row, 1st column), but could also be confused by the reflected room boundaries in the mirror as shown in~\figref{fig:matterport_eval} (6th row, 4th column).
{LayoutNet v2} tends to ignore the thin layout structures like the bumped out wall as shown in~\figref{fig:matterport_eval} (7th row, 1st column).
{DuLa-Net v2} is able to estimate the occluded portion of the scene, utilizing cues from the 2D ceiling view as shown in~\figref{fig:matterport_eval} (8th row, 2nd column), but could also be confused by ceiling edges as shown in~\figref{fig:matterport_eval} (8th row, last column).
\input{tbl_eval_matterport.tex}
\paragraph{Quantitative Evaluation.} \tabref{tab:eval_matterport} shows the quantitative comparison of three methods on estimating general Manhattan layout using the {MatterportLayout} dataset.
We consider the 3D IoU, 2D IoU and two depth accuracy measurements (\textit{i}.\textit{e}.,~ rmse and $\delta_1$) for the performance evaluation.
Overall, among the three methods, HorizonNet shows the best performance while {LayoutNet v2} has similar performance on 2D IoU and 3D IoU with cuboid room shape. {DuLa-Net v2} performs better than {LayoutNet v2} for non-cuboid shapes, while being slightly worse than HorizonNet. Although these three methods show competitive performance in the overall 2D and 3D IoU metric, the performance gap in the depth metrics is more obvious. This is because the depth metrics can quantify the detailed local geometric differences: predicting an ``L''- shape room with a small concave corner as a cuboid room can have less impact on 2D or 3D IoU, but the depth error will increase. This indicates the value of our newly proposed metrics.
\subsection{Discussions}
\label{exp:discuss}
\subsubsection{Why do {LayoutNet v2} and HorizonNet perform differently on different datasets?}\label{exp:discuss_conf}
On PanoContext dataset and Stanford 2D-3D dataset, {LayoutNet v2} outperforms the other two methods. However, on {MatterportLayout} dataset, HorizonNet stands to be the clear winner. We believe this is due to the different design of network decoder and the different representation of network's outputs, making each method performs differently for cuboid layout and non-cuboid layout, as discussed below.
{LayoutNet v2} relies more on the global room shape context, \textit{i}.\textit{e}.,~ it can predict one side of the wall given the prediction of the other three walls. This is benefited from the two-branch network prediction of room boundaries and corners, and the corner prediction is guided by the room boundaries: boundaries will also get gradients from error predicted corners during training. However, because of the reliance on the global context for {LayoutNet v2}, it is harder to generalize the reasoning process from one layout type to another. For example, learning how to use the global context on images with cuboid layout might not help in a room with 16 corners.
This gap is lesser for HorizonNet and {DuLa-Net v2} since they predict local outputs that can be generalized to arbitrary layouts. This is because HorizonNet and {DuLa-Net v2} emphasize more on local edge and corner responses, \textit{e}.\textit{g}.,~ predict whether this column has a corner, and the position of floor and ceiling in this column. A direct evidence is that, by training on Stanford 2D-3D dataset which has all cuboid shapes, {LayoutNet v2} predicts cuboid shape only, while HorizonNet has $10\%$ non-cuboid outputs. These characteristics are also reflected in the qualitative results shown in~\figref{fig:matterport_eval}. As we discussed in~\secref{exp:matterport_quali}, {LayoutNet v2} often misses thin layout structures such as pipes, while HorizonNet can be more sensitive to those thin structures.
We also show in~\figref{fig:cf} the confusion matrix on correctly estimating the number of corners of the 3D layouts for each method.
For the cuboid layout~(4 corners), {LayoutNet v2} shows the highest recall rate. However, {LayoutNet v2} also tends to predict some non-cuboid layouts~(\textit{e}.\textit{g}.,~ 6 corners, 8 corners, 10 corners) to be cuboid.
On the other hand, {DuLa-Net v2} and HorizonNet shows better and comparable performance for estimating the non-cuboid room layouts. Therefore, the error in layout type prediction is the major cause of error for {LayoutNet v2} in 3D reconstruction on the MatterportLayout dataset.
Moreover, HorizonNet differs from {LayoutNet v2} and {DuLa-Net v2} in the decoder architecture. HorizonNet uses a 1D RNN, while {LayoutNet v2} and {DuLa-Net v2} use 2D convolutions. We believe that this is also one of the reasons why HorizonNet's performance in the PanoContext dataset lags behind {LayoutNet v2} and {DuLa-Net v2}: 2D convolutions can better localize low level details like corners, lines and curves.
\subsubsection{Analysis and Future Improvements for {DuLa-Net v2}}
{DuLa-Net v2} is sensitive to the parameter of FOV (introduced in the E2P projection in~\secref{subsec:preprocess}). A smaller FOV (\textit{e}.\textit{g}.,~ $160^{\circ}$) can lead to higher quality predictions for most of the rooms, but some larger rooms could be clipped by the image plane after projection. A larger FOV (\textit{e}.\textit{g}.,~ $165^{\circ}$, $171^{\circ}$) could produce fewer clipped rooms after projection, but the prediction quality for some rooms may decrease, due to the down-scaled ground truth 2D floor plan in ceiling view. In this paper, we use the setting of FOV=$160^{\circ}$, but we suggest to improve the prediction quality by combining the prediction of multiple networks trained with different FOVs in the future work. To give an idea of the potential improvement, we report the numbers for {MatterportLayout} dataset by removing the rooms that are too big to be clipped by the boundary of the projection under the setting of FOV=$160^{\circ}$. For this case, the 3D IoU improves from 74.53 to 76.82.
\section{Related Work}
\label{sec:relatedwork}
There are numerous papers that propose solutions for estimating a 3D
room layout from a single image. The solutions differ in the layout shapes~(\textit{i}.\textit{e}.,~ cuboid layout \vs general Manhattan layout), inputs~(\textit{i}.\textit{e}.,~ perspective \vs panoramic image), and methods to predict geometric features and fit model parameters.
In terms of room layout assumptions, a popular choice is the ``Manhattan world" assumption~\cite{coughlan1999manhattan}, meaning that all walls are aligned with a canonical coordinate system~\cite{coughlan1999manhattan,ramalingam2013manhattan}. To make the problem easier, a more restrictive assumption is that the room is a cuboid~\cite{hedau2009recovering, dasgupta2016delay, lee2017roomnet}, \textit{i}.\textit{e}.,~ there are exactly four room corners in the top-down view.
Recent state-of-the-art methods~\cite{zou2018layoutnet,yang2019dula,sun2019horizonnet} adopt the Manhattan world assumption but allow for room layout with arbitrary complexity.
In terms of the type of input images, the images may differ in the FoV (field of view) - ranging from being monocular (\textit{i}.\textit{e}.,~ taken from a standard camera) to 360$^\circ$ panoramas, and whether depth information is provided. The methods are then largely depending on the input image types. It is probably most difficult problem when only a monocular RGB image is given. Typically, geometric (\textit{e}.\textit{g}.,~ lines and corners)~\cite{lee2009geometric, hedau2009recovering, ramalingam2013lifting} and/or semantic (\textit{e}.\textit{g}.,~ segmentation into different regions~\cite{hoiem2005geometric, hoiem2007recovering} and volumetric reasoning~\cite{gupta2010estimating}) "cues" are extracted from the input image, a set of room layout hypotheses is generated, and then an optimization or voting process is taken to rank and select the best one among the hypotheses.
Traditional methods treat the task as an optimization problem. Flint~\textit{et~al.}~\cite{flint2010dynamic} propose a dynamic programming approach for the room layout vectorization. They further apply a graphical model in the context of a moving camera. A more recent work by Delage ~\textit{et~al.}~\cite{delage2006dynamic} fit floor/wall boundaries in a perspective image taken by a level camera to create a 3D model under the Manhattan world assumption using dynamic Bayesian networks. Most methods are based on finding best-fitting hypotheses among detected line segments~\cite{lee2009geometric}, vanishing points~\cite{hedau2009recovering}, or geometric contexts~\cite{hoiem2005geometric}. Subsequent works follow a similar approach, with improvements to layout generation~\cite{schwing2012efficient,schwing2012efficient_eccv,ramalingam2013manhattan}, features for scoring layouts~\cite{schwing2012efficient_eccv,ramalingam2013manhattan}, and incorporation of object hypotheses~\cite{hedau2010thinking,gupta2010estimating,del2012bayesian,del2013understanding,zhao2013scene} or other context.
Recently, neural network-based methods took stride in tackling this problem. There exist methods that train deep network to classify pixels into layout surfaces~(\textit{e}.\textit{g}.,~ walls, floor, ceiling)~\cite{dasgupta2016delay,izadinia2017im2cad}, boundaries~\cite{mallya2015learning}, corners~\cite{lee2017roomnet}, or a combination~\cite{ren2016coarse}. A trend is that the neural networks generate higher and higher levels of information - starting from line segments~\cite{mallya2015learning, stpio}, surface labels~\cite{dasgupta2016delay}, to room types~\cite{lee2017roomnet} and room boundaries and corners~\cite{zou2018layoutnet}, to faciliate the final layout generation process. Recent methods push the edge further by using neural networks to directly predict a 2D floor plan~\cite{yang2019dula} or as three 1D vectors that concisely encode the room layout~\cite{sun2019horizonnet}. In both cases, the final room layouts are reconstructed by a simple post-processing step.
Another line of works aims to leverage the extra depth information for room model reconstruction, including utilizing single depth image for 3D scene reconstruction~\cite{zhang2013estimating, zou2019complete, liu2016layered}, and scene reconstructions from point clouds~\cite{newcombe2011kinectfusion, monszpart2015rapter, liu2018floornet, cabral2014piecewise}.
Liu~\textit{et~al.}~\cite{liu2015rent3d} present Rent3D, which takes advantage of a known floor plan. Note that neither estimated depths nor reconstructed 3D scenes necessarily equate a clean room layout as such inputs may contain clutters.
\paragraph{360$^\circ$ panorama:}
The seminal work by Zhang~\textit{et~al.}~\cite{zhang2014panocontext} advocates the use of 360$^\circ$ panoramas for indoor scene understanding, for the reason that the FOV of 360$^\circ$ panoramas is much more expansive. Work in this direction flourished, including methods based on optimization approaches over geometric~\cite{Fukano2016RoomRF,pintore2016omnidirectional,yang2016efficient,yang2016efficient,xu2017pano2cad} and/or semantic cues~\cite{xu2017pano2cad,automatic} and later based on neural networks~\cite{lee2017roomnet,zou2018layoutnet}. Most methods rely on leveraging existing techniques for single perspective images on samples taken from the input panorama.
The LayoutNet introduced by Zou~\textit{et~al.}~\cite{zou2018layoutnet} was the first approach to predict room layout directly on panorama, which led to better performance.
Yang~\textit{et~al.}~\cite{yang2019dula} and Pintore~\textit{et~al.}~\cite{pintore2016omnidirectional} follow the similar idea and propose to predict directly in the top-down view converted from input panorama.
In this manner, the vertical lines in the panorama become radial lines emanated from the image center. An advantage of this representation is that the room layout becomes a closed loop in 2D that can be extracted more easily. As mentioned in~\cite{yang2019dula}, the ceiling view is arguably better as it provides a clutter-free view of the room layout.
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1510.06883
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\section{Introduction}\label{sec:intro}
Enumerating inclusion-wise maximal vertex-sets of complete bipartite subgraphs (maximal bicliques) in bipartite graphs is a challenging theoretical and computational problem~\cite{Eppstein,AACFHS,MakinoUno} related to several classical problems in combinatorial optimization, theoretical computer
science~\cite{Amil,CornazFon,Agarwal,Berry07} and bioinformatics~\cite{Liliuliwong,Zhang} (and the references cited therein). The problem has been shown to be \#P-complete by Kuztnetsov~\cite{kuz} and there have been active efforts to bound and estimate the number of maximal bicliques as well as efficiently computing and listing such bicliques both in general and in restricted classes of bipartite graphs~\cite{prisner,Amil}. There are two non-trivial classes of bipartite graphs admitting polynomially many maximal bicliques: the class of bipartite domino-free graphs~\cite{Amil} and the class of $C_6$-free graphs~\cite{prisner} (in particular, the class of chordal-bipartite graphs): $O(m)$ in the former case, $m$ being the size of the graph, and $O((n_1\times n_2)^2)$ in the latter case, $n_1$ and $n_2$ being the number of vertices in the two color classes. In these cases, the interest is clearly on designing efficient algorithms to count the number of maximal bicliques, list all the maximal bicliques, and solving related computations. However, besides its own interest, what makes the problem even more appealing even in special cases, is the intimate relationship with the problem of building \emph{concept lattices} (also known as Galois lattices) of a formal context in Formal Concept Analysis, a well established (though still flourishing) topic in Applied Lattice Theory~\cite{gw}.
For our purposes, a formal context is a bipartite graph $G$ with color classes $X$ and $Y$. In Formal Concept Analysis, $X$ is interpreted as a set of objects and $Y$ as a set of attributes, while $G$ encodes the incidence binary relation between attributes and objects: object $x\in X$ has attribute $y\in Y$ if and only if $xy$ is an edge of $G$. A formal concept is an ordered pair $(X_0,Y_0)$, where $X_0$ is a subset of objects, $Y_0$ is a subset of attributes, and all the objects in $X_0$ share all the attributes in $Y_0$ in such a way that any other object $x\in X\setminus X_0$ fails to have at least one of the attributes in $Y_0$ and any other attribute $y\in Y\setminus Y_0$ is not possessed by at least one object in $X_0$. The sets $X_0$ and $Y_0$ are called the intent and the extent of the formal concept. Concepts can be (partially) ordered from the more specific to the more general: the more objects share a common set of attributes the less specific is the concept, e.g.\ ``mammal'' is less specific than ``dog'', the extent of the concept ``mammal'' contains the extent of ``dog'' as well as the extent of ``cats'' for instance, and dually the intent of ``mammal'', namely the set of attributes defining ``mammal'', is contained in the intent of ``dog''. It is convenient to assume the existence of the most specif concept of a context, namely the concept whose intent is the set of all attributes, as well as the most general concept, namely the concept whose extent consists of all objects. As proved by Ganter and Wille, according to the basic Theorem of concept lattices, the set of formal concepts of a given context, hierarchically ordered, is actually a lattice called the \emph{concept lattice} of the context $G$ (also known as the Galois lattice of $G$), with the most specific and the most general concepts as bottom and top, respectively.
From a graph-theoretical point of view, formal concepts can be identified with the maximal bicliques $B$ of $G$, hence if $\B(G)$ denotes the collection of the maximal bicliques of $G$, then $\mathcal{L}(G)=(\maxbic(G)\cup\{\bot,\top\},\preceq)$ is the Galois lattice of $G$, where, $\bot$ and $\top$ are two dummy maximal bicliques consisting, respectively, of the color class $X$ alone and the color class $Y$ alone (unless there are universal vertices in $G$) and the partial order $\preceq$ is defined by
$$B\preceq B' \Leftrightarrow X\cap B\subseteq X\cap B'.$$
Equivalently, the same partial order can be defined as
$$B\preceq B' \Leftrightarrow Y\cap B\supseteq Y\cap B'$$
since $X\cap B\subseteq X\cap B' \Leftrightarrow Y\cap B\supseteq Y\cap B'$ for any pair of maximal bicliques $B$ and $B'$.
Hence, with any bipartite graph there is an associated lattice on its collection of maximal bicliques and the shape of such a lattice can be characteristic of particular classes of bipartite graphs.
For instance \emph{Bipartite Distance Hereditary} graphs (BDH for shortness) have been investigated in \cite{acfDAM,acfIWOCA}. Recall that a graph is \emph{Distance Hereditary} if the distance between any two of its vertices is the same in every connected induced subgraph containing them. A graph is \emph{Bipartite Distance Hereditary} if it is both bipartite and distance hereditary.
In \cite{acfDAM}, BDH graphs have been characterized as the class of bipartite graphs whose Galois lattice is tree-like. More precisely, it has been shown that the Hasse diagram $\mathbf{H}^\circ(G)$ of the poset obtained by removing the top and bottom elements from the Galois lattice $\LL(G)$ of a bipartite graph $G$ is a tree if and only if $G$ is a BDH graph. This implies that the linear dimension of the Galois lattice of a BDH graph is at most 2. Anyway, no efficient algorithms for computing the Galois lattice of a BDH graph have been proposed, though special classes of graphs inducing efficiently computable Galois lattices (much more efficiently than in the general case) have been investigated \cite{Amil,BerryMSS06}. In particular, an $O(m\times n)$ worst case-time algorithm has been given in \cite{Amil} for computing the Galois lattice for the more general class of domino-free graphs with $m$ edges and $n$ vertices.
\mybreak
Bandelt and Mulder \cite{bm} proved that BDH graphs are exactly all the graphs that can be constructed starting from a single vertex by a sequence of adding
pending vertices and false twins of existing vertices. This is a special case of what happens for (not necessarily bipartite) distance hereditary graphs, that are characterized as graphs that can be built starting from a single vertex and a sequence of additions of pending vertices, false twins and true twins (see Section~\ref{sec:prelim} for the definition of false and true twins). This sequence is referred to as an \emph{admissible sequence} in~\cite{bm}, and it is the reverse of what is called a \emph{pruning sequence} in \cite{hammer}. Damiand \emph{et al.}~\cite{damiand} proposed
an optimal $O(m)$ worst case time algorithm for computing a pruning sequence of a distance hereditary graph $G$, where $m$ is the number of edges in $G$, using a cograph recognition algorithm in~\cite{corneil}. Obviously, the same algorithm computes a pruning sequence of a BDH graph.
\mybreak
In this paper we show that, for any BDH graph $G$ with $n$ vertices and $m$ edges:
\begin{itemize}
\item $G$ contains at most $n-2$ maximal bicliques. This improves, for BDH graphs, the more general $O(m)$ bound given in~\cite{Amil} for domino-free bipartite graphs and the $O(n^4)$ bound in~\cite{prisner} for $C_6$-free graphs;
\item the total size of $\LL(G)$, i.e., the sum of the number of vertices over all maximal bicliques of $G$, is $O(m)$.
\item it is possible to compute $\mathbf{H}^\circ(G)$, i.e., the Hasse diagram of the Galois lattice of $G$ in worst case time $O(m)$. This improves by a factor of $n$ (the number of vertices of $G$) the $O(m\times n)$ worst case time algorithm given in \cite{Amil} for the larger class of domino-free graphs.\ The construction we propose also finds meet-irreducible and join-irreducible elements in the Galois lattice, also known as \emph{introducers} (see next section), and provides an explicit representation of all maximal bicliques in $G$.
This result is based on a simpler constructive proof that BDH graphs have a tree-like Galois lattice (i.e., the \emph{if part} of the characterization in~\cite{acfDAM}).
\item it is possible to compute an arborescence $A$ such that $G$ is the arcs/paths incidence graph of a set of paths in $A$. The arborescence can be computed in worst case time $O(n)$, starting from a pruning sequence of $G$, and gives an $O(n)$ space representation of both neighborhoods in $G$ and maximal bicliques. This result provides a simpler and constructive proof of the maximal bicliques encoding proposed in~\cite{acfDAM};
\item relying on the arborescence representation above, it is possible to compute an $O(n)$ space representation of $\mathbf{H}^\circ(G)$.\ The compact representation is obtained in $O(n)$ time starting from a pruning sequence of $G$, yielding an overall $O(m+n)$ algorithm to compute $\mathbf{H}^\circ(G)$.
\end{itemize}
\section{Definitions and preliminaries}\label{sec:prelim}
Graphs dealt with in this paper are simple (no loops nor parallel edges).\ The neighborhood in $G=(V,E)$ of a vertex $v \in V$ is the set $N_G(v) = \{u\ |\ uv \in E\}$, and the number of vertices in $N_G(v)$ is denoted by $\deg_G(v)$ and it is called the \emph{degree of $v$ in $G$}.
A vertex $v$ is said to be a \emph{pending vertex in $G$} if $\deg_G(v) = 1$.
A vertex $v$ is said to be a \emph{false twin} \emph{in $G$} if a vertex $u \not= v$ exists such that $N_G(v) = N_G(u)$ and $v \not\in N_G(u)$, while $v$ is said to be a \emph{true twin} \emph{in $G$} if a vertex $u \not= v$ exists such that $N_G(v) = N_G(u)$ and $v \in N_G(u)$. Since we are dealing with bipartite graphs, and bipartite graphs cannot contain true twins, we will refer to false twins simply by \emph{twins}.
For ease of notation, we usually omit the subscript referring to $G$ when no confusion can arise. Occasionally, we denote the edge-set (arc-set) of a (directed) graph $G$ by $E(G)$.
An \emph{arborescence} is a directed tree with a
single special node distinguished as the \emph{root} such that,
for each other vertex, there is a dipath from the root to that
vertex. An arborescence $T$ induces a partial order $\leqslant_T$ on $E(T)$, the \emph{arborescence order}, as follows: $e\leqslant_T f$ if the unique path from the root of $T$ which ends with $f$ contains $e$. So we can think of $T$ as the partially ordered set $(E(T),\leqslant_T)$. The arborescence order allows us to identify paths $F$ of $T$ with intervals of the form $[\alpha(F),\beta(F)]$, where $\alpha(F)$ is the arc of $F$ closest to the root of $T$ (the $\leqslant_T$-least element of $F$) and $\beta(F)$ is the arc of $F$ farthest from the root of $T$ (the $\leqslant_T$-greatest element of $F$).
The color classes of a bipartite graph $G$ are referred to as the \emph{shores} of $G$. If the bipartite graph $G$ has shores $X$ and $Y$, we denote such a graph by $G=(X,Y,E)$. A \emph{complete bipartite graph} is a bipartite graph $(X,Y,E)$ where edge $xy \in E$ for each $x \in X$, $y\in Y$.
A \emph{biclique} $B$ in $G$ is a set of vertices of $G$ that induces a complete bipartite subgraph $(X',Y',E')$ with $X' \not= \emptyset$ and $Y' \not= \emptyset$. Such a biclique will be identified with the pair $(X',Y')$ of the shores of the graph it induces, and the shores of a biclique $B$ will be denoted by $X(B)$ and $Y(B)$. A biclique in $G$ is a \emph{maximal biclique} if it is not properly contained in any biclique of $G$.
\mybreak
The \emph{transitive reduction} of a partially ordered set $(S,\leqslant)$ is the directed acyclic graph on
$S$ where there is an arc leaving $x\in S$ and entering $y\in S$ if and
only if $x\leqslant y$ and there is no $z\in S \setminus \{x,y\}$ such that $x\leqslant z\leqslant y$. With some abuse of terminology, we refer to the transitive reduction of a partially ordered set as to its \emph{Hasse diagram}. Arcs $(x,y)$ in the Hasse diagram of $(S,\leqslant)$, will be denoted by $x\dot < y$ and will be referred to as \emph{cover pairs}. If $x\dot < y$ is a cover pair in the Hasse diagram of some poset we say that $y$ \emph{covers} $x$ or that $x$ \emph{is covered} by $y$.
\mybreak
As above, the set of maximal bicliques in $G$ is denoted by $\B(G)$, and $\mathcal{L}(G)=(\maxbic(G)\cup\{\bot,\top\},\preceq)$ is the associated Galois lattice. Moreover, we let $\mathcal{L}^\circ(G)=(\maxbic(G),\preceq)$, namely, $\mathcal{L}^\circ(G)$ is the partially ordered set obtained from $\mathcal{L}(G)$ after removing $\bot$ and $\top$. The symbols $\mathbf{H}(G)$ and $\mathbf{H}^\circ(G)$ denote the Hasse diagrams of $\mathcal{L}(G)$ and $\mathcal{L}^\circ(G)$, respectively.
\mybreak
It is known (see~\cite{gw}) that for any vertex $v \in X$ (resp., $v \in Y$) of a bipartite graph $G=(X,Y,E)$ there is a maximal biclique in $G$ (hence an element in $\B(G)$) of the form $(\bigcap_{x \in N(v)}N(x), N(v))$ (resp., $(N(v), \bigcap_{x \in N(v)}N(x))$). Conforming to concept lattice terminology, such an element of $\LL(G)$ is referred to as \emph{object concept} (resp., \emph{attribute concept}), while it is called the \emph{introducer} of $v$ in~\cite{Berry13}. The introducer of $v$ is the lowest (resp., highest) maximal biclique containing $v$ in $\LL(G)$, and will be denoted by $\introd(v)$. It can also be shown that irreducible elements in $\LL(G)$ are introducers---recall that in a partially ordered set (in particular in a lattice) an element $r$ is \emph{meet-irreducible} (resp., \emph{join-irreducible}) if $r$ is not the least upper bound (resp., the greatest lower bound) of any two other distinct elements $s$ and $t$-- however, we do not use such notions here.
\mybreak
Given a BDH graph $G$, we assume that the reverse of a pruning sequence for $G$ has been computed as in~\cite{bm}, for example applying the $O(m)$ algorithm in~\cite{damiand}. Hence, we know that $G$ can be built starting from a sigle vertex $v_1$, and adding a sequence of pending vertices and twin vertices $v_2, v_3, \ldots, v_n$. For $1 \leqslant i \leqslant n$, we denote by $G_i$ the subgraph of $G$ induced by $v_1, v_2, \ldots, v_i$.
The neighborhood of a vertex $v_j$ in $G_i$, for $j \leqslant i$, is denoted by $N_i(v_j)$, and the degree of $v_j$ in $G_i$ is denoted by $\deg_i(v_j) = |N_i(v_j)|$.
The number of maximal bicliques in $G_i$ containing vertex $v$ is denoted by $b_i(v)$.
Actually, the reverse of the pruning sequence in \cite{damiand,hammer} is defined for (not necessarily bipartite) distance hereditary graphs, and consists in a sequence $S = [s_2, s_3, \ldots, s_{n}]$ of triples, where $s_i = (v_i, C_i, v_k)$, and the value $C_i \in \{P,F,T\}$, for $k < i$, specifies whether $v_i$ is a pending vertex ($P$) of $v_k$ in $G_i$, or $v_i$ is a false twin ($F$) of $v_k$ in $G_i$, or $v_i$ is a true twin ($T$) of $v_k$ in $G_i$. In the case of bipartite distance hereditary graphs, the pruning sequence contains only pending vertices and false twins.
So, for $2 \leqslant i \leqslant n$, either $|N_i(v_i)| = 1$ (i.e., $v_i$ is a pending vertex in $G_i$), or a vertex $v_k$ exists, with $k<i$, such that $N_i(v_i) = N_i(v_k)$ (i.e., $v_i$ is a false twin of $v_k$ in $G_i$).
\section{Incremental construction of the Galois lattice of a BDH graph}
We can now describe how the Galois lattice of a BDH graph evolves during the Bandelt and Mulder construction. The following result holds for general bipartite graphs.
\begin{theorem}\label{th:induction}
If a bipartite graph $G_{i}=(X_i,Y_i,E_i)$ is obtained from a bipartite graph $G_{i-1}$ by adding a twin of an existing vertex or a pending vertex, then either $\mathbf{H}^\circ(G_{i})$ is isomorphic to $\mathbf{H}^\circ(G_{i-1})$ or $\mathbf{H}^\circ(G_i)$ is obtained from $\mathbf{H}^\circ(G_{i-1})$ by adding a pending vertex.
\end{theorem}
\begin{proof}
We distinguish two cases, depending on the added vertex $v_i$ being a pending vertex or a twin vertex. Let us initially assume $v_i \in X_i$.
\begin{description}
\item[$v_i$ is a twin vertex:] let $v_i$ be a twin of $v_k$ in $G_{i}$, with $k < i$. Since $N_i(v_i) = N_i(v_k)$, for each maximal biclique $(X',Y')$ in $G_{i-1}$, with $v_k \in X'$, there is a maximal biclique $(X' \cup \{v_i\},Y')$ in $G_{i}$.
Maximal bicliques in $G_{i-1}$ not containing $v_k$ remain unchanged in $G_i$. This changes do not alter the order relation among bicliques. Hence, $\mathbf{H}^\circ(G_{i})$ is isomorphic to $\mathbf{H}^\circ(G_{i-1})$.
\item[$v_i$ is a pending vertex:] let $v_i$ be a pending vertex of $v_j$, so $v_j \in Y$ and $N_{i}(v_j)=N_{i-1}(v_j) \cup \{v_i\}$. The only maximal biclique in $G_i$ containing $v_i$ is $(N_{i}(v_j), \{v_j\})$.
We distinguish two cases: either $(N_{i-1}(v_j), \{v_j\})$ is a maximal biclique in $G_{i-1}$, or not.
\begin{itemize}
\item $(N_{i-1}(v_j), \{v_j\})$ is a maximal biclique in $G_{i-1}$: the maximal biclique $(N_{i-1}(v_j), \{v_j\})$ in $\LLC(G_{i-1})$ is replaced in $\LLC(G_{i})$ by the maximal biclique $(N_{i-1}(v_j) \cup \{v_i\}, \{v_j\})$. So, no new maximal bicliques are created and the order relation among existing maximal bicliques is unchanged. Hence, $\mathbf{H}^\circ(G_{i})$ is isomorphic to $\mathbf{H}^\circ(G_{i-1})$.
\item $(N_{i-1}(v_j), \{v_j\})$ is not a maximal biclique in $G_{i-1}$: $\LLC(G_i)$ is obtained from $\LLC(G_{i-1})$ by adding the maximal biclique $B = (N_{i-1}(v_j) \cup \{v_i\}, \{v_j\})$ and a cover pair $B' \dot\prec B$, where $B'$ is the greatest maximal biclique containing $v_j$ in $\LLC(G_{i-1})$. The new biclique $B$ is the introducer of $v_j$ in $\LLC(G_{i})$, while $B'$ is the introducer of $v_j$ in $\LLC(G_{i-1})$. It is immediate to see that $B' \dot\prec B$ is the only new cover pair in $\LLC(G_i)$ with respect to $\LLC(G_{i-1})$. Hence, $\mathbf{H}^\circ(G_{i})$ is obtained by adding a pending vertex to $\mathbf{H}^\circ(G_{i-1})$, which is a maximal element in $\LLC(G_i)$.
\end{itemize}
\end{description}
\noindent
In case $v_i \in Y_i$, the only differences in the above arguments consist in swapping the shores in the maximal bicliques, and, as for the latter case, in adding a new cover pair $B \dot\prec B'$ (instead of $B' \dot\prec B$)---thus a new minimal element is added to $\LLC(G_i)$ instead of a new maximal element.
\end{proof}
Theorem~\ref{th:induction} provides the induction step to prove that the Galois lattice of a BDH graph is a tree.
\begin{corollary}\label{co:isatree}
If $G$ is a BDH graph, then $\mathbf{H}^\circ(G)$ is a tree.
\end{corollary}
\begin{proof}
Any BDH graph $G$ can be built by a sequence of vertex additions as in Theorem~\ref{th:induction}, starting from a single vertex. Graph $G_3$ is isomorphic to $K_{1,2}$, hence $G_3$ contains only one maximal biclique, and $\mathbf{H}^\circ(G_3)$ is a tree consisting in a single vertex. After each addition of pending vertices or twins of existing vertices, $\mathbf{H}^\circ(G_{i-1})$ is turned into $\mathbf{H}^\circ(G_{i})$ which is either isomorphic to $\mathbf{H}^\circ(G_{i-1})$ or is obtained from $\mathbf{H}^\circ(G_{i-1})$ by the addition of a pending vertex by Theorem~\ref{th:induction}. Therefore $\mathbf{H}^\circ(G_{i})$ is either the same tree as $\mathbf{H}^\circ(G_{i-1})$ or it is obtained from a tree by adding a pending vertex. Since adding a pending vertex to a tree always results in a tree the thesis follows.
\end{proof}
Note that Corollary~\ref{co:isatree} provides a simpler and constructive proof of the \emph{if part} of Theorem 1 in \cite{acfDAM}.
\section{Bounding the size of the Galois lattice}
Another consequence of Theorem~\ref{th:induction} is that BDH graphs have few maximal bicliques.
\begin{corollary}\label{co:linearbiclique}
The number of maximal bicliques in a BDH graph on $n$ vertices is at most $n - 2$.
\end{corollary}
\begin{proof}
Graph $G_3$ contains only one maximal biclique, since it is isomorphic to $K_{1,2}$.
By Theorem~\ref{th:induction}, the number of maximal bicliques in $G_i$ is at most the number of maximal bicliques in $G_{i-1}$ plus one, for $4 \leqslant i \leqslant n$.
\end{proof}
Corollary~\ref{co:linearbiclique} shows that the number of maximal bicliques of a BDH graph with $n$ vertices and $m$ edges is always smaller than $n$.\ The class of BDH graphs is the intersection of bipartite domino-free graphs and bipartite $C_{2k}$-free graphs, $k\geqslant 3$, namely \emph{chordal bipartite graphs}. The best known bound for the number of maximal bicliques for both classes is $O(m)$ (see~\cite{Amil,KloKra}).
Moreover, we can bound the number of maximal bicliques containing a given vertex.
\begin{theorem}\label{th:grado}
Each vertex $v$ of a BDH graph $G$ is contained in at most $2 \cdot \deg(v) - 1$ maximal bicliques, where $\deg(v)$ is the degree of $v$ in $G$.
\end{theorem}
\begin{proof}
Assume without loss of generality that $v \in X$, and let $\mathcal{Y}_v=\left(Y(B) \ |\ v\in X(B),\, B\in \maxbic\right)$ be the family of the $Y$-shores of the maximal bicliques containing $v$. Each member of $\mathcal{Y}_v$ is a subset of $N(v)$. We show that $\mathcal{Y}_v$ is a \emph{laminar family}, namely it has the property that for any two members $Y_1$ and $Y_2$ either such two members are disjoint or one is included in the other. Since it is well-known (see \cite[Chapter 2.2]{korte}) that a laminar family consisting of subsets of a common ground set of $k$ elements contains at most $2k-1$ sets, the thesis follows once we prove that $\mathcal{Y}_v$ is indeed laminar. Suppose to the contrary that there are $Y_1$ and $Y_2$ in $\mathcal{Y}_v$ such that all the following conditions hold: $Y_1 \cap Y_2 \not= \emptyset$, $Y_1 \not\subseteq Y_2$ and $Y_2 \not\subseteq Y_1$. Hence there are maximal bicliques $B_0$, $B_1$, $B_2$ and $B_3$ such that $Y(B_1)=Y_1$, $Y(B_2)=Y_2$, $B_0\leqslant B_i$, $i=1,2$ and $B_i\leqslant B_3$, $i=1,2$: just choose for $B_0$ the introducer of $v$ and for $B_3$ the smallest maximal biclique such that $Y(B_3)\supseteq Y_1\cap Y_2$. If one picks $v_1 \in X_1 \setminus X_2$, $v_2 \in X_2 \setminus X_1$, $w \in Y_1 \cap Y_2$, $w_1 \in Y_1 \setminus Y_2$, $w_2 \in Y_2 \setminus Y_1$, then $\{v,v_1,v_2,w,w_1,w_2\}$ induces a domino in $G$, contradicting that $G$ is a BDH graph.
\end{proof}
In view of Theorem~\ref{th:grado}, we can bound the total size of the Galois lattice of a BDH graph $G$, i.e., the sum of the number of vertices in each maximal biclique in $\LL(G)$.
\begin{corollary}\label{co:size}
The Galois lattice of a BDH graph $G$ has total size $O(m)$, where $m$ is the number of edges in $G$.\end{corollary}
\begin{proof}
Let $X$ and $Y$ be the shores of $G$. By Theorem~\ref{th:grado}, each vertex $v$ in $X$ appears in at most $2\cdot\deg_G(v)-1$ maximal bicliques. Therefore $\sum_{B \in \B(G)} |X(B)| \leqslant 2m-n$. Analogously, $\sum_{B \in \B(G)} |Y(B)| \leqslant 2m-n$.
\end{proof}
\section{An $O(m)$ algorithm for computing the Galois lattice of a BDH graph}
The computation of the Galois lattice of a BDH graph $G$ starts from the reverse $v_1, v_2, \ldots, v_n$ of a pruning sequence for $G$ (following the terminology in \cite{damiand,hammer}). The pruning sequence can be computed in $O(m)$ time for general (not necessarily bipartite) distance hereditary graphs, as shown by Damiand \emph{et al.} in \cite{damiand}, where the authors provide a fix for a previous algorithm presented by Hammer and Maffray \cite{hammer}.
The basic ideas to compute the Hasse diagram $\mathbf{H}^\circ(G)$ of $\LLC(G)$, starting from the pruning sequence, are given in the proof of Theorem~\ref{th:induction}. We describe here how that approach leads to an $O(m)$ time algorithm.
Note that, thanks to Corollaries~\ref{co:isatree} and \ref{co:size}, an explicit description of $\mathbf{H}^\circ(G)$, containing an exhaustive listing of the set of vertices in each maximal biclique and all cover pairs, can be given in $O(m)$ space.
The algorithm is incremental, i.e., for each $1\leqslant i \leqslant n$, the Hasse diagram $\mathbf{H}^\circ(G_i)$ of the graph $G_i$ induced by $v_1, v_2, \ldots, v_i$ is computed by updating $\mathbf{H}^\circ(G_{i-1})$. Note that each $G_i$ is a BDH graph as well.
Our algorithm also computes, for each vertex $v \in X$ (resp., $v \in Y$), its introducer $\introd(v)$, i.e., the lowest (resp., highest) maximal biclique in $\mathbf{H}^\circ(G)$ containing $v$. So, it is possible to retrieve all the $p$ maximal bicliques containing $v$ in time $O(p)$, by means of a simple upwards (resp., downwards) traversal in the tree-like $\mathbf{H}^\circ(G)$ starting from $\introd(v)$.
The algorithm is shown in Figure~\ref{fi:algom}.
For each maximal biclique $B=(X(B), Y(B))$ in $\mathbf{H}^\circ(G_i)$, we maintain the following information:
\begin{itemize}
\item the list of vertices in $X(B)$;
\item the list of vertices in $Y(B)$;
\item the list of maximal bicliques covered by $B$ in $\mathbf{H}^\circ(G_i)$;
\item the list of maximal bicliques that cover $B$ in $\mathbf{H}^\circ(G_i)$.
\end{itemize}
Moreover, for each vertex $v$ in $G_i$ we store a reference to $\introd(v)$ in $\mathbf{H}^\circ(G_i)$.
In case a twin vertex $v_i$ of $v_k$ is added, $v_i$ behaves exactly in the same way as $v_k$, so we just add $v_i$ to all the maximal bicliques containing $v_k$. In order to retrieve all these maximal bicliques we start from $\introd(v_k)$ and follow all upward arcs (if $v_i \in X$) or all downward arcs (if $v_i \in Y$). We also set $\introd(v_i)$ to $\introd(v_k)$.
In case a pending vertex $v_i$ of $v_k$ is added, then two cases may occur, depending on $B = (N_i(v_k), \{v_k\})$ (resp., $B =(\{v_k\}, N_i(v_k))$ if $v_i \in Y$) being a maximal biclique in $\mathbf{H}^\circ(G_{i-1})$ or not.
In case $B$ is a maximal biclique in $\mathbf{H}^\circ(G_{i-1})$, then we just add $v_i$ to $B$, and set $\introd (v_i)$ to $\introd(v_k)$.
Otherwise, if $B$ is not a maximal biclique in $\mathbf{H}^\circ(G_{i-1})$, then $\mathbf{H}^\circ(G_i)$ contains one more maximal biclique with respect to $\mathbf{H}^\circ(G_{i-1})$, the biclique $(N_i(v_k), \{v_k\})$ (or $(\{v_k\}, N_i(v_k))$), which also is the introducer of $v_i$ and $v_k$ in $\mathbf{H}^\circ(G_i)$.
\begin{figure}[ht]
\noindent\hrulefill%
\begin{prog}{pr:simplehasse}
Given:\\
-- a BDH graph $G$,\\
-- the reverse of a pruning sequence for $G$ $(v_2, C_2, v_{k_2}), (v_3, C_3, v_{k_3}), \ldots, (v_n, C_n, v_{k_n})$, where $C_i \in \{P, T\}$,\\
compute $\mathbf{H}^\circ(G)$.\\
W.l.o.g., we assume $v_1 \in X$ and $v_2 \in Y$\\
\vspace{0.3cm}\\
\refstepcounter{linecount}\thelinecount. \> $H \leftarrow$ a single biclique $(\{v_1\},\{v_2\})$\\
\refstepcounter{linecount}\thelinecount. \> $\introd(v_1) \leftarrow (\{v_1\},\{v_2\})$\\
\refstepcounter{linecount}\thelinecount. \> $\introd(v_2) \leftarrow (\{v_1\},\{v_2\})$\\
\\
\refstepcounter{linecount}\thelinecount. \> \key{for}$i=3$ to $n$\\
/*\\
we assume $v_i \in X$. Changes in case $v_i \in Y$ are straightforward,\\
except the change in Line \ref{li:x1} (see Line~\ref{li:y1})\\
*/\\
\refstepcounter{linecount}\thelinecount. \> \> \key{if}$v_i$ is a twin vertex in $G_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{let}$v_k$ be the twin vertex of $v_i$ in $G_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{let}$B = \introd(v_k)$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{add}$v_i$ to $X(B)$\\
\NL{li:traversal} \> \> \key{for each}maximal biclique $B'$ in $H$ such that $B \prec B'$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> \key{add}$v_i$ to $X(B')$\\
\refstepcounter{linecount}\thelinecount. \> \> \> $\introd(v_i) \leftarrow \introd(v_k)$\\
\refstepcounter{linecount}\thelinecount. \> \> \key{else} /* $v_i$ is a pending vertex in $G_i$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{let}$v_k$ be the vertex $v_{k_i}$ adjacent to $v_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{let}$B = \introd(v_k)$\\
\NL{li:test} \> \> \key{if}$B = (N_{i-1}(v_k), \{v_k\})$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> \key{add}$v_i$ to $X(B)$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\introd(v_i) \leftarrow \introd(v_k)$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{else} /* $\introd(v_k) \not= (N_{i-1}(v_k), \{v_k\})$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> \key{create}a new maximal biclique $B' = (N_i(v_k), \{v_k\})$\\
\NL{li:x1} \> \> \> \key{add}$B'$ to $H$ so that $B \prec B'$\\
\NL{li:y1} \> \> \> /* in case $v_i \in Y$: \key{add}$B'$ to $H$ so that $B' \prec B$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\introd(v_k) = B'$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\introd(v_i) = B'$\\
\refstepcounter{linecount}\thelinecount. \> \key{end for}\\
\refstepcounter{linecount}\thelinecount. \> \key{return}$H$\\
\end{prog}
\caption{Algorithm \texttt{ComputeBDHDiagram}.}
\protect\label{fi:algom}
\noindent\hrulefill%
\end{figure}
\begin{theorem}
Algorithm \texttt{ComputeBDHDiagram}\ requires $O(m)$ worst case time.
\end{theorem}
\begin{proof}
The test in Line~\ref{li:test} can be performed in constant time, by just checking whether the appropriate
shore in $B$ has size one.
The loop in Line~\ref{li:traversal} is performed by a simple traversal of $H$, requiring overall linear time in the number of maximal bicliques in which $v_i$ must be added, since the traversed portion of $H$ is a tree. Thus, the overall time complexity is given by the number of vertices that are added to each maximal biclique, which is $O(m)$ by Corollary~\ref{co:size}.
\end{proof}
\section{Compact representation of neighborhoods and maximal bicliques}\label{se:arbo}
A BDH graph may contain up to $\Theta(n^2)$ edges---for example, any complete bipartite graph is a BDH graph. Anyway, the neighborhood of each vertex can be conveniently encoded by a compact representation. The following theorem, proved in~\cite{acfDAM}, shows that a BDH graph always is the incidence graph of arcs of an arborescence and a set of paths in the arborescence.
\begin{theorem}[see Apollonio \emph{et al.} \cite{acfDAM}, Theorem 5]\label{th:arbo}
Let $G$ be a BDH graph with shores $X$ and $Y$. There exist arborescences $T_X$ and $T_Y$ and bijections $\psi_X: X\rightarrow E(T_X)$ and $\psi_Y:Y\rightarrow E(T_Y)$ such that $\psi_X(N(y))$ is a directed path in $T_X$ for each $y \in Y$ and $\psi_Y(N(x))$ is a directed path in $T_Y$ for each $x \in X$.
\end{theorem}
By Theorem~\ref{th:arbo}, it is possible to store a pair of arborescences $T_X$ and $T_Y$ so that the neighborhood of each vertex in $X$ (resp., $Y$) is implicitly represented by the two extremes of a dipath in $T_Y$ (resp., $T_X$). Such an implicit representation still requires overall $O(n)$ worst case space and
generalizes to an arborescence what is possible on a path for \emph{convex bipartite graphs}, namely, bipartite graphs $G=(X,Y,E)$ for which a linear order $L$ on $Y$ exists such that $N(x)$ is an interval in $L$ for each $x \in X$. For convex bipartite graphs, each neighborhood can be implicitly represented by the extremes of the corresponding interval in $L$. Nonetheless, BDH graphs are not convex, neither they are $c$-convex (in the sense of~\cite{albano}) in general, even for small $c \in \mathbb{N}$.
\mybreak
The pair of arborescences mentioned in Theorem~\ref{th:arbo}, along with the corresponding bijections, can be computed by specializing the algorithm of Swaminathan and Wagner~\cite{SwaWA} that runs Bixby and Wagner's
algorithm~\cite{BixWA} for the \emph{Graph Realization Problem}---roughly: the recognition problem for graphic matroids--as a subroutine. The ensuing running time is $O(\alpha(\nu, m)\times m)$ where $\nu$ is the number of vertices in shore $Z$, $Z\in \{X,Y\}$, while $m$ is the number of edges of $G$ and
$\alpha(\cdot,\cdot)$ is a function which grows very slowly and behaves essentially as a constant even for
large values of both its arguments.
\mybreak
We give here a simpler constructive proof of Theorem~\ref{th:arbo}, that also provides a much simpler and more efficient (though much less general) algorithm to compute the arborescence representation.
\vspace{2mm}
\begin{proof} \textbf{(of Theorem~\ref{th:arbo}, constructive)}
Since the role of the shores of $G$ is symmetrical, it suffices to prove the existence of an arborescence $T_Y$ and a bijection $\psi_Y$ fulfilling the thesis.\ The proof is carried out by induction on graphs $G_i$'s in the Bandelt and Mulder construction sequence of $G$. Let $X_i$ and $Y_i$, be the shores of $G_i$. For ease of notation we set $T_i=T_{Y_i}$ and $\psi=\psi_{Y_i}$. Hence we identify $\psi_{Y_i}$ with the restriction of $\psi$ on the vertices of $Y_i$.
We assume, w.l.o.g., that $v_1 \in X$ and $v_2 \in Y$.
\mybreak
Graph $G_2$ is necessarily isomorphic to $K_{1,1}$.\ Thus the thesis trivially holds for $G_2$: $T_2$ consists of a single arc $e$, with $\psi(v_2) = e$.\ The neighborhood $Y_2$ of the unique vertex in $X_2$ is mapped into a path consisting of the unique arc $e \in E(T_2)$.
\mybreak
Assume the thesis holds for $G_{i-1}$, with $i > 2$. The neighborhood $N_{i-1}(v_k)$, for each $v_k \in X_{i-1}$, is mapped by $\psi$ into a dipath in $T_{i-1}$.
When adding vertex $v_i$ we distinguish four cases, since $v_i$ can be added either to shore $X$ or to shore
$Y$, and it can be either a pending vertex or a twin vertex of an existing vertex.
\begin{enumerate}[(i)]
\item $v_{i}$ is a twin vertex in shore $X$: let $v_j\in X$ be a twin of $v_i$. The arborescence is unchanged. Since $N_i(v_{i}) = N_i(v_j) = N_{i-1}(v_j)$, and $\psi(N_{i-1}(v_j))$ was a path in $T_{i-1}$, then $\psi (N_i(v_i))$ is a path in $T_i$.
\item\label{case:twinY} $v_{i}$ is a twin vertex in shore $Y$: let $v_j\in y$ be a twin of $v_i$. We subdivide arc $\psi(v_j)$ into two consecutive arcs $\psi(v_i)$ and $\psi(v_j)$, by adding a new vertex to the arborescence. If $v_j \in N_i(x)$ for some $x \in X_{i}$, then also $v_i \in N_i(x)$, hence $\psi(N_i(x))$ contains both arc $\psi(v_j)$ and arc $\psi(v_i)$. Any path containing $\psi(v_j)$ is thus extended to a path containing $\psi(v_i)$. Therefore, $\psi(N_i(x))$ is still a path in $T_i$, for each $x \in X_i$.
\item $v_{i}$ is a pending vertex in shore $X$: the arborescence is unchanged. The neighborhood $N_i(v_i)$ is a single vertex, so $\psi (N_i(v_i))$ is a path consisting of a single arc.
\item\label{case:pendingY} $v_{i}$ is a pending vertex in shore $Y$: let $N_i(v_{i}) = \{v_j\}$. Only the neighborhood of $v_j$ is changed, with $N_i(v_j) = N_{i-1}(v_j) \cup \{v_i\}$. We add a new vertex and a new arc $\psi(v_i) = e$ to the arborescence, so that arc $e$ is adjacent to the last arc in the path $\psi(N_{i-1}(v_j))$. Since $\psi(N_{i-1}(v_k))$ is a path in $T_{i-1}$, for $v_k \in X_{i-1} \setminus \{v_j\}$, then $\psi(N_i(v_k))$ is a path in $T_i$. Moreover, $\psi(N_{i-1}(v_j))$ is a path in $T_{i-1}$ as well; hence $\psi(N_i(v_j))$ is a path in $T_i$, consisting of the concatenation of $\psi(N_{i-1}(v_j))$ and $e$.
\end{enumerate}
The only cases in which arcs are added to $T_{i-1}$ are~(\ref{case:twinY}) and~(\ref{case:pendingY}). It is immediate to see that, in both cases, if $T_{i-1}$ is an arborescence then also $T_i$ is an arborescence.
\end{proof}
An example of the above construction is shown in Figure~\ref{fig:arborescence}.
\begin{figure}[h]
\noindent\hrulefill%
\begin{center}
\def15cm{15cm}
\input{arborescenceUpward.pdf_tex}
\caption{A BDH graph and its supporting arborescence $T_Y$, where $Y$ is the shore on the right. The arborescence $T_Y$ is obtained incrementally under the addition in the graph of pending vertices and twin vertices $v_1, v_2, \ldots, v_{14}$, as described in the proof of Theorem \ref{th:arbo}.\comment{, in the order $A, 1, 2, B, C, 3, D, 4, E, F, 5, 6, 7, 8$} Labels $\pend(v)$ and $\twin(v)$ in the graph denote the insertion of a pending vertex adjacent to $v$ or a twin vertex of $v$. Arc $\psi(v_i)$ in the arborescence is labeled by $e_i$. Dashed arcs are the arcs added to each arborescence. Observe that the neighborhood of each vertex in $Y$ is mapped to a dipath in $T_Y$. For example, $N(v_7)$ is mapped to the dipath from $\alpha(N(v_7)) = e_{2}$ to $\beta(N(v_7)) = e_{13}$, while $N(v_4)$ is mapped to the dipath from $\alpha(N(v_4)) = e_{6}$ to $\beta(N(v_4)) = e_{8}$.}
\protect\label{fig:arborescence}
\end{center}
\noindent\hrulefill%
\end{figure}
\mybreak
Since each shore of a maximal biclique is an intersection of neighborhoods, and the intersection of dipaths in an arborescence is itself a dipath, we can encode each maximal biclique $(X(B),Y(B))$ of a BDH graph by means of a dipath in $T_X$ and a dipath in $T_Y$. It is therefore convenient to introduce a unique map $\psi:X \cup Y\rightarrow E(T_X) \cup E(T_Y)$ as follows: $\psi(v) = \psi_X(v)$ if $v \in X$ and $\psi(v) = \psi_Y(v)$ if $v \in Y$. The following fact follows now straightforwardly.
\begin{corollary}\label{co:arbo}
Given a BDH graph $G$ with shores $X$ and $Y$, there exist two arborescences $T_X$ and $T_Y$, and a bijection $\psi:X \cup Y\rightarrow E(T_X) \cup E(T_Y)$, such that for each maximal biclique $B\in \LLC(G)$ we have that $\psi(X(B))$ is a directed path in $T_X$ and $\psi(Y(B))$ is a directed path in $T_Y$.
\end{corollary}
Let $G$ a BDH graph with shores $X$ and $Y$ and let $S\in\left(N(x),\, x\in X\right)\cup\left(N(y),\, y\in Y\right)$. Let $T\in \{T_X,T_Y\}$. Dipaths in $T$ are identified by intervals in the arborescence order induced by $T$ (recall Section~\ref{sec:prelim}).\ In particular, dipath $\psi(S)$ of $T$ is identified with $[\alpha(\psi(S)),\beta(\psi(S))]$.\ For ease of notation we set $\alpha(\psi(S))=\alpha(S)$ and $\beta(\psi(S))=\beta(S)$.\ Hence, a maximal biclique $B$ will be encoded by the two intervals $[\alpha(X(B)), \beta(X(B))]$ in $T_X$ and $[\alpha(Y(B)), \beta(Y(B))]$ in $T_Y$.
\section{An $O(n)$ time algorithm for computing a compact representation of the Galois lattice of a BDH graph}
The arborescence representation described in Theorem~\ref{th:arbo} and Corollary~\ref{co:arbo}, together with the $O(n)$ upper bound in Corollary~\ref{co:linearbiclique} on the number of
maximal bicliques in $\LLC(G)$, allows us to derive an $O(n)$ space encoding of the Galois lattice of a BDH graph. We show here how this encoding can be computed in $O(n)$ worst case time. An exhaustive listing of the $k$ vertices in each maximal biclique can still be obtained in optimal $O(k)$ time by traversing the compact representation.
Algorithm \texttt{FastComputeBDHDiagram}\ is listed in Figure~\ref{fi:fastcomputehasse}. Starting from the reverse of a pruning sequence of a BDH graph $G$, it computes the two supporting arborescences $T_X, T_Y$
in Theorem~\ref{th:arbo} and an implicit representation of $\mathbf{H}^\circ(G)$.
Each maximal biclique $B=(X(B), Y(B))$ in $\mathbf{H}^\circ(G)$ is implicitly represented by the two intervals
$[\alpha(X(B)), \beta(X(B))]$ and $[\alpha(Y(B)), \beta(Y(B))]$. The list of the $k$ arcs in an interval can be retrieved in $O(k)$ time by a simple walk in the arborescence, starting from $\beta(\cdot)$ and following parent pointers to $\alpha(\cdot)$.
Thus, the Hasse diagram of the Galois lattice can be represented in $O(n)$ space, also including the two arborescences needed to list vertices in maximal bicliques when required.
The algorithm we propose also computes, for each vertex $v \in X$ (resp., $v \in Y$), its introducer. This allows us to retrieve all the $p$ maximal bicliques containing $v$ in time $O(p)$.
Algorithm \texttt{FastComputeBDHDiagram}\ follows the same steps as Algorithm \texttt{ComputeBDHDiagram}\ but, when a vertex in the reverse of the pruning sequence is processed, in addition to updating $\mathbf{H}^\circ(G_{i-1})$ to $\mathbf{H}^\circ(G_i)$, also the two arborescences $T_X$ and $T_Y$ are updated according to the proof of Theorem~\ref{th:arbo}.
For each maximal biclique $B=(X(B), Y(B))$ in $\mathbf{H}^\circ(G_i)$, we maintain the following information:
\begin{itemize}
\item the set of vertices in $X(B)$, represented through the end-arcs $\alpha(X(B)),\beta(X(B))$ of the associated dipath in $T_X$;
\item the set of vertices in $Y(B)$, represented through the end-arcs $\alpha(Y(B)),\beta(Y(B))$ of the associated dipath in $T_Y$;
\item the list of maximal bicliques covered by $B$ in $\mathbf{H}^\circ(G_i)$;
\item the list of maximal bicliques that cover $B$ in $\mathbf{H}^\circ(G_i)$.
\end{itemize}
Moreover, for each vertex $v_k$ in $G_i$, we store a reference to $\introd(v_k)$ in $\mathbf{H}^\circ(G_i)$.
In the algorithm we only show how to process pending vertices and twin vertices in $X$, the algorithm and the data structures being completely symmetric with respect to swapping shore $X$ for shore $Y$.
\begin{figure}[ht]
\noindent\hrulefill%
\begin{prog}{pr:fasthasse}
given a BDH graph $G$ and the reverse of a pruning sequence\\
for $G$ $[(v_2, C_2, v_{k_2}), (v_3, C_3, v_{k_3}), \ldots, (v_n, C_n, v_{k_n})],$\\
compute:\\
-- $\mathbf{H}^\circ(G)$\\
-- the arborescences $T_X$ and $T_Y$ representing vertices in $X$ and $Y$ as in Theorem~\ref{th:arbo}\\
~~~in which $\psi(v_i)$ is the arc denoted by $e_i$\\
\vspace{0.3cm}\\
for each vertex $v$ in $G_i$ we maintain a reference to $\introd(v)$ in $\mathbf{H}^\circ(G_i)$;\\
for each maximal biclique $B = (X(B), Y(B)) \in \mathbf{H}^\circ(G_i)$ we maintain:\\
-- the list of maximal bicliques covered by $B$ in $\mathbf{H}^\circ(G_i)$;\\
-- the list of maximal bicliques that cover $B$ in $\mathbf{H}^\circ(G_i)$;\\
-- the end-arcs $\alpha(X(B)), \beta(X(B))$ of the dipath in $T_X$ representing $X(B)$;\\
-- the end-arcs $\alpha(Y(B)), \beta(Y(B))$ of the dipath in $T_Y$ representing $Y(B)$.\\
\vspace{0.3cm}\\
/* w.l.o.g., we assume $v_1 \in X$ and $v_2 \in Y$ */\\
\refstepcounter{linecount}\thelinecount. \> $H \leftarrow$ a single biclique $B = (\{v_1\},\{v_2\})$\\
\refstepcounter{linecount}\thelinecount. \> $\introd(v_1) \leftarrow B$\\
\refstepcounter{linecount}\thelinecount. \> $\introd(v_2) \leftarrow B$\\
\refstepcounter{linecount}\thelinecount. \> $T_X \leftarrow \mbox{a single arc } e_1$\\
\refstepcounter{linecount}\thelinecount. \> $T_Y \leftarrow \mbox{a single arc } e_2$\\
\refstepcounter{linecount}\thelinecount. \> $\alpha(X(B)) \leftarrow e_1$; $\beta(X(B)) \leftarrow e_1$\\
\refstepcounter{linecount}\thelinecount. \> $\alpha(Y(B)) \leftarrow e_2$; $\beta(Y(B)) \leftarrow e_2$\\
\\
\refstepcounter{linecount}\thelinecount. \> \key{for}$i=3$ to $n$\\
/*\\
for the sake of simplicity we assume $v_i \in X$\\
changes in case $v_i \in Y$ are straightforward, except the change in Line \ref{li:x} (see Line~\ref{li:y})\\
*/\\
\refstepcounter{linecount}\thelinecount. \> \> \key{if}$v_i$ is a twin vertex in $G_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{let}$v_k$ be the twin vertex of $v_i$ in $G_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{update}$T_X$ by splitting $e_k$ into two arcs $e_i, e_k$, with $e_i \prec_{T_X} e_k$\\
\NL{li:inizioblocco} \> \> \key{for each}maximal biclique $B'=(X(B'), Y(B'))$ in $H$ with $\alpha(X(B')) = e_k$\\
\NL{li:fineblocco} \> \> \> $\alpha(X(B')) \leftarrow e_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> $\introd(v_i) \leftarrow \introd(v_k)$\\
\refstepcounter{linecount}\thelinecount. \> \> \key{else} /* $v_i$ is a pending vertex in $G_i$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{let}$v_k$ be the vertex adjacent to $v_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{append}a new arc $e_i$ in $T_X$ as a leaf above $\beta(X(\introd(v_k)))$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{if} $\introd(v_k) = (N_{i-1}(v_k), v_k)$ /* i.e., $\alpha(Y(\introd(v_k))) = \beta(Y(\introd(v_k)))$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\beta(X(\introd(v_k))) \leftarrow e_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\introd(v_i) \leftarrow \introd(v_k)$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \key{else} /* $\introd(v_k) \not= (N_{i-1}(v_k), v_k)$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> \key{create}a new maximal biclique $B'$\\
\NL{li:x} \> \> \> \key{add}$B'$ to $H$ as a leaf so that $\introd(v_k) \prec B'$\\
\NL{li:y} \> \> \> /* in case $v_i \in Y$: \key{add}a leaf $(v_k,N_{i}(v_k))$ so that $(v_k,N_{i}(v_k)) \prec \introd(v_k)$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\alpha(X(B')) \leftarrow \alpha(X(\introd(v_k)))$ /* because $B' = (N_{i-1}(v_k) \cup \{v_i\}, v_k)$ */\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\beta(X(B')) \leftarrow e_i$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\alpha(Y(B')) \leftarrow e_k$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\beta(Y(B')) \leftarrow e_k$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\introd(v_k) \leftarrow (N_{i}(v_k), v_k)$\\
\refstepcounter{linecount}\thelinecount. \> \> \> \> $\introd(v_i) \leftarrow (N_{i}(v_k), v_k)$\\
\refstepcounter{linecount}\thelinecount. \> \key{end for}\\
\refstepcounter{linecount}\thelinecount. \> \key{return}$H$\\
\end{prog}
\caption{Algorithm \texttt{FastComputeBDHDiagram}.}
\protect\label{fi:fastcomputehasse}
\noindent\hrulefill%
\end{figure}
\begin{theorem}
Starting from the pruning sequence of a BDH graph $G$ on $n$ vertices, an implicit representation of its Galois lattice can be computed in $O(n)$ worst case time and space. Retrieving the $p$ vertices in each maximal biclique requires $O(p)$ worst case time, and retrieving the $k$ maximal bicliques containing a given vertex requires $O(k)$ worst case time.
\end{theorem}
\begin{proof}
It is immediate to see that each step in Algorithm \texttt{FastComputeBDHDiagram}, except for the loop in Lines~\ref{li:inizioblocco} and \ref{li:fineblocco}, needs constant time per vertex to update the two arborescences $T_X$, $T_Y$. This because the Hasse diagram $\mathbf{H}^\circ(G)$ contains, for each maximal biclique $B=(X', Y')$, its implicit representation $\alpha(X')$, $\beta(X')$, $\alpha(Y')$ and $\beta(Y')$.
Concerning Lines~\ref{li:inizioblocco} and \ref{li:fineblocco}, instead of updating the value of $\alpha(X(B'))$ for each maximal biclique in $H$ with $\alpha(X(B')) = e_k$, we store a single reference to the $\alpha(\cdot)$ value for all maximal bicliques sharing the same value of $\alpha(\cdot)$ (analogously for $\alpha(Y(B'))$), thus the set of updates in Lines~\ref{li:inizioblocco} and \ref{li:fineblocco} can be performed in constant time by just substituting that reference.
The set of vertices in the $X$ shore of a maximal biclique $B$ can be listed by traversing $T_X$ starting from $\beta(X(B))$, following parent pointers, until $\alpha(X(B))$ is reached,
and analogously for the $Y$ shore on $T_Y$. The set of maximal bicliques containing vertex $v \in X$ (resp., $v \in Y$) can be reached by traversing $\mathbf{H}^\circ(G)$ upward (resp., downward) starting from $\introd(v)$. Since $\mathbf{H}^\circ(G)$ is a tree, each maximal biclique is reached only once during the traversal.
\end{proof}
\bibliographystyle{plain}
|
hep-lat/0509096
|
\section{Introduction}
Perturbation theory can be a frustrating tool for field theorists. Sometimes, it provides
extremely accurate answers, sometimes it is not even qualitatively correct. In recent
years, our main goal has been to construct
modified perturbative series which are converging and accurate. As briefly reviewed in Section \ref{sec:large}, our approach consists in removing large field configurations in a way that preserves the closeness to the correct answer.
In the case of quenched $QCD$, there are several questions that are relevant for this approach and that have been addressed. How sensitive is the average plaquette $P$ to a large field cutoff \cite{effects04}? How does $P$ behave when the coupling becomes negative \cite{gluodyn04}? How does $P$ differ from its weak coupling expansion \cite{burgio97,rakow2002}? Are all the derivatives of $P$ with respect to $\beta$ continuous in the crossover region?
The analysis \cite{rakow2002,third} of the weak series for $P$ up to order 10 \cite{direnzo2000} suggests an (unexpected) singularity in the second derivative of $P$,
or in other words in the third derivative of the free energy. In the following, we report our recent attempts to find this singularity. As all the technical details regarding
this question have just appeared in a preprint \cite{third}, we will only summarize the
main results leaving room for more discussion regarding the difference between series and the numerical values of $P$.
\section{Large field configurations and perturbation theory}
\label{sec:large}
The reason why perturbation theory sometimes fail is well understood for scalar field
theory. Large field configurations have little effect on commonly used observables but
are important for the average of large powers of the field and dominate the large order
behavior of perturbative series.
A simple way to remove the large field configurations consists in
restricting the range of integration for the scalar fields.
\begin{equation}
\prod_x \int_{-\phi_{max}}^{\phi_{max}}d\phi_x \ .
\nonumber\end{equation}
For a generic observable $Obs.$ in a $\lambda \phi^4$ theory, we have then
\begin{equation}
Obs.(\lambda )\simeq\sum_{k=0}^{K}a_k(\phi_{max})\lambda^k
\nonumber\end{equation}
The method produces series which apparently converge
in nontrivial cases such as the anharmonic oscillator and $D=3$ Dyson hierarchical model
\cite{convpert,tractable}.
The modified theory with a field cutoff differs from the original theory. Fortunately,
it seems possible, for a fixed order in perturbation theory, to adjust the field cutoff to an optimal value $\phi_{max}(\lambda,K)$ in order to minimize or eliminate the discrepancy with the (usually unknown) correct value of the observable in the original theory. In a simple example\cite{optim},
the strong coupling can be used to calculate approximately this optimal $\phi_{max}(\lambda,K)$. This method provides an approximate treatment of
the weak to strong coupling crossover and we hope it can be extended to gauge theory where this crossover \cite{kogut80} is a difficult problem.
The calculation of the modified coefficients remains a challenge, however approximately universal features of the transition between the small and large field cutoff limits
for the modified coefficients of the anharmonic oscillator \cite{asymp}, suggest the existence of simple analytical
formulas to describe the field cutoff dependence of large orders coefficients.
This method needs to be extended to the case of lattice gauge theories.
Important differences with the scalar case need to be understood. For compact groups such as $SU(N)$,
the gauge fields are not arbitrarily large. Consequently, it is possible to define a
sensible theory at negative $\beta=2N/g^2$. However, the average plaquette
tends to two different values in the two limits $g^2\rightarrow \pm 0$ \cite{gluodyn04}.
This precludes the existence of a regular perturbative series about $g^2=0$.
A first order phase transition near $\beta =-22$, was also observed \cite{gluodyn04} for $SU(3)$.
The impossibility of having a convergent perturbative series about $g^2=0$
is well understood \cite{plaquette} in the case of the partition function
for a single
plaquette which after gauge fixing to the identity on three links reads.
\begin{equation}
Z=\int dU {\rm e} ^{-\beta(1-\frac{1}{N}Re TrU)}\ ,
\end{equation}
If we expand the group element $U=e^{igA}$ with $A=A^aT^a$ and the Haar measure in powers of $g$, we obtain a converging sum that allows us to calculate $Z$ accurately, however, the ``coefficients'' are $g$-dependent.
This comes from the finite bounds of integration of the gauge fields that
are proportional to $1/g$. If $g^2$ is small and positive, we can extend the range of integration to infinity with errors that seem controlled by $\rm{e}^{-2\beta}$.
By ``decompactifying'' the gauge fields, we have transformed a converging sum into a power series in $g$ with constant
coefficients growing factorially with the order.
The situation is now resemblant to the scalar case and can be treated using this analogy.
We can introduce a gauge invariant field cutoff that is treated as a $g$ independent quantity.
For a given order in $g$, one can use the strong coupling expansion to determine
the optimal value of this cutoff. This provides a significant improvement in regions where
neither weak or strong coupling is adequate \cite{plaquette}.
This program can in principle be extended to LGT on $D$-dimensional lattices, however the calculation of the modified coefficients is difficult. An appropriately modified version of the
stochastic method seems to be the most promising for this task.
As the technology for completing this task is being developed, we will discuss several
questions about the average plaquette and its perturbative expansion.
\section{The average plaquette and its perturbative expansion in quenched $QCD$}
We now consider a $SU(3)$ lattice gauge theory in 4 dimensions without quarks (quenched $QCD$). We use the Wilson action without improvement.
Our main object will be the average plaquette action denoted $P$ and can be
expressed as $-\partial (\rm{ln}(Z)/6L^4)/\partial \beta$.
The effect of a gauge invariant field cutoff is very small but of a different size below, near or above $\beta=5.6$ (see Fig. 6 of Ref. \cite{effects04}).
This is in agreement with the idea that modifying the weight of the large
field configurations affects the crossover behavior \cite{mack78}. The weak coupling series for $P$ has been calculated up to order 10 in Ref. \cite{direnzo2000}:
\begin{equation}\nonumber
P_W(1/\beta)=\sum_{m=1}^{10} b_m \beta^{-m} +\dots.
\nonumber\end{equation}
The coefficients are given in table 1.
The values corresponding to the series and the numerical data calculated on a $16^4$ lattice
is shown in Fig. \ref{fig:pade}. A discrepancy becomes visible below $\beta = 6$.
The situation can be improved by using Pad\'e approximants, however, they do not show any change in curvature and often have poles near $\beta=5.2$. For comparison, Pad\'e
approximant for the strong coupling expansion \cite{balian74err} depart visibly from the
numerical values when $\beta$ becomes slightly larger than 5. In conclusion, it is not clear that by combining the two series we can get a complete information regarding the
crossover behavior.
\begin{figure}
\label{fig:pade}
\includegraphics[width=2.8in,angle=0]{wpade2.eps}
\includegraphics[width=2.8in,angle=0]{strpade2.eps}
\caption{Regular weak series (blue) and 4/6 weak Pad\'e (red) for the plaquette (left);
7/7 strong Pad\'e (right) }
\end{figure}
The difference between the weak coupling expansion $P_W$ and the numerical data $P$ can be further analyzed. From the example of the one-plaquette model \cite{plaquette}, one could infer that by adding the tails of integration, we should make errors of order
$\rm{e}^{-C\beta}$, for some constant $C$. Consistently with this argument, the difference
should scale as a power of the lattice spacing, namely
\begin{equation}
P_{Non Pert.}=(P-P_W)\propto a^A \propto \left({\rm e}^{-\frac{4\pi^2}{33}\beta} \right)^A \ .
\end{equation}
A case for
$A=2$ has been made in Ref. \cite{burgio97} based on a series of order 8.
Another analysis supports $A=4$ (the canonical dimension of $F_{\mu \nu}F^{\mu \nu }$) \cite{rakow2002,rakowthese}. Fig. \ref{fig:apower} shows fits at different orders
and in different regions that support each of these possibilities. It would be interesting to study cases where long series are available and non-perturbative effects
well understood in order to define a prescription to extract the power properly.
\begin{figure}
\label{fig:apower}
\includegraphics[width=2.9in,angle=0]{sd8.eps}
\includegraphics[width=2.9in,angle=0]{sd10bis.eps}
\caption{$Log_{10}|P-P_W|$ for order 8 (left) and 10 (right, in a different range of $\beta$); the constant is fitted asumming
$a^2$ (blue) or $a^4$ (red). }
\end{figure}
The series $P_W$ has another intriguing feature:
$r_m=b_m/b_{m-1}$, the ratio of two
successive coefficients seem to extrapolates near 6 when $m\rightarrow\infty$ when $m$ becomes large \cite{rakow2002}. This suggests a behavior of the form
\begin{equation}\nonumber
P=(1/\beta _c -1/\beta )^{-\gamma } (A_0 + A_1 (\beta _c -\beta)^{
\Delta } +....)\ ,
\label{eq:convpar} \nonumber\end{equation}
as encountered in the study of the critical behavior of spin models.
We have reanalyzed \cite{third} the series using estimators \cite{nickel80} known as the
the extrapolated ratio ($\widehat{R}_m$)
and the
extrapolated slope ($\widehat{S}_m$)
in order to estimate $\beta_c$ and $\gamma $. We found that the weak series suggests
\begin{equation}
\label{eq:critical}
P\propto (1/5.74-1/\beta)^{1.08} \ .
\end{equation}
These estimators are sensitive to small variations in the coefficients and show
a remarkable stability when the volume is increased from $8^4$ to $24^4$. The numbers are in good agreement with
the estimates of Ref. \cite{rakow2002} with other methods.
A finite radius of convergence is
not expected and one does not expect any singularity between
the limits where
confinement and asymptotic freedom hold.
It may simply be that the series is too short to draw conclusion about its asymptotic behavior. A simple example where this happens \cite{third} is
\begin{equation}
Q(\beta)=\int_0^{\infty}dt {\rm e}^{-t}t^{\alpha}[1-t\beta_c/(\alpha \beta)]^{-\gamma} \ ,
\end{equation}
with $\alpha$ sufficiently large.
If $m<<\alpha$,
$r_m\simeq \beta_c(1+(\gamma -1)/m), $
For $m>>\alpha$ we have $r_m \propto m$ and the coefficients grow factorially.
If we take Eq. (\ref{eq:critical}) seriously, it implies that the second derivative of $P$ diverges near $\beta =5.7$. We have searched for such a singularity \cite{third}.
We have shown that the peak in the third derivative of the free energy present
on $4^4$ lattices disappears if the size of the lattice is increased isotropically
up to a $10^4$ lattice.
On the other hand, on $4\times L^3$ lattices, a jump in the third
derivative persists when $L$ increases. Its location coincides with the onset of
a non-zero average for the Polyakov loop and seems consequently related to the finite
temperature transition. It should be noted that the possibility of a third-order phase transition
has been discussed for effective theories of the Polyakov's loop \cite{pisarski}.
A few words about the tadpole improvement \cite{lepage92} for the weak series. If we consider the resummation
\begin{equation}
P_W(1/\beta)=\sum_{m=1}^{K} e_m \beta_R^{-m} + O(\beta_R^{-K-1})
\end{equation}
with $\beta_R=\beta (1-\sum_{m=1} b_m \beta^{-m})$,
the ratios $e_{m}/e_{m-1}$ stay close to -1.5 for $m$ up to 7, but seem to start oscillating more for large $m$.
\begin{table}[h]
\begin{tabular}{||c||c|c|c|c|c|c|c|c|c|c||}
\hline
$m$&1&2&3&4&5&6&7&8&9&10\cr
\hline
$b_m$& 2 & 1.2208 & 2.9621 & 9.417 &
34.39 & 136.8 & 577.4 & 2545 &
11590 &54160 \cr
$e_m$&2 & -2.779 &3.637 &-3.961 &4.766 &
-3.881 & 6.822 &
-1.771 & 17.50 &
48.08 \cr
\hline
\end{tabular}
\caption{$b_m$: regular coefficients; $e_m$: tadpole improved coefficients}
\end{table}
This research was supported in part by the Department of Energy
under Contract No. FG02-91ER40664. We thank G. Burgio, F. di Renzo and P. Rakow for interesting discussions.
\providecommand{\href}[2]{#2}\begingroup\raggedright
|
math/0509316
|
\section{Introduction}\label{SecIntro}
In June 2005, Michael Somos \cite{Somos05} observed that
the $12$-th root of the theta series of
Nebe's extremal $3$-modular even lattice
in $24$ dimensions
(\cite{Nebe95}, \cite{Nebe98}, \cite{Quebbemann95},
sequence A004046 in \cite{OEIS})
appeared to have integer coefficients.
This led us to consider analogous questions
for other lattices, and we discovered that
the cube root of the theta series of the $6$-dimensional lattice $E_6$,
the eighth root of the theta series of the $8$-dimensional lattice $E_8$,
and
the $24$th root of the theta series of the $24$-dimensional Leech
lattice $\Lambda _{24}$
also appeared to have integer coefficients.
Although it seemed unlikely (and still seems unlikely!)
that these results were not already known,
they were new to us, and so we considered the following
general question.
Let $\ensuremath{\ZZ[[x]]}$ denote the ring of formal power series in $x$
with integer coefficients, let $\ensuremath{\ZZ[[x]]^{\ast}}$ denote the
subset of $\ensuremath{\ZZ[[x]]}$ with constant term $\pm 1$
(that is, the set of units in $\ensuremath{\ZZ[[x]]}$),
and let $\RT \subseteq \ensuremath{\ZZ[[x]]^{\ast}}$ be the elements with constant term $1$.
If ${\cal{P}}_n$ denotes the set $\{ g^n \mid g \in \RT \}$,
when is a given $f \in \RT $ an element of ${\cal{P}}_n$ with $n \ge 2$?
In Section \ref{SecThms} we give some general conditions which
ensure that a series belongs to ${\cal{P}}_n$.
In Section \ref{SecTheta} we study the theta series of lattices
and establish some general theorems which explain all the
above observations. We also state some conjectures which would provide
converses to these theorems.
Section \ref{SecCodes} deals with the weight enumerators of codes.
Surprisingly (in view of the usual parallels between self-dual codes
and unimodular lattices, cf. \cite{SPLAG}, \cite{Elkies}, \cite{NRS06}),
there do not seem to be any exact analogues of the theorems
for theta series.
We show that the weight enumerator of
the $r$th order Reed-Muller code
of length $2^m$ is in ${{\cal{P}}_{2^r}}$ for $r=0, 1, \ldots, m$,
and make an analogous conjecture for extended BCH codes.
Similarly, we show that the theta series of the Barnes-Wall lattice
in $\mathbb R ^{2^m}$ is in ${{\cal{P}}_{2^m}}$.
In Section \ref{SecSq} we consider the special case of series that
are squares, and report on a search for possible squares
in the {\em On-Line Encyclopedia of Integer Sequences} \cite{OEIS}.
This search led us to Paul Hanna's sequences, which are the subject of the final section.
It is worth mentioning that $\ensuremath{\ZZ[[x]]}$ is known to be a unique
factorization domain \cite{Samuel61},
although we will make no explicit use of this since we are concerned
only with the multiplicative group of units in $\ensuremath{\ZZ[[x]]}$.
\paragraph{Notation:}
If the formal power series $f(x) \in {\cal{P}}_n$ we will say that $f(x)$, or
its sequence of coefficients, is ``an $n$th power''.
For a prime $p$, $|~|_p$ denotes the $p$-adic
valuation ($|0|_p := 0$; if $0 \ne r \in {\mathbb Q} \, , \,
r = p^a \, \frac{b}{c}
\mbox{~with~} a, b, c \in {\mathbb Z}, c \neq 0, \mbox{~and~}
\gcd(p,b) = \gcd(p,c) = 1,
\mbox{~then~} |r|_p := a$).
We will use the facts that
$|r!|_p < r/(p-1)$ for $r>0$,
$|\binom{p^i}{j}|_p = |p^i|_p-|j|_p$
(cf. \cite{Gouvea93}).
\section{ Conditions for $f$ to be an $n$th power }\label{SecThms}
We first show that, for investigating whether $f \in \RT $ is an $n$th power,
it is enough to consider $f$ mod $\mu_n$,
where
$$
\mu_n := n \prod_{p | n} p \, .
$$
\begin{theorem}\label{ThTaylor}
For $f \in \RT , f \in {\cal{P}}_n$
if and only if
$f \pmod{ \mu_n} \in {\cal{P}}_n $.
\end{theorem}
\paragraph{Proof.}
We will show that, for $k \ge 1$,
the coefficients in $f^{1/n}$ are integers if and only if
the coefficients in $(f + \mu_n x^k)^{1/n}$ are integers.
Let $\phi (f) := f^{1/n}$. By Taylor's theorem,
\begin{eqnarray}
\phi(f + \mu_n x^k) & ~=~ & \sum_{r=0}^{\infty} \,
\frac{ (\mu_nx^k)^r}{r!} \, \phi^{(r)}(f) \nonumber \\
& ~=~ & \sum_{r=0}^{\infty} \,
\frac{ (\mu_nx^k)^r}{r!} \, r! \binom{\frac{1}{n}}{r}
f^{ 1/n - r }
\nonumber \\
& ~=~ & f^{1/n} \sum_{r=0}^{\infty} \mu_n^r \binom{\frac{1}{n}}{r}
\frac{ x^{kr}}{f^r} ~. \nonumber
\end{eqnarray}
Let $c := \mu_n^r \binom{\frac{1}{n}}{r}$.
For a prime $p$ dividing $n$,
$|c|_p = r|\mu_n|_p - r|n|_p - |r!|_p \ge 0$,
by definition of $\mu_n$. For a prime
$p$ not dividing $n$, $1/n$ is a $p$-adic unit and again
$|c|_p \ge 0$.
Hence $c \in {\mathbb Z}$.
Since $f \in \RT $, $f^{-r}$ has integer coefficients,
and so
$(f + \mu_n x^k)^{1/n} = f^{1/n}g$ for some $g \in \RT $.
Thus the coefficients in $(f + \mu_n x^k)^{1/n}$ are integers
if and only if the coefficients in $f^{1/n}$ are integers.~~~${\vrule height .9ex width .8ex depth -.1ex }$
Since $1 \in {\cal{P}}_n$, we have:
\begin{cor}\label{CorMod}
If $f \in \RT $ satisfies $ f \equiv 1 \pmod{ \mu_n}$,
then $f \in {\cal{P}}_n $.
\end{cor}
\begin{cor}\label{Cor3}
Suppose $f = 1 + f_1 x + f_2 x^2 + \cdots \in \RT $.
If $A$ and $B$ are positive integers such
that $\mu_n | AB$ and $\mu_n | A^2$,
then
$f(Ax) \in {\cal{P}}_n$ if $B | f_1$.
\end{cor}
This is an immediate consequence of Corollary \ref{CorMod}.
Similar conditions involving further coefficients
of $f$ can be obtained in the same way.
For example, if $n=2$, $f^{1/2}(4x)$ has
integer coefficients for any $f \in \RT $,
and $f^{1/2}(2x)$ has integer coefficients if
$2 | f_1$. (See Section \ref{SecSq} for more
about the case $n=2$.)
Furthermore, $n$th roots are unique mod $\mu_n/n$:
\begin{theorem}\label{ThMod2}
Given $f \in {\cal{P}}_n $, there is a unique
$g \in \RT $ mod $\mu_n/n$ such that $g^n \equiv f \pmod{\mu_n}$.
\end{theorem}
\paragraph{Proof.}
Given $f \in {\cal{P}}_n $, suppose $g \in \RT $ is such that
$g^n \equiv f \pmod{\mu_n}$.
We will show that,
for any $k \ge 1$,
$(g + \frac{\mu_n}{n} x^k)^n \equiv g^n \equiv f \bmod{\mu_n}$.
In fact,
$$
(g + \frac{\mu_n}{n} x^k)^n = g^n + \sum_{r=1}^{n} \binom{n}{r} \Big(\frac{\mu_n}{n}\Big)^r
x^{rk} g^{n-r} \, .
$$
Then for $r \ge 1$, $c := \binom{n}{r} \big(\frac{\mu_n}{n}\big)^r$
is divisible by $\mu_n$, because for primes $q$
not dividing $n$, $|c|_q = |\mu|_q = 0$,
while if $p$ divides $n$ then
$|c|_p \ge |n|_p - |r|_p +r
\ge |n|_p - |r|_p +p^{|r|_p}
\ge |\mu_n|_p = |n|_p +1$.
So we may reduce the coefficients of $g$ mod $\mu_n/n$.
Conversely, suppose $g^n \equiv h^n$ (mod $\mu_n$)
but $g \not\equiv h$ (mod $\mu_n/n$). Let $g$ and $h$
first differ at the $x^k$ term:
\begin{eqnarray}
g & = & 1 + g_1 x + \cdots + g_{k-1} x^{k-1} + \alpha x^k + \cdots ~,
\nonumber \\
h & = & 1 + g_1 x + \cdots + g_{k-1} x^{k-1} + \beta x^k + \cdots ~,
\nonumber
\end{eqnarray}
with $\alpha \not\equiv \beta$ mod $\mu_n/n$.
Equating coefficients of $x^k$ in $g^n \equiv h^n$ (mod $\mu_n$)
gives
$n \alpha \equiv n \beta$ (mod $\mu_n$),
which implies $\alpha \equiv \beta$ (mod $\mu_n/n$),
a contradiction. So $g$ is unique.~~~${\vrule height .9ex width .8ex depth -.1ex }$
In the other direction, associated with any
$g \in ({\mathbb Z}/\frac{\mu_n}{n}{\mathbb Z})[[x]]$ with constant term $1$
is a unique $f \in ({\mathbb Z}/\mu_n{\mathbb Z})[[x]] \cap {\cal{P}}_n$,
namely $f := g^n \bmod{\mu_n}$.
So the elements of ${{\cal{P}}_2}$, for example, are enumerated by
infinite binary strings beginning with $1$.
We also note the following useful lemma.
\begin{lem}\label{LemmaI}
For $r, s \ge 1$,
$$
{{\cal{P}}_r} \cap {{\cal{P}}_s} = {{\cal{P}}_{\lcm (r,s)}}.
$$
\end{lem}
\paragraph{Proof.}
Clearly
${{\cal{P}}_{\lcm (r,s)}} \subset {{\cal{P}}_r}, \, {{\cal{P}}_s}$.
On the other hand, suppose $f \in
{{\cal{P}}_r} \cap {{\cal{P}}_s}$. Let $a,b$ be integers such that
$ar+bs = \gcd(r,s)$, and define
$$
g := (f^{\frac{1}{r}})^b (f^{\frac{1}{s}})^a \, .
$$
Then $g \in \RT$ and
$g^{\lcm(r,s)} = g^{rs/\gcd(r,s)} = f$.~~~${\vrule height .9ex width .8ex depth -.1ex }$
\section{ Theta series of lattices }\label{SecTheta}
The theta series of an integral lattice $\Lambda$ in $\mathbb R^d$
(that is, a lattice in which all inner products are integers)
is
$$
\Theta_{\Lambda}(x) := \sum_{u \in \Lambda} x^{u \cdot u} \in \RT \, .
$$
The theta series of extremal lattices in various genera are
especially interesting in view of their connections with
modular forms and Diophantine equations
(\cite{SPLAG}, \cite{Scharlau99}, \cite{Serre88}).
\begin{lem}\label{LemmaAPF}
If $f\in 1+m x{\mathbb Z}[[x]]$ for some integer $m$, then for any integer $n$,
$$
f^n\in 1+m n' x{\mathbb Z}[[x]] \,,
$$
where $n' = \prod_{p|m} p^{|n|_p} ~(\mbox{or~} 0 \mbox{~if~} n=0)$.
\end{lem}
\paragraph{Proof.}
It suffices to consider the case $m=p^k$, $k>0$ and $n$ prime.
If $n\ne p$, the claim is trivial, while otherwise, if $f=1+mg$, then
$$
(f^p-1)/m = \sum_{i=1}^{p} m^{i-1} \binom{p}{i} g^i \,.
$$
Every term on the right is a multiple of $p$,
and thus the claim follows.~~~${\vrule height .9ex width .8ex depth -.1ex }$
\begin{theorem}\label{th:E1}
If $\Lambda$ is an extremal even unimodular lattice in $\mathbb R^d$,
$d$ a multiple of $8$,
then $\Theta_\Lambda (x) \in {\cal{P}}_n$, where $n$ is obtained from
$d$ by discarding any prime factors other than $2$, $3$ and $5$.
\end{theorem}
\paragraph{Proof.}
Suppose $d = 8t = 2^i 3^j 5^k 7^\ell \cdots$ (with $i \ge 3$),
and let $a = \lfloor d/24 \rfloor = \lfloor t/3 \rfloor$.
Then $n = 2^i 3^j 5^k$ and $\mu_n$ is a divisor of $30n$.
It is known that $\Theta_\Lambda (x)$ can be written in the form
\beql{Eq1}
\Theta_\Lambda (x) = \sum_{i=0}^{a} c_i \psi^{t-3i}(x) \Delta^i(x) \, ,
\end{equation}
where
\beql{Eq2a}
\psi(x) := \Theta_{E_8}(x) = 1 + 240 \sum_{m=1}^{\infty} \sigma_3(m) x^{2m}\, ,
\end{equation}
\beql{Eq2b}
\Delta(x) := x^2 \prod_{m=1}^{\infty} (1-x^{2m})^{24}\, ,
\end{equation}
$\sigma_3(m)$ is the sum of the cubes of the divisors of $m$,
and the coefficients $c_0 := 1, c_1, \ldots, c_a$ are such that
\beql{Eq4}
\Theta_{\Lambda} (x) = 1 + O(x^{2a+2}) \, .
\end{equation}
We will show that
\beql{Eq5}
\Theta_\Lambda(x) \equiv 1 \pmod{30n}\, \, ,
\end{equation}
which by Corollary \ref{CorMod} implies the desired result.
We apply Lemma \ref{LemmaAPF},
taking $f=\psi, m=240, n=t, n'=2^{i-3} 3^j 5^k$, obtaining
$\psi^t(x) \equiv 1 \pmod{30n}$.
By equating $\eqref{Eq1}$ and $\eqref{Eq4}$,
we obtain an upper triangular system of equations for the $c_i$
with diagonal entries equal to $1$;
this implies inductively that for $i \ge 1$, $c_i \equiv 0 \pmod{30n}$,
and $\eqref{Eq5}$ follows.~~~${\vrule height .9ex width .8ex depth -.1ex }$
The theta series mentioned in Theorem \ref{th:E1}
is a modular form of weight $w=d/2 \equiv 0$ mod~$4$ for the full
modular group $\SL _2({\mathbb Z})$. More generally, we have:
\begin{theorem}\label{th:EMF}
Let $f(x)$ be the extremal modular form
of even weight $w$ for $\SL _2({\mathbb Z})$ (cf. \cite{Mallows75}).
Then $f(x) \in {\cal{P}}_n$,
where $n$ is obtained from $2w$ by discarding
all primes $p$ such that $p-1$ does not divide $w$.
\end{theorem}
\paragraph{Proof.}
To show that the extremal modular form of weight $w$ is in
${\cal P}_n$, it suffices to construct {\em any} modular form of weight $w$
congruent to $1 \mod \mu_n$; this form may even have denominators, as long
as they are prime to $\mu_n$.
Indeed, the difference between such a form and the extremal form
will be a cusp form with all leading coefficients a multiple of $\mu_n$;
it follows as in the proof of Theorem \ref{th:E1} that such a cusp form has {\em
all} coefficients a multiple of $\mu_n$.
In particular, one may consider the Eisenstein series.
Every nonconstant coefficient of $E_w$ for $w$ even is a
multiple of $(-2w)/B_w$, where $B_w$ is a Bernoulli number,
so it suffices to show that $\mu_n$ divides the denominator of $B_w/(2w)$.
By a result of Carmichael \cite{Carmichael},
$m$ divides this denominator
if and only if the exponent of ${\mathbb Z}_m^*$ divides $w$.
In particular, $2^{k+2}$ divides the denominator
if and only if $2^k$ divides $w$,
while for odd primes, $p^{k+1}$ divides
the denominator if and only if $p^k(p-1)$ divides $w$.
The stated rule for $n$ follows.~~~${\vrule height .9ex width .8ex depth -.1ex }$
For $2$- and $3$-modular
lattices, we take powers of $\Theta_{D_4}$ and $\Theta_{A_2}$
respectively to determine $\mu_n$. Presumably these results could be
improved by using the respective Eisenstein series instead.
\begin{theorem}\label{th:E2}
If $\Lambda$ is an extremal $2$-modular lattice in $\mathbb R^d$,
$d$ a multiple of $4$,
then $\Theta_\Lambda (x) \in {\cal{P}}_n$, where $n$ is obtained from
$d$ by discarding any prime factors other than $2$ and $3$.
\end{theorem}
\begin{theorem}\label{th:E3}
If $\Lambda$ is an extremal $3$-modular lattice in $\mathbb R^d$,
$d$ a multiple of $2$,
then $\Theta_\Lambda (x) \in {\cal{P}}_{n/2}$, where $n$ is obtained from
$d$ by discarding any prime factors other than $2$ and $3$.
\end{theorem}
It is a consequence of Theorems
\ref{th:E1}, \ref{th:E2} and \ref{th:E3} that
that the theta series of the following
lattices are in ${\cal{P}}_d$, where $d$ (the subscript)
is the dimension of the lattice:
$D_4$ [sequence A004011 in \cite{OEIS}],
$E_8$ [A004009],
$BW_{16}$ [A008409],
$\Lambda_{24}$ [A008408]
and
Quebbemann's $Q_{32}$ [A002272].
Also, the theta series of the
Coxeter-Todd lattice $K_{12}$ [A004010]
is in ${\cal{P}}_6$,
and the theta series of Nebe's $24$-dimensional
lattice [A004006]
is in ${\cal{P}}_{12}$, establishing
Somos's conjecture mentioned in Section \ref{SecIntro}.
In the next section we will show more generally that the
theta series of the Barnes-Wall lattice $BW_{2^m}$
is in ${{\cal{P}}_{2^m}}$ for all $m \ge 1$.
The coefficients of the $n$th roots in these
examples in general will not be the coefficients of
any modular form (at least, not in the sense of being
associated to any Fuchsian group).
$\Theta_{E_8}(e^{2 \pi i z})$, for example,
has a zero in the open upper half-plane,
and so its eighth root has an algebraic singularity
in the upper half plane, and the coefficients have
exponential growth.
The coefficients of the $n$th roots
also do not appear to have
any particular combinatorial significance.
For example, the theta series of the $D_4$ lattice is
$$
1+24\,{x}^{2}+24\,{x}^{4}+96\,{x}^{6}+24\,{x}^{8}+144\,{x}^{10}+96\,{x}^{12}+ \cdots \, ,
$$
in which the coefficient of $x^{2m}$ is the number
of ways of writing $2m$ as a sum of four squares,
while its fourth root [A108092] is
\begin{eqnarray*}
& 1 & + ~ 6\,{x}^{2}-48\,{x}^{4}+672\,{x}^{6}-10686\,{x}^{8}+185472\,{x}^{10}-3398304\,{x}^{12} \\
&&{} ~ +64606080\,{x}^{ 14} -1261584768\,{x}^{16}+25141699590\,{x}^{18}-509112525600\,{x}^{20}\\
&&{} ~ +10443131883360\,{x}^{22} -216500232587520\,{x}^{24}+4528450460408448\,{x}^{26} \\
&&{} ~-95438941858567104\,{x}^{28}+ 2024550297637849728\,{x} ^{30} - \cdots \, .
\end{eqnarray*}
Do these coefficients have any other interpretation?
\paragraph{Further examples.}
The extremal {\em odd} unimodular lattices
have been completely classified (cf. \cite{Conway78}, \cite[Chap. 19]{SPLAG}),
and the ${\cal{P}}_n$ to which their theta series belong are as follows:
$\Theta_{{\mathbb Z}^d} ~ (1 \le d \le 7) \in {\cal{P}}_d$,
$\Theta_{D_{12}^{+}}$ [A004533] $\in {\cal{P}}_4$,
$\Theta_{E_{7}^{2+}}$ [A004535] $\in {\cal{P}}_2$,
while the theta series of $A_{15}^{+}$ [A004536] and
the odd Leech lattice [A004537]
are only in ${\cal{P}}_1$.
This is a straightforward verification
since the theta series are known explicitly.
The theta series of both $E_6$ [A004007] and
its dual $E_6^{\ast}$ [A005129]
are $\equiv 1 \bmod{9}$ (this follows from
\cite[p. 127, Eqs. (121), (122)]{SPLAG}),
and so are in ${\cal{P}}_3$.
Michael Somos \cite{Somos05} has also pointed out that
$x j(x) \in {\cal{P}}_{24}$, where
$j(x)$ is the modular function
$\frac{1}{x} + 744 + 196884 x + \cdots$ (\cite{Schoeneberg}).
This follows from $x j(x) = \psi(x)^3/\Delta(x)$.
We believe that the values of $n$ in Theorems \ref{th:E1} -- \ref{th:E3}
are best possible as far as the primes 2, 3, 5 and 7 are concerned.
For example, it is easy to check that
the theta series of the extremal even unimodular lattice in $\mathbb R^{56}$
[A004673]
belongs to ${\cal{P}}_8$ but not ${\cal{P}}_{56}$.
The following conjecture also seems very plausible,
although again we do not have a proof:
\begin{conj}
Let $\Theta_{\Lambda}(x)$ be the theta series
of a $d$-dimensional lattice.
If $\Theta_{\Lambda}(x) \in {\cal{P}}_n$ then $n \le d$.
$($In fact, we have not found any counterexample to
the stronger conjecture that $\Theta_{\Lambda}(x) \in {\cal{P}}_n$
implies that $n$ divides $d$.$)$
\end{conj}
Note that, considered as a formal power series,
$\Theta_{\Lambda}(x)$ determines the dimension $d$
(see \cite[p. 47, Eq. (42)]{SPLAG})---in Conway's terminology \cite{Conway97},
the dimension is an ``audible'' property.
\section{ Weight enumerators of codes }\label{SecCodes}
The weight enumerator of an $[n, \, k, \, d \, ]_q$ code (that is,
a linear code of length $n$, dimension $k$
and minimal Hamming distance $d$ over the field ${\mathbb F} _q$) is
$$
W_C(x) := \sum_{c \in C} x^{\wt(c)} \, ,
$$
where $\wt$ denotes Hamming weight
(\cite{LintI}, \cite{MS77}).
Although the weight enumerators are polynomials,
the roots, if they exist, are normally infinite series.
There does not seem to be an analogue of Theorem \ref{th:E1}
for extremal doubly-even binary self-dual codes, since
the weight enumerator of the $[24,12,8]_2$ Golay code,
$$
1 + 759 x^8 + 2576 x^{12} + 759 x^{16} + x^{24} \, ,
$$
is not in ${\cal{P}}_n$ for any $n > 1$.
However, the weight enumerator of the $[8,4,4]_2$ Hamming code,
$1+14x^4+x^8$, is in ${\cal{P}}_2$
since it is congruent to $1+2x^4+x^8 \bmod{4}$,
although it is not in ${\cal{P}}_n$ for any $n > 2$.
This Hamming code is also the Reed-Muller code $RM(1,4)$
(cf. \cite{LintI}, \cite{MS77}).
More generally, we have:
\begin{theorem}\label{ThRM}
Let $W_{r,m}(x)$ denote the weight enumerator of the
$r$th order Reed-Muller code $RM(r,m)$, for $0 \le r \le m$,
and let $W_{r,m}(x) := W_{m,m}(x) = (1+x)^{2^m}$ for $r>m$.
Then for $r \le m$,
\beql{EqRM1}
W_{r,m}(x) \equiv ( 1 ~+~ x^{2^{m-r}} ) ^ {2^r} \, \pmod{2^{r+1}} \, ,
\end{equation}
and so by Theorem \ref{ThTaylor} is in ${\cal{P}}_{2^r}$.
\end{theorem}
We will deduce Theorem \ref{ThRM} from the following result:
\begin{theorem}\label{ThRM2}
For $0 \le r \le m+1$,
\beql{EqRM3}
W_{r,m+1}(x)-W_{r,m}(x^2) \equiv 0 \pmod{2^{m+1}} \, .
\end{equation}
\end{theorem}
\paragraph{Proof.}
Reed-Muller codes may be built up recursively from
\beql{EqRMu}
RM(r,m+1) = \{ (u,u+v) \mid
u \in RM(r,m), \, v \in RM(r-1,m) \} \, ,
\end{equation}
for $1 \le r \le m$,
with $RM(0,m+1) = \{ 0^{2^{m+1}}, 1^{2^{m+1}} \}$,
$RM(m+1,m+1) = \{ 0,1 \}^{2^{m+1}}$
(\cite[Chap. 13, Theorem 2]{MS77}).
Let $G$ be the group $({\mathbb F}_2^+)^{m+1}$
in its natural action on $C:=RM(r,m+1)$
(consisting of the diagonal action of $({\mathbb F}_2^+)^m$ on
$RM(r,m)$ and $RM(r-1,m)$ together with the involution swapping the two halves).
If $O(x)$ is the generating function for $G$-orbits, indexed by the
weight of the elements of the orbit, then by Burnside's Lemma,
$$
|G| \, O(x) = \sum_{g\in G} W_{\Fix_g(C)}(x) \, ,
$$
where $W_{\Fix_g(C)}(x)$ is the weight enumerator
of the subcode fixed by $g$.
For nonzero $g$, $W_{\Fix_g(C)}=W_{r,m}(x^2)$, from \eqref{EqRMu}.
Therefore
$$
|G| \, O(x) = W_{r,m+1}(x) + (|G|-1) W_{r,m}(x^2) \, .
$$
Since $|G|=2^{m+1}$, the result follows immediately.~~~${\vrule height .9ex width .8ex depth -.1ex }$
Theorem \ref{ThRM} now follows from Theorem \ref{ThRM2}
by induction on $m$.
Another consequence of Theorem \ref{ThRM2} is:
\begin{cor}
For any dyadic rational number $\lambda$
$($i.e., any element of ${\mathbb Z}[1/2])$
satisfying $0\le \lambda \le 1$,
and any integer $r\ge 0$, the sequence
\beql{EqRM4}
f_{r,m}(\lambda ) = |\{u\in RM(r,m) \mid \wt(u) = \lambda 2^m \}|
\, , \quad m = r, r+1, r+2, \ldots \, ,
\end{equation}
converges $2$-adically as $m\to\infty$.
\end{cor}
Special cases of this were already known, but
in view of the many investigations of
weight enumerators of Reed-Muller codes
(\cite{Kasami76},
\cite[\S6.2]{LintL},
\cite{MS77},
\cite{Sloane70},
\cite{Sugino71},
\cite{Sugita96},
etc.),
it is worth putting the general remark on record.
For example, in the special case $\lambda = \frac{1}{2^r}$ it follows from
\cite[Chap. 13, Theorem 9]{MS77} that the limit in \eqref{EqRM4} is
$2^r / \prod_{i=1}^r (1-2^i)$.
Other special cases may be deduced from the results
in \cite{Sloane70} (or \cite[Chap. 15, Theorem 8]{MS77})
and \cite{Kasami76}.
The Nordstrom-Robinson, Kerdock and Preparata codes
are closely related to Reed-Muller codes
(\cite{Hammons94}, \cite{LintK}, \cite{MS77}).
The weight enumerator of the Nordstrom-Robinson code
of length $16$ is in ${\cal{P}}_2$, and more generally so is that
of the Kerdock code of length $4^m$, $m \ge 2$
(this follows immediately from \cite[Fig. 15.7]{MS77}).
It appears, although we do not have a proof,
that the weight enumerator of the Preparata code
of length $4^m$ is in ${\cal{P}}_{2^{m-3}}$.
There is a conjectural analogue of Theorem \ref{ThRM} for BCH codes:
\begin{conj}\label{ConjBCH}
Let $C$ be obtained by adding an overall parity check
to the primitive BCH code of length $2^m-1$
and designed distance $2t-1$, so that $C$
has length $n=2^m$ and minimal distance $d \ge 2t$.
We conjecture that the weight enumerator of $C$
is in ${\cal{P}}_{2^m/d'}$,
where $d'$ is the smallest power of $2 \ge 2t$.
\end{conj}
We have verified this for $m \le 6$.
Here are three further examples.
The Hamming weight enumerator of the $[12,6,6]_3$
ternary Golay code [A105683] is in ${\cal{P}}_4$,
and that of the $[18,9,8]_4$ extremal self-dual code $S_{18}$ over ${\mathbb F}_4$
(\cite{Cheng90}, \cite{MacWilliams78}, A014487)
is in ${\cal{P}}_{18}$.
A more unlikely example is the weight enumerator of the $[48,23,8]_2$
Rao-Reddy code (\cite{Rao73}, \cite{MS77}, [A031137]),
\begin{eqnarray*}
1 & + & 7530\,{x}^{8}+92160\,{x}^{10}+1080384\,{x}^{12}
\nonumber \\
& & {} ~
+7342080\,{x}^{14}+34408911\,{x}^{16}+111507456\,{x}^{18}
\nonumber \\
& & {} ~
+255566784\,{x}^{20}+417404928\,{x}^{22}+492663180\,{x}^{24}
\nonumber \\
& & {} ~
+417404928\,{x}^{26}+255566784\,{x}^{28}+111507456\,{x}^{30}
\nonumber \\
& & {} ~
+34408911\,{x}^{32}+7342080\,{x}^{34}+1080384\,{x}^{36}
\nonumber \\
& & {} ~
+92160\,{x}^{38}+7530\,{x}^{40}+{x}^{48} \, ,
\nonumber
\end{eqnarray*}
which is a square since it is congruent to $(1+x^8+x^{16}+x^{24})^2 \pmod{4}$.
(The square root is given in A108179.)
Barnes-Wall lattices are also closely related to Reed-Muller codes
\cite{SPLAG}, \cite{NRS02}, \cite{NRS06}.
It will be convenient here to normalize these lattices so that
the $2^m$-dimensional Barnes-Wall lattice $BW_{2^m}$
has minimal norm $2^{m-1}$ (making $BW_{2^m}$ a $2^{m-1}$-modular
lattice, cf. \cite{Quebbemann95}, in which all norms
are multiples of $2^{\lceil \frac{m}{2} \rceil } $).
Thus the first few instances are
$$
BW_{2} = {\mathbb Z} ^2, \,
BW_{4} = D_4, \,
BW_{8} =\sqrt{2} \, E_8, \,
BW_{16} =\sqrt{2}\, \Lambda _{16}, \ldots \,.
$$
In particular, we see that for $m=1, \ldots, 4$,
$BW_{2^m}$ is in ${{\cal{P}}_{2^m}}$.
In fact, we have:
\begin{theorem}
The theta series of $BW_{2^m}$ in $\mathbb R ^{2^m}$
is congruent to $1 \pmod{2^{m+1}}$ for $m \ge 1$,
and is thus in ${{\cal{P}}_{2^m}}$.
More precisely, for $m \ge 2$, we have
\beql{EqBWTh}
\frac{ \Theta _{BW_{2^m}} (x) - 1}{2^{m+1}}
~\equiv~
(1-2^{m-1}) \, \frac{\Theta _{BW_{2^{m-1}}}(x^2) - 1}{2^{m}}
\pmod {2^{m}} \, .
\end{equation}
\end{theorem}
\paragraph{Proof.}
For $m=1$, $BW_{2} = {\mathbb Z} ^2$ and
$\Theta _{BW_{2}} (x) \equiv 1 \pmod{4}$.
The automorphism group $G$ of $BW_{2^m}$
contains as a normal subgroup the extraspecial
group $2_{+}^{1+2m}$ (cf. \cite{NRS02}).
For $m \ge 2$ the extraspecial group
consists of four conjugacy classes of $G$, with representatives, sizes and
fixed sublattices as shown in Table \ref{TT1} (here $n = 2^m$):
\renewcommand{\arraystretch}{2.1}
\begin{table}[htb]
\caption{ }
$$
\begin{array}{ccc}
\hline
\mbox{~Representative~} & \mbox{~Size~} & \mbox{~Fixed sublattice~} \\
\hline
1 & 1 & BW_{2^m} \\
-1 & 1 & 0 \\
\begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix} & 2^{2m} + 2^m -2 & \sqrt{2} \, BW_{2^{m-1}} \\
\begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix} & 2^{2m} - 2^m & 0 \\
\hline
\end{array}
$$
\label{TT1}
\end{table}
\renewcommand{\arraystretch}{1.0}
\noindent Then Burnside's Lemma gives us the congruence
$$
\Theta _{BW_{2^m}} (x) + (2^{2m}-2^m+1)
+ (2^{2m}+2^m-2) \, \Theta _{BW_{2^{m-1}}} (x^2)
~ \equiv ~ 0 \pmod{2^{2m+1}} \, ,
$$
which implies \eqref{EqBWTh}.~~~${\vrule height .9ex width .8ex depth -.1ex }$
This in particular implies that, for any dyadic rational
$\lambda \ge 1$, the coefficient of $x^{\lambda 2^{m-1}}$
(that is, the number of lattice vectors of norm equal to
$\lambda$ times the minimal norm) in
$$
\frac{ \Theta _{BW_{2^m}} (x) - 1}{2^{m+1}}
$$
converges to a $2$-adic limit.
For the kissing number itself, i.e. for $\lambda = 1$,
the limit is
$\prod_{i=1}^\infty (1+2^i)$.
We end this section with a question:
Is there a simple way to test if a code has a
weight enumerator which is an $m$-th power?
\section{ Squares }\label{SecSq}
We know from Theorem \ref{ThTaylor} that to test if a given
$f(x) \in \RT$ is a square,
it is enough to consider $f(x) \bmod 4$,
and from Theorem \ref{ThMod2} that
if $f(x)$ {\em is} a square then
there is a unique binary series $g(x)$ associated with it.
There is a simple necessary and
sufficient condition for
$f(x)$ to be a square.
\begin{theorem}\label{ThSquare}
Given $f(x) := 1 + \sum_{r \ge 1} f_r x^r \in \RT$,
let $\bar{f}(x) := 1 + \sum_{r \ge 1} \bar{f_r} x^r$
be obtained by reducing the coefficients of $f(x)$ mod $4$.
If $\bar{f}_{2t} -g_t$ and $\bar{f}_{2t+1}$
are even for all $t \ge 0$,
where $g_0:=1, \, g_1, \ldots \in {\mathbb Z}/2{\mathbb Z}$ are
defined recursively by
\begin{eqnarray}\label{EqRec1}
\frac{\bar{f}_{2t} -g_t}{2} & \equiv & g_{2t} +
\sum_{r=1}^{t-1} g_r g_{2t-r} \pmod{2} \, , \nonumber \\
\frac{\bar{f}_{2t+1}}{2} & \equiv & g_{2t+1} +
\sum_{r=1}^{t} g_r g_{2t+1-r} \pmod{2} \, ,
\end{eqnarray}
then $f(x) \in {\cal{P}}_2$ and
\beql{EqRec2}
f(x) \equiv \bar{f}(x) \equiv g^2(x) := (1+\sum_{r=1}^{\infty} g_r x^r)^2 \pmod{4} \, .
\end{equation}
Conversely, if for some $t$ either $\bar{f}_{2t} -g_t$ or $\bar{f}_{2t+1}$
fails to be even, then $f(x) \notin {\cal{P}}_2$.
\end{theorem}
There is a simple {\em necessary} condition for $f(x)$ to
be a square, which generalizes to $p$th powers
for any prime $p$.
\begin{theorem}\label{ThSquare2}
Let $p$ be a prime.
If $f(x) := 1 + \sum_{r \ge 1} f_r x^r \in {\cal{P}}_p$,
say $f(x) = g(x)^p$, then
\beql{EqPP0}
f_r \equiv 0 \pmod{p} \mbox{ unless } p \mbox{ divides } r \, ,
\end{equation}
\beql{EqPP1}
g(x) \equiv 1 + f_p x + f_{2p} x^{2} + f_{3p} x^{3} + \cdots \pmod{p}
\end{equation}
and
\beql{EqPP2}
f(x) \equiv (1 + f_p x + f_{2p} x^{2} + f_{3p} x^{3} + \cdots)^p \pmod{p^2} \, .
\end{equation}
\end{theorem}
\paragraph{Proof.}
This follows immediately from Theorem \ref{ThMod2} and the fact that
$$
(1 + g_1 x + g_2 x^{2} + g_3 x^{3} + \cdots)^p
\equiv
1 + g_1 x^p + g_2 x^{2p} + g_3 x^{3p} + \cdots \pmod{p} \, .
$$
${\vrule height .9ex width .8ex depth -.1ex }$
The {\em On-Line Encyclopedia of Integer Sequences}
\cite{OEIS} is a database containing over $100,000$
number sequences.
We tested the corresponding formal power series to see which
were -- or at least appeared to be -- in ${\cal{P}}_2$.
As a first step we used the symbolic language {\em Maple}
\cite{Maple}
to weed out any series which did not begin $1 + \cdots$
or which had an obviously non-integral square root.
This produced $3030$ possible members of ${\cal{P}}_2$.
To reduce this number we discarded those series
which appeared to be congruent to 1 mod $4$, which left
$905$ candidates.
More detailed examination of these $905$ showed that most
of them could be grouped into one of the following
(not necessarily disjoint) classes.
(1) Sequences which are obviously squares, usually with
a square generating function.
These are often described as ``self-convolutions'' of other sequences.
For example, A008441, which gives the number of
ways of writing $n$ as the sum
of two triangular numbers,
with generating function $x^{-1/4}\eta(x^2)^4/\eta(x)^2$,
where $\eta(x)$ is the Dedekind eta function.
(2) Sequences which reduce mod $4$ to a square.
For example, periodic sequences of the form
$$
1,2,3, \ldots, k,
1,2,3, \ldots, k,
1,2,3, \ldots, k, \ldots \, ,
$$
are squares if and only if $k$ is a multiple of $4$.
More generally, any sequence which reduces mod $4$ to
$1,2,3,4,5,6,\ldots $ is a square.
(3) Theta series of lattices and weight enumerators of codes,
as discussed in the preceding sections.
(4) McKay-Thompson series associated with
conjugacy classes in the Monster simple group
(\cite{Ford94}, \cite{McKay90}, e.g. A101558).
As with the modular function $j(x)$ mentioned above,
the fact that these series are squares follows at once
from known properties.
(5) Sequences with an exponential generating function
involving trigonometric, inverse trigonometric, exponential, etc.,
functions.
One example out of many will serve as an illustration.
Vladeta Jovovi\'c's sequence A088313 \cite{Jovovic03}:
$$
1,2,7,36,241,1950,18271,193256,2270017,\ldots
$$
gives the number of ``sets of lists'' with an odd number of
lists, that is, the number of partitions of $\{1, \ldots, n\}$ into an
odd number of ordered subsets (cf. Motzkin \cite{Motzkin71}).
There is no apparent reason why this should be a square.
The analogous sequences for an even number of lists
(A088312) or with any number of lists (A000292) are not squares.
Jovovi\'c's sequence has exponential generating function
$$
\mbox{sinh}~\big(\frac{x}{1-x}\big) ~=~
x+\frac{2}{2!}{x}^{2}+{\frac {7}{3!}}{x}^{3}+
{\frac {36}{4!}}{x}^{4}+
{\frac {241}{5!}}{x}^{5}+{\frac{1950}{6!}}{x}^{6}+
{\frac { 18271}{7!}}{x}^{7}+ \, \cdots \, ,
$$
and is a square, since an elementary calculation shows that if
$$
\mbox{sinh}~\big(\frac{x}{1-x}\big) ~=~
\sum_{k=1}^{\infty} \, c_k \, \frac{x^k}{k!} \, ,
$$
then $c_k \equiv k \pmod{4}$.
(6) Paul Hanna's sequences, discussed in the following section.
These were the most interesting examples
that were turned up by our search. We were disappointed
not to find other sequences as challenging as these.
(7) Sequences whose square root proved to have a non-integral
coefficient once further terms were computed.
\section{ Paul Hanna's sequences }\label{SecHanna}
In May 2003, Paul D. Hanna \cite{Hanna2} contributed a family
of sequences to \cite{OEIS}.
For $k \ge 1$, the
$k$th Hanna sequence
${H}_k := (1, h_1, h_2, \ldots)$ is
defined as follows: for all $n\ge 1$, $h_n$ is the smallest
number from the set $\{1, \ldots, k\}$ such that
$(1+h_1 x + h_2 x^2 + \cdots )^{1/k}$ has integer
coefficients.
He asked if the sequences are well-defined and unique
for all $k$, and if they are eventually periodic.
For example, ${H}_2$ [A083952] is
$$
1,2,1,2,2,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,1,2,1,\ldots \, ,
$$
and the coefficients of its square root [A084202] are
$$
1,1,0,1,0,1,-1,2,-2,4,-6,10,-16,27,-44,75,-127,218,-375,650,-1130,
\ldots \, .
$$
The sequence ${H}_3$ [A083953] is
$$
1,3,3,1,3,3,3,3,3,3,3,3,1,3,3,2,3,3,2,3,3,1,3,3,2,3,3,3,3,3,2,3,3,3,3,
\ldots \, ,
$$
and the coefficients of its cube root [A084203] are
$$
1,1,0,0,1,-1,2,-2,2,0,-4,12,-24,38,-46,33,29,-176,443,-827,1222,-1310,
\ldots \, .
$$
\begin{theorem}\label{ThHanna0}
For all $k \ge 1$, ${H}_k$ is well-defined and is unique.
\end{theorem}
\paragraph{Proof.}
Suppose $f(x) := 1 + h_1 x + h_2 x^2 + \cdots = g(x)^k$,
where $g(x) := 1 + g_1 x + g_2 x^2 + \cdots$.
Then for $n \ge 1$, $h_n = k g_n + \Phi(g_1, \ldots, g_{n-1})$,
for some function $\Phi(g_1, \ldots, g_{n-1})$.
Write $\Phi(g_1, \ldots, g_{n-1}) = qk + r, ~ 0 \le r < k$.
If $r=0, h_n = k $ and $g_n = -(q-1)$,
while if $r>0$, $h_n=r$ and $g_n =-q$.~~~${\vrule height .9ex width .8ex depth -.1ex }$
We will analyze ${H}_2$ and ${H}_3$ in detail,
find generating functions for them,
and show that they are not periodic.
We know from Section \ref{SecThms} that to study the $k$th root
$({H}_k)^{1/k}$ it is enough to look at its values mod $\mu_k/k$.
The square root of ${H}_2$ read mod $2$ gives
the binary sequence
$$
{S}_2 := (1,1,0,1,0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0,0,0,0,1,1,\ldots)
$$
[A108336], and the cube root of ${H}_3$ read
mod $3$ gives
$$
{S}_3 := (1,1,0,0,1,2,2,1,2,0,2,0,0,2,2,0,2,1,2,1,1,1,1,1,0,1,1,\ldots)
$$
[A104405].
\begin{theorem}\label{ThHanna2}
The generating function $g(x) := 1 + x + x^3 + x^5 + x^6 + \cdots$
for ${S}_2$ satisfies $g(0)=1$ and
\beql{EqHan2}
g(x^2) + g(x)^2 \equiv \frac{2}{1-x} \pmod{4} \, .
\end{equation}
\end{theorem}
\paragraph{Proof.}
If $f(x)$ is the generating function for ${H}_2$,
we have $f(x) \equiv g(x)^2 \pmod{4}$. It follows (compare
Theorem \ref{ThSquare}) that $f_{2t}=1$ if $g_t=1$,
$f_{2t}=2$ if $g_t=0$, and $f_{2t+1}=2$.
Thus $f_{2t} \equiv 3 g_t + 2$ mod $4$. Hence
\beql{EqHGf2}
f(x) \equiv 3g(x^2) +\frac{2}{1-x^2} + \frac{2x}{1-x^2} \pmod{4} \, ,
\end{equation}
and \eqref{EqHan2} follows.~~~${\vrule height .9ex width .8ex depth -.1ex }$
\begin{cor}
${H}_2$ is not periodic.
\end{cor}
\paragraph{Proof.}
${H}_2$ is periodic if and only if ${S}_2$ is.
Suppose ${S}_2$ is periodic with period $\pi$.
Then $g(x) = p(x)/(1-x^{\pi})$,
where $p(x)$ is a polynomial of degree $\le \pi -1$.
From \eqref{EqHan2},
\beql{EqHan2b}
\frac{p(x^2)}{1-x^{2 \pi}} +
\frac{p(x)^2}{(1-x^{\pi})^2} \equiv \frac{2}{1-x} \pmod{4} \, ,
\end{equation}
hence
$$
p(x^2)(1-x^{\pi}) + p(x)^2(1+x^{\pi}) \equiv
2 \frac{(1-x^{\pi})(1-x^{2\pi})}{1-x} \pmod{4} \, .
$$
The coefficient of $x^{3 \pi -1}$ is $0$ on the left,
$2$ on the right, a contradiction.~~~${\vrule height .9ex width .8ex depth -.1ex }$
Similar arguments apply to the ternary case; we
omit the details.
\begin{theorem}\label{ThHanna3}
Let $g(x) := 1 + x + x^4 + 2x^5 + 2x^6 + \cdots$
be the generating function for ${S}_3$,
and write it as $g(x) = g_{+}(x)+2g_{-}(x)$,
where $g_{+}(x)$ $($resp. $g_{-}(x))$ contains the
powers of $x$ with coefficient $1$ $($resp. $2)$. Then
$g(x)$ satisfies $g(0)=1$ and
\beql{EqHan3}
2g_{+}(x^3) + g_{-}(x^3) + g(x)^3 \equiv \frac{3}{1-x} \pmod{9} \, .
\end{equation}
The generating function for ${H}_3$ is given by
\beql{EqHGf3}
f(x) \equiv \frac{3}{1-x} -2g_{+}(x^3) - g_{-}(x^3) \pmod{9} \, .
\end{equation}
\end{theorem}
\begin{cor}
${H}_3$ is not periodic.
\end{cor}
We have not studied the sequences ${H}_k$ for $k \ge 4$.
Another sequence of Hanna's is worth mentioning.
This is the sequence $a_0, a_1, a_2, \ldots$
defined by $a_0=1$, and for $n>0$, $a_n$ is
the smallest positive number not already in the sequence
such that $(a_0 + a_1 x + a_2 x^2 + \cdots)^{1/3}$
has integer coefficients [A083349]:
$$
1,3,6,4,9,12,7,15,18,2,21,24,27,30,33,36,39,42,45,48,51,5,54,57,10,60,
\ldots \, .
$$
Although this sequence is similar in spirit to $H_3$,
there is no obvious relation between them.
Hanna \cite{Hanna1} has shown that this sequence is a permutation
of the positive integers.
No generating function is presently known.
\section*{ Postscript, Nov. 6, 2005 }
We cannot resist adding one further example, again a sequence
[A111983] studied by Paul Hanna. The series
$$
f(x) := \sum_{n=0}^{\infty} \, (2n+1) \, 8^n \, x^{ \frac{n(n+1)}{2} }
$$
is in ${\cal{P}}_{12}$. Proof:
Mod 9, $f(x) \equiv \sum_{0}^{\infty}
(-1)^n (2n+1) x^{n(n+1)/2} = \prod_{m=1}^{\infty} (1-x^m)^3$,
by an identity of Jacobi \cite[Th. 357]{Hardy},
so by Theorem \ref{ThTaylor} $f(x) \in {\cal{P}}_{3}$.
Mod 8, $f(x) \equiv 1$, so $f(x) \in {\cal{P}}_{4}$
by Corollary \ref{CorMod}, and then
$f(x) \in {\cal{P}}_{12}$ by Lemma \ref{LemmaI}.~~~${\vrule height .9ex width .8ex depth -.1ex }$
\section*{ Acknowledgments }
N.H. thanks the
AT\&T Labs Fellowship Program
for support during the summer of 2005 at Florham
Park, NJ, when this research was carried out.
We thank Michael Somos for telling us about his discovery
of the property of Nebe's lattice which prompted this work
and for further discussions about theta functions of lattices.
We also thank Andrew Granville for some helpful comments,
and Allan Wilks for some computations related to
Hanna's sequences.
|
math-ph/0509013
|
\section*{1. Introduction}
Firstly, we construct two linear differential equations
whose solutions behave at infinity as the so-called
subnormal Thom\'e solutions, in contrast to
solutions of a confluent and a
double-confluent Heun equations \cite{ronveaux}, from which
the former equations are obtained by a limit process. Secondly,
we provide solutions which afford the
expected asymptotic behavior for these equations.
Finally, we find that the Schr\"odinger equation
with inverse fourth and sixth-power potentials reduces to
particular instances of the double-confluent
Heun equation and its Ince limit, respectively.
In the first place, let us introduce the two equations under consideration.
Our starting point is the generalized spheroidal wave equation
(GSWE) in the form used by Leaver \cite{leaver1}, namely,
\begin{eqnarray}
\label{gswe1}
z(z-z_{0})\frac{d^{2}U}{dz^{2}}+(B_{1}+B_{2}z)
\frac{dU}{dz}+
\left[B_{3}-2\eta \omega(z-z_{0})+\omega^{2}z(z-z_{0})\right]U=0,
(\omega\neq 0)
\end{eqnarray}
where $B_{i}$, $\eta$ and $\omega$ are constants
(notice that, if $\omega=0$ and $\eta$ is fixed, the equation may
be transformed into a hypergeometric equation). The points
$z=0$ and $z=z_{0}$ are regular singularities
with indices ($0,1+B_{1}/z_{0}$)
and ($0,1-B_{2}-B_{1}/z_{0}$), respectively,
while the infinity is an irregular singularity in which the
behavior of $U(z)$, inferred from the normal Thom\'e solutions
\cite{olver}, is given by
\begin{eqnarray}\label{asym}
\lim_{z\rightarrow \infty}U(z)\sim e^{\pm i\omega z}z^{\mp i\eta-(B_{2}/2)}.
\end{eqnarray}
Since its parameters are not specified,
the above GSWE is equivalent to the
confluent Heun equation \cite{ronveaux}, an
equation that is more general than the original Wilson GSWE
\cite{wilson}. Furthermore, as noted by Leaver,
for $z_{0}=0$ we obtain a double-confluent Heun equation
(DCHE) having five parameters,
rather than four as in other contexts \cite{decarreau1,schmidt},
namely,
\begin{eqnarray}\label{dche}
z^{2}\frac{d^{2}U}{dz^{2}}+(B_{1}+B_{2}z)\frac{dU}{dz}
+\left[B_{3}-2\eta \omega z+\omega^{2}z^{2}\right]U=0,\
(\omega\neq 0,\ B_{1}\neq 0),
\end{eqnarray}
where the singular points $z=0$ and $z=\infty$ are both irregular.
For $\omega= 0$ and/or $B_{1}= 0$ (with $\eta$ fixed) this equation
degenerates into confluent hypergeometric equations (see Appendix A).
At infinity, the behavior of $U(z)$ is again given by
(\ref{asym}), while at $z=0$ we find in the usual way
\cite{olver} that
\begin{eqnarray}\label{z0}
\lim_{z\rightarrow 0}U(z)\sim 1 \ \ \mbox{or}\ \
\lim_{z\rightarrow 0}U(z)\sim e^{B_{1}/ z}z^{2-B_{2}} .
\end{eqnarray}
The Leaver procedure also allows us to obtain solutions to
the DCHE from solutions to the GSWE when $z_{0}$ goes
to zero. The known Leaver-type solutions \cite{leaver1,eu}
are appropriate to solve,
for instance, the Teukolsky equations for the extreme upper limit of
the rotation parameter \cite{leaver1}, the time dependence
of Dirac test fields in dust-dominated
Friedmann-Robertson-Walker spacetimes
and the Schr\"odinger equation with asymmetric double-Morse
potentials \cite{eu}. They are suitable either
for handling the Schr\"odinger equation with inverse fourth-power
potentials, as we will see.
Now, to get the equations we are interested in,
the Levear limit is combined with a limit
that Ince \cite{ince} had
used to derive the Mathieu equation from the Whittaker-Hill
equation. The Ince limit is obtained by taking
\begin{eqnarray}\label{limits}
\omega\rightarrow 0, \ \
\eta\rightarrow
\infty, \ \mbox{such that }\ \ 2\eta \omega =-q,
\end{eqnarray}
where $q$ is a constant. Thus, the Ince limit of the GSWE is
\begin{eqnarray}
\label{lindemann}
z(z-z_{0})\frac{d^{2}U}{dz^{2}}+(B_{1}+B_{2}z)
\frac{dU}{dz}+
\left[B_{3}+q(z-z_{0})\right]U=0,\ (q\neq0).
\end{eqnarray}
This is a generalization of the Mathieu equation for, by setting
\letra
\begin{eqnarray}\label{mathieu2}
z_{0}=1,\ B_{1}=-1/2, \ B_{2}=1,\ z=\cos^2{(\sigma u)},\
W(u)=U(z),
\end{eqnarray}
we obtain the equation
\begin{eqnarray}\label{mathieu}
\frac{d^2W}{du^2}+\sigma^2\left[2q-4B_{3}-2q
\cos(2\sigma u)\right]W=0,
\end{eqnarray}
that is, the Mathieu equation if $\sigma=1$,
and the modified Mathieu equation if $\sigma=i$
\cite{McLachlan}. In fact, inserting $ z_{0}=1$,
$\ B_{1}=-1/2$ and $\ B_{2}=1 $ into Eq. (\ref{lindemann}),
one recovers the algebraic Lindemann form
for the Mathieu equation \cite{lindemann}. Nevertheless, the trigonometric
form (\ref{mathieu}) with $4B_{3}=2q-a$ is useful to
verify that our solutions for the Ince limit of the GSWE give
solutions already known for the Mathieu equation.
On the other hand, the Ince limit of the DCHE -- or Leaver limit of
the Eq. (\ref{lindemann}) -- is the equation
\antiletra
\begin{eqnarray}
\label{lindemann2}
z^2\frac{d^{2}U}{dz^{2}}+(B_{1}+B_{2}z)
\frac{dU}{dz}+
\left(B_{3}+qz\right)U=0,\ (q\neq0, \ B_{1}\neq 0)
\end{eqnarray}
which degenerates into simpler equations if $q=0$
and/or $B_{1}=0$ (see Appendix A). Solutions are obtained
for this special DCHE by taking the Leaver limit
of solutions for the Eq. (\ref{lindemann}). By the way, we shall
see that the Schr\"odinger equation for
an inverse sixth-power potential is a particular case
of Eq. (\ref{lindemann2}), as stated in the first paragraph.
We emphasize that the Ince limits of
the GSWE and DCHE, unlike the original GSWE and DCHE,
require solutions behaving according to
the subnormal Thom\'e solutions \cite{olver}, that is,
\begin{eqnarray}\label{thome2}
\lim_{z\rightarrow \infty}U(z)\sim
e^{\pm 2i\sqrt{qz}}z^{(1/4)-(B_{2}/2)}.
\end{eqnarray}
Despite this, our main mathematical issue consists in deriving
pairs of series solutions to Eqs. (\ref{lindemann}) and (\ref{lindemann2})
-- having the behavior stipulated above at the singular points --
from pairs of solutions to the GSWE. For this we shall
again employ the Ince and Leaver limits. The solutions in each pair
have the same series coefficients and these satisfy
three-term recurrence relations.
In section 2, a pair of solutions for the Ince limit of the GSWE
is obtained by taking the Ince limit (\ref{limits})
of a known pair of solutions for the GSWE. One solution is
given by an expansion in series of hypegeometric functions and
converges for any finite $z$; the other solution is
given by an expansion in series of modified
Bessel functions and converges for
$|z|>|z_{0}|$. Other
pairs are generated by using transformation
rules. These rules result from variable
substitutions that preserve the form of
the differential equations but
modify their parameters and/or arguments.
In section 3, we find pairs of solutions for the Ince
limit of the DCHE by taking the
Leaver limit ($z_{0}\rightarrow0$) of solutions for the
Ince limit of the GSWE. Solutions in series
of irregular confluent hypergeometric functions result
from expansions in series of hypergeometric functions
and converge for any finite $z$. The other solution in each pair
is given by a series of modified Bessel functions and
converges for $\vert {z}\vert>0$.
In both of these sections we deal with solutions with and
without a phase parameter $\nu$. In general, this $\nu$ is introduced
in order to assure the convergence of the series when
there is no free constant in the differential
equation, as in some scattering problems
or in equations where $z$ is a variable related to
the time \cite{leaver1,eu}.
Solutions with a phase parameter are
two-sided in the sense that the summation index $n$
runs from $-\infty$ to $\infty$. However, if there is an arbitrary
parameter in the equation, we can
truncate the series by requiring that
$n\geq 0$. In this manner, we obtain
$\nu$ in terms of parameters of
the differential equation.
In section 4, we show that the Schr\"odinger
equation with inverse fourth and sixth-power
potentials in fact leads to the DCHE and its Ince limit.
Some additional considerations are provided in section 5,
while in Appendix A we discuss the degenerate cases of the
DCHEs, in Appendix B we present an alternative derivation
of the expansions
in Bessel functions, and in Appendix C we rewrite the Leaver-type
solutions for the DCHE in a form appropriate to solve
the Schr\"odinger equation with an inverse fourth-power potential.
\section*{2. Ince's limits for the generalized spheroidal wave equation }
In this section we use transformation rules that permit us to generate
new solutions from a given solution for the Ince limit of the GSWE.
The rules $T_{1}$, $T_{2}$ and $T_{3}$ below
are derived from the ones valid for the GSWE
\cite{eu} and can be checked by
substitution of variables.
If $U(z)=U(B_{1},B_{2},B_{3};
z_{0},q;z)$ denotes one solution for Eq.
(\ref{lindemann}), the effects of
these rules are as follows
\letra
\begin{eqnarray}\begin{array}{l}
T_{1}U(z)=z^{1+B_{1}/z_{0}}
U(C_{1},C_{2},C_{3};z_{0},q;z),\ \ z_{0}\neq0,
\vspace{3mm}\\
T_{2}U(z)=(z-z_{0})^{1-B_{2}-B_{1}/z_{0}}U(B_{1},D_{2},D_{3};
z_{0},q;z), \ \ z_{0}\neq0,
\vspace{3mm}\\
T_{3}U(z)=
U(-B_{1}-B_{2}z_{0},B_{2},
B_{3}-q z_{0};z_{0},-q;z_{0}-z),
\end{array}
\end{eqnarray}
where
\begin{eqnarray}\begin{array}{l}
C_{1}=-B_{1}-2z_{0}, \ \
C_{2}=2+B_{2}+\frac{2B_{1}}{z_{0}},\ C_{3}=B_{3}+
\left(1+\frac{B_{1}}{z_{0}}\right)
\left(B_{2}+\frac{B_{1}}{z_{0}}\right),
\vspace{.3cm}\\
D_{2}=2-B_{2}-\frac{2B_{1}}{z_{0}},\ D_{3}=B_{3}+
\frac{B_{1}}{z_{0}}\left(\frac{B_{1}}{z_{0}}
+B_{2}-1\right).
\end{array}
\end{eqnarray}
We use only $T_{1}$ and
$T_{2}$. The rule $T_{3}$ exchange
the position of the regular singular points $z=z_{0}
\leftrightarrow z=0$ and may be used to get
an alternative representation for the solutions,
but these are not proper
for getting the limit $z_{0}\rightarrow 0$.
In section 2.1 we derive two pairs of solutions for the Ince
limit of the GSWE -- denoted by $ (U_{i\nu}^{0},
U_{i\nu}^{\infty})$, $i=1,2$ -- with a phase parameter $\nu$.
The superscript `zero' indicates
that the series converges in any finite part of the complex
plane, while the superscript `infinity' indicates convergence
for $\vert z\vert>\vert z_{0}\vert$.
The second pair of solutions results from the first by
means of the rule $T_{2}$.
In section 2.2, we truncate these series by taking $n\geq0$
and obtain four pairs of solutions without
phase parameter.
\subsection*{2.1. Solutions with a phase parameter}
Denoting by $b_{n}$ the series coefficients of a solution, their
recurrence relations will have the general form
\antiletra\letra
\begin{eqnarray}\label{rec}
\alpha_{n}b_{n+1}+\beta_{n}b_{n}+\gamma_{n}b_{n-1}=0,\ (-\infty<n<\infty)
\end{eqnarray}
where $\alpha_{n}$, $\beta_{n}$, $\gamma_{n}$
and $b_{n}$ depend on a phase parameter $\nu$
which may be determined from a characteristic
equation given as a sum of two infinite continued fractions, namely,
\begin{eqnarray}
\beta_{0}=\frac{\alpha_{-1}\gamma_{0}}{\beta_{-1}-} \frac{\alpha_{-2}
\gamma_{-1}}{\beta_{-2}-}\frac{\alpha_{-3}\gamma_{-2}}
{\beta_{-3}-}\cdots+\frac{\alpha_{0}\gamma_{1}}{\beta_{1}-}
\frac{\alpha_{1}\gamma_{2}}
{\beta_{2}-}\frac{\alpha_{2}\gamma_{3}}{\beta_{3}-}\cdots .
\end{eqnarray}
For a specific pair of solutions we add
a superscript in each of these quantities.
The first pair of solutions for the Ince limit of the GSWE comes from
the following pair of solutions of the GSWE \cite{eu}
\antiletra\letra
\begin{eqnarray}
\label{eu} \begin{array}{l}
U_{1\nu}^{0}(z)=
e^{i\omega z}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}F\left(\frac{B_{2}}{2}-n-\nu-1,
n+\nu+\frac{B_{2}}{2};B_{2}+\frac{B_{1}}{z_{0}};
1-\frac{z}{z_{0}}\right),
\vspace{3mm}\\
U_{1\nu}^{\infty}(z) =e^{i\omega z}z^{1-(B_{2}/2)}
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}
(-2i\omega z)^{n+\nu}
\Psi(n+\nu+1+i\eta,2n+2\nu+2;-2i\omega z),
\end{array}\end{eqnarray}
where $F(a,b;c;y)$ and $\Psi(a,b;y)$ denote, respectively,
the hypergeometric functions and the irregular confluent
hypergeometric functions \cite{abramowitz,erdelyi1}. The solution
$U_{1\nu}^{0}$ converges for any finite $z$, whereas
$U_{1\nu}^{\infty}$ converges for $\mid z\mid>\mid z_{0}\mid$.
In the recurrence relations (\ref{rec}) for $b_{n}^{(1)}$ we have
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{(1)} = i\omega z_{0}\frac{\left(n+\nu+2-\frac{B_{2}}{2}\right)
\left(n+\nu+1-\frac{B_{2}}{2}-\frac{B_{1}}{z_{0}}\right)
\left(n+\nu+1-i\eta\right)}
{2(n+\nu+1)\left(n+\nu+\frac{3}{2}\right)},
\vspace{3mm} \\
\beta_{n}^{(1)} = -B_{3}-\eta \omega z_{0}-(n+\nu+1-\frac{B_{2}}{2})
(n+\nu+\frac{B_{2}}{2})
-\frac{\eta \omega z_{0}\left(\frac{B_{2}}{2}-1\right)
\left(\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)}
{(n+\nu)(n+\nu+1)},
\vspace{.3cm} \\
\gamma_{n}^{(1)} = -i\omega z_{0}\frac{\left(n+\nu+\frac{B_{2}}{2}-1\right)
\left(n+\nu+\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)
(n+\nu+i\eta)}
{2\left(n+\nu-\frac{1}{2}\right)(n+\nu)}.
\end{array}
\end{eqnarray}
Note that $\nu$
cannot be integer or half-integer in order
to avoid vanishing denominators in
the coefficients of the recurrence relations.
Moreover, for an integer or half-integer $\nu$,
we would have two equal hypergeometric or
confluent hypergeometric
functions (for different values of $n$),
contrary to the hypothesis that all the terms
of the series are independent. On the other
hand, the hypergeometric functions
are not defined if $B_{2}+(B_{1}/z_{0})$
is zero or a negative integer. Nonetheless,
a transformation rule supplies another solution
which is valid for these values of $B_{2}+(B_{1}/z_{0})$.
The three-term recurrence relations (\ref{rec}) constitute
a infinite system of homogeneous linear equations for which
nontrivial solutions for the coefficients $b_{n}$ demand
that the determinant of respective tridiagonal matrix vanishes.
Equivalently, the characteristic equation must be satisfied and
this is a condition necessary also to assure the convergence of the series
by means of a Poincar\'e-Perron theorem \cite{gautschi}.
However, there are two possibilities to satisfy this requirement.
On the one hand, if there is some free constant in the differential
equation, that constant must be determined so that the characteristic
equation is fulfilled for the admissible values of $\nu$ (that is,
neither integer nor half-integer). In this case, the freedom of
choosing $\nu$ may be used in two different ways: (i) to
obtain two-sided solutions ($-\infty<n<\infty$) by ascribing
appropriate values for $\nu$, or (ii) to obtain one-sided
solutions by choosing $\nu$ such that $n\geq 0$. At the end
of the present section, we use the first
alternative to rederive some Poole's solutions \cite{poole,poole2} for
the Mathieu equation, having period $2\pi m$, where $m$ is any integer
equal or greater than 2. In section 2.2, we use the second
alternative for the general case.
These latter solutions afford solutions with period $\pi$ or $2\pi$
for the Mathieu equation, in contrast to the solutions obtained
in the first alternative.
On the other hand, if there is no arbitrary parameter in the
differential equation, the parameter $\nu$ takes the
role of free parameter in the sense that it must be
adjusted to ensure the validity of the characteristic equation
and, consequently, the convergence of the series. By this
reason, $\nu$ is also called characteristic index or parameter
\cite{buhring2}. Examples of equations requiring a phase
parameter are discussed in section 4.
From the above pair of solutions for the GSWE,
by using the Ince limit (\ref{limits}), we readily find the solution
$U_{1\nu}^{0}(z)$ written in the first pair below.
To get the Ince limit of the solution $U_{1\nu}^{\infty}(z)$,
we define $c_{n}$ as
\begin{eqnarray*}
b_{n}^{(1)}=(i\eta)^{n+\nu}\Gamma(i\eta-n-\nu)c_{n}.
\end{eqnarray*}
This imply that
\begin{eqnarray*}
U_{1\nu}^{\infty}(z) =e^{i\omega z}z^{1-(B_{2}/2)}
\displaystyle \sum_{n=-\infty}^{\infty}c_{n}\Gamma(i\eta-n-\nu)
(-qz)^{n+\nu}
\Psi\left(n+\nu+1+i\eta,2n+2\nu+2;-\frac{qz}{i\eta}\right),
\end{eqnarray*}
where $q=-2\eta \omega$. The recurrence relations for $c_{n}$
are
\begin{eqnarray*}
\bar{\alpha}_{n}c_{n+1}+\beta_{n}^{(1)}c_{n}+\bar{\gamma}_{n}c_{n-1}=0,\ (-\infty<n<\infty)
\end{eqnarray*}
with
\antiletra
\begin{eqnarray*}
\bar{\alpha}_{n}=\frac{i\eta}{i\eta-n-\nu-1}\alpha_{n}^{(1)}, \
\bar{\gamma}_{n}=\frac{i\eta-\nu-1}{i\eta}\gamma_{n}^{(1)}.
\end{eqnarray*}
On the other hand, we have \cite{erdelyi1}
\begin{eqnarray}
\lim_{a\rightarrow \label{K} \infty}[\Gamma(a+1-c)\Psi(a,c;x/a)]=
2x^{(1-c)/2}K_{c-1}(2\sqrt{x})
\end{eqnarray}
where $K_{\lambda}(\xi)$ denotes the
modified Bessel function of the second
kind \cite{luke} whose definition in terms of
irregular confluent hypergeometric functions
is \cite{erdelyi1}
\begin{eqnarray}\label{2.8}
K_{\lambda}(\xi)=K_{-\lambda}(\xi)=\sqrt{\pi}\ e^{-\xi}(2\xi)^{\lambda}
\Psi\left(\lambda+\frac{1}{2},2\lambda+1;2\xi\right).
\end{eqnarray}
Then, using (\ref{K}) we find that for $i\eta\rightarrow \infty$
($n$ fixed and $q=$constant)
\begin{eqnarray*}\begin{array}{l}
\Gamma(i\eta-n-\nu)
(-qz)^{n+\nu}
\Psi\left(n+\nu+1+i\eta,2n+2\nu+2;-\frac{qz}{i\eta}\right)
\rightarrow 2(-qz)^{1/2}K_{2n+2\nu+1}(\pm 2i\sqrt{qz}),
\vspace{3mm}\\
\lim\bar{\alpha}_{n}\rightarrow \lim\alpha_{n}^{(1)},\ \
\lim\bar{\gamma}_{n}\rightarrow \lim\gamma_{n}^{(1)}\Rightarrow
\lim c_{n}\rightarrow \lim b_{n}^{(1)}.
\end{array}\end{eqnarray*}
Using these results, we find the Ince limit of $U_{1\nu}^{\infty}$,
written in the first pair below. Although this is a formal derivation, the
solution may be checked by inserting it into Eq. (\ref{lindemann})
(see Appendix B). In addition, from the relation \cite{luke}
\begin{eqnarray}\label{21}
\lim_{\mid \xi\mid\rightarrow \infty}K_{\lambda}(\xi)\sim
\sqrt{\frac{\pi}{2\xi}}\ e^{-\xi}, \ \ -\frac{3\pi}{2}<\arg\xi<
\frac{3\pi}{2}
\end{eqnarray}
we see that the expansions in series of
Bessel functions have the behavior given by
\begin{eqnarray*}
\lim_{z\rightarrow \infty}U_{j\nu}^{\infty}(z)\sim
e^{\pm 2i\sqrt{qz}}z^{(1/4)-(B_{2}/2)}, \ \ -\frac{3\pi}{2}
<\arg(\pm2i\sqrt{qz})<
\frac{3\pi}{2},\ \ (j=1,2)
\end{eqnarray*}
in accordance with Eq. (\ref{thome2}).
The second pair of solutions follows
from the first one through the rule $T_{2}$, as mentioned before.
Moreover, solutions for the Mathieu
equation are obtained
by using Eqs. (\ref{mathieu2}) and by
noting that in this case the hypergeometric
functions can be rewritten in terms of trigonometric functions.\\
\noindent
{\bf First pair}
\letra
\begin{eqnarray}
\label{s1}
\begin{array}{l}
U_{1\nu}^{0}(z)=
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}F\left(\frac{B_{2}}{2}-n-\nu-1,
n+\nu+\frac{B_{2}}{2};B_{2}+\frac{B_{1}}{z_{0}};
1-\frac{z}{z_{0}}\right),
\vspace{.3cm}\\
U_{1\nu}^{\infty}(z) =z^{(1-B_{2})/2}
\displaystyle \sum_{n=-\infty}^{\infty} b_{n}^{(1)}
K_{2n+2\nu+1}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
where in the recurrence relations (\ref{rec})
\begin{eqnarray}\label{apB}
\begin{array}{l}
\alpha_{n}^{(1)} = q z_{0}\frac{\left(n+\nu+2-\frac{B_{2}}{2}\right)
\left(n+\nu+1-\frac{B_{2}}{2}-\frac{B_{1}}{z_{0}}\right)}
{(n+\nu+1)\left(n+\nu+\frac{3}{2}\right)},
\vspace{.2cm} \\
\beta_{n}^{(1)} = 4B_{3}-2q z_{0}+4\left(n+\nu+1-\frac{B_{2}}{2}\right)
\left(n+\nu+\frac{B_{2}}{2}\right)
-2q z_{0}\frac{\left(\frac{B_{2}}{2}-1\right)
\left(\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)}
{(n+\nu)(n+\nu+1)},
\vspace{.3cm} \\
\gamma_{n}^{(1)} = q z_{0}\frac{\left(n+\nu+\frac{B_{2}}{2}-1\right)
\left(n+\nu+\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)}
{\left(n+\nu-\frac{1}{2}\right)(n+\nu)}.
\end{array}
\end{eqnarray}
If $B_{2}+(B_{1}/z_{0})$ is zero or a negative
integer we have the solution $U_{2\nu}^{0}$
instead of $U_{1\nu}^{0}$.
For the Mathieu equation we use Eqs. (\ref{mathieu2}) and
the formula \cite{abramowitz}
\antiletra
\begin{eqnarray*}
\label{hyper1}
F\left[-a,a; (1/2);\sin^2(\sigma u)\right]=\cos(2a\sigma u).
\end{eqnarray*}
Thence, we obtain even solutions with respect to
$u$, namely,
\letra
\begin{eqnarray}
\begin{array}{ll}
W_{1\nu}^{0}(u)=
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}\cos[(2n+2\nu+1)\sigma u],& \
|\cos(\sigma u)|< \infty,
\vspace{.3cm}\\
W_{1\nu}^{\infty}(u) =
\displaystyle \sum_{n=-\infty}^{\infty} b_{n}^{(1)}
K_{2n+2\nu+1}\left[\pm2i\sqrt{q}\cos(\sigma u)\right],&
|\cos(\sigma u)|> 1,
\end{array}
\end{eqnarray}
with the simplified recurrence relations ($a=2q-4B_{3}$)
\begin{eqnarray}
qb_{n+1}^{(1)}+\left[\left(
2n+2\nu+1\right)^2-a\right]b_{n}^{(1)}+qb_{n-1}^{(1)}=0.
\end{eqnarray}
\noindent
{\bf Second pair}
\antiletra
\letra
\begin{eqnarray}
\begin{array}{l}
U_{2\nu}^{0}(z)=(z-z_{0})^{1-B_{2}
-\frac{B_{1}}{z_{0}}}\ z^{1+\frac{B_{1}}{z_{0}}}
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(2)}\times
\vspace{.3cm}\\
\hspace{1.5cm}F\left(-n-\nu-\frac{B_{2}}{2}+1,
n+\nu+2-\frac{B_{2}}{2};
2-B_{2}-\frac{B_{1}}{z_{0}};1-\frac{z}{z_{0}}\right),
\vspace{.3cm}\\
U_{2\nu}^{\infty}(z)=(z-z_{0})^{1-B_{2}-\frac{B_{1}}{z_{0}}}
\ z^{\frac{B_{1}}{z_{0}}+\frac{B_{2}}{2}-\frac{1}{2}}
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(2)}
K_{2n+2\nu+1}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
where
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{(2)}= qz_{0}\frac{\left(n+\nu+\frac{B_{2}}{2}\right)
\left(n+\nu+1+\frac{B_{2}}{2}+
\frac{B_{1}}{z_{0}}\right)}{(n+\nu+1)
\left(n+\nu+\frac{3}{2}\right)},
\ \beta_{n}^{(2)}= \beta_{n}^{(1)},
\vspace{.3cm} \\
\gamma_{n}^{(2)}= q z_{0}\frac{\left(n+\nu+1-\frac{B_{2}}{2}\right)
\left(n+\nu-\frac{B_{2}}{2}-
\frac{B_{1}}{z_{0}}\right)}
{\left(n+\nu-\frac{1}{2}\right)(n+\nu)},
\end{array}
\end{eqnarray}
in the recurrence relations (\ref{rec}) for $b_{n}^{(2)}$.
If $B_{2}+(B_{1}/z_{0})$ is a positive integer
equal or greater than $2$ we have the
solution $U_{1\nu}^{0}$ instead of
$U_{2\nu}^{0}$. Note that, in writing the solution
$U_{2\nu}^{0}$, we have used the relation
\antiletra
\begin{eqnarray}\label{hyp2}
F(a,b;c;y)=(1-y)^{c-a-b}F(c-a,c-b;c;y).
\end{eqnarray}
For the Mathieu equation we use the relation
\cite{abramowitz}
\begin{eqnarray*}
\label{hyper2}
F\left(a,1-a;\frac{3}{2};
\sin^2(\sigma u)\right)=\frac{\sin[(2a-1)\sigma u]}
{(2a-1)\sin(\sigma u)}
\end{eqnarray*}
and, in addition, define
$c_{n}$ as $b_{n}^{(2)}=(2n+2\nu+1)c_{n}$.
So, we find that the recurrence relations
for $c_{n}$ become identical to the ones
for $b_{n}^{(1)}$, giving the odd solutions
\begin{eqnarray}
\begin{array}{ll}
W_{2\nu}^{0}(u)=
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}\sin[(2n+2\nu+1)\sigma u],
\vspace{.3cm}\\
W_{2\nu}^{\infty}(u) =\tan{(\sigma u)}
\displaystyle \sum_{n=-\infty}^{\infty}\left(2n+2\nu+1
\right) b_{n}^{(1)}
K_{2n+2\nu+1}\left[\pm2i\sqrt{q}\cos(\sigma u)\right],
\end{array}
\end{eqnarray}
where $|\cos(\sigma u)|< \infty$ and
$|\cos(\sigma u)|> 1$, respectively.
As we have explained earlier, if there is a free parameter
in the differential equation, it is possible to satisfy the
characteristic equation for any noninteger or half-integer $\nu$.
We use this fact to rederive some Poole's solutions \cite{poole,poole2}
to the Mathieu equation. For this, in the previous $W_{1\nu}^{0}(u)$
and $W_{1\nu}^{0}(u)$ we take
%
\begin{eqnarray}
2\nu+1=l/m, \ \ \sigma=1,
\end{eqnarray}
where $l$ and $m$ are integers prime to one another, $l<m$.
Then, we find the two-sided Poole solutions $W_{1}^{P}(u)$
and $W_{1}^{P}(u)$ given by
%
\begin{eqnarray}
W_{1}^{P}(u)=
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}\cos
\left[ \left( 2n+\frac{l}{m}\right) u\right] ,\ \
W_{2}^{P}(u)=
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}\sin
\left[ \left( 2n+\frac{l}{m}\right) u\right] .
\end{eqnarray}
The first is even with respect to $u$ and the second is odd,
and both of them have period $2\pi m$, $m>1$. Since they have
the same series coefficients, we can combine them to
find another Poole solution, that is,
\begin{eqnarray}
W^{P}(u)=
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}\exp
\left[ i\left( 2n+\frac{l}{m}\right) u\right] .
\end{eqnarray}
Furthermore, for an arbitrary $\nu$ we find
\begin{eqnarray}
W(u)=
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}\exp
\left[ i\left( 2n+2\nu+1\right) u\right] ,
\end{eqnarray}
which is also a solution already known in the literature
\cite{abramowitz, poole2}.
\subsection*{2.2. Solutions without phase parameter}
Now we truncate the solutions obtained in section 2.1
by taking $n\geq 0$. This gives $\nu$ in terms of some parameters
of the differential equation. The resulting solutions are
convergent only if there is a free parameter to be determined
from the characteristic equation.
This truncation reverses the procedure by which the
solution $U_{1\nu}^{0}(z)$ for the GSWE, given
in Eq. (\ref{eu}), was obtained. Indeed, that solution
was constructed \cite{eu2} as a generalization of an one-sided series
of Jacobi polynomials, constructed by Fackerell and Crossman
\cite{fackerell} to solve the angular Teukolsky equations of
the relativistic astrophysics.
Despite this, the truncated solutions found in \cite{eu} are more general
than the Fackerell-Crossman ones because no particular
values are attached to the parameters of the GSWE
and also because the truncation was extended to the Leaver
expansion $U_{1\nu}^{\infty}(z)$. I addition, these one-sided
series are suitable to get solutions in finite series,
the so called quasi-polynomial solutions. In effect, a solution
whose coefficients $b_{n}$ obey recurrence relations as
\begin{eqnarray*}
\alpha_{n}b_{n+1}+\beta_{n}b_{n}+\gamma_{n}b_{n-1}=0,
\ \ n\geq0,\ \ b_{-1}=0
\end{eqnarray*}
becomes a quasi-polynomial solution with $0\leq n\leq N-1$ whenever
$\gamma_{N}=0$ for some $n=N$ \cite{arscott2} .
For the truncated solutions -- denoted by
($U_{i}^{0},U_{i}^{\infty}$), $i=1,2,3,4$ --
the recurrence relations and the
characteristic equations have one of
the three forms written below. The first
case ($\alpha_{-1}=0$) is the general
one and the others ($\alpha_{-1}\neq 0$)
may occur only for special cases.
\begin{eqnarray}
&&\left.
\begin{array}{l}
\alpha_{0}b_{1}+\beta_{0}b_{0}=0,
\vspace{.2cm} \\
\alpha_{n}b_{n+1}+\beta_{n}b_{n}+
\gamma_{n}b_{n-1}=0\ (n\geq1),
\end{array}\right\} \Rightarrow
\beta_{0}=\frac{\alpha_{0}\gamma_{1}}{\beta_{1}-}\ \frac{\alpha_{1}
\gamma_{2}}
{\beta_{2}-}\ \frac{\alpha_{2}\gamma_{3}}{\beta_{3}-}\cdots.
\label{r1a}
\end{eqnarray}
\begin{eqnarray}
\left.
\begin{array}{l}
\alpha_{0}b_{1}+\beta_{0}b_{0}=0,
\vspace{.2cm} \\
\alpha_{1}b_{2}+\beta_{1}b_{1}+\left[
\alpha_{-1}+\gamma_{1}\right]b_{0}=0,
\vspace{.2cm} \\
\alpha_{n}b_{n+1}+\beta_{n}b_{n}+\gamma_{n}
b_{n-1}=0\ (n\geq2),
\end{array}\right\}\Rightarrow
\beta_{0}=\frac{\alpha_{0}\left[\alpha_{-1}+\gamma_{1}
\right]}
{\beta_{1}-}
\ \frac{\alpha_{1}\gamma_{2}}{\beta_{2}-}\ \frac{\alpha_{2}\gamma_{3}}
{\beta_{3}-}\cdots .
\label{r2a}
\end{eqnarray}
\begin{eqnarray}
\left.
\begin{array}{l}
\alpha_{0}b_{1}+\left[\beta_{0}+\alpha_{-1}
\right]b_{0}=0,
\vspace{.2cm} \\
\alpha_{n}b_{n+1}+\beta_{n}b_{n}
+\gamma_{n}b_{n-1}=0\ (n\geq1),
\end{array}\right\}\Rightarrow
\beta_{0}+\alpha_{-1}=\frac{\alpha_{0}\gamma_{1}}
{\beta_{1}-}\ \frac{\alpha_{1}\gamma_{2}}{\beta_{2}-}
\ \frac{\alpha_{2}\gamma_{3}}
{\beta_{3}-}\cdots .
\label{r3a}
\end{eqnarray}
Note that we have
$n\geq -1$ in $\alpha_{n}$, $n\geq0$ in $\beta_{n}$ and
$n\geq 1$ in $\gamma_{n}$.
These forms for the recurrence relations are the same
that appear in truncation of the expansions (\ref{eu})
for the GSWE \cite{eu}. As a matter of fact,
the solutions of the present section are the Ince limit of
solutions for the GSWE given in section 3 of Ref. \cite{eu}.
However, in order to illustrate how these recurrence relations are
obtained, we insert the solution $U_{1\nu}^{\infty}$ given in (\ref{s1})
into the Ince limit of the GSWE. Then, for $n\geq 0$, from
Eq. (\ref{2.10}) we find that
\begin{eqnarray*}
\displaystyle \sum_{n=0}^{\infty}
\alpha_{n-1}^{(1)}b_{n}^{(1)}
K_{2n+2\nu-1}(\xi)+
\displaystyle \sum_{n=0}^{\infty}
\beta_{n}^{(1)}b_{n}^{(1)}
K_{2n+2\nu+1}(\xi)+
\displaystyle \sum_{n=0}^{\infty}
\gamma_{n+1}^{(1)}b_{n}^{(1)}
K_{2n+2\nu+3}(\xi)=0.
\end{eqnarray*}
Setting $m=n-1$, $m=n$ and $m=n+1$ in the first, second and
third terms, respectively, this equation becomes
\begin{eqnarray}\label{recorrencia}
&&\alpha_{-1}b_{0}K_{2\nu-1}(\xi)+
\left[\alpha_{0}b_{1}+\beta_{0}
b_{0}\right]K_{2\nu+1}(\xi)+
\left[\alpha_{1}b_{2}+\beta_{1}
b_{1}+\gamma_{1}b_{0}\right]K_{2\nu+3}(\xi)
+\nonumber\\
&&
\displaystyle \sum_{n=2}^{\infty}\left[\alpha_{n}b_{n+1}
+\beta_{n}b_{n}+
\gamma_{n}b_{n-1}\right]K_{2n+2\nu+1}(\xi)=0,
\end{eqnarray}
where we have dropped the upper suffixes.
Therefore, if we can choose $\nu$ so
that $\alpha_{-1}=0$, we find the first set
of recurrence relations. However, notice that
\begin{eqnarray*}
\alpha_{-1}=\frac{qz_{0} \left(\nu+1-\frac{B_{2}}{2}\right)
\left(\nu-\frac{B_{1}}{x_{0}}-\frac{B_{2}}{2}\right)
}{2\nu(\nu+1/2)}=0,\ \mbox{if}
\left\{
\begin{array}{l}
\nu=\frac{B_{2}}{2}-1
\ \mbox{for}\ B_{2}\neq 1,2;
\vspace{3mm}\\
\nu=\frac{B_{1}}{x_{0}}+\frac{B_{2}}{2} \
\ \mbox{for} \ \frac{B_{1}}{x_{0}}+\frac{B_{2}}{2}\neq0,
\frac{1}{2}.
\end{array}
\right.
\end{eqnarray*}
Hence we see that there are two possible choices
for $\nu$ and, for each of them we have two
cases in which $\alpha_{-1}$ may differ from
zero. Let us consider only the case $\nu=(B_{2}/2)-1$.
Then, for the exceptional case $B_{2}=1$ ($\nu=-1/2$),
we find $K_{2\nu-1}=K_{2\nu+3}=K_{2}$ (since
$K_{\lambda}=K_{-\lambda}$) and therefore the Bessel
functions in the first and third terms of Eq.
(\ref{recorrencia}) are equal, giving the
recurrence relations
(\ref{r2a}). Similarly, if $B_{2}=2$ ($\nu=0$)
we find $K_{2\nu-1}=K_{2\nu+1}=K_{1}$ and
this leads to the recurrence relations (\ref{r3a}).
In this manner we obtain the first pair given
below. The remaining can be derived from this
by using the transformations rules $T_{1}$ and $T_{2}$ as
\begin{eqnarray*}
\left(U_{1}^{0},U_{1}^{\infty}\right)
\stackrel{T_{1}}{\longleftrightarrow}
\left(U_{2}^{0},U_{2}^{\infty}\right)
\stackrel{T_{2}}{\longleftrightarrow}
\left(U_{3}^{0},U_{3}^{\infty}\right)
\stackrel{T_{1}}{\longleftrightarrow}
\left(U_{4}^{0},U_{4}^{\infty}\right)
\stackrel{T_{2}}{\longleftrightarrow}
\left(U_{1}^{0},U_{1}^{\infty}\right).
\end{eqnarray*}
The condition on each pair is imposed
in order to assure that the special functions
are independent in both solutions;
it guarantees either that there is no
vanishing denominator in the recurrence
relations. Furthermore, we have additional restrictions
on the parameters of the solutions
$U_{i}^{0}$. Thus, if $B_{2}+(B_{1}/z_{0})$
is zero or a negative integer, the
hypergeometric functions are not defined in
$U_{1}^{0}$ and $U_{2}^{0}$ but are defined
in $U_{3}^{0}$ and $U_{4}^{0}$, and vice-versa.
The results for the Mathieu
equations are already known \cite{McLachlan},
but note that the recurrence relations for this case come
from the three Eqs. (\ref{r1a}-\ref{r3a}) above.\\
\noindent
{\bf First pair}: $B_{2}\neq 0,-1,-2,\cdots$. This first pair
corresponds to $\nu=(B_{2}/2)-1$ in
($U_{1\nu}^{ 0},U_{1\nu}^{\infty }$).
\letra
\begin{eqnarray}
\begin{array}{l}
U_{1}^{0}(z)=
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(1)}F\left(-n,
n+B_{2}-1;B_{2}+\frac{B_{1}}{z_{0}};
1-\frac{z}{z_{0}}\right),
\vspace{.3cm}\\
U_{1}^{\infty}(z) =z^{(1-B_{2})/2}
\displaystyle \sum_{n=0}^{\infty} b_{n}^{(1)}
K_{2n+B_{2}-1}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
with the following coefficients
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{(1)} = \frac{q z_{0}\left(n+1\right)
\left(n-\frac{B_{1}}{z_{0}}\right)}
{\left(n+\frac{B_{2}}{2}\right)\left(n+\frac{B_{2}}{2}+\frac{1}{2}
\right)},
\vspace{.2cm} \\
\beta_{n}^{(1)} = 4B_{3}-2q z_{0}+4n\left(n+B_{2}-1\right)
-\frac{2q z_{0}\left(\frac{B_{2}}{2}-1\right)
\left(\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)}
{\left(n+\frac{B_{2}}{2}-1\right)
\left(n+\frac{B_{2}}{2}\right)},
\vspace{.3cm} \\
\gamma_{n}^{(1)} = \frac{q z_{0}\left(n+B_{2}-2\right)
\left(n+B_{2}+\frac{B_{1}}{z_{0}}-1\right)}
{\left(n+\frac{B_{2}}{2}-\frac{3}{2}\right)
\left(n+\frac{B_{2}}{2}-1\right)},
\end{array}
\end{eqnarray}
in the recurrence relations for the
$b_{n}^{(1)}$, namely: Eqs. (\ref{r1a}) if
$B_{2}\neq 1,2$; Eqs. (\ref{r2a}) if $B_{2}=1$;
Eqs. (\ref{r3a}) if $B_{2}=2$.
For the Mathieu equation we find solutions
\antiletra
\letra
\begin{eqnarray}
\begin{array}{ll}
W_{1}^{0}(u)=
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(1)}\cos(2n\sigma u),& |\cos(\sigma u)|< \infty,
\vspace{.3cm}\\
W_{1}^{\infty}(u) =
\displaystyle \sum_{n=0}^{\infty} b_{n}^{(1)}
K_{2n}\left[\pm2i\sqrt{q}\cos(\sigma u)\right],&
|\cos(\sigma u)|> 1,
\end{array}
\end{eqnarray}
with the simplified recurrence relations
\begin{eqnarray}\begin{array}{l}
qb_{1}^{(1)}-ab_{0}^{(1)}=0,
\ \ qb_{2}^{(1)}+\left[4-a\right]b_{1}^{(1)}+2qb_{0}^{(1)}=0,\vspace{.3cm}\\
qb_{n+1}^{(1)}+\left[
4n^2-a\right]b_{n}^{(1)}+qb_{n-1}^{(1)}=0\ (n\geq2).
\end{array}
\end{eqnarray}
These solutions are even with respect to $u$ and, for $\sigma=1$,
the solution $W_{1}^{0}(u)$ has period $\pi$.\\
\noindent
{\bf Second pair}: $(B_{2}/2)+(B_{1}/z_{0})\neq-1,
-3/2,-2,-5/2\cdots$. This pair of solutions can also be obtained by
taking $\nu=(B_{2}/2)+(B_{1}/z_{0})$ in
($U_{1\nu}^{ 0},U_{1\nu}^{0 }$).
\antiletra
\letra
\begin{eqnarray}
\begin{array}{l}
U_{2}^{0}(z)=z^{1+(B_{1}/z_{0})}
\displaystyle \sum_{n=0}^{\infty}b_{n}^{ (2)}
F\left(-n,n+1+B_{2}+\frac{2B_{1}}{z_{0}};B_{2}+
\frac{B_{1}}{z_{0}};1-\frac{z}{z_{0}}\right),
\vspace{.3cm}\\
U_{2}^{\infty}(z)=z^{(1-B_{2})/2}
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(2)}
K_{2n+1+B_{2}+(2B_{1}/z_{0})}\left(\pm2i\sqrt{qz}
\right),
\end{array}
\end{eqnarray}
where
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{^{(2)}}= \frac{qz_{0} (n+1)
\left(n+2+\frac{B_{1}}{x_{0}}\right)}
{\left(n+1+\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)
\left(n+\frac{3}{2}+\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}
\right)},
\vspace{.3cm} \\
\beta_{n}^{^{(2)}}= 4B_{3}-2q z_{0}+4\left(n+1+\frac{B_{1}}{x_{0}}\right)
\left(n+B_{2}+\frac{B_{1}}{x_{0}}\right)
-\frac{2q z_{0}\left(\frac{B_{2}}{2}-1\right)
\left(\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)}
{\left(n+\frac{B_{2}}{2}+\frac{B_{1}}{x_{0}}\right)
\left(n+1+\frac{B_{2}}{2}+\frac{B_{1}}{x_{0}}\right)},
\vspace{.2cm} \\
\gamma_{n}^{^{(2)}}= \frac{q z_{0}\left(n+B_{2}+
\frac{B_{1}}{x_{0}}-1\right)
\left(n+B_{2}+\frac{2B_{1}}{z_{0}}\right)}
{\left(n-\frac{1}{2}+\frac{B_{2}}{2}+\frac{B_{1}}
{z_{0}}\right)
\left(n+\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)},
\end{array}
\end{eqnarray}
in the recurrence relations for $b_{n}^{(2)}$:
Eqs. (\ref{r1a})
if $(B_{2}/2)+(B_{1}/z_{0})\neq 0,
-1/2$; Eqs. (\ref{r2a}) if
$(B_{2}/2)+(B_{1}/z_{0})=
-1/2$; Eqs. (\ref{r3a}) if
$(B_{2}/2)+(B_{1}/z_{0})=0$.
For the Mathieu equation we again have even solutions
\antiletra
\letra
\begin{eqnarray}
\begin{array}{ll}
W_{2}^{0}(u)=
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(2)}\cos[(2n+1)\sigma u],& |\cos(\sigma u)|< \infty,
\vspace{.3cm}\\
W_{2}^{\infty}(u) =
\displaystyle \sum_{n=0}^{\infty} b_{n}^{(2)}
K_{2n+1}\left[\pm2i\sqrt{q}\cos(\sigma u)\right],&
|\cos(\sigma u)|> 1,
\end{array}
\end{eqnarray}
with the recurrence relations
\begin{eqnarray}\begin{array}{l}
qb_{1}^{(2)}+\left[q+1-a\right]b_{0}^{(2)}=0, \vspace{.3cm}\\
qb_{n+1}^{(2)}+\left[\left(
2n+1\right)^2-a\right]b_{n}^{(2)}+qb_{n-1}^{(2)}=0 \ (n\geq 1).
\end{array}
\end{eqnarray}
If $\sigma=1$ the solution $W_{2}^{0}(u)$
has period $2\pi$.\\
\noindent
{\bf Third pair}: $B_{2}\neq 4,5,6,\cdots$.
This corresponds to $\nu=1-(B_{2}/2)$ in ($U_{2\nu}^{0}, U_{2\nu}^{\infty})$.
\antiletra
\letra
\begin{eqnarray}
\begin{array}{l}
U_{3}^{0}(z)=
(z-z_{0})^{1-B_{2}-\frac{B_{1}}{z_{0}}}
z^{1+\frac{B_{1}}{z_{0}}}
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(3)}
F\left(-n,n+3-B_{2};2-B_{2}-\frac{B_{1}}{z_{0}};
1-\frac{z}{z_{0}}\right),
\vspace{3mm}\\
U_{3}^{\infty}(z) =(z-z_{0})^{1-B_{2}-
\frac{B_{1}}{z_{0}}}z^{\frac{B_{1}}{z_{0}}+\frac{B_{2}}{2}-\frac{1}{2}}
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(3)}
K_{2n+3-B_{2}}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
with the coefficients
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{ (3)} = \frac{qz_{0}\
(n+1)
\left(n+2+\frac{B_{1}}{z_{0}}\right)}
{\left(n+2-\frac{B_{2}}{2}\right)\left(n+\frac{5}{2}-\frac{B_{2}}{2}\right)},
\vspace{.2cm} \\
\beta_{n}^{ (3)} = 4B_{3}-2q z_{0}+4(n+1)(n+2-B_{2})
-\frac{2q z_{0}\left(\frac{B_{2}}{2}-1\right)
\left(\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)}
{\left(n+1-\frac{B_{2}}{2}\right)\left(n+2-
\frac{B_{2}}{2}\right)},
\vspace{.3cm} \\
\gamma_{n}^{ (3)} =
\frac{q z_{0}\ \left(n+2-B_{2}\right)
\left(n+1-B_{2}-\frac{B_{1}}{z_{0}}\right)}
{\left(n+\frac{1}{2}-\frac{B_{2}}{2}\right)\left(n+1-\frac{B_{2}}{2}\right)}.
\end{array}
\end{eqnarray}
in the recurrence relations: Eqs. (\ref{r1a}) if
$B_{2}\neq 2,3$; Eqs. (\ref{r2a}) if $B_{2}=3$;
Eqs. (\ref{r3a}) if $B_{2}=2$.
For the Mathieu equation we redefine the coefficients
$b_{n}^{(3)}$ as $b_{n}^{(3)}\rightarrow (2n+2)b_{n}^{(3)}$. Then
we find the odd solutions
\antiletra
\letra
\begin{eqnarray}
\begin{array}{ll}
W_{3}^{0}(u)=
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(3)}\sin[(2n+2)\sigma u],& |\cos(\sigma u)|< \infty,
\vspace{.3cm}\\
W_{3}^{\infty}(u) =\tan{(\sigma u)}
\displaystyle \sum_{n=0}^{\infty}\left(2n+2
\right) b_{n}^{(3)}
K_{2n+2}\left[\pm2i\sqrt{q}\cos(\sigma u)\right],&
|\cos(\sigma u)|> 1,
\end{array}
\end{eqnarray}
with the recurrence relations
\begin{eqnarray}\begin{array}{l}
qb_{1}^{(3)}+\left[4-a\right]b_{0}^{(3)}=0,
\vspace{.3cm}\\
qb_{n+1}^{(3)}+\left[4\left(
n+1\right)^2-a\right]b_{n}^{(3)}+qb_{n-1}^{(3)}=0,
\ (n\geq 1).
\end{array}
\end{eqnarray}
For $\sigma=1$ the solution $W_{3}^{0}(u)$
has period $\pi$.\\
\noindent
{\bf Fourth pair}: $(B_{2}/2)+(B_{1}/z_{0})\neq1,
3/2,2,5/2\cdots$. This can also be obtained by
setting $\nu=-(B_{2}/2)-(B_{1}/z_{0})$ in
($U_{2\nu}^{ 0},U_{2\nu}^{ \infty}$)
\antiletra
\letra
\begin{eqnarray}
\begin{array}{l}
U_{4}^{0}=(z-z_{0})^{1-B_{2}-\frac{B_{1}}{z_{0}}}
\displaystyle \sum_{n=0}^{\infty}b_{n}^{ (4)}
F\left(-n,n+1-B_{2}-\frac{2B_{1}}{z_{0}};2-B_{2}
-\frac{B_{1}}{z_{0}};1-\frac{z}{z_{0}}\right), \vspace{5mm}\\
U_{4}^{\infty} =(z-z_{0})^{1-B_{2}-\frac{B_{1}}{z_{0}}}
z^{\frac{B_{1}}{z_{0}}+\frac{B_{2}}{2}-\frac{1}{2}}\displaystyle \sum_{n=0}^{\infty}b_{n}^{ (4)}
K_{2n+1-B_{2}-(2B_{1}/z_{0})}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
with coefficients
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{ (4)} = \frac{qz_{0}\ (n+1)
\left(n-\frac{B_{1}}{z_{0}}\right)}
{\left(n+1-\frac{B_{2}}{2}-\frac{B_{1}}{z_{0}}\right)
\left(n+\frac{3}{2}-\frac{B_{2}}{2}-\frac{B_{1}}{z_{0}}
\right)},
\vspace{.2cm} \\
\beta_{n}^{ (4)} = 4B_{3}-2q z_{0}+4
\left(n-\frac{B_{1}}{z_{0}}\right)
\left(n-B_{2}+1-\frac{B_{1}}{z_{0}}\right)
-\frac{2q z_{0}\left(\frac{B_{2}}{2}-1\right)
\left(\frac{B_{2}}{2}+\frac{B_{1}}{z_{0}}\right)}
{\left(n-\frac{B_{2}}{2}-\frac{B_{1}}{z_{0}}\right)
\left(n+1-\frac{B_{2}}{2}-\frac{B_{1}}{z_{0}}\right)},
\vspace{.3cm} \\
\gamma_{n}^{ (4)} = \frac{q z_{0}\
\left(n+1-B_{2}-\frac{B_{1}}{z_{0}}\right)
\left(n-B_{2}-\frac{2B_{1}}{z_{0}}\right)}
{\left(n-\frac{1}{2}-\frac{B_{2}}{2}-\frac{B_{1}}
{z_{0}}\right)
\left(n-\frac{B_{2}}{2}-\frac{B_{1}}{z_{0}}\right)},
\end{array}
\end{eqnarray}
in the recurrence relations: Eqs. (\ref{r1a}) if
$(B_{2}/2)+(B_{1}/z_{0})\neq 0, 1/2$;
Eqs. (\ref{r2a}) if $(B_{2}/2)+(B_{1}/z_{0})=1/2$;
Eqs. (\ref{r3a}) if $(B_{2}/2)+(B_{1}/z_{0})=0$.
For the Mathieu equation we redefine
$b_{n}(4)$ according to $b_{n}^{(4)}\rightarrow (2n+1)b_{n}^{(4)}$
and find the odd solutions
\antiletra
\letra
\begin{eqnarray}
\begin{array}{ll}
W_{4}^{0}(u)=
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(4)}\sin[(2n+1)\sigma u],& |\cos(\sigma u)|< \infty,
\vspace{.3cm}\\
W_{4}^{\infty}(u) =\tan{(\sigma u)}
\displaystyle \sum_{n=0}^{\infty}\left(2n+1
\right) b_{n}^{(4)}
K_{2n+1}\left[\pm2i\sqrt{q}\cos(\sigma u)\right],&
|\cos(\sigma u)|> 1,
\end{array}
\end{eqnarray}
with the recurrence relations
\begin{eqnarray}\begin{array}{l}
qb_{4}^{(4)}+\left[1-q-a\right]b_{0}^{(4)}=0,
\vspace{.3cm}\\
qb_{n+1}^{(4)}+\left[\left(
2n+1\right)^2-a\right]b_{n}^{(4)}+qb_{n-1}^{(4)}=0 \ (n\geq1).
\end{array}
\end{eqnarray}
Now, for $\sigma=1$, $W_{4}^{0}(u)$ has period $2\pi$.
\section*{3. Ince's limits for the double-confluent Heun equation}
As in the case of the Ince limit of the GSWE, we have found no solution
in the literature for the Ince limit of the DCHE.
The solutions below are obtained by taking the limit
$z_{0}\rightarrow 0$ (Leaver limit) of the solutions
given in section 2 for the Ince limit of the GSWE.
For this we use the formulas \cite{erdelyi1}
\antiletra\letra
\begin{eqnarray}\label{rel}
&&\lim_{c\rightarrow \infty}F\left(a,b;c;1-\frac{c}{y}\right)=
y^a\Psi(a,a+1-b;y),\\
&&\lim_{\alpha\rightarrow \infty}\left(1+\frac{y}{\alpha}\right)^{\alpha}=e^{y}
\Rightarrow \lim_{z_{0}\rightarrow 0}\left(1-\frac{z_{0}}{z}\right)^{-B_{1}/
z_{0}}=e^{B_{1}/z}.
\end{eqnarray}
Actually, it is not necessary to use the second equation above,
since we can get one pair of solutions as the limit of the first
pair of section 2.1
and, then, generate the other pair by means of the transformation rule
\antiletra
\begin{eqnarray}
\tau U(z)=e^{{B_{1}}/{z}}z^{2-B_{2}}U(-B_{1},4-B_{2},
B_{3}+2-B_{2}; q;z),
\end{eqnarray}
where $U(z)=U(B_{1},B_{2},B_{3}; q;z)$ denotes known
solutions of Eq. (\ref{lindemann2}).
On the other hand, to check that the solutions $U_{i}^{0}(z)$ exhibit
the behavior given in Eq. (\ref{z0}) when $z\rightarrow 0$,
we may use the relation \cite{erdelyi1}
\begin{eqnarray}\label{tricomi}
\lim_{\vert y\vert\rightarrow \infty}\Psi(a,b;y)\sim y^{-a}[1+O(|y|^{-1}],
\ \ -\frac{3\pi}{2}<\arg y< \frac{3\pi}{2}.
\end{eqnarray}
\subsection*{3.1. Solutions with a phase parameter}
For the solution $U_{1\nu}^{0}$ of
section 2.1, we find that the limit
of the hypergeometric functions when $z_{0}$
tends to zero, $B_{2}$ and $B_{1}$ being fixed ($c=B_{2}+B_{1}/z_{0}
\rightarrow \infty$), is given by
\begin{eqnarray*}
&&\lim_{z_{0}\rightarrow 0}F\left(n+\nu+\frac{B_{2}}{2},-n-\nu-1+
\frac{B_{2}}{2};B_{2}+\frac{B_{1}}{z_{0}};
1-\frac{z}{z_{0}}\right)\\
&&\propto z^{-\nu-\frac{B_{2}}{2}}
\left(\frac{B_{1}}{z}\right)^{n}\Psi\left(n+\nu+\frac{B_{2}}{2},2n+2\nu+2;
\frac{B_{1}}{z}\right).
\end{eqnarray*}
Then, considering also the solution $U_{1\nu}^{\infty}$ and the
limits for the coefficients in the recurrence relations, we get the first pair
of solutions with a phase parameter $\nu$ (different
of integer or half-integer). The rule $\tau$ leads
to the second pair.\\
\noindent
{\bf First pair}
\letra
\begin{eqnarray}
\begin{array}{l}
U_{1\nu}^{0}(z)=
z^{-\nu-\frac{B_{2}}{2}}\displaystyle \sum_{n=-\infty}^{\infty}
b_{n}^{(1)}
\left(\frac{B_{1}}{z}\right)^{n}\Psi\left(n+\nu+
\frac{B_{2}}{2},2n+2\nu+2;
\frac{B_{1}}{z}\right),
\vspace{.3cm}\\
U_{1\nu}^{\infty}(z) =z^{(1-B_{2})/2}
\displaystyle \sum_{n=-\infty}^{\infty} b_{n}^{(1)}
K_{2n+2\nu+1}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
where in the recurrence relations (\ref{rec})
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{(1)} = - \frac{qB_{1}\left(n+\nu+2-\frac{B_{2}}{2}\right)}
{(n+\nu+1)\left(n+\nu+\frac{3}{2}\right)},
\vspace{.2cm} \\
\beta_{n}^{(1)} = 4B_{3}+4\left(n+\nu+1-\frac{B_{2}}{2}\right)
\left(n+\nu+\frac{B_{2}}{2}\right)
-\frac{q B_{1}\left(B_{2}-2\right)}
{(n+\nu)(n+\nu+1)},
\vspace{.3cm} \\
\gamma_{n}^{(1)} = \frac{qB_{1}\left(n+\nu+\frac{B_{2}}{2}-1\right)}
{\left(n+\nu-\frac{1}{2}\right)(n+\nu)}.
\end{array}
\end{eqnarray
\noindent
{\bf Second pair}
\antiletra
\letra
\begin{eqnarray}
\begin{array}{l}
U_{2\nu}^{0}(z)=e^{{B_{1}}/{z}}z^{-\nu-
\frac{B_{2}}{2}}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(2)}
\left(-\frac{B_{1}}{z}\right)^{n}\Psi\left(n+\nu+2-
\frac{B_{2}}{2},2n+2\nu+2;
-\frac{B_{1}}{z}\right),
\vspace{.3cm}\\
U_{2\nu}^{\infty}(z)=e^{{B_{1}}/{z}}z^{(1-B_{2})/2}
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(2)}
K_{2n+2\nu+1}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
where
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{^{(2)}}= \frac{qB_{1}\left(n+\nu+\frac{B_{2}}{2}\right)}
{(n+\nu+1)\left(n+\nu+\frac{3}{2}\right)}, \
\beta_{n}^{^{(2)}}= \beta_{n}^{(1)},\
\gamma_{n}^{^{(2)}}= -\frac{qB_{1}\left(n+\nu+1-\frac{B_{2}}{2}\right)}
{\left(n+\nu-\frac{1}{2}\right)(n+\nu)},
\end{array}
\end{eqnarray}
in the recurrence relations (\ref{rec}) for $b_{n}^{(2)}$.
\subsection*{3.2. Solutions without phase parameter}
These solutions may be derived by truncating the solutions
of section 3.1. In this case, we see that there is only one choice for
$\nu$ in each pair. Alternatively, the solutions can be found
by applying the Leaver procedure to the first and third pairs of
section 2.2.\\
\noindent
{\bf First pair}: $B_{2}\neq 0,-1,-2,\cdots$.
This corresponds to $\nu=(B_{2}/2)-1$ in $\left(U_{1\nu}^{ 0},
U_{1\nu}^{\infty }\right)$.
\antiletra
\letra
\begin{eqnarray}
\begin{array}{l}
U_{1}^{0}(z)=z^{1-B_{2}}
\displaystyle \sum_{n=0}^{\infty}b_{n}^{(1)}\left(\frac{B_{1}}
{z}\right)^n\Psi\left(n+B_{2}-1,2n+B_{2};
\frac{B_{1}}{z}\right),
\vspace{.3cm}\\
U_{1}^{\infty}(z) =z^{(1-B_{2})/2}
\displaystyle \sum_{n=0}^{\infty} b_{n}^{(1)}
K_{2n+B_{2}-1}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
with the following coefficients
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{(1)} = - \frac{q B_{1}\left(n+1\right)}
{\left(n+\frac{B_{2}}{2}\right)\left(n+\frac{B_{2}}{2}+\frac{1}{2}
\right)},
\vspace{.2cm} \\
\beta_{n}^{(1)} = 4B_{3}+4n\left(n+B_{2}-1\right)
-\frac{q B_{1}\left(B_{2}-2\right)}
{\left(n+\frac{B_{2}}{2}-1\right)
\left(n+\frac{B_{2}}{2}\right)},
\vspace{.3cm} \\
\gamma_{n}^{(1)} = \frac{q B_{1}\left(n+B_{2}-2\right)}
{\left(n+\frac{B_{2}}{2}-\frac{3}{2}\right)
\left(n+\frac{B_{2}}{2}-1\right)}.
\end{array}
\end{eqnarray}
in the recurrence relations for the
$b_{n}^{(1)}$: Eqs. (\ref{r1a}) if
$B_{2}\neq 1,2$; Eqs. (\ref{r2a}) if
$B_{2}=1$; Eqs. (\ref{r3a}) if $B_{2}=2$.\\
\noindent
{\bf Second pair}: $B_{2}\neq 4,5,6,\cdots$.
It corresponds to $\nu=1-(B_{2}/2)$ in $\left(U_{2\nu}^{ 0},
U_{2\nu}^{\infty }\right)$ but can also be obtained from the first
pair via the rule $\tau$.
\antiletra
\letra
\begin{eqnarray}
\begin{array}{l}
U_{2}^{0}(z)=e^{{B_{1}}/{z}}z^{-1}\displaystyle \sum_{n=0}^{\infty}b_{n}^{(2)}
\left(-\frac{B_{1}}{z}\right)^{n}\Psi\left(n+3-B_{2},2n+4-B_{2};
-\frac{B_{1}}{z}\right),
\vspace{.3cm}\\
U_{2}^{\infty}(z)=e^{{B_{1}}/{z}}z^{(1-B_{2})/2}\displaystyle \sum_{n=0}^{\infty}b_{n}^{(2)}
K_{2n+3-B_{2}}\left(\pm2i\sqrt{qz}\right),
\end{array}
\end{eqnarray}
where
\begin{eqnarray}
\begin{array}{l}
\alpha_{n}^{ (3)} = \frac{qB_{1}\
(n+1)}
{\left(n+2-\frac{B_{2}}{2}\right)\left(n+\frac{5}{2}-\frac{B_{2}}{2}\right)},
\vspace{.2cm} \\
\beta_{n}^{ (3)} = 4B_{3}+4(n+1)(n+2-B_{2})
-\frac{qB_{1}\left(B_{2}-2\right)}
{\left(n+1-\frac{B_{2}}{2}\right)\left(n+2-
\frac{B_{2}}{2}\right)},
\vspace{.3cm} \\
\gamma_{n}^{ (3)} =
-\frac{q B_{1}\ \left(n+2-B_{2}\right)}
{\left(n+\frac{1}{2}-\frac{B_{2}}{2}\right)\left(n+1-\frac{B_{2}}{2}\right)}.
\end{array}
\end{eqnarray}
\antiletra
in the recurrence relations for $b_{n}^{(2)}$: Eqs. (\ref{r1a}) if
$B_{2}\neq 2,3$; Eqs. (\ref{r2a}) if $B_{2}=3$;
Eqs. (\ref{r3a}) if $B_{2}=2$.
\section*{4. Potential applications}
As we have mentioned, the Schr\"odinger equation
with inverse fourth and sixth-power potentials can be
reduced, respectively, to the double-confluent Heun
equation (\ref{dche}) and its Ince limit (\ref{lindemann2}).
Singular potentials like these have appeared in the description
of intermolecular forces \cite{frank} and in the
scattering of ions by polarizable atoms. For the sake
of illustration, we consider the last problem.
Before discussing these examples, let us present the so called
normal forms of the DCHE, that is, the forms in which there is no
first-order derivative terms in the differential equations. The general
procedure for this, consists in writing the equation as
\begin{eqnarray*}
\frac{d^2 U}{dz^2}+p(z)\frac{dU}{dz}+q(z)U=0.
\end{eqnarray*}
Then, the substitution
\begin{eqnarray*}
U(z)=F(z)\exp{\left(-\frac{1}{2}\int p(z)dz\right)}
\end{eqnarray*}
gives a first normal form, namely,
\begin{eqnarray*}
\frac{d^2 F}{dz^2}+I(z)F=0,\ \ I(z)=q(z)-\frac{1}{2}\frac{dp(z)}{dz}
-\frac{1}{4}[p(z)]^2.
\end{eqnarray*}
From this, other normal forms are obtained by the transformations
\begin{eqnarray*}
z=h(\vartheta),\ \ F(z)=\sqrt{\frac{dh}{d\vartheta}}\ G(\vartheta)
\end{eqnarray*}
which yield
\begin{eqnarray*}
\frac{d^2 G}{d\vartheta^2}+J(\vartheta)G=0,\ \ J(\vartheta)=I[h(\vartheta)]\left( \frac{dh}{d\vartheta}\right) ^2+
\frac{1}{2}\frac{d^3h}{d\vartheta^3}/
\frac{dh}{d\vartheta}-
\frac{3}{4}\left( \frac{d^2h}{d\vartheta^2}
/ \frac{dh}{d\vartheta}\right)^2 .
\end{eqnarray*}
By employing this procedure, Lemieux and Bose \cite{lemieux}
have derived several normal forms for the general Heun equation
and its confluent cases, excepting the triconfluent equation.
These forms are useful to recognize whether a given equation
belongs to the Heun class. Nevertheless, to find the solutions
for the equation, we have to come back to the form for which
the solutions were established, as below.
The three Lemieux-Bose normal forms for the DCHE, together
with the transformations of variables, are the following:
\begin{eqnarray}\label{N1}
\begin{array}{l}
U(z)=z^{-B_{2}/2}e^{B_{1}/(2z)}F(z),
\vspace{3mm}\\
\frac{d^2F}{dz^2}+\left[\omega^{2}-\frac{2\eta \omega}{z}+\frac{1}{z^2}
\left(B_{3}-\frac{B_{2}^{2}}{4}+\frac{B_{2}}{2}\right)+\frac{B_{1}}{z^3}
\left(1-\frac{B_{2}}{2}\right)
-\frac{B_{1}^2}{4z^4}\right]F=0;
\end{array}
\end{eqnarray}
\begin{eqnarray}\label{N2}
\begin{array}{l}
z= \rho^2, \ \ U(z)=\rho^{(1-2B_{2})/2}e^{B_{1}/(2\rho^2)}G(\rho)
\Leftrightarrow G(\rho)=z^{(2B_{2}-1)/4} e^{-B_{1}/(2z)}U(z),
\vspace{3mm}\\
\frac{d^2G}{d\rho^2}+\left[4\omega^{2}\rho^2-8\eta \omega+\frac{4}{\rho^2}
\left(B_{3}-\frac{B_{2}^{2}}{4}+\frac{B_{2}}{2}-\frac{3}{16}\right)+
\frac{4B_{1}}{\rho^4}
\left(1-\frac{B_{2}}{2}\right)
-\frac{B_{1}^2}{\rho^6}\right]G=0;
\end{array}
\end{eqnarray}
\begin{eqnarray}\label{N3}
\begin{array}{l}
z=e^{\lambda u},\ \
U(z)=H(u)\exp\left[ {\frac{1}{2}\lambda(1-B_{2}) u+\frac{B_{1}}{2}e^{-\lambda u}}\right] \ \Leftrightarrow
\ H(u)=z^{(B_{2}-1)/2}e^{-B_{1}/(2z)}U(z),
\vspace{3mm}\\
\frac{d^2H}{du^2}+
\lambda^2\left[ B_{3}-\left( \frac{1-B_{2}}{2}\right)^2
-\frac{B_{1}^2}{4}e^{-2\lambda u} -
B_{1}\left( \frac{B_{2}}{2}-1\right)e^{-\lambda u}-
2\eta \omega e^{\lambda u} +\omega^2e^{2\lambda u}\right] H=0,
\end{array}
\end{eqnarray}
where $\lambda$ is a constant at our disposal, for example,
$\lambda=1$ or $\lambda=i$. Note that, since these transformations
involve neither $\eta$ nor $\omega$, their Ince limits
are obtained by putting $\omega^2=0$ and $2\eta\omega=-q$.
Now we proceed with the scattering problem. The radial
part $R(r)=\chi(r)/r$ of the wave function for the
Schr\"{o}dinger equation in three dimensions,
for a particle with mass $\mu$ and energy $E$, is
\begin{eqnarray}\label{radial}
\frac{d^2\chi(r)}{dr^2}+\left[k^2-\frac{l(l+1)}{r^2}-
\frac{2\mu}{\hbar^{2}}V(r)
\right]\chi(r)=0,
\end{eqnarray}
where $k^2=2\mu E/\hbar^2$, $l$ is the angular momentum and $V(r)$
is the potential. Now, according to Kleinman,
Hahn and Spruch \cite{kleinman}, for the interaction of a light
particle of charge $e'$ with a fixed atom of charge $Z\overline{e}$
containing $z'$ electrons, we have
\begin{eqnarray}
V(r)=\frac{(Z-z')\overline{e}e'}{r}-\frac{\alpha_{1}'{(e')^{2}}}{2r^4}-
\left(\alpha_{2}'-6a_{0}\beta_{1}'\right)
\frac{(e')^2}{2r^6},
\end{eqnarray}
where $r$ is the distance from the incident ion to the atom,
$a_{0}=\hbar^2/(\mu\overline{e}^{2} )$ is the Bohr radius,
$\alpha_{1}'$ and $\alpha_{2}'$ are,
respectively, the electric dipole and quadrupole polarizabilities
of the atom and $\beta_{1}'$ is a parameter
resulting from a nonadiabatic correction ($\alpha_{1}'$, $\alpha_{2}'$
and $\beta_{1}'$ are constants which describe the properties
of the target only). For this potential, the Schr\"odinger
equation becomes
\begin{eqnarray}\label{radial2}
\frac{d^2\chi}{dr^2}+\left[k^2-\frac{2\mu(Z-z')
\overline{e}e'}{\hbar^{2} r}
-\frac{l(l+1)}{ r^2}
+\frac{\mu\alpha_{1}'{(e')^{2}}}{\hbar^{2}r^4}+
\frac{\mu\left(\alpha_{2}'-6a_{0}\beta_{1}'
\right)(e')^2}{\hbar^{2} r^6}\right]\chi=0.
\end{eqnarray}
Therefore, for neutral targets ($Z=z'$) this is a particular
case of the Ince limit of the DCHE, as we see from Eq. (\ref{N2})
with $\omega^2=0$, $2\eta\omega=-q$ and $z=\rho^2=r^2$.
On the hand, if the inverse sixth-power term vanishes
($\alpha_{2}^{'}=6a_{0}\beta_{1}^{'}$),
this radial Schr\"odinger equation is a particular
case of the DCHE as seen from Eq. (\ref{N1}) with $B_{2}=2$,
for neutral or ionized targets.
In both cases the energy of the incident particle ($k^2$) is given
and, consequently, there is no free parameter in these
equations since the other constants are also fixed.
Then, convergent solutions require
a phase parameter $\nu$, analogously to the scattering by
the field of an electric dipole \cite{leaver1}.
To obtain the radial dependence $R(r)$ we must convert
Eq. (\ref{radial2}) into the DCHE (\ref{dche}) and its
limit (\ref{lindemann2}). Below we discuss only the asymptotic
behaviors of the solutions for the each case. By this reason,
we do not write the recurrence relations for the coefficients.\\
\noindent
{\bf Potential with inverse fourth and sixth-power terms}. Eq. (\ref{N2})
suggests the substitutions
\begin{eqnarray*}
\begin{array}{l}
z= r^2, \ \ \chi(r)=e^{-B_{1}/(2r^2)}r^{B_{2}-(1/2)}U(z=r^2)\ \
\mbox{with}
\vspace{3mm}\\
B_{1}=\pm\frac{e'}{\hbar}\sqrt{\mu(6a_{0}\beta_{1}^{'}-
\alpha_{2}^{'})}, \ \ B_{2}=2-\frac{\alpha_{1}^{'}(e')^2}{2\hbar^2
B_{1}}, \ \ (6a_{0}\beta_{1}^{'}\neq\alpha_{2}^{'})
\end{array}
\end{eqnarray*}
which transform the Schr\"odinger equation (\ref{radial2}) into
\begin{eqnarray*}
z^2\frac{d^{2}U}{dz^{2}}+(B_{1}+B_{2}z)
\frac{dU}{dz}+
\left[\left( \frac{B_{2}}{2}-\frac{1}{4}\right)
\left( \frac{B_{2}}{2}-\frac{3}{4}\right)-\frac{l(l+1)}{4}+
\frac{k^2}{4}z-\frac{\mu}{2}(Z-z')\sqrt{z}\right]U=0.
\end{eqnarray*}
Then, for $Z\neq z'$, the Schr\"odinger
equation is more general than the Ince limit of DCHE.
However, assuming a neutral target, we may form two
pairs of solutions according to
\begin{eqnarray}
R_{i\nu}(r)=\frac{1}{r}
\chi_{i\nu}(r) =e^{-B_{1}/(2r^2)}r^{B_{2}-(3/2)}
U_{i\nu}(z=r^2) \ (i=1,2)
\end{eqnarray}
where on the right-hand side the $U_{i\nu}$ represent the solutions
with a phase parameter for the Ince limit of the DCHE,
given in section 3.1. Then, taking into account that for this
case $q=k^2/4=\mu E/(2\hbar^2)$ and $z=r^2$, we find
\begin{eqnarray}
\begin{array}{l}
R_{1\nu}^{0}(r)=e^{-B_{1}/(2r^2)}
r^{-2\nu-\frac{3}{2}}\displaystyle \sum_{n=-\infty}^{\infty}
b_{n}^{(1)}
\left(\frac{B_{1}}{r^2}\right)^{n}\Psi\left(n+\nu+
\frac{B_{2}}{2},2n+2\nu+2;
\frac{B_{1}}{r^2}\right),
\vspace{.3cm}\\
R_{1\nu}^{\infty}(r) =e^{-B_{1}/(2r^2)}r^{-1/2}
\displaystyle \sum_{n=-\infty}^{\infty} b_{n}^{(1)}
K_{2n+2\nu+1}\left(\pm ikr\right);
\end{array}
\end{eqnarray}
\begin{eqnarray}
\begin{array}{l}
R_{2\nu}^{0}(r)=e^{{B_{1}}/{(2r^2)}}r^{-2\nu-
\frac{3}{2}}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(2)}
\left(-\frac{B_{1}}{r^2}\right)^{n}\Psi\left(n+\nu+2-
\frac{B_{2}}{2},2n+2\nu+2;
-\frac{B_{1}}{r^2}\right),
\vspace{.3cm}\\
R_{2\nu}^{\infty}(r)=e^{{B_{1}}/{(2r^2)}}r^{-1/2}
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(2)}
K_{2n+2\nu+1}\left(\pm ikr \right).
\end{array}
\end{eqnarray}
From these expressions we obtain
\begin{eqnarray}
\lim_{r\rightarrow\infty}R_{1\nu}^{\infty}(r)\propto
\lim_{r\rightarrow\infty}R_{2\nu}^{\infty}(r)
\sim\frac{e^{\mp ikr}}{r}, \ \ -\frac{3\pi}{2}<\arg{(\pm ikr)}<
\frac{3\pi}{2}
\end{eqnarray}
where we have employed the limit (\ref{21}) for the modified Bessel
functions. Thus, when $r\rightarrow\infty$,
the solutions $R_{i\nu}^{\infty}$ are bounded even if
$k$ is a pure imaginary, since in this case $\exp(ikr)$ or $\exp(-ikr)$
goes to zero. At $r=0$, Eq. (\ref{tricomi}) implies that
\begin{eqnarray}\begin{array}{l}
\displaystyle\lim_{r\rightarrow0}R_{1\nu}^{0}(r)
\sim e^{-B_{1}/r^2}r^{B_{2}-(3/2)}, \ \ -\frac{3\pi}{2}<\arg{\frac{B_{1}}{r^2}}<
\frac{3\pi}{2},
\vspace{3mm}\\
\displaystyle\lim_{r\rightarrow0}R_{2\nu}^{0}(r)
\sim e^{B_{1}/r^2}r^{(5/2)-B_{2}}, \ \ -\frac{3\pi}{2}<\arg{\left( -\frac{B_{1}}{r^2}\right) }<
\frac{3\pi}{2}.
\end{array}
\end{eqnarray}
Thence, if $B_{1}$ is a positive real number, the first limit goes to
zero; if $B_{1}$ is a negative real number, the second limit goes to
zero. However, if $B_{1}$ is a pure imaginary, we write
\begin{eqnarray*}
B_{1}=iC, \ \ B_{2}=2+\frac{i\alpha_{1}^{'}(e')^2}{2\hbar^2C}
\end{eqnarray*}
where $C $ is real. Thus we find
\begin{eqnarray*}
\vert R_{1\nu}^{0}(r)\vert\propto\vert R_{2\nu}^{0}(r)\vert\sim\sqrt{r}\rightarrow 0.
\end{eqnarray*}
Therefore, it is possible to find at least one pair of solutions
for which both the solutions are bounded at the singularities.\\
\noindent
{\bf Potential without inverse sixth-power term}.
From Eq. (\ref{N1}) we find that the substitutions
\begin{eqnarray*}
z= r, \ \ \chi(r)=e^{-B_{1}/(2r)}r^{B_{2}/2}U(z=r)\ \
\mbox{with}\ \
\hbar^2B_{1}^{2}=-4\mu(e')^2, \ \ B_{2}=2
\end{eqnarray*}
transform the Schr\"odinger equation (\ref{radial2}) into
\begin{eqnarray}
r^2\frac{d^{2}U}{dr^{2}}+(B_{1}+2r)
\frac{dU}{dr}+
\left[-l(l+1)-
\frac{2\mu}{\hbar^2}(Z-z')\overline{e}e'r+k^2 r^ 2+
\frac{\mu(\alpha_{2}^{'}-6a_{0}\beta_{1}^{'})(e')^2}
{\hbar^2r^4}\right]U=0.
\end{eqnarray}
Thus, in absence of the inverse sixth-power term,
the radial Schr\"odinger equation, even if we have a Coulomb term
in the potential, may be solved by
\begin{eqnarray}
R_{i\nu}(r)=e^{-B_{1}/(2r)}U_{i\nu}(z=r)
\end{eqnarray}
where $U_{i\nu}(z=r)$ are solutions with a phase parameter
for the DCHE with $z=r$ and $B_{2}=2$ (see Appendix B). As
\begin{eqnarray}
\omega=\pm k\leftrightarrow
\pm\eta=\pm\frac{\mu}{k\hbar^2}(Z-z')\bar{e}e',\ \ k=\frac{\sqrt{2\mu E}}{\hbar},
\end{eqnarray}
those solutions give
\begin{eqnarray}
\begin{array}{l}
R_{1\nu}^{0} (r) =e^{\pm ikr-\frac{B_{1}}{2r}}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}
\left(\frac{B_{1}}{r}\right)^{n+\nu+1}\Psi\left(n+\nu+1,2n+2\nu+2;
\frac{B_{1}}{r}\right),
\vspace{0.3cm}\\
R_{1\nu}^{\infty}(r) =e^{\pm ikr-\frac{B_{1}}{2r}}
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}(\mp2ikr)^{n+\nu}\Psi(n+\nu+1\pm
i\eta,2n+2\nu+2;\mp 2ik r);
\end{array}
\end{eqnarray}
\begin{eqnarray}
\begin{array}{l}
R_{2\nu}^{0} (r) =e^{\pm ik r+\frac{B_{1}}{2r}}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}
\left(-\frac{B_{1}}{r}\right)^{n+\nu+1}\Psi\left(n+\nu+1,2n+2\nu+2;
-\frac{B_{1}}{r}\right),
\vspace{3mm}\\
R_{2\nu}^{\infty}(r) =e^{\pm ik r+\frac{B_{1}}{2r}}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}
(\mp 2ik r)^{n+\nu}\Psi(n+\nu+1\pm
i\eta,2n+2\nu+2;\mp 2ik r).
\end{array}
\end{eqnarray}
Using Eq. (\ref{tricomi}), we find
\begin{eqnarray}
\lim_{r\rightarrow\infty}R_{1\nu}^{\infty}(r)\propto
\lim_{r\rightarrow\infty}R_{2\nu}^{\infty}(r)
\sim r^{\mp i\eta}\ \frac{e^{\pm ikr}}{r}, \ \ -\frac{3\pi}{2}<\arg{(\mp ikr)}<
\frac{3\pi}{2}
\end{eqnarray}
Thus, when $r\rightarrow\infty$,
the solutions $R_{i\nu}^{\infty}$ are bounded even if
$k$ is a pure imaginary number, since in this case the behavior
of $\exp(ikr)$ or $\exp(-ikr)$
predominates over the other factor. At $r=0$, by using
Eq. (\ref{tricomi}) we get
\begin{eqnarray}\begin{array}{l}
\displaystyle\lim_{r\rightarrow0}R_{1\nu}^{0}(r)
\sim e^{-B_{1}/(2r)}, \ \ -\frac{3\pi}{2}<\arg{\frac{B_{1}}{r}}<
\frac{3\pi}{2}
\vspace{3mm}\\
\displaystyle\lim_{r\rightarrow0}R_{2\nu}^{0}(r)
\sim e^{B_{1}/(2r)}, \ \ -\frac{3\pi}{2}<\arg{\left( -\frac{ B_{1}}{r}\right) }<
\frac{3\pi}{2}.
\end{array}
\end{eqnarray}
As $B_{1}$ is a pure imaginary number, we find that
\begin{eqnarray*}
\vert R_{1\nu}^{0}(r)\vert\propto\vert R_{2\nu}^{0}(r)\vert\sim 1
\end{eqnarray*}
Therefore, in this case we can form two pairs
of solutions which are regular at the singular points, both pairs
having the same series coefficients. For neutral targets
($\eta=0$) the previous results have already been found by B\"uhring
who has treated the Schr\"odinger equation as a
DCHE \cite{buhring2,buhring1}. Before this author, the Schr\"odinger
equation (for neutral targets and
an inverse fourth-power polarization potential) had been
transformed into a Mathieu equation \cite{vogt,holzwarth}.
Thus, the B\"uhring approach is profitable since it works
for ionized targets, too. In addition, as we have seen,
for inverse sixth-power polarization potential, the Schr\"odinger
equation may be transformed to the Ince limit of the DCHE,
provided that the target is neutral.
\section*{5. Final remarks}
We have constructed the differential equation (\ref{lindemann})
by applying the Ince limit, defined in Eq. (\ref{limits}), to a generalized
spheroidal wave equation (GSWE). The Leaver limit
($z_{0}\rightarrow 0$) of that equation
has afforded Eq. (\ref{lindemann2}) that turns out to be the Ince limit of
a double-confluent Heun equation (DCHE) as well. The subnormal
Thom\'e behavior at $z=\infty$, for the solutions of the
these Ince limits of the GSWE and DCHE, distinguishes such
equations from the original GSWE and DCHE hitherto considered
in the literature.
In section 2, a pair of solutions (with a phase parameter)
for the Ince limit of the GSWE has been found
as the Ince limit of a pair of solutions for
the original GSWE. One solution is given by a series of
hypergeometric functions and the other by
a series of modified Bessel functions of the second
kind. Both solutions in that pair have
the same series coefficients but different regions of convergence,
as in solutions for the Mathieu
equations. Other pair has followed from the first one by means of
a transformation rule. Hence, four pairs of solutions without phase
parameter have resulted from the truncation of the
series with a phase parameter, that is, by restricting the summation
index of the series to $n\geq0$.
In section 3, solutions for the Ince limit of the DCHE have
been established by taking the Leaver limit
of solutions for the Ince limit of the GSWE. These solutions,
given by series of irregular confluent hypergeometric functions
and modified Bessel functions, present the
appropriate behavior at the irregular singularities
$z=0$ and $z=\infty$. Note, nonetheless, that in sections 3 and 4
we have dealt with expansions in
series of modified Bessel functions only.
Other possibilities may be investigated,
specially solutions in series of Bessel function products,
as these could have important properties as regards
the convergence of the series.
In the solutions without phase parameter for the Ince limits
of the GSWE and DCHE, there are three possible forms to the
recurrence relations for the series coefficients. This fact is
relevant in itself and, in particular, is essential to
recover solutions for the Mathieu equation from the ones for the
Ince limit of the GSWE.
The solutions we have obtained for the Mathieu equation
are already known and exhibit the usual parity
and periodicity properties. This includes also
the solutions found by Poole, given by two-sided series
($-\infty<n<\infty$) and having period $2\pi m$, where $m$ is any integer
greater than $1$. However, we note that other types of
solutions for the Mathieu equations (and also for the
Whittaker-Hill equations)
are possible, since these equations may be considered
as particular cases of both the GSWE and
double-confluent Heun equations as well \cite{decarreau2}.
At last, notice that we have point out no
application for Ince limit of the GSWE. Nevertheless, in section
4 we have seen that
the Schr\"odinger equation (\ref{radial2}) for the scattering
of low-energy particles by polarizable targets leads to an
DCHE and its Ince limit.
The exception is the Schr\"odinger equation with
Coulomb and inverse sixth-power terms which requires solutions
for a more general equation, possibly similar to
an equation considered by Kurth and Schmidt in \cite{kurth}.
I thank Herman J. Mosquera Cuesta for his careful reading
of this manuscript and valuable suggestions. I also thank the
participants of the ICRA-BR ``Pequenos Semin\'arios" for discussions
and insight on potential physical applications of the results of the
present investigation.
\section*{Appendix A: Degenerate DCHEs}
\protect\label{A}
\setcounter{equation}{0}
\renewcommand{\theequation}{A\arabic{equation}}
Let us show that DCHE
\begin{eqnarray*}
z^2\frac{d^{2}U}{dz^{2}}+(B_{1}+B_{2}z)\frac{dU}{dz}+
\left(B_{3}-2\eta\omega z+\omega^{2}z^2\right)U=0,\ \ (B_{1}\neq 0,\
\ \omega\neq 0),
\end{eqnarray*}
for $B_{1}=0$
and/or $\omega=0$ degenerates into a confluent
hypergeometric equation or an equation with
constant coefficients. Thus, if $B_{1}=0$
and $\omega\neq 0$, the
substitutions
\begin{eqnarray*}
y=-2i\omega z, \ \ U(z)=e^{-y/2}y^{\alpha}f(y), \ \
\alpha^2-(1-B_{2})\alpha+B_{3}=0
\end{eqnarray*}
give the confluent hypergeometric equation
\begin{eqnarray*}
y\frac{d^2f}{dy^2}+[(2\alpha+B_{2})-y]\frac{df}{dy}-
\left(i\eta+\alpha+\frac{B_{2}}{2}\right)f=0.
\end{eqnarray*}
If $B_{1}\neq 0$ and $\omega= 0$, the change of variables
\begin{eqnarray*}
y=B_{1}/z, \ \ U(z)=y^{\beta}g(y), \ \ \beta^2-(B_{2}-1)\beta+B_{3}=0
\end{eqnarray*}
leads to
\begin{eqnarray*}
&&y\frac{d^2g}{dy^2}+[(2\beta+2-B_{2})-y]\frac{dg}{dy}-\beta g=0.
\end{eqnarray*}
If $B_{1}=\omega=0$, we find an equation with constant coefficients by
taking $z=\exp{y}$.
Now let us show that the Ince limit of the DCHE
\begin{eqnarray*}
z^2\frac{d^{2}U}{dz^{2}}+(B_{1}+B_{2}z)\frac{dU}{dz}+
\left(B_{3}+q z\right)U=0,\ (q\neq0,\ B_{1}\neq 0)
\end{eqnarray*}
also gives degenerate cases if $q\neq0$ and/or $\ B_{1}\neq 0$.
In fact, if $q=0$ and $B_{1}\neq 0$, this equation is equivalent
to the DCHE with $\omega= 0$ and $B_{1}\neq 0$.
If $q\neq0$ and $B_{1}= 0$, the
substitutions
\begin{eqnarray*}
\xi=\pm 2i\sqrt{qz}
,\ \ U(z)=\xi^{1-B_{2}}T(\xi)
\end{eqnarray*}
reduces the equation to the modified Bessel equation
\begin{eqnarray*}
\xi^2\frac{d^2T}{d\xi^2}+\xi\frac{dT}{d\xi}-
\left[(1-B_{2})^2-4B_{3}+\xi^2\right]T=0.
\end{eqnarray*}
Finally, for $q=B_{1}= 0$, we find again an equation with constant
coefficients by taking $z=\exp{y}$.
\section*{Appendix B: The solutions in series of Bessel functions}
\protect\label{B}
\setcounter{equation}{0}
\renewcommand{\theequation}{B\arabic{equation}}
The solution $U_{1\nu}^{\infty}(z)$ in
series of Bessel functions can also be constructed as follows.
We perform the substitutions
\begin{eqnarray}\label{2.2}
\xi=\pm2i\sqrt{qz},\ \ U(z)=\xi^{1-B_{2}}Y(\xi)
\end{eqnarray}
in the Ince limit of the GSWE (\ref{lindemann}). This yields
\begin{eqnarray}\label{2.4}
\xi^2\frac{d^2Y}{d\xi^2}+\xi\frac{dY}{d\xi}-
\xi^2Y=-4qz_{0}\frac{d^2Y}{d\xi^2}-
\frac{4q(z_{0}-2B_{1}-2B_{2}z_{0})}{\xi}
\frac{dY}{d\xi} \nonumber\\
+\left[4q(1-B_{2})\frac{2B_{1}+B_{2}z_{0}+z_{0}}
{\xi^2}+(1-B_{2})^2+4qz_{0}-4B_{3}\right]Y.
\end{eqnarray}
Now we expand $Y(\xi)$ according to
\begin{eqnarray}\label{2.6}
Y(\xi)=\displaystyle \sum_{n=-\infty}^{\infty}b_{n}^{(1)}
K_{\lambda}(\xi), \ \ \lambda=2n+2\nu+1,
\end{eqnarray}
where $K_{\lambda}(\xi)$ denotes the
modified Bessel function of the second
kind \cite{luke}. The last equation and (\ref{2.2})
afford the solution $U_{1\nu}^{\infty}(z)$.
When we insert (\ref{2.6}) into (\ref{2.4}),
we use some difference-differential relations derived
from the properties of $K_{\lambda}$ \cite{luke}.
Thus, we have
\begin{eqnarray*}
\xi^2\frac{d^2K_{\lambda}(\xi)}{d\xi^2}+\xi
\frac{dK_{\lambda}(\xi)}{d\xi}-
\xi^2K_{\lambda}(\xi)=\lambda^2K_{\lambda}(\xi)
\end{eqnarray*}
on the left-hand side and
\begin{eqnarray*}
4\frac{d^2K_{\lambda}(\xi)}{d\xi^2}=
K_{\lambda+2}(\xi)+2K_{\lambda}(\xi)+
K_{\lambda-2}(\xi),\ \
\frac{4}{\xi}\frac{dK_{\lambda}(\xi)}{d\xi}=
-\frac{4\lambda}{\xi^2}K_{\lambda}(\xi)+
\frac{2}{\lambda-1}\left[K_{\lambda-2}(\xi)-
K_{\lambda}(\xi)\right]
\end{eqnarray*}
on the right-hand side. This gives
\begin{eqnarray*}
&&qz_{0}\displaystyle \sum_{n=-\infty}^{\infty}
\left[1+\frac{2[1-2B_{2}-(2B_{1}/z_{0})]}{\lambda-1}\right]b_{n}^{(1)}
K_{\lambda-2}(\xi)\nonumber\\
&&+\displaystyle \sum_{n=-\infty}^{\infty}
\left[\lambda^2+4B_{3}-2qz_{0}-(1-B_{2})^2-
\frac{2qz_{0}[1-2B_{2}-(2B_{1}/z_{0})]}{\lambda-1}
\right]b_{n}^{(1)}
K_{\lambda}(\xi)
\nonumber\\
&&+qz_{0}\displaystyle \sum_{n=-\infty}^{\infty}
b_{n}^{(1)}K_{\lambda+2}(\xi)\nonumber\\
&&=qz_{0}\displaystyle \sum_{n=-\infty}^{\infty}
\left[\left(1-2B_{2}-\frac{2B_{1}}{z_{0}}
\right)\lambda+(1-B_{2})\left(
1+B_{2}+\frac{2B_{1}}{z_{0}}\right)\right]b_{n}^{(1)}
\frac{4K_{\lambda}(\xi)}{\xi^2}.
\end{eqnarray*}
To remove the term $4K_{\lambda}(\xi)/\xi^2$
on the right-hand side we use the relation
\begin{eqnarray*}
\frac{4K_{\lambda}(\xi)}{\xi^2}=
\frac{K_{\lambda-2}(\xi)}{\lambda(\lambda-1)}-
\frac{2K_{\lambda}(\xi)}{(\lambda-1)(\lambda+1)}+
\frac{K_{\lambda+2}(\xi)}{\lambda(\lambda+1)}.
\end{eqnarray*}
Then, reminding that $\lambda=2n+2\nu+1$, we
find
\begin{eqnarray}\label{2.10}
\displaystyle \sum_{n=-\infty}^{\infty}
\alpha_{n-1}^{(1)}b_{n}^{(1)}
K_{2n+2\nu-1}(\xi)+
\displaystyle \sum_{n=-\infty}^{\infty}
\beta_{n}^{(1)}b_{n}^{(1)}
K_{2n+2\nu+1}(\xi)+
\displaystyle \sum_{n=-\infty}^{\infty}
\gamma_{n+1}^{(1)}b_{n}^{(1)}
K_{2n+2\nu+3}(\xi)=0,
\end{eqnarray}
where the coefficients
$\alpha_{n}^{(1)}$, $\beta_{n}^{(1)}$
and $\gamma_{n}^{(1)}$ are
just the ones given in equations (\ref{apB}).
To get the recurrence relations with the form
given in (\ref{rec}), we change $n\rightarrow m+1$ and
$n\rightarrow m-1$ in the first and third
terms, respectively. After this, we equate to zero
the coefficients of each independent
$K_{2m+2\nu+1}(\xi)$.
On the other hand, to study the convergence of the series, we
apply a Perron-Kreuser theorem \cite{gautschi}
for the minimal solutions of the recurrence relations for
$b_{n}^{(1)}$ and obtain (if $z_{0}\neq0$)
\begin{eqnarray}\label{2.12}
\lim_{n\rightarrow \infty}\frac{b_{n+1}^{(1)}}{b_{n}^{(1)}}=
\lim_{n\rightarrow -\infty}\frac{b_{n-1}^{(1)}}{b_{n}^{(1)}}=
-\frac{qz_{0}}
{4n^2}.
\end{eqnarray}
Using also the relation \cite{luke}
\begin{eqnarray*}
\lim_{\lambda\rightarrow \infty}K_{\lambda}({\xi})=
\frac{1}{2}\Gamma({\lambda})
\left(\frac{\xi}{2}\right)^{-\lambda}
\end{eqnarray*}
and $K_{-\lambda}({\xi})=K_{\lambda}({\xi})$, we get
\begin{eqnarray*}
\lim_{n\rightarrow \infty}
\frac{K_{2n+2\nu+3}({\xi})}{K_{2n+2\nu+1}({\xi})}=
\lim_{n\rightarrow -\infty}
\frac{K_{2n+2\nu-1}({\xi})}{K_{2n+2\nu+1}({\xi})}=
-\frac{4n^2}{qz}.
\end{eqnarray*}
Hence, we have
\begin{eqnarray*}
\lim_{n\rightarrow \infty}\frac{b_{n+1}^{(1)}K_{2n+2\nu+3}({\xi})}
{b_{n}^{(1)}K_{2n+2\nu+1}({\xi})}=
\lim_{n\rightarrow -\infty}\frac{b_{n-1}^{(1)}K_{2n+2\nu-1}({\xi})}
{b_{n}^{(1)}K_{2n+2\nu+1}({\xi})}=
\frac{z_{0}}{z}.
\end{eqnarray*}
Therefore, by the ratio test the series converges for
$|z|>|z_{0}|$. In (\ref{2.12}) we have supposed
that $z_{0}\neq0$ but, if $z_{0}=0$, we find
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty}\frac{b_{n+1}^{(1)}}{b_{n}^{(1)}}=
\lim_{n\rightarrow -\infty}\frac{b_{n-1}^{(1)}}{b_{n}^{(1)}}=
-\frac{B_{1}}
{4n^3}\ \Rightarrow\\
&&
\lim_{n\rightarrow \infty}\frac{b_{n+1}^{(1)}K_{2n+2\nu+3}({\xi})}
{b_{n}^{(1)}K_{2n+2\nu+1}({\xi})}=
\lim_{n\rightarrow -\infty}\frac{b_{n-1}^{(1)}K_{2n+2\nu-1}({\xi})}
{b_{n}^{(1)}K_{2n+2\nu+1}({\xi})}=
\frac{B_{1}}{nz}.
\end{eqnarray*}
Thus, in this limit the series converges for $|z|>0$
and per se this result is already included in
$\vert z\vert>\vert z_{0}\vert$.
\section*{Appendix C: Solutions for the DCHE of section 4}
\protect\label{C}
\setcounter{equation}{0}
\renewcommand{\theequation}{C\arabic{equation}}
The Leaver-type solutions for the DCHE (\ref{dche}) present
some simplifications for $B_{2}=2$.
The solutions given in Ref. \cite{eu} are expansions
in series of regular and irregular confluent hypergeometric functions.
However, to obtain the expected behavior at the singular points
$z=0$ and $z=\infty$, we have to choose the irregular functions.
Then, by using the same notation of sections 2.1 and 3.1, we find that
for $B_{2}=2$ the first pair of solutions with a phase parameter
is given by
\begin{eqnarray}
\begin{array}{l}
U_{1\nu}^{0} (z) =e^{i\omega z}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}
\left(\frac{B_{1}}{z}\right)^{n+\nu+1}\Psi\left(n+\nu+1,2n+2\nu+2;
\frac{B_{1}}{z}\right),
\vspace{0.3cm}\\
U_{1\nu}^{\infty}(z) =e^{i\omega z}
\displaystyle \sum_{n=-\infty}^{\infty}b_{n}(-2i\omega z)^{n+\nu}\Psi(n+\nu+1+
i\eta,2n+2\nu+2;-2i\omega z),
\end{array}
\end{eqnarray}
and the second pair takes the form
\begin{eqnarray}
\begin{array}{l}
U_{2\nu}^{0} (z) =e^{i\omega z+\frac{B_{1}}{z}}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}
\left(-\frac{B_{1}}{z}\right)^{n+\nu+1}\Psi\left(n+\nu+1,2n+2\nu+2;
-\frac{B_{1}}{z}\right),
\vspace{3mm}\\
U_{2\nu}^{\infty}(z) =e^{i\omega z+\frac{B_{1}}{z}}\displaystyle \sum_{n=-\infty}^{\infty}b_{n}
(-2i\omega z)^{n+\nu}\Psi(n+\nu+1+
i\eta,2n+2\nu+2;-2i\omega z).
\end{array}
\end{eqnarray}
Then, we see that the two pairs have the same
series coefficients $b_{n}$ and the coefficients in the recurrence
relations (\ref{rec}) are simply
\begin{eqnarray}
\begin{array}{l}
\alpha_{n} = i\omega B_{1}\left( \frac{n+\nu+1-i\eta}
{2n+2\nu+3}\right) ,\ \
\beta_{n} = B_{3}+(n+\nu)(n+\nu+1),
\gamma_{n}=i\omega B_{1} \left( \frac{n+\nu+
i\eta}{2n+2\nu-1}\right) .
\end{array}
\end{eqnarray}
In these solutions $\nu$ cannot be integer or half-integer and
the $U_{i\nu}^{0}$ converge for any finite $z $,
whereas the $U_{i\nu}^{\infty}$ converge for $\vert z\vert>0$.
Note, moreover, that the irregular confluent hypergeometric functions
that appear in $U_{i\nu}^{0}$ could be rewritten in terms of
modified Bessel of the second kind by using the definition (\ref{2.8}).
In the solutions $U_{i\nu}^{\infty}$ the confluent hypergeometric
functions could be rewritten in terms of the Hankel functions
$H_{\rho}^{(1)}$ but only if $\eta=0$ (neutral target, in
the problem of section 4). For this we have to use the relation
\cite{erdelyi1}
\begin{eqnarray*}
\Psi\left(\rho+\frac{1}{2},2\rho+1;-2ix\right)=
\frac{i}{2\sqrt{\pi}}
e^{i(\rho \pi-x)}H_{\rho}^{(1)}(x), \ \rho=n+\nu+\frac{1}{2}.
\end{eqnarray*}
The asymptotic behaviors of the solutions given in (C1-2) may
be found by using Eq. (\ref{tricomi}).
|
hep-th/0509194
|
\section{Introduction}
The study of brane collisions has recently gained a special interest as
it may provide a new scenario for the creation of the hot big-bang
Universe \cite{CU, Bubble}. Motivated by heterotic M-theory and the
Randall Sundrum (RS) model \cite{RSI}, the collision between two
orbifold branes has been explored, leading to a five-dimensional
singularity \cite{Khoury:2001bz}. When the boundary branes are close, an effective theory
can be derived for this scenario
and is hence valid just before or just after the collision \cite{collision, CL, CL2, CL3, CU}.
In this paper, we extend this analysis to the case where a brane is
present in the bulk. This regime is of interest as it allows us to
study a bulk/orbifold brane collision
in a situation where the five-dimensional
geometry remains regular similarly as in the first Ekpyrotic scenario
\cite{CU}. What is particularly interesting about the effective theory
we will develop, is that it is capable of describing the regime where
the branes have large velocities, something which the usual low energy
effective theory cannot do.
Although similar work has been derived for close boundary branes, \cite{CL, CL2,
CL3} it relied strongly on the presence of a
$\mathbb{Z}_2$ orbifold symmetry which is generically
broken for bulk branes.
This work will hence give us a general formalism for the derivation of an effective theory on a
generic non $ \mathbb{Z}_2 $-brane. Such branes are interesting to study as
they represent more realistic candidates for cosmology and at high-energies, their behaviour
is expected to be strongly modified \cite{Davis:2000jq}.
In order to get some insight on the brane geometry one should in
principal solve the full higher-dimensional theory exactly before
being able to infer the geometry on the brane. Unfortunately, this
is only possible in very limited cases, and for more general
situations, one should in practice either rely on numerical
simulation or work in some specific regime where effective theories
may be derived. This is the approach which is generally undertaken in
order to derive a {\it low-energy} effective theory. Assuming a
low-energy regime, it is possible to express the geometry on
the brane as the lowest order of a gradient expansion \cite{CU,GE,
Cotta, Moduli}. In this
paper, we use a similar method, but choose instead to work in a
close-brane regime, where we only consider terms of leading order
in the distance between the branes. This method allow us to
highlight the presence of ``asymmetric" terms on the bulk brane
(generic to the absence of $\mathbb{Z}_2 $-symmetry) which are
negligible at low-energies and are usually discarded. As far as we
are aware,
this is the first effective-theory that models these
terms in a covariant way beyond the low-energy limit.\\
\indent The rest of this paper is organized as follows. In Sec. II, we
consider three branes and derive the effective theory on the
asymmetric bulk brane. In that theory, two scalar dynamical degrees
of freedom are present, namely the distance between the bulk brane
and each of the boundary branes. We point out the low-energy limit
of this theory, and check its consistency with previous results. In
particular, we show that the theory on the bulk brane is a standard
scalar-tensor theory of gravity coupled with two scalar fields.
In Sec. III, we apply our effective theory to cosmology and compare
our result with solutions from the five-dimensional theory. We show
that for large velocities, the matter on the orbifold branes do not
affect the bulk brane. As a specific example, we present the
derivation of tensor perturbations. As expected, at large velocities
the perturbations on the bulk brane decouple from the stress energy on
the boundary branes. Finally, we summarize our
study and present some possible extensions as future works in Sec. IV.
\section{Effective theory for Three close branes}
Motivated by M-theory and the Randall-Sundrum model \cite{RSI}, we
consider spacetime to be effectively five-dimensional with
the extra-dimension compactified on an $S^1/ \mathbb{Z}_2$-orbifold. Two
orbifold branes are located at the fixed point of the symmetry, and
we consider a third brane in the bulk.
In this paper, we shall be interested in the
limit where the three branes are close to each other, i.e.,
when they are either about to collide or have just emerged
from such a collision.
In this paper we use the index conventions that Greek
indices $\mu,\nu=0,\cdots,3$ are four dimensional, labeling the transverse $x^\mu$
directions, while Roman capital indices $M,N=0,\cdots,4$ are fully five dimensional
and lower cap Roman indices $i,j=1,2,3$ designate the spatial transverse
directions.
Without loss of generality, we use the
following metric ansatz
\begin{eqnarray}
ds^2=g_{MN}dx^M dx^N=e^{2\varphi_\pm(y,x)}dy^2+g_{\mu\nu}(y,x) dx^\mu dx^\nu
\end{eqnarray}
and we suppose that the branes are located at $y=y_+,y_0,y_-$.
The branes located at $y=y_\pm$ are the orbifold branes and they are
subject to a $ \mathbb{Z}_2 $-reflection symmetry.
The brane at $y=y_+$ is a positive tension brane, whereas the one at
$y=y_-$ has a negative tension. The brane located at $y=y_0$ is a
bulk brane and no symmetry is imposed.
In what follows, we denote by $\mathcal{R}_+$ (resp. $\mathcal{R}_-$)
the region between the bulk brane and the positive (resp. negative)
boundary brane. All through this paper we use the notation that an
index $'+'$ (resp. $'-'$) represents a quantity evaluated in the
region $\mathcal{R}_+$ (resp. $\mathcal{R}_-$), as shown in Fig. 1.
\begin{figure}
\centering
\includegraphics[width=.9\columnwidth]{Pic3.eps}
\caption{Two-brane Randall-Sundrum model with an asymmetric brane present in the bulk.}
\end{figure}
In particular, the Anti-de Sitter(AdS) length scale on each region will be denoted
as $\ell_\pm$ and for any quantity $Q$, $^\pm Q(y_0)=\lim _{\epsilon
\rightarrow 0}Q(y_0\mp \epsilon)$. We denote by $^{(i)}T^\mu_\nu$
the stress-energy tensor for matter fields confined on the brane at
$y=y_i$, for $i=+,-,0$. We assume all branes to have a tension
$\sigma_i$ fine-tuned to their canonical value and absorb any
departure in their stress-energy: $\sigma_\pm=\pm 6/\kappa^2
\ell_\pm$ and $\sigma_0=3\left(\ell_-^{-1}-\ell_+^{-1}\right)/\kappa^2$
where $\kappa^2/8\pi$ is the five-dimensional gravitational constant.
The aim of this work is to derive an effective theory for the asymmetric
bulk brane. We will hence work on the bulk brane frame throughout
this paper unless otherwise specified. We will first decompose the
extrinsic curvature on the bulk brane, in terms of a quantity that may
be determined from the Isra\"el matching condition and another
``asymmetric'' quantity which needs to be determined by other
means. Working in the close-brane limit, we may express this
``asymmetric'' term as an expansion in terms of the extrinsic
curvature on the orbifold branes which are uniquely determined by the
junction conditions. The rest follows as in \cite{CL2, CL3}. In
particular, we
use the close-brane approximation to express the derivative of the extrinsic
curvature in terms of the extrinsic curvature on the bulk brane as
well as the one on the orbifold brane. This allows us to specify all unknown
quantities in the modified Einstein equation of the bulk brane and
hence obtain an effective theory for this brane.
\subsection{Expression of the ``asymmetric'' term}
For the boundary branes, the junction conditions are simply
\begin{eqnarray}
K^\mu_\nu(y=y_\pm) = - \frac{1}{\ell_\pm} \delta^\mu_\nu
\mp \frac{\kappa^2}{2} \Bigl({}^{(\pm)}T^\mu_\nu -\frac{1}{3}\delta^\mu_\nu
{}^{(\pm)}T \Bigr),
\end{eqnarray}
where $K^\mu_\nu$ is the extrinsic curvature.
Whereas for the bulk brane, due to the absence of any
$\mathbb{Z}_2$-reflection symmetry, the extrinsic curvature cannot
be uniquely determined by the junction conditions
\begin{eqnarray}
\Delta K^\mu_\nu(y_0) & := & \hspace{-5pt} {}^-K^\mu_\nu(y_0)-{}^+K^\mu_\nu (y_0)
\nonumber \\
\hspace{-5pt} & = & \hspace{-5pt
\left(\frac{1}{\ell_+}-\frac{1}{\ell_-}\right)\delta^\mu_\nu
- \kappa^2 \Bigl({}^{(0)}T^\mu_\nu -\frac{1}{3}\delta^\mu_\nu {}^{(0)}T
\Bigr).\hspace{10pt} \label{junction bulk brane}
\end{eqnarray}
Since we are interested in the close-brane limit, we assume in
what follows, that the proper distance between the branes is much
shorter than the AdS curvature scale. In this case, the following
recursive relation is valid \cite{CL2,CL3}:
\begin{eqnarray}
{}^{\pm}K^{\mu (n)}_\nu (y_0)= \hat O_\pm K^{\mu (n-2)}_\nu
(y_0)(y_\pm-y_0)^{-2}, \label{rec}
\end{eqnarray}
where ${}^{\pm}K^{\mu (n)}_\nu (y_0)\equiv \partial_y^{\, n}\,{}^{\pm}\!K^{\mu}_\nu (y_0)$
and the
action of the operator $\hat O_\pm$ is defined by
\begin{eqnarray}
\hat O_\pm S^\mu_\nu = D^\mu d_\pm D_\alpha d_\pm S^\alpha_\nu +
D_\nu d_\pm D^\alpha d_\pm S^\mu_\alpha -(D d_\pm)^2
S^\mu_\nu \hspace{5pt}
\end{eqnarray}
for any symmetric tensor $S_{\mu\nu}=S_{(\mu\nu)}$.
The proper distance between the branes is
\begin{eqnarray}
d_\pm :=\pm e^{\varphi_\pm (y_0,x)} (y_0-y_\pm),
\end{eqnarray}
in the gauge where $\varphi$ is independent of $y$. In Ref. \cite{CL2},
it is shown that the result is independent of this gauge choice.
Working in such a gauge, and using the five-dimensional Einstein
equations, the Gauss equation on
the brane is (Cf. Ref. \cite{SMS}),
\begin{multline}
{}^{\pm}R^\mu_\nu (y_0) = -\frac{4}{\ell_\pm^2}\delta^\mu_\nu
+{}^{\pm}K(y_0) {}^{\pm}K^\mu_\nu (y_0) \\
+ e^{-\varphi_\pm}{}^{\pm}K'^\mu_{\ \nu} (y_0) +d_\pm^{-1}D^\mu D_\nu d_\pm. \label{ricci}
\end{multline}
The remaining task for the derivation of the effective theory is the
evaluation of both ${}^{\pm}K^\mu_\nu(y_0)$ and ${}^{\pm}K'^\mu_{\
\nu} (y_0)$. We hence decompose ${}^{\pm}K^\mu_\nu(y_0)$ into a
``known" contribution $\Delta K^\mu_\nu$, and an undetermined part
which represents the asymmetry across the brane:
\begin{eqnarray}
{}^{\pm}K^\mu_\nu (y_0) = \mp \frac{1}{2}\Delta K^\mu_\nu (y_0)
+\bar
K^\mu_\nu (y_0), \label{Kbulk}
\end{eqnarray}
where $\bar K^\mu_\nu (y_0) :=\frac 1 2 \left({}^{+}K^\mu_\nu (y_0)+
{}^{-}K^\mu_\nu (y_0) \right) $ and where the Codacci equation
holds for both quantities independently: $D_\mu \left(\Delta
K^\mu_\nu-\Delta K \delta^\mu_\nu\right) =D_\mu \left( \bar
K^\mu_\nu-\bar K \delta^\mu_\nu\right)=0 $. For a $ \mathbb{Z}_2$-symmetric
brane, $\bar{K}^\mu_\nu=0$.
Writing the extrinsic curvature on the orbifold branes as a Taylor
expansion in terms of the one on the bulk brane, we have:
\begin{eqnarray}
K^\mu_\nu (y_\pm) & = & \sum_{n \geq 0}^\infty \frac{1}{n!}{}^{\pm} K^{\mu (n)}_\nu (y_0)
(y_\pm-y_0)^n
\nonumber \\
& = & \frac{{\rm sinh}{\sqrt {\hat O}_\pm}}{{\sqrt {\hat
O}_\pm}}{}^{\pm}K'^\mu_{\ \nu}(y_0) (y_\pm-y_0) \label{taylor}
\\
& & +{\rm cosh}{\sqrt {\hat O}_\pm}{}^{\pm} K^\mu_\nu(y_0).\nonumber
\end{eqnarray}
This provides an expression for the derivative of the extrinsic
curvature
\begin{multline}
{}^{\pm}K'^\mu_{\ \nu} (y_0) = \frac{1}{y_\pm -y_0}
\frac{{\sqrt {\hat O_\pm}}}{{\rm sinh}{\sqrt {\hat O_\pm}}}
\Biggl(K^\mu_\nu (y_\pm) \label{derivative} \\
-{\rm cosh} {\sqrt {\hat O_\pm}} {}^{\pm}K^\mu_\nu (y_0)
\Biggr).
\end{multline}
Substituting this expression into Eq. \eqref{ricci} and recalling that the
metric should be continuous across the bulk brane, ie.
${}^{+}R^\mu_\nu (y_0)-{}^{-}R^\mu_\nu(y_0)=0$, we obtain the
constraint
\begin{eqnarray}
&&\sum_{i=\pm}-\frac{i}{d_i}D^\mu D_\nu d_i+
\frac{4i}{\ell_i^2}\delta^\mu_\nu
-i\ {}^{i}\!K(y_0)\, {}^{i}\!K^\mu_\nu(y_0) \\
&&+
\frac{1}{d_i} \frac{{\sqrt {\hat O_i}}}{{\rm sinh}{\sqrt
{\hat O_i}}}
\Biggl(K^\mu_\nu (y_i) -{\rm cosh} {\sqrt {\hat O_i}} \
{}^{i}K^\mu_\nu (y_0) \Biggr)=0. \nonumber
\end{eqnarray}
Since the last two terms of the first line are of order $d^0$ and are hence
negligible compared to the other ones which are of order $d^{-1}$, we can solve the above
equation for the unknown part of ${}^{\pm}K^\mu_\nu (y_0)$:
\begin{eqnarray}
\bar K^\mu_\nu (y_0)= \hat L \ Z^\mu_\nu, \label{K bar}
\end{eqnarray}
where the operator $\hat L$ is
\begin{eqnarray}
\hat L = -\left[\frac{1}{d_-}\frac{{\sqrt {\hat O_-}}}{{\rm tanh}{\sqrt {\hat O_-}}}
+\frac{1}{d_+}\frac{{\sqrt {\hat O_+}}}{{\rm tanh}{\sqrt {\hat O_+}}} \right]^{-1}
\end{eqnarray}
and
\begin{multline}
Z^\mu_\nu :=\sum_{i=\pm}\frac{i}{d_i}D^\mu D_\nu
d_i-\frac{1}{d_i}\frac{\sqrt{\hat{O}_i}}{\sinh \sqrt{\hat{O}_i}}\
K^\mu_\nu(y_i) \label{average}\\
-\frac{i}{2 d_i}
\frac{\sqrt{\hat{O}_i}}{\tanh \sqrt{\hat{O}_i}}\
\Delta K^\mu_\nu(y_0).
\end{multline}
It is worth pointing out that if a reflection symmetry was imposed
across the bulk brane, we would have $d_+=d_-$, $\hat O_+=\hat O_-$
and $K^\mu_\nu(y_+)=-K^\mu_\nu(y_-)$. The tensor $Z^\mu_\nu$
would hence vanish, and so would $\bar{K}^\mu_\nu$.\vspace{10pt}
\subsection{Effective-theory}
In the previous subsection, we have derived an expression for the
``asymmetric'' part of the extrinsic curvature in terms of quantities
that can be determined from the Isra\"el matching conditions and in
terms of the first derivative of the extrinsic curvature.
As far as we are aware this is the first derivation of the ``asymmetric''
term beyond the low-energy regime. Knowing the extrinsic curvature on
the brane, we may use its expression in the Taylor expansion
\eqref{taylor}, to get an expression for the derivative
\eqref{derivative} which can finally be substituted into the modified
Einstein equation \eqref{ricci}.
But first, we may express the equation of
motion for the radions, which can be derived from the traceless
property of the Weyl tensor ${}^{\pm}E^\mu_\nu (y_0)$. Using the
result of Ref.
\cite{CL,SMS}, the Weyl tensor
can be formally expressed as
\begin{eqnarray}
{}^{\pm}E^\mu_\nu (y_0) = -e^{-\varphi_\pm} {}^{\pm}K'^\mu_{\ \nu}
(y_0)
-\frac{1}{d_\pm}D^\mu
D_\nu d_\pm,
\end{eqnarray}
to leading order in $d_\pm$.
Since $E^\mu_\mu (y_0)=0$, this leads to the following Klein Gordon
equation for the distance between the branes:
\begin{eqnarray}
\hspace{-10pt}D^2 d_\pm =\hspace{-3pt}\pm\, \delta^\nu_\mu \frac{\sqrt{\hat{O}_\pm}}{\sinh
\sqrt{\hat{O}_\pm}}\Biggl[
K^\mu_\nu(y_\pm)\label{Box d}-\cosh \sqrt{\hat{O}_\pm}\, {}^{\pm}\!K^\mu_\nu(y_0)
\Biggr],
\end{eqnarray}
where the right-hand side can be computed from Eqs.
\eqref{Kbulk} and \eqref{K bar}.
The tracelessness of the Weyl
tensor together with the continuity constraint of the Ricci scalar
across the brane, also implies the supplementary constraint on the
asymmetric term $\bar K^\mu_\nu$:
\begin{eqnarray}
\Delta K \bar{K}-\Delta K^\alpha_\beta \bar{K}^\beta_\alpha
=6\left(\frac{1}{\ell_-^2}-\frac{1}{\ell_+^2}\right).
\label{HamConstraint}
\end{eqnarray}
The formal expression for the Ricci scalar on the bulk brane is therefore:
\begin{eqnarray}
{}^{(4)}R&=&-6\left(\frac{1}{\ell_+^2}+\frac{1}{\ell_-^2}\right)
+\bar{K}^2-\bar K^\alpha_\beta \bar{K}^\beta_\alpha \label{R}\\
&&+\frac 1 4 \Delta K^2-\frac 1 4\Delta K^\alpha_\beta \Delta
K^\beta_\alpha. \nonumber
\end{eqnarray}
We may now express the effective gravitational equation on the bulk brane. As
expected, it can be described by two scalar fields non-trivially
coupled to gravity:
\begin{multline}
{}^{(4)}G^\mu_\nu (y_0) = \frac{1}{d_\pm} (D^\mu D_\nu d_\pm -
\delta^\mu_\nu D^2 d_\pm)
\\ \mp
\frac{1}{d_\pm}
\left(\delta^\mu_\beta \delta^\alpha_\nu-\delta^\mu_\nu \delta^\alpha_\beta\right)
\frac{{\sqrt {\hat O_\pm}}}{{\rm
sinh}{\sqrt {\hat O_\pm}}}\\
\times
\Biggl[ K^\beta_\alpha (y_\pm)
-{\rm cosh}{\sqrt {\hat O_\pm}}{}^{\pm} K^\beta_\alpha (y_0)
\Biggr]. \label{G eff}
\end{multline}
The following formulae will be useful to rewrite the above equation in a more convenient way
\begin{eqnarray}
&\hat O \delta^\mu_\nu = 2D ^\mu d D _\nu d -\delta^\mu_\nu (D d)^2
\label{form1}\\
&\hat O^2 \delta^\mu_\nu =(D d)^4 \delta^\mu_\nu \label{form2}\\
&{\hat O}^2 S^\mu_\nu = 2D ^\mu d D _\nu d D ^\alpha d
D _\beta d
S^\beta_\alpha
-(D d)^2\hat O S^\mu_\nu
\label{form4}\\
&{\hat O}^{2n}S^\mu_\nu = (D d)^{4(n-1)} \hat O^2
S^\mu_\nu~~(n\geq 1)
\label{form5}\\
&{\hat O}^{2n+1}S^\mu_\nu = (D d)^{4n} \hat O S^\mu_\nu
\label{form6}
\end{eqnarray}
\begin{widetext}
\begin{eqnarray}
{\rm cosh}{\sqrt {\hat O}}S^\mu_\nu = S^\mu_\nu + \frac{{\rm
cos}|D d|-1 }{|D d|^2} \hat O S^\mu_\nu + \frac{{\rm
cosh}|D d|+ {\rm cos}|D d|-2}{|D d|^4}
D ^\mu d D _\nu d D _\alpha d D ^\beta d S^\alpha_\beta
\end{eqnarray}
\begin{eqnarray}
\frac{{\sqrt {\hat O}}}{{\rm sinh}{\sqrt {\hat O}}} S^\mu_\nu
= S^\mu_\nu +\Biggl(\frac{|D d|}{{\rm sin}|D d|} -1 \Biggr)|D d|^{-2} \hat O S^\mu_\nu
+\Biggl(\frac{|D d|}{{\rm sin}|D d|} +\frac{|D d|}{{\rm sinh}|D d|}-2 \Biggr)
|D d|^{-4} D ^\mu d D _\nu d D _\alpha d
D ^\beta d S^\alpha_\beta
\end{eqnarray}
\begin{eqnarray}
\frac{{\sqrt {\hat O}}}{{\rm tanh}{\sqrt {\hat O}}} S^\mu_\nu =
S^\mu_\nu +\Biggl(\frac{|D d|}{{\rm tan}|D d|} -1
\Biggr)|D d|^{-2} \hat O S^\mu_\nu
+\Biggl(\frac{|D d|}{{\rm tan}|D d|} +\frac{|D d|}{{\rm tanh}|D d|}-2 \Biggr)
|D d|^{-4} D ^\mu d D _\nu d D _\alpha d
D ^\beta d S^\alpha_\beta.
\end{eqnarray}
\end{widetext}
\subsection{Low-energy limit}
In the slow-velocity limit, the effective theory simplifies greatly. We
neglect the coupling of the radions to matter and neglect any terms
beyond second order in derivatives. In that case, the expression for
the asymmetric tensor $\bar{K}^\mu_\nu$ takes the form
\begin{eqnarray}
&&\hspace{-5pt}\bar K^\mu_\nu (y_0) =-\frac{1}{d}\left[d_-D^\mu
D_\nu d_+-d_+D^\mu D_\nu d_-\right]\\
&&\hspace{-5pt}-\frac{\kappa^2}{2d}\left[
d_- {}^{(+)}\tilde{T}^\mu_\nu-d_+ {}^{(-)}\tilde{T}^\mu_\nu
+\left(d_--d_+\right) {}^{(0)}\tilde{T}^\mu_\nu
\right]\nonumber\\
&&\hspace{-5pt}+\frac{1}{2\ell_+\ell_-}\left[
-\left(\ell_-+\ell_+\right)+\frac{1}{d}\left(\ell_+d_+\hat{O}_-+\ell_-d_-\hat{O}_+\right)
\right]\delta ^\mu_\nu , \nonumber
\end{eqnarray}
where $d:=d_++d_-$ and $\tilde{T}^\mu_\nu=T^\mu_\nu-\frac 1 3 T
\delta^\mu_\nu$. Using this expression in the modified Einstein
equation \eqref{G eff}, we obtain the induced Einstein tensor on the brane:
\begin{multline}
{}^{(4)}G^\mu_\nu (y_0)=\frac{1}{d}(D^\mu D_\nu d -\delta^\mu_\nu
D^2d )
+\frac{\kappa^2}{d}\ {}^{(\text{eff})}T^\mu_\nu\\
+\frac{1}{d}\Biggl[
\frac{1}{\ell_-} D ^\mu d_- D _\nu d_-+\frac{1}{2\ell_-}\left( D
d_-\right)^2 \delta^\mu_\nu\\
-\frac{1}{\ell_+} D ^\mu d_+ D _\nu d_+-\frac{1}{2\ell_+}\left( D
d_+\right)^2 \delta^\mu_\nu
\Biggr],
\end{multline}
where
${}^{(\text{eff})}T^\mu_\nu=(1/2){}^{(+)}T^\mu_\nu+(1/2){}^{(-)}T^\mu_\nu+{}^{(0)}T^\mu_\nu$.
This is precisely the close-brane limit of the low-energy theory
derived in \cite{Cotta}, and is hence a good consistency check.
The equations of motion for the two scalar fields can be derived
from Eqs. \eqref{Box d} and \eqref{HamConstraint}. Although
Eq. \eqref{Box d} appears as two different equations, they are not
independent and only give rise to the same following constraint for $ D^2 d$:
\begin{eqnarray}
D^2 d=\frac{\kappa^2}{3}\ {}^{(\text{eff})}T
+\frac{1}{\ell_-}\left(D d_-\right)^2
-\frac{1}{\ell_+}\left(D d_+\right)^2.
\end{eqnarray}
Using this result together with the continuity constraint
\eqref{HamConstraint}, we obtain the decoupled Klein-Gordon
equations for the two scalar fields at low-energy
\begin{eqnarray}
D^2 d_\pm=\frac{\kappa^2}{6}\left({}^{(\pm)}T\mp\frac{2\ell_\pm}{\ell_--\ell_+}{}^{(0)}T\right)
\mp\frac{1}{\ell_\pm}\left(D d_\pm\right)^2.
\end{eqnarray}
As another check, we can verify that this result is consistent with the
usual four-dimensional low-energy theory \cite{GE} if a reflection symmetry
was imposed across the brane. In that case $d_+=d_-$ and only one
scalar field is coupled to gravity.
\section{Applications}
In this section, we apply our effective theory
to cosmology and perturbations.
For the background solution, it is possible to solve the
Einstein equation exactly. We may therefore use this feature to
compare the exact five-dimensional result with our effective theory in
the close-brane limit. This provides us a useful check.
We will then use the effective theory in order to study cosmological
perturbations around this background.
\subsection{Cosmology}
\subsubsection{Five-dimensional solution}
In this subsection, we
first solve the five-dimensional Einstein equation exactly assuming
cosmological symmetry (i.e. we assume the spacetime to be homogeneous
and isotropic along the three spatial directions tangent to the
branes). Working in the frame where the bulk is static, one may use
the Birkhoff's theorem to derive easily the exact form of the
solution. But in order to compare this solution with our effective
theory, it will be useful to work instead in the frame where the
branes are static. Such a change of frame is in general difficult to
perform, but working in the close-brane regime, and neglecting higher
order terms in the distance between the branes, the change of frame
may perform easily as it has been shown in Refs. \cite{CL2,CL3}. We
will hence use the result of these papers to infer the geometry on the brane.
In the frame where the bulk
is static, the geometry on both regions $\mathcal{R}_\pm$ is simply
Schwarzschild-Anti-de Sitter (SAdS) with black-hole mass parameter
$\mathcal{C}_\pm$:
\begin{eqnarray}
ds^2_{\mathcal{R}_\pm}&=&-n_\pm^2 dT_\pm^2+dY_\pm^2 +a_\pm^2 d\mathbf{x}^2\\
a_\pm^2&=&e^{-2 Y_\pm/\ell_\pm}+\frac{\mathcal{C}_\pm}{4}e^{2 Y_\pm/\ell_\pm}\nonumber \\
n_\pm^2&=&a^2_\pm-\frac{\mathcal{C}_\pm}{a_\pm^2}.\nonumber
\end{eqnarray}
It is important to notice that in this frame, the branes are not
static, as in the previous section, and we will assume the branes to
have loci $Y=Y_i(T)$. In particular, the bulk brane has loci
$Y_+=Y_0^{(+)}(T_+)$ with respect of the region $\mathcal{R}_+$, and loci
$Y_-=Y_0^{(-)}(T_-)$ as measured from an observed in the static bulk frame $\mathcal{R}_-$.
The induced line element on the bulk brane can be read off as:
\begin{eqnarray}
ds^2_{0}&=&-\left[n_0^2 - \left(\frac{d Y_0^{(\pm)}}{d
T_\pm}\right)^2\right]dT_\pm^2+a_0^2 \, d\mathbf{x}^2\\
&=&-d t^2+a_0^2 \, d\mathbf{x}^2,
\end{eqnarray}
where $a_0$ is the induced scale factor on the bulk brane:
$a_0(t)=a_+\left(Y_+=Y_0^{(+)}(T_+(t))\right)=a_-\left(Y_-=Y_0^{(-)}(T_-(t))\right)$,
and similarly for $n_0$.
The physical time $t$ on the bulk brane, may be expressed in terms
of the five-dimensional time coordinate $T_\pm$:
\begin{eqnarray}
d t^2 =\left[n_0^2 - \left(\frac{d Y_0^{(\pm)}}{d
T_\pm}\right)^2\right]dT_\pm^2.
\end{eqnarray}
In order to derive the Friedmann equation on the branes, we may use the
Isra\"el junction conditions \eqref{junction bulk brane}
\begin{eqnarray}
\Delta K^i_j(y_0)=-\frac{\kappa^2}{3}\left(\sigma_0+\rho_0\right)
\delta^i_j,
\label{junction cosmology}
\end{eqnarray}
where $\rho_0$ is the energy density of matter fields located on the
brane and the extrinsic curvature is
\begin{eqnarray}
^{\pm}K^i_j(y_0)=\delta^i_j\
\left(1-\frac{\dot{Y}_0^{(\pm) \, 2}}{n_0^2}\right)^{-1/2}\left.\frac{d a_\pm(Y)}{a_\pm
d Y_\pm}\right|_{Y_0^{(\pm)}},
\end{eqnarray}
where $\dot{Y}_0^{(\pm)}=d Y_0^{(\pm)}(T_\pm)/d T_\pm$. We may now re-express
the extrinsic curvature in terms of the Hubble parameter on the
brane. In particular we use the relation
\begin{eqnarray}
\dot{Y}_0^{(\pm)}=\frac{d a_0}{d t}\frac{d t}{d T_\pm}
\left(\left. \frac{d a_\pm(Y_\pm)}{dY_\pm}\right|_{Y_0^{(\pm)}}\right)^{-1}.
\end{eqnarray}
Using the fact that $d a_\pm/dY_\pm=-n_\pm/\ell_\pm$, we have:
\begin{eqnarray}
\dot{Y}_0^{(\pm)\, 2}=\frac{\ell_\pm^2 a^2_0 H^2}{1+ \ell^2_\pm \, a_0^2
\, H^2/n_0^2}, \label{ydot}
\end{eqnarray}
where $H$ is the Hubble parameter on the brane $H =(da_0/dt)/a_0$. The
extrinsic curvature on each side of the bulk brane can therefore be
expressed in terms of the Hubble
parameter as:
\begin{eqnarray}
^{\pm}K^i_j(y_0)&=&-\delta^i_j\
\sqrt{\frac{n_0^2}{\ell_\pm^2 a_0^2}+H
^2},\notag \\
&=&-\delta^i_j\
\sqrt{\frac{1}{\ell_\pm^2}-\frac{\mathcal{C}_\pm}{\ell_\pm^2 a_0^4}+H
^2}.\label{K cosmology}
\end{eqnarray}
Having an expression for the extrinsic curvature in terms of the
Hubble parameter on each side of the bulk brane, we can therefore use the
Isra\"el matching condition Eq. \eqref{junction cosmology} to
express the Hubble parameter on the brane in terms on the energy
density and the black-hole mass parameters.
Substituting Eq. \eqref{K cosmology} into \eqref{junction cosmology}, we find the modified
Friedmann equation on the asymmetric bulk brane:
\begin{multline}
H ^2=\frac{T_b^2}{4}
+\frac{1}{4T_b^2}\left[\frac{1}{\ell_+^{2}}-\frac{1}{\ell_-^{2}}-\frac{1}{a_0^{4}}
\left(\frac{\mathcal{C}_+}{\ell_+^2}-\frac{\mathcal{C}_-}{\ell_-^2}\right)
\right]^2 \\
-\frac{1}{2}\left[\frac{1}{\ell_+^2}+\frac{1}{\ell_-^2}-\frac{1}{a_0^4}
\left(\frac{\mathcal{C}_+}{\ell_+^2}+\frac{\mathcal{C}_-}{\ell_-^2}\right)\right],
\label{Friedmann non Z2}
\end{multline}
with $T_b=\frac{\kappa^2}{3}\left(\sigma_0+\rho_0\right)$.
In this modified Friedmann equation, one might think that the
parameters $\mathcal{C}_\pm$ are arbitrary, but in
what follows, we show that they depend strongly on the brane
velocities and find the precise relation between them. This is important
as it will allow us to compare this result with the one obtained from
the effective close-brane theory which gives a direct relation with
the brane velocities.
\subsubsection{Expression for the velocity of the branes.}
Without loss of generality, we work in the specific situation where
the bulk brane is about to collide with the positive boundary brane
($\dot d _+<0$),
and moves away from the negative boundary brane ($\dot d_->0$).
The velocity of both boundary branes should therefore be
positive while the velocity of the bulk brane should be negative.
In the case where the bulk brane has a positive canonical tension, $\ell_+>\ell_-$
the Hubble constant on the bulk brane will be positive.
We may now use the results of Refs. \cite{CL2,CL3} where
the the following
relation for the radion velocity holds in the close-brane regime:
\begin{eqnarray}
\dot{d}_\pm= \mp
\left(\tanh^{-1}\left(\frac{v_\pm}{n_0}\right)+\tanh^{-1}\left(\frac{^{\pm}v_0}{n_0}\right)
\right),
\end{eqnarray}
where $v_\pm$ are the absolute value of velocities of the boundary branes with respect
to five-dimensional physical time at the collision and
$^{\pm}v_0=\left|d Y_0^{(\pm)}/d T_{\pm}\right|$
is the absolute value of the velocity of the bulk brane as measured by a static observer
in region $\mathcal{R}_\pm$. In what follows, a dot designates the
derivative with respect to the physical time $t$ on the bulk brane.
Using the result of Ref. \cite{CL3}, one has:
\begin{eqnarray}
\left(\frac{v_\pm}{n_0}\right)=
\sqrt{1-\frac{1-\mathcal{C}_\pm/a_0^4}{\left(1\pm\frac16\kappa^2\ell_\pm\rho_\pm\right)^2}},
\end{eqnarray}
where $\rho_\pm$ is the energy density on the $\pm$-brane.
Using the expression \eqref{ydot}, one has the expression for the bulk
brane velocity
\begin{eqnarray}
\left(\frac{^{\pm}v_0}{n_0}\right)=\frac{\ell_\pm H }{\sqrt{\ell_\pm^2
H ^2+1-\mathcal{C}_\pm/a^4_0}}.
\end{eqnarray}
The radions' velocities
$\dot d_\pm$ can hence be expressed in terms of
the black-hole mass parameter $\mathcal{C}_\pm$
\begin{multline}
\dot d_\pm=\mp\tanh ^{-1}\sqrt{1-\frac{1-\mathcal{C}_\pm/a_0^4}{\left(1\pm\frac16
\kappa^2 \ell_\pm\rho_\pm\right)^2}}\\
\mp\tanh^{-1}\left(\frac{\ell_\pm
H }{\sqrt{\ell_\pm^2
H ^2+1-{\mathcal{C}_\pm}/{a_0^4}}}\right).
\end{multline}
We may use these equations to find an expression for the constants
$\mathcal{C}_\pm$ in terms of the brane velocities:
\begin{multline}
\frac{\mathcal{C}_\pm}{a_0^4}=\left[\frac{\pm\, \ell_\pm H +\sinh \dot d_\pm
\left(1\pm \frac16\kappa^2\ell_\pm\rho_\pm\right)}{\cosh \dot
d_\pm}\right]^2 \label{C and H} \\
\mp\frac16{\kappa^2\ell_\pm}\rho_\pm\left(
2\pm\frac16{\kappa^2\ell_\pm}\rho_\pm
\right).
\end{multline}
Using this relation for $\mathcal{C}_-$, one has
\begin{eqnarray}
H^2+\frac{1-\mathcal{C}_-/a_0^4}{\ell_-^2}=
\left(\frac{1-\frac16{\kappa^2\ell_-}\rho_- +\ell_- H \sinh \dot
d_-}{\ell_-\cosh \dot
d_-}\right)^2. \label{H and C}
\end{eqnarray}
But from Eq. \eqref{Friedmann non Z2}, one has as well:
\begin{multline}
H^2+\frac{1-\mathcal{C}_-/a_0^4}{\ell_-^2}=\\
\left[
\frac{T_b}{2}-\frac{1}{2 T_b}
\left(\frac{1}{\ell_+^2}-\frac{1}{\ell_-^2}+
\frac{\mathcal{C}_-}{a_0^4\ell_-^2}-\frac{\mathcal{C}_+}{a_0^4\ell_+^2}\right)
\right]^2.
\end{multline}
Substituting the expression \eqref{C and H} for $\mathcal{C}_\pm$ into the last line, we
therefore have an equation for $H $ in terms of $\dot d_\pm$,
which has for solution:
\begin{eqnarray}
H =
\Lambda^{\text{eff}}+\frac16{\alpha \kappa^2}\rho_0^{\text{eff}},
\label{H and
dot d}
\end{eqnarray}
with the notation
\begin{align}
\rho_0^{\text{eff}}&=\frac{\rho_-}{2\cosh \dot
d_-}+\frac{\rho_+}{2\cosh \dot d_+}+\rho_0,\nonumber\\
\Lambda^{\text{eff}}&=\frac{\alpha}{2\ell_-}\left(1-\frac{1}{\cosh \dot
d_-}\right)
-\frac{\alpha}{2\ell_+}\left(1-\frac{1}{\cosh \dot d_+}\right), \nonumber\\
\alpha&=2 \left(\tanh \dot d_++\tanh \dot d_-\right)^{-1}. \nonumber
\end{align}
We may now compare this result with what is obtained from the
effective theory. But first, we might make some important remarks.
At low-energy, the contribution from the matter on each brane has an
equal weight: $\rho_0^{\text{eff}}\simeq
\left(\rho_-/2+\rho_+/2+\rho_0\right)$.
This is due to the fact, that
at low-energy, when the branes are close, the bulk geometry is
almost Minkowski and each brane has an equal contribution.
This result is however not obvious from the usual Friedmann equation (\ref{Friedmann non
Z2}), where only the matter on the bulk brane seems to contribute,
but one should take into account the expression of $\mathcal{C}_\pm$ in terms of $\rho_\pm$.
However, at high-velocities, the situation is radically
different: The geometry on the bulk brane decouples entirely from
the matter content on the orbifold branes (we may point out that
this result is valid for the two-brane case as well Cf. Ref. \cite{CL3}).
In that limit, we indeed have $\rho_0^{\text{eff}}\rightarrow \rho_0$, with an effective
cosmological constant $({1}/{2})\left|{1}/{\ell_+}-{1}/{\ell_-}\right|$.
Its contribution vanishes when the asymmetry
across the brane is maximal: $\ell_+=\ell_-$. In that case the bulk
geometry is almost unperturbed by the brane and the Friedmann
equation on the bulk brane couples quadratically to its own matter
$H^2_0={\kappa^4}\rho_0^2/36$. This is an exact result
arising from the five-dimensional equations of motions in the close
brane and high-velocity limit.
\subsubsection{Cosmology in the effective theory}
We now wish to compare this exact result with the predictions from
the close-brane effective theory.
The modified Einstein equation \eqref{G eff} on the
bulk brane reads:
\begin{eqnarray}
G^i_j(y_0)&=&\left(H ^2-\frac{1}{3}R^2\right)\delta^i_j \sim d^0 \nonumber\\
&=& \frac{1}{d_-}D^iD_jd_-+\frac{(y_--y_0)}{d_-}\, {}^-K^{\prime\,
i}_{\, \, j}\sim d^{-1}.\ \
\end{eqnarray}
Since the first line is of higher order in the distance between the
brane, the second line should vanish. The expression for the
derivative of the extrinsic curvature can be found in
\eqref{derivative}, and
using the relation $\hat{O}_\pm Z^i_j=\dot{d}_\pm^2 Z^i_j$ valid for
the background, we therefore have the equation:
\begin{multline}
0=-H \dot d_-\delta^i_j+\frac{\dot d_-}{\sinh \dot d_-}
K^i_j(y_-) \label{eq H}\\
-\frac{\dot d_-}{\tanh \dot d_-}\left(\frac{1}{2}\Delta
K^i_j(y_0)+\bar{K}^i_j(y_0)\right) ,
\end{multline}
where $\Delta K^i_j(y_0)$ is given in Eq. \eqref{junction cosmology},
\begin{eqnarray}
K^i_j(y_\pm)=\left(-\frac{1}{\ell_\pm}\mp
\frac{\kappa^2}{6}\rho_\pm\right)\delta^i_j, \label{Kij orbifold}
\end{eqnarray}
and $\bar{K}^i_j$ is given by:
\begin{eqnarray}
\hspace{-10pt}\bar{K}^i_j\hspace{-2pt}&=&\hspace{-2pt}-
\Biggl[
-\frac{\dot d_-}{d_-\sinh \dot d_-}K^i_j(y_-)
-\frac{\dot d_+}{d_+\sinh \dot d_+}K^i_j(y_+)\label{K bar
cosmology}\\
\hspace{-2pt}&+&\hspace{-2pt}\frac{1}{2}\left(\frac{\dot d_-}{d_-\tanh \dot d_-}-\frac{\dot
d_+}{d_+\tanh \dot d_+}\right)\Delta K^i_j(y_0)\notag\\
\hspace{-2pt}&+& \hspace{-2pt} H \left(\frac{\dot d_-}{d_-}-\frac{\dot
d_+}{d_+}\right)\delta^i_j\Biggr]
\left(\frac{\dot d_+}{d_+ \tanh \dot d_+}+\frac{\dot d_-}{d_- \tanh \dot
d_-}\right)^{-1}.\notag
\end{eqnarray}
Using these expressions \eqref{junction cosmology}, \eqref{Kij
orbifold} and \eqref{K bar cosmology} in the Eq. \eqref{eq H} for
$H $, we finally obtain the relation between the Hubble parameter on
the asymmetric
brane and the radions' velocities:
\begin{eqnarray}
H =
\Lambda^{\text{eff}}+\frac16{\alpha \kappa^2}\rho_0^{\text{eff}},
\end{eqnarray}
with the same notations as for Eq. \eqref{H and dot d}. This
corresponds precisely to what was obtained from the exact
five-dimensional theory and represents an important consistency
check.
We may also wonder whether this theory is capable of reproducing
the expression \eqref{Friedmann non Z2} for the Hubble parameter.
This Friedmann equation is a simple consequence of the tracelessness
of the Weyl tensor as we shall see. Using this property for the Weyl
tensor, we have indeed obtained in Eq. \eqref{R} an expression for
the Ricci scalar in terms of $\bar{K}^\mu_\nu$ and $\Delta
K^\mu_\nu$. The expression of $\Delta K^\mu_\nu$ is in general
complicated, but for cosmological solutions the Eq. \eqref{HamConstraint} imposes
the constraint:
\begin{eqnarray}
\bar{K}^0_0=\left(-2-\frac{\Delta K^0_0}{\Delta K^1_1}\right)\bar{K}^1_1
+\frac{2}{\Delta K^1_1}\left(\frac{1}{\ell_+^2}-\frac{1}{\ell_-^2}\right),
\end{eqnarray}
which we may reexpress as
\begin{eqnarray}
\bar{K}^0_0=\frac{-\kappa^2p_0-3x}{x-\kappa^2\rho_0/3}\bar{K}^1_1
-\frac{2xy}{x-\kappa^2\rho_0/3},
\end{eqnarray}
where for simplicity we wrote $x=\ell_+^{-1}-\ell_-^{-1}$ and
$y=\ell_+^{-1}+\ell_-^{-1}$. Furthermore, from the Codacci equation,
we have the relation
\begin{eqnarray}
\partial_a \bar{K}^1_1(a)=a \left(\bar{K}^0_0-\bar{K}^1_1\right).
\end{eqnarray}
Using this result together with the conservation of energy condition
$p_0=-a\rho_0'(a)/3-\rho_0$, we may solve this
differential equation for $\bar{K}^1_1$ and obtain
\begin{eqnarray}
\bar{K}^1_1=-\frac{xy}{2\left(x-\kappa^2\rho_0/3\right)}
+\frac{C_{\text{A}}}{a_0^4\left(x-\kappa^2\rho_0/3\right)},
\end{eqnarray}
where the ``asymmetric" constant $C_{\text{A}}$ appears as an
integration constant. We may now use this expression in Eq. \eqref{R}:
\begin{eqnarray}
R&=&6aHH'(a)+12H^2 \nonumber\\
&=& \left[-\frac{\left({2C_{\text{A}}}/{a^4}-x y\right)^2}{2
T_b^3}+\frac 1 2 T_b^2\right]a\, \rho'_0(a)\\
&+&\frac{1}{T_b^2}\left[-12
\left(\frac{C_{\text{A}}}{a^4}\right)^2+\left(y^2-T_b^2\right)\left(x-T_b\right)\rho_0(a)
\right],\nonumber
\end{eqnarray}
where $T_b=T_b(a)=(1/3){\kappa^2}\left(\sigma_0+\rho_0(a)\right)$. This
is simply a first order differential equation for $H(a)$, of which
solution is
\begin{equation}
H^2=\frac{T_b^2}{4}+\frac{1}{4T_b^2}\left(
-xy+2\frac{C_{\text{A}}}{a_0^4}
\right)^2
-\frac 1 4 \left(x^2+y^2+\frac{C_S}{a_0^4}\right).
\end{equation}
The parameter $C_{\text{S}}$ appears as an integration
constant as well. This expression corresponds precisely to the Friedmann
equation \eqref{Friedmann non Z2} obtained by solving the
five-dimensional geometry, and we can now relate the black-hole mass
parameters to the integration constants $C_{\text{A}}$ and
$C_{\text{S}}$
\begin{eqnarray}
C_{\text{A}}&=&\frac{1}{2}\left(\frac{\mathcal{C}_+}{\ell_-^2}
-\frac{\mathcal{C}_-}{\ell_+^2}\right)\\
C_{S}&=&-2\left(\frac{\mathcal{C}_+}{\ell_-^2}+\frac{\mathcal{C}_-}{\ell_+^2}\right).
\end{eqnarray}
In particular, the contribution from $C_{\text{S}}$ should be
expected in a general case and is responsible for the dark energy
term. The contribution of $C_{\text{A}}$ is specific to the
asymmetric brane and should cancel when $\bar{K}^\mu_\nu=0$, this is
indeed the case when one has opposite AdS scales on each regions
$\mathcal{R}_\pm$: $\ell_-=-\ell_+$, and when the black-hole mass
parameters are the same: $\mathcal{C}_-=\mathcal{C}_+$. In a general
case it is not possible to deduce the expression of the asymmetric
term by solving the five-dimensional theory, and our theory provides
a useful alternative. This is for instance the case for the study of
perturbations.
\subsection{Cosmological perturbations}
The aim of this paper is to provide an effective theory capable of
describing a bulk brane geometry in a consistent way, beyond the
low-energy approximation.
In this paper we will hence not extend
this study to a large analysis of perturbations, which would be a
subject on its own, but we may point out some useful comments which
will be relevant for such a study. In particular, we have seen in the
previous section, that at high-velocities, the geometry on the bulk
brane decouples from the matter content of the orbifold branes. This
result was valid for cosmology and we may check the effect of
tensor perturbations.
First we may stress that the operators $\hat O_+$ and $\hat O_-$ do
not commute in general. For a symmetric tensor $S_{\mu \nu}=S_{\left(\mu
\nu\right)}$,
\begin{multline}
\left[\hat O_-,\hat O_+\right]S^\mu_\nu
= \Big[
\left(D^\mu d_-D_\beta d_+-D^\mu d_+D_\beta d_-\right)S^\beta_\nu \\
+\left(D_\nu d_-D^\beta d_+-D_\nu d_+D^\beta d_-\right)S^\mu_\beta \Big]
\left(D_\alpha d_-\, D^\alpha d_+\right).
\end{multline}
So apart for cosmology and for tensor, vector perturbations or if we
work in a gauge where both $\delta d_-$ and $\delta d_+$ may be set
to zero, the
order of the action of the different operators in the effective
theory is important. In particular we expect this feature to have
consequences on the evolution of non-linear perturbations.
As a specific simple example, one might study here the evolution of
tensor perturbations:
\begin{eqnarray}
ds^2_0=a^2\left[-d\tau^2+\left(\delta_{ij}+h_{ij}\right)dx^i
dx^j\right],
\end{eqnarray}
where $\tau$ is the conformal time on the bulk brane and we write $h^i_j=\delta^{ik}h_{kj}$.
For tensor
perturbations, the right hand side of the modified Einstein equation
\eqref{G eff} is hence:
\begin{multline}
\delta G^i_j=\delta R^i_j=-\frac{({\dot d_+}/{2
d_+})X_-+({\dot d_-}/{2
d_-})X_+}{X_-+X_-}\ \dot{h}^i_j\\
+\kappa^2\, \frac{X_-X_+}{X_-+X_+}\ {}^{(\text{eff})}\!\delta
T^i_j ,
\end{multline}
with the effective matter contribution:
\begin{eqnarray}
\hspace{-15pt}{}^{(\text{eff})}\!\delta T^i_j=
\frac{1}{2 \cosh \dot d_-}\,{}^{(-)}\!\delta T^i_j
+\frac{1}{2 \cosh \dot d_+}\,{}^{(+)}\!\delta T^i_j
+\,{}^{(0)}\!\delta T^i_j,
\end{eqnarray}
where $^{(k)}\!\delta T^i_j$ is the tensor part of matter
perturbations on the brane at $y=y_k$.
For simplicity, we wrote $X_\pm=\dot d_\pm /\left(d_\pm \tanh \dot
d_\pm\right)$. The only non-negligible part in the perturbation of the
Ricci tensor is $\delta R ^i_j=-\frac{\bar{\partial^2}}{2 a_0^2}\, h^i_j$,
the evolution of the tensor perturbations, is hence controlled by
\begin{equation}
\hat{\boxdot} \, h^i_j=-a_0^2\,\Omega\kappa^2\ {}^{(\text{eff})}\!\delta T^i_j,
\end{equation}
with
\begin{align}
\hat{\boxdot}&=
\left[\bar{\partial^2}
-\frac{d'_+d'_-}{d_+d_-}\frac{\tanh \dot d_++\tanh \dot d_-}
{\frac{d'_+}{d_+}\tanh \dot d_++\frac{d'_-}{d_-}\tanh \dot
d_-}\partial_\tau
\right],\nonumber \\
\Omega&=2\left(\frac{d_+ \tanh \dot d_+}{\dot d_+}
+\frac{d_- \tanh \dot d_-}{\dot d_-}\right)^{-1},\nonumber
\end{align}
where a prime designates derivative with respect to
the conformal time $\tau$ and the operator $\bar{\partial^2}$ is the
Laplacian in Minkowski space. One may note that in the close brane
limit, if $d_\pm \sim \dot d_\pm t$, the damping term
$\frac{d'_+d'_-}{d_+d_-}\frac{\tanh \dot d_++\tanh \dot d_-}
{({d'_+}/{d_+})\tanh \dot d_++({d'_-}/{d_-})\tanh \dot d_-}$
simply goes as $\tau^{-1}$, which is what is expected from a
usual four-dimensional theory. The expression for the effective
four-dimensional Newtonian constant is on the other hand slightly
affected: $\kappa_{(4d)}^2=\kappa^2\Omega$, which is similar to the
result obtained in Ref. \cite{CL3}. For more sophisticated analysis,
we however expect the result to be more interesting, especially
when the operators $\hat O_\pm$ do not commute.
However, we may point out that the remarks formulated for the
background remain valid at the level of perturbations. Namely, for
large brane velocities, the effective matter contribution on the
bulk brane is
\begin{eqnarray}
{}^{(\text{eff})}\!\delta T^i_j\rightarrow\,{}^{(0)}\!\delta
T^i_j,
\end{eqnarray}
and perturbations are not sensitive to the matter content of
the orbifold branes.
This provides a braneworld scenario, where the branes
could be close (and hence the Kaluza Klein modes difficult to
excite and to affect the brane), and yet the geometry on the bulk
brane would decouple from the other ones. \vspace{5pt}
\section{Summary and Discussion}
In this paper, an effective theory describing the gravitational
behaviour of a bulk brane has been derived. The absence of any
reflection symmetry across a generic bulk brane makes its behaviour
especially interesting to study. In the ``light"-brane limit, ie.
when the five-dimensional geometry is almost unaffected by the
presence of the brane, the asymmetry on the brane itself is
important and affects its own behaviour. In this work, we have
developed a four-dimensional effective theory capable of describing
this ``asymmetry" in a covariant way. For that, we have
considered a close-brane approximation, where we assumed the bulk
brane to be close to both orbifold branes.
Using this approximation, we obtained a resulting theory of gravity coupled in
a non-trivial way with two scalar fields representing the distance
between the bulk brane and each of the orbifold branes. This
four-dimensional theory can be tested in several limits, such as at
low-energy, when a reflection symmetry is imposed by hand and for
cosmology. In all these regimes, predictions from the close-brane
theory agree perfectly with the expected results. The case of
cosmology is of special interest, at high-velocity the bulk geometry
is not sensitive to the matter present on the orbifold branes, and
this result remains valid for tensor perturbations. In the limit
where the AdS length scale is the same on both side of the brane,
ie. the bulk is not perturbed by the brane, we show that the Hubble
parameter couples linearly to the energy density on the bulk brane.
This is an interesting result, which might strongly affect the
gravitational behaviour on such a brane.
This effective theory could as well be derived on the orbifold
branes in the presence of such a brane in the bulk. We may point out
that the asymmetric tensor we derived on the bulk brane depends on the bulk
brane metric. It will hence be necessary to find its expression in
terms of the orbifold brane metric before being able to derive an
effective theory for these orbifold branes. This is left for a future study.
A straightforward extension of this model, would be the scenario where the bulk brane
is close to only one of the orbifold branes, and the radion
representing the distance with the other orbifold brane is moving
slowly. One side of the theory would hence be modeled by the
low-energy effective theory while the close-brane theory would be a
good description for the other side. Such a model would be of
interest if one considers the collision of the bulk brane with one
of the orbifold branes. Such a process might produce a phase
transition which could have some interesting consequences from a
cosmological point of view. This is also left for a future study.
\section*{Acknowledgements}
The work of TS was supported by Grant-in-Aid for Scientific
Research from Ministry of Education, Science, Sports and Culture of
Japan(No.13135208, No.14102004, No. 17740136 and No. 17340075).
CdR is supported by DAMTP and was invited by a FGIP visiting
program. CdR wishes to thank TITECH for its hospitality.
|
hep-th/0509096
|
\section*{Appendix \thesection\protect\indent \parbox[t]{11.715cm} {#1}}
\addcontentsline{toc}{section}{Appendix \thesection\ \ \ #1}
}
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\newcommand{\begin{equation}}{\begin{equation}}
\newcommand{\end{equation}}{\end{equation}}
\newcommand{\begin{aligned}}{\begin{aligned}}
\newcommand{\end{aligned}}{\end{aligned}}
\newcommand{{1\over 2}}{{1\over 2}}
\newcommand{{\it i.e.}}{{\it i.e.}}
\newcommand{{\it e.g.}}{{\it e.g.}}
\newcommand{y^{(1)}}{y^{(1)}}
\newcommand{y^{(2)}}{y^{(2)}}
\newcommand{\mathfrak{sl}(2)}{\mathfrak{sl}(2)}
\newcommand{\mathfrak{su}(2)}{\mathfrak{su}(2)}
\makeatletter
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\begin{document}
\thispagestyle{empty}
\begin{flushright}\footnotesize
\texttt{hep-th/0509096}\\
\texttt{AEI-2005-145}\\
\texttt{DESY-05-163}\\
\texttt{ZMP-HH/05-17}\\
\vspace{0.8cm}
\end{flushright}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\setcounter{footnote}{0}
\begin{center}
{\Large\textbf{\mathversion{bold}
Stringy sums and corrections to the \\
quantum string Bethe ansatz
}\par}
\vspace{1.5cm}
\textrm{Sakura Sch\"afer-Nameki$^{\alpha}$ and Marija Zamaklar$^{\beta}$ } \vspace{8mm}
\textit{$^{\alpha}$ II. Institut f\"ur Theoretische Physik der Universit\"at Hamburg\\
Luruper Chaussee 149, 22761 Hamburg, Germany} \\
\texttt{sakura.schafer-nameki@desy.de}
\vspace{3mm}
\textit{$^{\alpha}$ Zentrum f\"ur Mathematische Physik, Universit\"at Hamburg\\
Bundesstrasse 55, 20146 Hamburg, Germany} \vspace{3mm}
\textit{$^{\beta}$ Max-Planck-Institut f\"ur Gravitationsphysik, AEI\\
Am M\"uhlenberg 1, 14476 Golm, Germany}\\
\texttt{marzam@aei.mpg.de}
\vspace{3mm}
\par\vspace{1cm}
\textbf{Abstract}\vspace{5mm}
\end{center}
\noindent
We analyze the effects of zeta-function regularization on the
evaluation of quantum corrections to spinning strings.
Previously, this method was applied in the $\mathfrak{sl}(2)$ subsector and yielded
agreement to third order in perturbation
theory with the quantum string Bethe ansatz.
In this note we discuss
related sums and compare zeta-function regularization against
exact evaluation of the sums, thereby showing that the zeta-function
regularized
expression misses out perturbative as well as non-perturbative
terms. In particular, this may imply corrections to the
proposed quantum string Bethe equations.
This also explains the previously observed discrepancy between the
semi-classical string and the quantum string Bethe ansatz in the
regime of large winding number.
\vspace*{\fill}
\newpage
\setcounter{page}{1}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\tableofcontents
\section{Introduction and Summary}
Explicit checks of the AdS/CFT correspondence beyond the supergravity
approximation have been obstructed by the disjointness of the regimes
in which gauge theory and string theory are understood in perturbation
theory. Exact quantization of string theory on $AdS_5\times S^5$ may help
overcoming this problem and has therefore been the focus of much
recent investigations.
Key progress in this direction was triggered by the insight
gained from studying the AdS/CFT correspondence in
specific limits, as initiated by \cite{Berenstein:2002jq},
\cite{Gubser:2002tv}, and in \cite{FrolovTseytlinI, Frolov:2003qc,Frolov:2003tu,Frolov:2003xy,Arutyunov:2003uj,Arutyunov:2003za, Frolov:2004bh}.
Further insight was obtained by identifying the integrable
structures both in gauge and string theory.
On the gauge theory side, this was deduced from the identification of the planar
one-loop dilatation operator of $N=4$ SYM
with the Hamiltonian of an integrable (super) spin chain
\cite{Minahan:2002ve, Beisert:2003yb}, solvable by means of
a Bethe ansatz. The extension of the integrable structure to higher
loops was subsequently shown in \cite{Beisert:2003tq, Beisert:2003ys,
Serban:2004jf}\footnote{Altough integrability breaks down beyond the
planar limit, some remnants of it persist and can be used to study
decays of semi-classical strings
\cite{Peeters:2004pt}.}. On the other
hand, integrability of the string sigma model on $AdS_5 \times S^5$
\cite{Metsaev:1998it} was observed in
\cite{Bena:2003wd}, and then utilised to test the AdS/CFT
correspondence \cite{Beisert:2003xu, Beisert:2003ea, Engquist:2003rn}\footnote{For reviews and further references see
\cite{Tseytlin:2003ii, Beisert:2004ry, Zarembo:2004hp, Tseytlin:2004xa,
Plefka:2005bk}. }.
An important step linking the two integrable structures on more general
grounds was made in \cite{Kazakov:2004qf} by the
construction of a set of Bethe equations for the classical string
sigma-model\footnote{See also \cite{Arutyunov:2003rg, Arutyunov:2004xy,
Arutyunov:2005nk, Mikhailov:2005wn} which identified the infinite
tower of conserved charges on both sides.
The classical string sigma-model reduces in the
large spin limit to the effective action of the spin-chain, as was
first observed in \cite{Kruczenski:2003gt}.
}.
These were then compared to the gauge theory Bethe equations in the
thermodynamic limit, first for various subsectors and then the full
$N=4$ SYM and $AdS_5\times S^5$ superstring \cite{Kazakov:2004qf,
Beisert:2004ag, Schafer-Nameki:2004ik, Beisert:2005bm, Alday:2005gi,
Beisert:2005di}.
Inspired by the classical Bethe equations, a proposal was put forward
for the description of quantum strings on $AdS_5\times S^5$
\cite{Arutyunov:2004vx,Staudacher:2004tk,Beisert:2005fw}. It was
conjectured that the string spectrum
can be described by a new type of quantum string Bethe equations, which
diagonalize some underlying string chain, and which are obtained by
discretizing the classical string Bethe equations
\cite{Kazakov:2004qf}.
The conjectured quantum string Bethe equations were rigorously tested at infinite
$\lambda $. However, they could potentially receive $1/\sqrt{\lambda
}$ corrections \cite{Arutyunov:2004vx}.
To further test the proposal of
\cite{Arutyunov:2004vx,Staudacher:2004tk,Beisert:2005fw},
a detailed comparison between the one-loop worldsheet
correction to the energy of a particular string configuration (which
was computed
semi-classically) to the finite size
corrections following from the quantum
string Bethe ansatz was recently performed \cite{Schafer-Nameki:2005tn}.
The configuration studied was a circular string
spinning in $AdS_3\times S^1$ \cite{Park:2005ji}. In this case the
correction to the classical energy depends on two
parameters $\mathcal{J}$ and $k$ ($\mathcal{J}^2 =1/\lambda' =
J^2/\lambda$), where $k$ is the
string winding number and $J$ is the spin in the $S^1$ direction. In
\cite{Schafer-Nameki:2005tn} the comparison between
semi-classical strings and Bethe ansatz was studied in the following two regimes:
large $\mathcal{J}$ (and finite $k$) and large
$k$ (and finite $\mathcal{J}$).
In the first instance, due to the high complexity of the sums for the
semi-classical string corrections, the analysis was performed by first
expanding the summands in the parameter $1/{\mathcal J}$ (assuming
that the summation index $n$ is smaller than $\mathcal{J}$) and
subsequent resummation. This procedure clearly breaks down for
$n\geq\mathcal{J}$, and thus yields divergent expressions at each
order in $1/\mathcal{J}^{2l}$. However upon zeta-function
regularisation these agree with the Bethe ansatz in the first three
orders in $1/\mathcal{J}^2$ \cite{Schafer-Nameki:2005tn}. This
extended the leading order agreement previously found in
\cite{Beisert:2005mq, Hernandez:2005nf}. Other
discussions of $1/J$ corrections have appeared in
\cite{Beisert:2003xu, Freyhult:2004iq, Freyhult:2005fn, Fuji:2005ry, Minahan:2005ux}.
In the second case of large winding number $k$, exact evaluation of
the sum (which did not involve
zeta-function regularization) resulted in a disagreement with the
prediction of the
string Bethe ansatz already at leading order in $1/k$
\cite{Schafer-Nameki:2005tn}. A similar mismatch was observed numerically.
As a possible explanation for the incompatibility of these results
it was proposed that zeta-function regularization may not correctly
sum the semi-classical string result \cite{Schafer-Nameki:2005tn}.
A numerical analysis was performed to confirm this conjecture, but due to the
insufficient numerical precision it was not possible to deduce a firm
conclusion in its favour.
In this note we further examine this issue. We find strong evidence that
zeta-function regularization does not give the correct answer for the
sums in question. We first consider a simple toy example of a sum
which has the same divergence problems when expanded in
$1/\mathcal{J}$ as the sum in \cite{Schafer-Nameki:2005tn}. We then
discuss the case of the folded string in the $\mathfrak{sl}(2)$ subsector and
circular string in the $\mathfrak{su}(2)$ subsector
\cite{Frolov:2003qc,Frolov:2003tu}. We evaluate the sums in
question first by zeta-function regularization and then exactly, using
various methods developed in \cite{Lucietti:2003ki, Lucietti:2004wy,
Schafer-Nameki:2005tn}. These results confirm that zeta-function
regularization does not reproduce the full sum. The explicit analysis
(in the $\mathfrak{su}(2)$ subsector) shows that although the coefficients of
$1/\mathcal{J}^{2n}$ in the expansion are correctly reproduced by the
zeta-function regularisation, the coefficients of
$1/\mathcal{J}^{2n+1}$ are not present, as well as the possibly
non-vanishing non-perturbative contributions ({\it i.e.} of order
$e^{-\mathcal{J}}$). Both types of terms do not follow from the
quantum string Bethe equations, explaining thus the mismatch
in the large $k$ regime found in \cite{Schafer-Nameki:2005tn}. In particular
the oscillatory behaviour observed in the large $k$ limit in
\cite{Schafer-Nameki:2005tn}, is hidden in the exponential terms,
which are entirely missed by zeta-function regularization.\footnote{We are
grateful to K. Zarembo for this remark.}
One important outcome of this analysis is that the terms in the string
sums which are not captured by the quantum Bethe equations are non-analytic
in the coupling, being proportional to $(\sqrt{\lambda'})^{2n+1}$ for
integral $n$ and $e^{- 1/\sqrt{\lambda'}}$.
It would be important to modify the S-matrix of
\cite{Arutyunov:2004vx,Staudacher:2004tk,Beisert:2005fw} to
incorporate these effects. Some of these issues are discussed in
\cite{BT}, where the terms with odd powers of $1/\mathcal{J}$ were also
found in the $\mathfrak{su}(2)$ subsector and the relation to the Bethe
ans\"atze in \cite{Arutyunov:2004vx,Staudacher:2004tk,Beisert:2005fw}
was discussed.
The plan of this note is as follows. We first discuss two relatively
simple sums (a toy model, as well as the folded string solution), which can be
evaluated both exactly and by zeta-function regularization.
In both cases zeta-function regularization fails to reproduce the exact sum.
In section 4 we apply an approximation method, replacing the sum by an
integral.
Comparison with the exact expression for the sums, shows that the
approximate evaluation correctly reproduces the terms
missing in the zeta-function regularized result.
We then apply this method to the $\mathfrak{su}(2)$ string and by comparing it with the
zeta-function evaluated result, identify the missing terms.
\section{Folded string solution}
In this section we consider the one-loop
energy shift for the folded rigid string, which rotates with a single
spin $S$ in $AdS_3$ and no spin in $S^5$. This correction was
computed in \cite{FrolovTseytlinI}, and is (in approximation) given by
\begin{equation} \label{ERot}
\kappa \delta E_{fold} = \sum_{n=1}^\infty \sqrt{n^2 + 4 \kappa^2} +2
\sqrt{n^2 + 2 \kappa^2}+ 5 n - 8 \sqrt{n^2 + \kappa^2} \, ,
\end{equation}
where $\kappa \sim \log \mathcal{S}, \, \mathcal{S} = S/\sqrt{\lambda}$.
We wish to evalute this sum for large values of the parameter
$\kappa$.\footnote{We thank A.~Tseytlin for the suggestion to consider
this sum.} Recall, that the asymptotic value for the sum, obtained in
\cite{FrolovTseytlinI} by replacing the sum with an integral is
\begin{equation}
\label{FTAsymp}
\delta E_{fold}^{FT} = - 3 \log 2 \ \kappa+
O(\kappa^0) \,.
\end{equation}
In the following sections we shall evaluate the sum (\ref{ERot}) first
by naive zeta-function regularization and then by various exact
evaluation methods. This will show that zeta-function fails to
reproduce the correct sum.
\subsection{Zeta-function regularization}
Let us first evaluate the sum along the lines of the zeta-function
regularization applied
in \cite{Schafer-Nameki:2005tn}. In order to do so, we pull the
large-$\kappa$ limit into the sum, {\it i.e.} expand each summand in
$1/\kappa$ assuming that the summation index $n$ is smaller than
$\kappa$. This expansion is obviously incorrect when $n \geq
\mathcal{\kappa}$, which reflects itself in the divergence of the resulting sums at each order
in $1/\kappa$ -- despite the fact that the initial sum is convergent.
We regularize these divergences using the zeta-function $\zeta (z)$
analytically continued to negative integers. This can in fact be done
to all orders in $1/\kappa$ and results in
\begin{equation} \begin{aligned}
\label{zeta-3}
\delta E_{fold}
&= \sum_n 2 (\sqrt{2}-3) + {1\over \kappa} \sum_n 5n +
O\left({1\over \kappa^2}\right) \cr
&= (3-\sqrt{2}) - {5\over 12} {1\over \kappa} + O(e^{-\kappa})\, .
\end{aligned}
\end{equation}
Here we used that $\zeta (-1) = -B_2/2 = -1/12$ and each higher term
is a sum over $n^{2l}$, and thus vanishes in the zeta-function
prescription. This clearly contradicts the asymptotics in
(\ref{FTAsymp}) by missing out the crucial linear term in
$\kappa$. The result (\ref{FTAsymp}) was obtained by an
approximative method, so it would be desirable to have independent
checks of the sum to confirm the failure of zeta-function
regularization. We shall subsequently present three methods which will
be in agreement with (\ref{FTAsymp}), as well as produce subleading
terms obtained in (\ref{zeta-3}) (up to exponentially small corrections).
\subsection{Asymptotic evaluation}
A method to asymptotically
evaluate sums of the type (\ref{ERot}) was obtained in appendix B of
\cite{Lucietti:2003ki} in the context of plane-wave string field
theory. The main idea is to represent the square root
terms using the integral representation of the Gamma-function
\begin{equation}\label{GammaInt}
{1\over x^z} = {1\over \Gamma(z)} \int_0^\infty dt t^{z-1} e^{-xt} \,,
\end{equation}
which is valid for $x, z>0$. For this to be applicable, we first act with
${\partial\over \partial \kappa }\left({1\over \kappa}
{\partial\over \partial \kappa} \right )$
on the sum (\ref{ERot}), which reduces to the expression
\begin{equation}
R= -8\kappa \sum_{n=1}^\infty \bigg( {2 \over (n^2 + 4 \kappa^2)^{3/2}}
+{1 \over (n^2 + 2 \kappa^2)^{3/2}}
-{1 \over (n^2 + \kappa^2)^{3/2}} \bigg) \,.
\end{equation}
Each partial sum is now absolutely convergent and can be asymptotically evaluated
separately using (\ref{GammaInt}). The relevant asymptotics derived in
\cite{Lucietti:2003ki}\footnote{Similar sums are discussed in
\cite{Foerger:1998kw, Bigazzi:2003jk, Bertoldi:2004rn}.} are
\begin{equation}\label{LSSAsymp}
\begin{aligned}
\sum_{n=1}^\infty {1\over (\mathcal{J}^2 + n^2)^{3/2}}
&= {2\over \sqrt{\pi} \mathcal{J}^3} \int_0^\infty ds s^{1/2} e^{-s}
(\theta (s/(\pi \mathcal{J}^2)) -1)\cr
&= {1\over \mathcal{J}^2} -{1\over 2 \mathcal{J}^3} +
O\left(e^{-\mathcal{J}}\right)\,.
\end{aligned}
\end{equation}
Here $\theta (t)= \sum_{n\in\mathbb{Z}} e^{-\pi n^2 t}$ and we modular
transformed and used the asymptotics $\theta(t)\rightarrow 1$ as
$t\rightarrow \infty$. Applied to the present case we obtain
\begin{equation}
R ={1\over \kappa^2} (3-\sqrt{2}) + O(e^{-\kappa}) \,,
\end{equation}
which after repeated integration results in
\begin{equation}\label{RotAsymp}
\delta E_{fold} = (3-\sqrt{2}) + {c_1 \kappa \over 2} + {c_0\over
\kappa} + O(e^{-\kappa})\,,
\end{equation}
where $c_i$ are integration constants, which need to be determined in
some other way. In particular, this is in accord with
\cite{FrolovTseytlinI}, as there are choices for $c_i$, for which the
sums can be made to agree. The integration constants can be derived in
the way done in \cite{Lucietti:2004wy}, but we shall present two
alternative methods to compute the sum exactly.
\subsection{Bessel function evaluation}
The energy shift can be likewise evaluated using the following
integral representation obtained in \cite{Schafer-Nameki:2005tn}
eq. (2.7) and (2.10). Recall that
\begin{equation}
\sum_{n=1}^\infty
\left(
\sqrt{(n+ \gamma)^2 + \alpha^2} + \sqrt{(n-\gamma)^2 + \alpha^2} - 2n
-{\alpha^2\over n}\right)
= \gamma^2 - \sqrt{\gamma^2 +\alpha^2} + F(\{\gamma\}, \alpha) \,,
\end{equation}
where we defined the function
\begin{equation}\label{FDef}
F(\beta ,\alpha )\equiv
\sqrt{\alpha ^2+\beta ^2}-\beta ^2+\alpha ^2\int_{0}^{\infty }
\frac{d\xi }{e^\xi -1}\left(
\frac{2J_1(\alpha \xi )}{\alpha \xi }\,\cosh \beta \xi -1
\right).
\end{equation}
For large $\alpha$ the asymptotic behaviour of this function is
\begin{equation}\label{FAsymp}
F(\beta ,\alpha )=-\alpha ^2\ln\left(\frac{e^{C-1/2}}{2}\,
\alpha \right)+\frac{1}{6}+
O\left(e^{-\alpha }\right) \, ,
\end{equation}
where $C=0.5772\ldots $ is the Euler constant.
Applying this to (\ref{ERot}) results in
\begin{equation}\label{sum-asy}
\delta E_{fold}=-3\ln 2\,\kappa +3-\sqrt{2}-\frac{5}{12\kappa }
+O(e^{-\kappa }),
\end{equation}
in agreement with \cite{FrolovTseytlinI} and implying that the
integration constants in (\ref{RotAsymp}) are $c_0=-5/12$ and $c_1= -6
\log (2)$. Note that this also calculates all
subleading terms up to exponential (powerlike in $1/\mathcal{S}$ as
$\kappa \sim \log \mathcal{S}$)
corrections.
\subsection{Generalized zeta-function evaluation}
The result obtained with Bessel functions in the last subsection can
be confirmed by the following analytic continuation argument.
Consider a generalization of the Riemann zeta-function
\begin{equation}
\zeta(s,\kappa) = \sum_{n=1 }^\infty {1\over (n^2+ \kappa^2)^s} \,.
\end{equation}
This is to begin with not well-defined for the choice $s=-1/2$ that we
are interested in, but the generalized zeta function can be
analytically continued to this value. Again, representing the summand
using the Gamma function integral representation as (\ref{LSSAsymp})
derived in appendix B of \cite{Lucietti:2003ki},
it follows that the large $\kappa$ asymptotics of this expression is
\begin{equation}\label{GeneralizedZeta}
\zeta (s, \kappa) = -{1\over 2} \kappa^{2 s} +
{1\over 2 \kappa^{2s-1}} {\Gamma(1/2) \Gamma (s-1/2)
\over \Gamma (s)} + O(e^{-\kappa}) \,.
\end{equation}
Note now, that this would have been obtained likewise by approximating
the sum by an integral, namely setting $u = n/\kappa$ in the large
$\kappa$ limit
\begin{equation}\label{IntegralRep}
\zeta (s, \kappa)
\sim {1\over \kappa^{2s-1}}\int_0^\infty du {1\over (1+ u^2)^s}
= {1\over 2 \kappa^{2s-1} } {\Gamma (1/2) \Gamma (s-1/2) \over
\Gamma (s)} \,.
\end{equation}
Applying this to $\delta E_{fold}$ for $s=-1/2 + \alpha$ for $\alpha
\rightarrow 0$ and that the Riemann zeta-function analytically
continued gives $\zeta
(-1)=-1/12$, we arrive at
\begin{equation}
\delta E_{fold} = -3 \log2 \ \kappa + (3-\sqrt{2}) -{5\over 12\kappa} + O(e^{-\kappa}) \,,
\end{equation}
in agreement with the above Bessel function evaluation and
\cite{FrolovTseytlinI}.
This method is quite general and also explains why zeta-function
regularization does not always work. Namely, zeta-function
regularization drops the term that comes from the Gamma-functions in
(\ref{GeneralizedZeta}).
\subsection{Exponential corrections}
So far we have refrained from working out explictly the exponential
corrections at $O(e^{-\kappa})$. These may however turn out to be
crucial for comparison to the quantum string Bethe ansatz. We shall now prove that
in the simpler case of the folded string these terms are indeed
non-vanishing and find explicit formulas for these terms.
As the starting point, consider the asymptotic evaluation method presented
earlier. Recall that
\begin{equation}
\kappa \sum_{n=1}^\infty {1\over (n^2 + \kappa^2 a^2)^{3/2}}
= -{1 \over 2 a^3 \kappa^2} + {1\over a^2 \kappa}+ {2 \over a^2
\kappa} \int_0^\infty dt e^{-t} \sum_{n=1}^\infty \left(e^{-\pi^2 n^2 \kappa^2 a^2
/t}\right) \,.
\end{equation}
The last term is the exponential correction term and can be further evaluated
\begin{equation}
\begin{aligned}
R_{exp}
&= {2 \over a^2\kappa} \sum_{n=1}^\infty \int_0^\infty dt e^{-t -\pi^2 n^2
a^2 \kappa^2/t} \cr
&= {4\over a^2} \sum_{n=1}^\infty 2 \pi a n K_1(2 \pi n a\kappa) \,.
\end{aligned}
\end{equation}
Note that $\partial_\kappa K_0(2
\pi n a \kappa) = - 2\pi n a K_1 (2 \pi n a \kappa)$. So, already
integrating up once with respect to $\kappa$ yields
\begin{equation}
\int d\kappa R_{exp} = -{4 \over a^2} \sum_{n=1}^\infty K_0(2 \pi n a
\kappa) \,.
\end{equation}
Then apply the integral represetation (see also appendix D of
\cite{Lucietti:2004wy})
\begin{equation}
K_0(z \kappa ) = \int_0^\infty dt {e^{-z \sqrt{t^2 + \mu^2}} \over
\sqrt{t^2 + \mu^2}} \,,
\end{equation}
and perform the sum, which yields
\begin{equation}
\begin{aligned}
\kappa \int d \kappa R_{exp}
&=-{4 \over a^2}\kappa \int_0^\infty dt
{1\over \sqrt{t^2 + \kappa^2}} {1\over e^{2 \pi a \sqrt{t^2 +
\kappa^2}} -1} \cr
&= -{2 \over a^2} \kappa \int_1^{\infty} dr {\coth (a \kappa \pi r )-1
\over \sqrt{s^2 -1}} \,.
\end{aligned}
\end{equation}
Integrating repeatedly with respect to $\kappa$, we arrive at
\begin{equation}
\int d\kappa \kappa \int d\kappa R_{exp}
= -{2 \over a^2}
\int_1^\infty dr
\left[
{\kappa \log \left(1-e^{-2 \pi a \kappa r}\right)
\over a r
\sqrt{r^2 -1}}
+{(r^3 + r^2 -1) {\rm Li}_2 \left( e^{-2\pi a \kappa r}\right)
\over 2 a^2 \pi^2 r^2 \sqrt{r^2 -1}}
\right] \,.
\end{equation}
This is a closed formula for the exponential correction term we were looking for.
Adding up the contributions with the various choices for $a$ of each summand in
(\ref{ERot}) produces the complete correction term for the folded string.
If one is interested in obtaining the first correction term in
$e^{-\kappa} O(\kappa^0)$ explicitly, one can proceed as follows.
Note that $\int d\kappa \kappa K_0(b \kappa) = -\kappa K_1(b \kappa)/b$. So
we obtain
\begin{equation}
\int d\kappa \kappa \int d\kappa R_{exp} = {2 \kappa \over \pi a^3}
\sum_{n=1}^\infty {K_1 (2 \pi n a \kappa) \over n} \,.
\end{equation}
With the asymptotics $K_1 (z) = \sqrt{\pi/2 z } e^{-z} (1+ O(1/z))$ we
obtain that the first exponential correction terms are
\begin{equation}
\int d\kappa \kappa \int d\kappa R_{exp}
= {\kappa \over \pi a^3}\sum_{n=1}^\infty
e^{- 2 \pi n a \kappa} {1\over n} {\sqrt{1 \over n a \kappa }}
\left[1+ O\left({1\over \kappa }\right)\right] \,.
\end{equation}
Adding together the terms with the correct prefactors and choices for
$a$ gives the correction to (\ref{ERot}).
In summary we have shown in this section that the exponential
corrections do not vanish for the folded string case. It would of
course be interesting to see, whether they contribute in more
complicated sums than (\ref{ERot}), such as the one-loop energy shift
for the $\mathfrak{su}(2)$ and $\mathfrak{sl}(2)$ subsectors.
\section{Toy model}
As a second test case consider the situation of two bosonic and two
fermionic frequencies with the energy shift given by
\begin{equation}\label{ToyE}
\delta E_{toy}= \sum_{n=1}^\infty
\sqrt{1+ (n+\gamma)^2/\mathcal{J}^2} +
\sqrt{1+ (n-\gamma)^2/\mathcal{J}^2} - 2 \sqrt{1+ n^2/ \mathcal{J}^2} \,,
\end{equation}
where $\gamma$ is a constant independent of $\mathcal{J}$ and the sum
is convergent in the same sense as for the $\mathfrak{su}(2)$ and $\mathfrak{sl}(2)$
spinning strings.
Again we compare zeta function regularization with the
exact evaluation of the sum in the large $\mathcal{J}$ limit
and find disagreement.
\subsection{Zeta-function regularization}
For the naive perturbative evaluation of (\ref{ToyE}), pull the large
$\mathcal{J}$ limit through the sum.
As each term in the $1/\mathcal{J}$ expansion is of order $n^0$ or
higher, using zeta-function regularization the sum evaluates to
\begin{equation}
\label{zeta-regu}
\delta E^\zeta_{toy} = -{1\over 2} \left(-2 + 2 \sqrt{1+ {\gamma^2\over
\mathcal{J}^2}}\right) \,.
\end{equation}
Expanding this in $1/\mathcal{J}$ yields the energy shift at
arbitrary loop orders as obtained from this prescription.
\subsection{Asymptotic evaluation of sums}
Alternatively, in this simple case, one can evaluate the sum exactly
(up to terms $e^{-\mathcal{J}}$) using the method in \cite{Lucietti:2003ki}.
Consider the sum
\begin{equation}
\delta E_{toy} \mathcal{J}= S
= \sum_{n=1}^\infty
\sqrt{(n+\gamma)^2 + \mathcal{J}^2 } +
\sqrt{(n-\gamma)^2 + \mathcal{J}^2} -2
\sqrt{n^2 + \mathcal{J}^2 } \,.
\end{equation}
Then following the strategy in \cite{Lucietti:2003ki}, act with
${\partial\over \partial \mathcal{J}}\left({1\over \mathcal{J}}
{\partial\over \partial \mathcal{J}} \right )$ to obtain
\begin{equation}
R= -\mathcal{J}
\sum_{n=1}^\infty
{1 \over ((n+\gamma)^2 + \mathcal{J}^2 )^{3/2}}
+ {1 \over ((n-\gamma)^2 + \mathcal{J}^2)^{3/2}}
-2{1 \over (\mathcal{J}^2 + n^2)^{3/2}} \,.
\end{equation}
Now each part of the sum is absolutely convergent by itself and can be evaluated and
later on integrated up to give the result for the complete sum.
The last summand is easiest and is evaluated the same way as in
appendix B of \cite{Lucietti:2003ki}, {\it i.e.} (\ref{LSSAsymp}).
The remaining two terms are computed likewise. First recall the definition of the
generalized theta-functions
\begin{equation}
\theta\bigg[\begin{aligned} a\cr b \end{aligned}\bigg] (t) = \sum_{n=-\infty}^\infty e^{\pi t
(n+a)^2 + 2 \pi n b i } \,,
\end{equation}
which satisfies the modular transformation law, shown by Poisson resummation,
\begin{equation}
\theta\bigg[\begin{aligned} a\cr b \end{aligned}\bigg] (t) = {1\over \sqrt{-t}}
\theta\bigg[\begin{aligned} b\cr -a \end{aligned}\bigg] (1/t)\,.
\end{equation}
So in particular we can write
\begin{equation}
\theta \bigg[\begin{aligned} \gamma \cr 0\end{aligned} \bigg] (-t/\pi) = e^{-\gamma^2 t } +
\sum_{n=1}^\infty \left(e^{-(n+\gamma)^2 t} + e^{-(n-\gamma)^2 t}\right)\,.
\end{equation}
This allows the evaluation of the remaining two terms in the sum,
again asymptotically for large $\mathcal{J}$
\begin{equation}\label{Comput}
\begin{aligned}
\sum_{n=1}^\infty
& {1 \over ((n+\gamma)^2 + \mathcal{J}^2 )^{s}}
+ {1 \over ((n-\gamma)^2 + \mathcal{J}^2)^{s}} \cr
& \qquad=
{1\over \Gamma (s)} \int_0^\infty dr r^{s-1} e^{-\mathcal{J}^2 r}
\sum_{n=1}^\infty \left(e^{-(n+\gamma)^2 r} + e^{-(n-\gamma)^2
r}\right)\cr
& \qquad=
{1\over \Gamma(s)\mathcal{J}^{2s}} \int_0^\infty dt t^{s-1} e^{- t}
\left(\theta \bigg[\begin{aligned} \gamma \cr 0\end{aligned} \bigg] (-t/(\pi\mathcal{J}^2))-
e^{-\gamma^2 t/\mathcal{J}^2} \right) \cr
& \qquad= -{1\over( \mathcal{J}^2 + \gamma^2)^{s}}
+ {\sqrt{\pi}\over \Gamma(s) \mathcal{J}^{2s-1}}
\int_0^\infty dt t^{s-3/2}e^{-t} \theta \bigg[\begin{aligned} 0 \cr -\gamma\end{aligned} \bigg]
(-\pi \mathcal{J}^2/t)\cr
& \qquad= -{1\over( \mathcal{J}^2 + \gamma^2)^{s}}
+ { \sqrt{\pi} \Gamma (s-1/2) \over \Gamma (s) \mathcal{J}^{2s-1}}
+ {\sqrt{\pi} \over \mathcal{J}^{2s-1}}
\int_0^\infty dt t^{s-3/2}
e^{-t} \left(\theta \bigg[\begin{aligned} 0 \cr -\gamma\end{aligned} \bigg]
(-\pi \mathcal{J}^2/t) -1 \right) \,.
\end{aligned}
\end{equation}
For $s=3/2$ the last term is of order $e^\mathcal{-J}$, which can be
seen by changing to $u=\mathcal{J}^N t$.
So in summary we obtain
\begin{equation}
\sum_{n=1}^\infty
{1 \over ((n+\gamma)^2 + \mathcal{J}^2 )^{3/2}}
+ {1 \over ((n-\gamma)^2 + \mathcal{J}^2)^{3/2}}
= {2\over \mathcal{J}^2} -{1\over( \mathcal{J}^2 + \gamma^2)^{3/2}}
+ O\left(e^{-\mathcal{J}}\right) \,.
\end{equation}
Thus we obtain that
\begin{equation}
R= -{\mathcal{J}} \left({1\over \mathcal{J}^3} - {1\over
(\mathcal{J}^2 + \gamma^2)^{3/2}}\right) \,.
\end{equation}
Integrating up, we obtain
\begin{equation}
\delta E_{toy} = {1\over \mathcal{J}} \left(\mathcal{J} - \sqrt{\gamma^2 +
\mathcal{J}^2} \right) + c_0 \mathcal{J}+ {c_1 \over \mathcal{J}}
+ O(e^{-\mathcal{J}}) \,,
\end{equation}
which for vanishing integration constants agrees up to terms $O(e^{-\mathcal{J}})$ with the
perturbative zeta-function regularized expression $\delta E^\zeta$.
In order to determine the integration constants, derive with respect
to $\gamma$ and then evaluate the large ${\mathcal J}$ in analogy to
\cite{Lucietti:2004wy}. However, we shall determine these using the Bessel
and generalized zeta-function methods introduced earlier.
\subsection{Bessel function evaluation}
Consider now the evaluation using Bessel functions.
First split the sum into two partial sums which both converge absolutely
\begin{eqnarray}
\delta E_{toy} \mathcal{J}= S
&=& S_1 + S_2 \nonumber \\
\label{S1}
S_1 &=& \sum_{n=1}^\infty
\sqrt{(n+\gamma)^2 + \mathcal{J}^2 } +
\sqrt{(n-\gamma)^2 + \mathcal{J}^2} - 2n - {\mathcal{J}^2\over n} \nonumber \\
\label{S2}
S_2 &=& -2 \sqrt{n^2 + \mathcal{J}^2 }+ 2n + {\mathcal{J}^2\over n} \,.
\end{eqnarray}
The representation (\ref{FDef}) implies
\begin{eqnarray}
\label{S12}
S_1 &=& \gamma^2 - \sqrt{\gamma^2 + {\mathcal J}^2 } + F(\{ \gamma \}, {\mathcal J}) \nonumber \\
S_2 &=& {\mathcal J} - F(0, \mathcal{J})\,.
\end{eqnarray}
The large $\mathcal{J}$ asymptotics follow from (\ref{FAsymp}), so that
\begin{eqnarray}
\label{sums12}
S_1 &=& \gamma^2 - \sqrt{\gamma^2 + \mathcal{J}^2 } - \mathcal{J}^2 \ln \mathcal{J} - \mathcal{J}^2 \ln \left( {e^{C- {1\over 2}} \over 2} \right) + {1\over 6} + O\left(e^{-\mathcal{J} }\right)
\nonumber \\
\label{S2}
S_2 &=& \mathcal{J}+ \mathcal{J}^2 \ln \mathcal{J} + \mathcal{J}^2
\ln \left( {e^{C- {1\over 2}} \over 2} \right) - {1\over 6} +
O\left(e^{-\mathcal{J} }\right) \,,
\end{eqnarray}
and thus the asymptotic expansion for the energy is up to
exponentially small corrections
\begin{equation} \label{EtoyBessel}
\delta E_{toy}
= {1\over \mathcal{J}}\left(\gamma^2+\mathcal{J} - \sqrt{\gamma^2 + \mathcal{J}^2}
\right) + O\left(e^{-\mathcal{J} }\right) \,,
\end{equation}
This is in agreement with the asymptotic evaluation and determines the
integration constants as $c_0=0$ and $c_1=\gamma^2$.
\subsection{Generalized zeta-function evaluation}
To confirm the result from the last section, we apply analytic
continuation to the following generalized zeta-function
\begin{equation}
\zeta (s, \gamma,\mathcal{J}) = \sum_{n=1}^\infty {1\over ((n+ \gamma)^2 +
\mathcal{J}^2)^s}\,.
\end{equation}
Then by analytic continuation to $s=-1/2$ we can compute the sums in
$\delta E$. The asymptotics for large values of $\mathcal{J}$ follow
using (\ref{Comput}) in
the last section using generalized theta functions and setting $s=-1/2$
\begin{equation}
\zeta (s, \gamma,\mathcal{J}) + \zeta (s, -\gamma,\mathcal{J}) =
-{1\over (\gamma^2 + \mathcal{J}^2)^{s}}
+ {\sqrt{\pi} \Gamma (s-1/2)\over \Gamma(s)\mathcal{J}^{2s-1}} +
\gamma^2 + O(e^{-\mathcal{J}})\,.
\end{equation}
The last term in (\ref{Comput}) for $s=-1/2$ is not exponentially
suppressed and is extracted by performing the integral yielding $\sum
{a_n/(n\mathcal{J})^2} K_1(\pi \mathcal{J} n)$, which has the given asymptotics.
Up to exponential corrections we obtain that the sum has large
$\mathcal{J}$ behaviour given by
\begin{equation}
\begin{aligned}
S&= \lim_{\alpha\rightarrow 0}\left\{
\zeta (-1/2 + \alpha, \gamma, \mathcal{J})+
\zeta (-1/2 + \alpha, -\gamma, \mathcal{J})
- 2 \zeta (-1/2 +\alpha , 0, \mathcal{J}) \right\}\cr
&= \lim_{\alpha\rightarrow 0}\left\{
-(\gamma^2 + \mathcal{J}^2)^{1/2 -
\alpha} + {\sqrt{\pi} \Gamma (-1 +\alpha)\over
\Gamma(-1/2)}\mathcal{J}^2 + \gamma^2
- 2 \left(-{1\over 2\mathcal{J}} + {\mathcal{J}^2\over 2}
{\Gamma(1/2) \Gamma (-1+ \alpha) \over \Gamma(-1/2 + \alpha) }\right) \right\} \cr
&= \gamma^2 + \mathcal{J} - \sqrt{\gamma^2 + \mathcal{J}^2} \,.
\end{aligned}
\end{equation}
This is again in agreement with the two independent methods of
evaluation presented earlier and confirms the incompleteness of the
evaluation by means of
zeta-function regularization.
\section{Zeta-function regularization versus exact summation}
In the previous sections we have performed exact, analytic evaluations of
the sums (\ref{ERot}) and (\ref{ToyE}) using several methods. These were
compared to the zeta function regularized expressions
(\ref{zeta-3}) and (\ref{zeta-regu}) and were found to disagree with
them. We would now like to determine the origin of this
disagreement\footnote{Some of the ideas in this section arose in
discussions with A.~Tseytlin.
Similar observations have recently appeared in \cite{BT}.}.
The nature of this section is more experimental and it would be
important to understand this in full generality, {\it e.g.} in
relation with the observation in (\ref{IntegralRep}). In particular,
it should be possible to extend this to the case of the $\mathfrak{sl}(2)$
subsector.
To proceed, we split the infinite sum into a finite sum, where
zeta-function regularization applies, and another part, which will be
approximated by simply replacing the sum by an integral. The
correction terms that are computed by the Euler-Maclaurin summation
formula will be discussed below.
More precisely
\begin{eqnarray}
\label{sp}
S(\eta) &=& \sum_{n=1}^K f( n, \eta) + \sum_{n=K}^\infty f( n, \eta) \nonumber \\
&=& S^{\rm I} (K, \eta) + S^{\rm II} (K, \eta) \, , \quad \quad K \gg 1 \,,
\end{eqnarray}
where we have denoted the large parameters $\kappa$ and ${\mathcal J}$ in
(\ref{ERot}) and (\ref{ToyE}) by $\eta$. Since $K\gg
1$ the second sum $S^{\rm II}(\eta)$ can be replaced with an
integral, which will be denoted by $\tilde{S}^{\rm II}$.
Further let us assume that
\begin{equation}
\label{con2}
1 \ll K\ll \eta \,.
\end{equation}
Then the second sum ({\it i.e.} integral) $\tilde{S}^{{\rm II}}(\eta)$ can be
expanded in $1/\eta$.
On the other hand, for the zeta-function regularization used in
\cite{Schafer-Nameki:2005tn} one first expands $f(n,\eta)$ in
$1/\eta$ and then resums the expanded series. It is clear that this
expansion fails, when $n>\eta$, inducing spurious divergences. These
were cured by introducing the zeta-function regularization, which
effectively means that one multiplies all terms in the sum with a
factor $e^{-\alpha \, n}$. Since $n \leq K$ in the first sum, the
expansion in $1/\eta$ is correct one, and zeta function
regularization does not affect this part of the result.
We thus focus only on the second sum. To compare the zeta-function
regularized results with the integrated sum $\tilde{S}^{\rm II}$, we
first need to determine the value of the zeta function that is cut-off
$K$ dependent, and approximated by the integral as when
evaluating the sum $S^{{\rm II}}$. For this, we use simply the replacement of the
sum by an integral, as in \cite{FrolovTseytlinI}.
More precisely, we use the right colum of the following equation
as the values of the zeta-function (taking $\alpha <1/K$)
\begin{eqnarray}
\label{cut-off-zeta}
\sum_{n=K}^{\infty} e^{- \alpha n} = {1\over \alpha} + \left(
{1\over 2} - K \right) + { O}(\alpha) \quad
&\rightarrow& \quad \int_{K}^\infty dn \, e^{-\alpha n} = {1\over
\alpha} - K + {O}(\alpha) \nonumber \\
\sum_{n=K}^{\infty} e^{- \alpha n} n = {1\over \alpha^2} + \left( -
{1\over 12} + {K\over 2} - {K^2 \over 2} \right) + {
O}(\alpha) \quad
&\rightarrow& \quad \int_{K}^\infty dn \, e^{-\alpha n} n = {1\over
\alpha^2} - {K^2 \over 2 } + { O}(\alpha) \nonumber \\
\sum_{n=K}^{\infty} e^{- \alpha n} n^2 = {2\over \alpha^3} - {1\over
6} \left( K - 3 K^2 + 2 K^3 \right) + { O}(\alpha) \quad
&\rightarrow& \quad \int_{K}^\infty dn \, e^{-\alpha n} n^2 = {2 \over
\alpha^3} - {K^3 \over 3 } + { O}(\alpha) \,.\nonumber \\
&&
\end{eqnarray}
Note that this method was also used in
\cite{FrolovTseytlinI}.
Comparing to the standard
Euler-Maclaurin summation formula yields that all extra tail and
boundary terms contribute subleading in $K$ and can be neglected.
However, we will see
that this heuristic method reproduces precisely the missing terms in the
zeta-function regularization.
We shall now compare the standard zeta-function regularized result
with the integral version zeta-function regularized expression using this
prescription.
Let us first apply both methods to compute the sum $S^{\rm II}$ for the
folded string (\ref{ERot}) and the toy model (\ref{ToyE}).
\subsection{Folded string and toy model}
Approximating the sums (\ref{ToyE}), (\ref{ERot}) with an integral,
and subsequently expanding in $1/\eta$, we obtain, respectively
\begin{eqnarray}
\label{expann}
S_{ toy}^{\rm II}(K, \mathcal{J})
&=& \gamma^2 - K \gamma^2 {1\over
{\mathcal J}} + \left( {1\over 2} K^3 \gamma^2 + {1\over 4} K
\gamma^4 \right) {1\over {\mathcal J}^3}
+ { O} \left( {K\over
{\mathcal J}^5} \right) \nonumber\\
S_{ fold}^{\rm II} (K,\kappa)
&=& - 3 \log 2 \, \kappa^2 - 2 (\sqrt{2} -3) K \kappa - {5\over 2}
K^2 - {1\over 6} \left(\sqrt{2} - {15 \over 2}\right) {K^3 \over \kappa} + { O}
\left( {K^5 \over \kappa^3} \right) \,.
\end{eqnarray}
On the other hand, expanding the summands $f(n, \eta)$ as done for the
zeta-function regularization leads to
\begin{eqnarray}
\label{expd}
f_{ toy}(n, \mathcal{J}) &=& \gamma^2 {1\over \mathcal{J}} +
\left(-
{3\over 2} n^2 \gamma^2 - {\gamma^4 \over 4} \right) {1\over {\mathcal J}^3}
+ O\left({1\over \mathcal{J}^4}\right) \nonumber \\
\label{expdd}
f_{fold}(n, \kappa) &=& 2 (\sqrt{2} - 3) \kappa + 5 n + \left({\sqrt{2}
\over 2} - {15\over 4}\right) n^2 {1\over \kappa} + O\left({1\over \kappa^2}\right) \, .
\end{eqnarray}
Comparing the expansions (\ref{expann}) with
(\ref{expd}), we note the absence of the leading,
$1/\mathcal{J}^0$ and $\kappa^2$ terms in the expansion of the
summands. Summing up the expanded terms (\ref{expd}) from
$(K,\infty)$ and using the zeta function results (\ref{cut-off-zeta})
we obtain the same results as in (\ref{expann}) except
for the $1/\mathcal{J}^0$ and $\kappa^2$ terms, which were absent from
the beginning in the expansion.
These terms, being cut-off $K$ independent parts of the sums, can be
obtaind by setting $K=0$ in the integral. So the difference between the
two results is given by
\begin{equation}
\Delta(\eta) = \int_0^\infty f(n, \eta) \, dn \,.
\end{equation}
\subsection{The circular string in the $\mathfrak{su}(2)$ subsector }
In this section we will consider the evaluation of the 1-loop energy
energy shift corresponding to the circular string which rotates in an
$S^3$ inside the $S^5$ with two equal spins $J_1 = J_2 =J/2$. The energy shift
takes the following form \cite{Frolov:2003tu, Frolov:2004bh, Beisert:2005mq}
\begin{equation} \delta E = \delta E^{(0)} +
\sum_{n=1}^\infty \delta E^{(n)} \,, \end{equation}
where
\begin{equation} \begin{aligned}
\label{SU2E}
\delta E^{(0)} &= 2 + \sqrt{1-{ 2 k^2\over \mathcal{J}^2 + k^2}} - 3
\sqrt{1-{k^2 \over \mathcal{J}^2 + k^2}} \cr \delta E^{(n)} &= 2
\sqrt{1+ {(n+ \sqrt{n^2 - 4 k^2 })^2 \over 4 (\mathcal{J}^2 + k^2
)}} + 2 \sqrt{ 1+ {n^2 - 2 k^2 \over \mathcal{J}^2 + k^2}} + 4
\sqrt{1 + {n^2 \over \mathcal{J}^2 + k^2}} \cr & \ \ - 8 \sqrt{1 +
{n^2 - k^2 \over \mathcal{J}^2 + k^2}} \,.
\end{aligned}
\end{equation}
The zeta-function regularized version of the sum is derived to all
orders in $1/\mathcal{J}$ in
appendix A. It is hard to exactly repeat the procedure from the previous section
for the sum (\ref{SU2E}) due to the complexity of the integral
$\tilde{S}^{\rm II}$. So let us instead first expand the sum
(\ref{SU2E}) in the small parameter $k$ and then repeat the
computation from the previous section order by order in $k$. Note
also, that although the winding number $k$ is in principle integer
valued, in the regime which we are interested, namely $\mathcal{J} \gg 1$,
$n>K\gg 1$, the expansion in small $k$ is justified.
The expansion of the summand (\ref{SU2E}) is
\begin{eqnarray}\label{ElargekExp}
\delta E^{(n)} &=& - {(\mathcal{J}^2 + 2 n^2) \over \mathcal{J} n^2 (\mathcal{J}^2 + n^2)^{3 \over 2}} k^4 + {- 2 \mathcal{J}^4 - 2 \mathcal{J}^2 n^2 + n^4 \over \mathcal{J}^3 n^4 (n^2 + \mathcal{J}^2)^{3\over 2}} k^6 + O(k^8) \, \nonumber \\
&\equiv& \delta E_1^{(n)} k^4 + \delta E_2^{(n)} k^6 + O(k^8) \, .
\end{eqnarray}
We can now repeat the procedure from the previous section for the sums $\delta E_1$ and $\delta E_2$.
Expansion of the first integral yields
\begin{equation}
\label{SUexpans}
\int_{K}^\infty dn\, \delta E_1^{(n)} = - {1\over \mathcal{J} K
\sqrt{\mathcal{J}^2 + K^2}} = - {1\over K} {1\over \mathcal{J}^2} +
{1\over 2} K {1\over \mathcal{J}^4} - {3\over 8} K^3 {1\over
\mathcal{J}^6} + O\left({1\over \mathcal{J}^8}\right)\, .
\end{equation}
The integrated function thus admits an integer power expansion in
${1\over \mathcal{J}^{2n}}$, and thus is analytic in $\lambda'$.
On the other hand, the naive expansion ({\it i.e.} the expansion where we assume that $n<\mathcal{J}$) of the integrand $\delta E_1^{(n)}$ gives
\begin{equation}
\label{f1}
\delta E_1^{(n)} = - {1\over n^2} {1\over \mathcal{J}^2} - {1\over 2}
{1\over \mathcal{J}^4} + {9 \over 8} n^2 {1\over
\mathcal{J}^6} + O\left({1\over \mathcal{J}^8}\right)\, .
\end{equation}
As expected, these terms yield divergent sums starting from
$1/\mathcal{J}^4$, however they appear
with powers $1/\mathcal{J}^{2k}$, {\it i.e.} the same powers of the expansion
in (\ref{SUexpans}). Integrating the
expression (\ref{f1}) and using the integral version of the
zeta-function
prescription (\ref{cut-off-zeta}), we reproduce
all terms in (\ref{SUexpans}).
The evaluation of the second order term $\delta E^{(n)}_2$ is
different
\begin{eqnarray}
\label{f2}
\int_{K}^\infty dn \, \delta E_2^{(n)} &=& {-2 \mathcal{J}^4 + 2
\mathcal{J}^2 K^2 + K^3 (K - \sqrt{\mathcal{J}^2 + K^2}) \over 3
\mathcal{J}^5 K^3 \sqrt{\mathcal{J}^2 + K^2}} \nonumber \\
&=& - {2\over 3} {1\over K^3} {1\over \mathcal{J}^2} + {1\over K}
{1\over \mathcal{J}^4} - {1\over 3} {1\over \mathcal{J}^5} -
{1\over 4}K {1\over \mathcal{J}^6} +
O\left({1\over \mathcal{J}^9}\right)\, .
\end{eqnarray}
The main difference to the former case is,
the presence of the term $1/\mathcal{J}^5$, which is non-analytic in
$\lambda'$ and
which appears as the cuf-off $K$ independent part of the integral. On
the other hand, the naive expansion of
$\delta E_2^{(n)}$ yields
\begin{eqnarray}
\label{Eexp}
\delta E_2^{(n)} = - {1\over n^4} {1\over \mathcal{J}^2} + {1\over
n^2}{1\over \mathcal{J}^4} + {1\over 4} {1\over \mathcal{J}^6} - {7
\over 8} n^2 {1\over \mathcal{J}^8} +O\left({1\over
\mathcal{J}^{10}}\right) \,,
\end{eqnarray}
where all terms are analytic in $\lambda'$.
Since the
zeta-function prescription
does not change the order in $1/\mathcal{J}$ in the expansion, it is thus
clear that the terms at order $1/\mathcal{J}^5$ in (\ref{f2}) can
never be reproduced by the zeta-function regularization of the expression
(\ref{Eexp}). The regular terms in (\ref{f2}) are on the other hand
easily reproduced using the
cut-off zeta-function regularization (\ref{cut-off-zeta}). Similar
analysis for the order $k^6$ and higher, yields the discrepancy
between the zeta-function regularization and the exact string
result at the orders $1/\mathcal{J}^{2k+1}$.
A more detailed analysis of the correction terms in the
Euler-Maclaurin summation formula
for the sums appearing in (\ref{ElargekExp}), shows that only
the coefficients of $
1/\mathcal{J}^{2k}$ are corrected, and also that all these corrections
are supressed with inverse powers of the cutoff. Thus, the approximate
integral evaluation of the coefficients of $1/\mathcal{J}^{2k+1}$ gives
the exact result\footnote{Recall that the sum $S^{I}$ in (\ref{sp})
only contributes to the even powers of $\mathcal{J}$.}.
It should be possible to resum the effect of these terms that are
missed by zeta-function regularization.
\section*{Acknowledgments}
\label{sec:acknowledgements}
We are especially grateful to A.~Tseytlin and K.~Zarembo for useful
discussions and comments. We also thank G.~Arutyunov, N.~Beisert,
S.~Frolov, T.~Klose, K.~Peeters, J.~Plefka, M.~Staudacher and
C.~Sonnenschein for discussions. The work of S.S.-N. was partially
supported by the DFG, DAAD, and European RTN Program
MRTN-CT-2004-503369.
\setcounter{section}{0}
|
0805.2265
|
\section{Introduction}
All spaces are assumed to be normal.
By a map we mean a continuous function.
We say that a compactum~$X$ is \emph{hereditarily indecomposable}
if for every two intersecting continua in~$X$ one is contained in the other.
The main result of this note is the following theorem.
\begin{theorem}\label{thm.1.1}
Let $f : X \to Y$ be a perfect map with hereditarily indecomposable fibers
from a separable metrizable space~$X$ onto a zero-dimensional separable
metrizable space~$Y$.
Then there are a hereditarily indecomposable metrizable
compactification~$X^{\star }$ of\/~$X$ with $\dim X^*= \dim X$ and
a zero-dimensional metrizable compactification~$Y^*$ of\/~$Y$ such
that $f$~can be extended to a map $f^*: X^* \to Y^*$.
\end{theorem}
Let us note that this result, combined with a pseudosuspension
method, yields a theorem of Ma\'{c}kowiak~\cite{M2} on the
existence of universal $n$-dimensional hereditarily
indecomposable continua.
This theorem was obtained by Ma\'{c}kowiak by a quite different method
based on a subtle use of inverse limits.
We comment on this in Corollary~\ref{cor.4.1}.
Rather unexpectedly, our proof uses, in an essential way, large nonmetrizable
compactifications and a considerable strenghtening of Marde\v{s}i\'c's
Factorization Theorem (see \cite{E}*{Theorem~3.4.1}).
This strengthening is a dual version of the L\"owenheim-Skolem theorem from
model theory; it appears as Theorem~3.1 in~\cite{MR1785837} and it
was put to good use in~\cite{HvMP} and~\cite{vdS}.
In Section~\ref{sec.2} we explain some general facts concerning this
technique and in section~\ref{sec.3} we show how our theorem follows from
these results.
Among other consequences of this technique is the following theorem, proved in
section~\ref{sec.3}.
\begin{theorem}\label{thm.1.2}
For every cardinal~$\tau$ and $n \in \{0,1,\ldots , \infty \}$
there exists a hereditarily indecomposable compactum $X(n,\tau)$ of
weight~$\tau$ and dimension~$n$ that contains a copy of every hereditarily
indecomposable compactum of weight not more than~$\tau$ and
dimension at most~$n$.
\end{theorem}
The following property of a space $X$ was formulated by
Krasinkiewicz and Minc~\cite{KM}:
\begin{KM}
For every two disjoint closed
sets $C$ and $D$ in $X$ and disjoint open sets $U$ and $V$ in $X$
with $C \subset U$ and $D \subset V$ there exist closed sets
$X_0$, $X_1$ and $X_2$ in $X$ such that
$X = X_0 \cup X_1 \cup X_2 $,
$C \subset X_0$,
$D \subset X_2$,
$X_0\cap X_1 \subset V$,
$X_1 \cap X_2 \subset U$ and
$X_0\cap X_2= \emptyset$.
\end{KM}
To avoid having to write down the six conditions each time we shall
call a triple $\langle X_0,X_1,X_2\rangle$ a \emph{fold of~$X$} for
the quadruple $\langle C,D,U,V\rangle$.
\begin{theorem}[\cite{KM}] \label{thm.1.3}
A compact space is hereditarily indecomposable
if and only if it has Property~(KM).
\end{theorem}
\section{A factorization method}
\label{sec.2}
The factorization method alluded to in the Introduction is based on a mix
of Model Theory and Set-Theoretic Topology.
It works best in the realm of compact Hausdorff spaces, as will become clear
shortly.
The first ingredient is Wallman's representation theorem, \cite{Wallman},
for distributive lattices: if $L$ is such a lattice then the set $wL$ of
ultrafilters on~$L$ carries a natural compact $T_1$-topology.
This topology has the family $\{\bar a:a\in L\}$ as a base for the closed
sets, where $\bar a=\{u\in wL:a\in u\}$.
If $X$ is compact and $T_1$ and $L$ is the family of closed subsets of~$X$,
with union and intersection as its lattice operations then
$x\mapsto u_x=\{a\in L:x\in a\}$ is a homeomorphism from~$X$ onto~$wL$;
this remains true if $L$~is a base for the closed sets of~$X$ that is closed
under unions and intersections.
See, e.g., \cite{Aarts} for a short introduction to Wallman representations.
For a normal space~$X$ one can obtain the \v{C}ech-Stone compactification,
$\beta X$, as the Wallman representation of the lattice of closed sets
of~$X$.
This is the key to the next theorem.
\begin{theorem}\label{thm.2.1}
If $X$ has Property~(KM) then so does its \v{C}ech-Stone
compactification~$\beta X$ and, in particular,
$\beta X$~is hereditarily indecomposable.
\end{theorem}
\begin{proof}
To begin: it should be clear that Property~(KM) can be
(re)for\-mu\-lated in terms
of closed sets only and that it is a finitary lattice-theoretic property;
one can express it as an implication involving seven variables.
Thus if $X$~has Property~(KM) then the canonical base, $\mathcal{B}$, for
the closed sets of~$\beta X$ satisfies this implication.
This does not automatically imply that $\beta X$ has Property~(KM), because
that means that the full family of closed sets of~$\beta X$ satisfies the
lattice-theoretic formula.
However, in the present case one can start with arbitrary
$C$, $D$, $U$ and~$V$ and use compactness and the fact that $\mathcal{B}$
is closed under finite unions and intersections to
find~$C'$, $D'$, $U'$ and~$V'$ such
that $C\subseteq C'\subseteq U'\subseteq U$ and
$D\subseteq D'\subseteq V'\subseteq V$, and such that
$C'$, $D'$, $\beta X\setminus U'$ and $\beta X\setminus V'$ belong
to~$\mathcal{B}$.
One can then find a fold $\langle X_0,X_1,X_2\rangle$ for
$\langle C',D',U',V'\rangle$ in~$\mathcal{B}$
and this will also be a fold for~$\langle C,D,U,V\rangle$.
\end{proof}
The second ingredient is the use of notions from Model Theory,
especially elementary substructures and the L\"owenheim-Skolem theorem.
In the context of lattices elementarity is perhaps best explained in terms
of solutions to equations.
One can interpret Property~(KM) as saying that certain equations should have
solutions: the quadruple $\langle C,D,U, V\rangle$ determines six equations
and a fold $\langle X_0,X_1,X_2\rangle$ is a solution to this system.
One calls $M$ an elementary sublattice of~$L$ if every lattice-theoretic
equation with constants from~$M$ that has a solution in~$L$ also has a
solution in~$M$.
To illustrate its use we prove the following lemma.
\begin{lemma}\label{lemma.elem.KM}
Assume $X$ is a hereditarily indecomposable compact space and let $L$ be
an elementary sublattice of the lattice of closed subsets of~$X$.
Then $wL$ is also hereditarily indecomposable.
\end{lemma}
\begin{proof}
By elementarity the lattice~$L$ satisfies Property~(KM):
if $C$, $D$, $X\setminus U$ and $X\setminus V$ belong to~$L$ then there is
a fold $\langle X_0,X_1,X_2\rangle$ in the full family of closed sets,
\emph{hence} there is also such a fold in~$L$.
Next, in $wL$ the same argument as in the proof of Theorem~\ref{thm.2.1}
applies: an arbitrary quadruple can be expanded to a quadruple from the base.
\end{proof}
The L\"owenheim-Skolem Theorem provides us with many elementary substructures:
given a lattice~$L$ and some subset~$A$ of~$L$ one can construct
an elementary sublattice~$L_A$ of~$L$ that contains~$A$ and whose
cardinality is at most $|A|\times\aleph_0$.
\begin{theorem}[\cites{MR1785837,vdS,HvMP}]\label{thm.2.2}
Let $f : X \to Y$ be a continuous surjection from a hereditarily
indecomposable compact space onto a compact space.
Then there are a compact space~$Z$ and continuous maps
$g:X \to Z$ and $h : Z \to Y$ such that
$Z$~is hereditarily indecomposable,
$\dim Z= \dim X$,
\ $w(Z)=w(Y)$ and $f=h \circ g$.
\end{theorem}
\begin{proof}
Let $\mathcal{B}$ be a base for the closed sets of~$Y$ of cardinality~$w(Y)$.
Via $B\mapsto f^{-1}[B]$ we can identify $\mathcal{B}$ with a sublattice
of the lattice~$\mathcal{D}$ of closed subsets of~$X$.
Apply the L\"owenheim-Skolem Theorem to find an elementary
sublattice~$\mathcal{C}$ of~$\mathcal{D}$ that contains~$\mathcal{B}$
and has the same (infinite) cardinality as~$\mathcal{B}$;
we let $Z=w\mathcal{C}$.
The two inclusions
$\mathcal{B}\subseteq\mathcal{C}\subseteq\mathcal{D}$ induce
continuous surjections $g:X\to Z$ and $h:Z\to Y$ that, as one readily shows,
satisfy $f=h\circ g$.
By Lemma~\ref{lemma.elem.KM} the space~$Z$ is hereditarily indecomposable.
The same argument shows that $\dim Z=\dim X$: one can use, for example,
the Theorem on Partitions, \cite{E}*{Theorem~1.7.9}, to turn the statement
$\dim X\le n$ into an equation~$\Phi_n$.
By elementarity $\mathcal{C}$ and $\mathcal{D}$ satisfy~$\Phi_n$ for exactly
the same values of~$n$.
The expansion trick applies in this case as well so that $\dim Z\le n$ for
exactly the same values of~$n$ for which $\mathcal{C}$ satisfies~$\Phi_n$.
\end{proof}
We refer to \cite{HodgesShorterModelTheory} for basic information on Model
Theory.
\begin{remark}
The thesis \cite{vdS} contains a systematic study of properties that are
preserved by continuous maps that are induced by elementary embeddings.
\end{remark}
\section{Proofs of the main results}
\label{sec.3}
We start with the following
\begin{theorem}\label{thm.3.1}
Let $f: E \to F$ be a perfect mapping from a space $E$ onto a
strongly zero-dimensional paracompact space~$F$ such that for
every $y \in F$ the fiber~$f^{-1}(y)$ is hereditarily indecomposable.
Then $E$~has Property~(KM).
\end{theorem}
\begin{proof}
Let $C$ and~$D$ be disjoint closed subsets of $E$ and let $U$ and~$V$
disjoint open subsets of $E$ around $C$ and $D$ respectively.
Let us fix $y \in F$.
We shall find a (clopen) neighbourhood~$O_y$ of~$y$ and a fold
of~$f^{-1}[O_y]$ for $\langle C\cap f^{-1}[O_y],D\cap f^{-1}[O_y],U,V\rangle$.
Since $f^{-1}(y)$ is compact and hereditarily indecomposable,
by Theorem~\ref{thm.1.3} it has Property~(KM) and hence there exists a
fold $\langle X_0,X_1,X_2\rangle$ of~$f^{-1}(y)$ for
$\langle C\cap f^{-1}(y),D\cap f^{-1}(y),U,V\rangle$.
Apply \cite{E}*{Theorem~3.1.1} to find a sequence
$B=\langle W_0,W_1, W_2, O_U, U_V\rangle$ of open sets
such that their closures form a swelling of the
sequence~$A=\langle C\cup X_0,X_1,D\cup X_2,X\setminus U,X\setminus V\rangle$,
which means that each term of~$A$ is a subset of the corresponding term~$B$
and whenever $I$ is such that $\bigcap_{i\in I}A_i=\emptyset$ then
$\bigcap_{i\in I}\cl B_i=\emptyset$.
Specifically this means that
\begin{enumerate}
\item $f^{-1}(y)\subseteq W_0\cup W_1\cup W_2$;
\item $\cl W_0\cap\cl W_1\subseteq V$;
\item $\cl W_1\cap\cl W_2\subseteq U$;
\item $\cl W_0\cap\cl W_2=\emptyset$.
\end{enumerate}
As the map~$f$ is perfect and the space~$F$ is zero-dimensional we can find
a clopen neighbourhood~$O_y$ of~$y$ such that
$f^{-1}[O_y]\subseteq W_0\cup W_1\cup W_2$.
It follows that $\langle\cl W_0,\cl W_1,\cl W_2\rangle$ is a fold
of~$f^{-1}[O_y]$ for
$\langle C\cap f^{-1}[O_y], D\cap f^{-1}[O_y],U,V\rangle$.
By strong zero-dimensionality and paracompactness we can find a disjoint
clopen refinement~$\mathcal{O}$ of~$\{O_y:y\in F\}$; it is then a routine
matter to combine the `local' folds into one `global' fold of~$E$
for~$\langle C,D,U,V\rangle$.
\end{proof}
We are now ready to prove the first main result.
\begin{proof}[Proof of Theorem~\ref{thm.1.1}]
To begin we construct a zero-dimensional compactification~$Y^*$ of~$Y$,
a compactification~$X_1$ of~$X$ and a continuous extension $f_1:X_1\to Y^*$.
One way of doing this is by assuming that $X$~is embedded in the Hilbert
cube~$I^{\aleph_0}$, that $Y$~is embedded in the Cantor set~$\{0,1\}^{\aleph_0}$
and then to identify $X$ with the graph of~$f$, i.e.,
$X$~is identified with
$G(f)=\bigl\{\bigl(x,f(x)\bigr):x\in X\bigr\}
\subseteq I^{\aleph_0}\times\{0,1\}^{\aleph_0}$
via $x\mapsto\bigl(x,f(x)\bigr)$.
After this identification $f$~is simply $\pi_2\restr G(f)$, where $\pi_2$~is
the projection onto the second factor of the product;
we can then let $X_1=\cl G(f)$ (in the product) and $Y^*=\cl Y$
(in the Cantor set), the desired extension~$f_1$ of~$f$ then is
$\pi_2\restr X_1$.
Next let $j:\beta X\to X_1$ be the natural map
(the extension of the inclusion of~$X$ into~$X_1$).
By Theorem~\ref{thm.3.1} $X$~has Property~(KM) so by Theorem~\ref{thm.2.1}
$\beta X$~is hereditarily indecomposable.
Apply Theorem~\ref{thm.2.2} to obtain a factorization of~$j$ consisting
of maps $g:\beta X\to X^*$ and $h:X^*\to X_1$ in which $X^*$~is hereditarily
indecomposable, second-countable and satisfies $\dim X^*=\dim \beta X=\dim X$.
Then $X^*$ is a metrizable compactification of~$X$ as $g\restr X$~is a
homeomorphism.
It remains to set $f^*=f_1\circ h$.
\end{proof}
Let us note that since $f$ is perfect and $X^*$ is a compactification of~$X$,
the extension $f^*$ satisfies $(f^*)^{-1}(y)=f^{-1}(y)$ for $y \in Y$.
To get universal hereditarily indecomposable compacta we use the
the factorization method again.
\begin{proof}[Proof of Theorem~\ref{thm.1.2}]
Let $\{X_{s}\}_{s\in S}$ be the family of all compact hereditarily
indecomposable subspaces of the Tychonoff cube~$I^{\tau}$ whose dimension is
not larger than~$n$, and let $i_{s}: X_{s}\to I^{\tau}$ be the inclusion.
Let $X = \bigoplus _{s \in S}X_{s}$ be the free union of the~$X_{s}$'s and
let $i : X \to I^{\tau}$ be defined by $i(x)=i_{s}(x)$ for $x \in X_{s}$.
Let $f : \beta X \to I^{\tau}$ be the extension of~$i$ over~$\beta X$.
Obviously, $X$~has Property~(KM), so by Theorem~\ref{thm.2.1}
$\beta X $~is hereditarily indecomposable.
By Theorem~\ref{thm.2.2} $f$~can be factored as $h \circ g$,
where $g:X \to Z$ and $h : Z \to Y$ and where $Z$~is hereditarily
indecomposable, $w(Z)\le\tau$ and $\dim Z = \dim X$.
We can take $X(n,\tau)=Z$.
\end{proof}
\section{Corollaries and Remarks}
\label{sec.4}
Let us note that as a corollary to either Theorem~\ref{thm.1.1} or
Theorem~\ref{thm.1.2}
one can obtain the following theorem of Ma\'{c}kowiak \cite{M2}.
\begin{corollary}\label{cor.4.1}
For every $n \in \{1,2,\ldots \infty \}$ there exists a
hereditarily indecomposable metric continuum $Z_{n}$ of dimension
$ n$ containing a copy of every hereditarily indecomposable metric
continuum of dimension at most~$n$.
\end{corollary}
\begin{proof}[Proof using Theorem~\ref{thm.1.1}]
Let $\mathcal{P}$ be the subset of the hyperspace $2^{I^{\aleph _0}}$
of the Hilbert cube consisting of all hereditarily indecomposable continua
of dimension~$n$ or less.
Then $\mathcal{P}$ is a $G_\delta$-subset of~$2^{I^{\aleph_0}}$
(see \cite{Ku}*{\S~45, IV, Theorem~4 and \S~48, V, Remark~5}).
Therefore there is a continuous surjection $\varphi:Y\to \mathcal{P}$,
where $Y$~is the space of the irrationals.
Then let $X$ be the following subset of $I^{\aleph_0} \times Y$:
$$
\bigl\{ (x,t): t \in Y \text{ and }x\in\varphi(t)\bigr\}
$$
and let $\pi: I^{\aleph_0}\times Y\to Y$ be the projection.
The restriction $f=\pi \restr X : X \to Y$ is a perfect map
(cf.~\cite{Ku1}*{\S~18} or~\cite{vM}*{Exercise~1.11.26})
with hereditarily indecomposable fibers.
By Theorem~\ref{thm.1.1} there exists a hereditarily indecomposable
$n$-dimensional compact space~$X^*$ that contains~$X$ and hence a copy
of every hereditarily indecomposable continuum of dimension~$n$.
The decomposition of~$X^*$ into its components yields a compact
zero-di\-men\-sional space.
The pseudo-arc $P$ contains a copy of this decomposition space (as indeed does
any uncountable compact metrizable space).
Let $q:X^*\to P$ be a map such that $A=q[X]$ is that decomposition space
and $q:X\to A$ is the quotient map.
By Theorem~15 of~\cite{M1} there exist a hereditarily indecomposable
continuum~$Z_{n}$ and an atomic mapping~$r$ from~$Z_{n}$ onto~$P$ such
that $r \restr r^{-1}(P\setminus A )$ is a homeomorphism and $r^{-1}(A)$~is
homeomorphic to~$X^*$
\ ($Z_{n}$~is a so-called pseudosuspension of~$X^*$ over~$P$ by~$q$).
Since $\dim Z_{n}\le n$ by the countable sum theorem and $Z_{n}$~contains
$X^*$ topologically, the space $Z_{n}$ has the required properties.
\end{proof}
\begin{proof}[Proof using Theorem~\ref{thm.1.2}]
Use the second half of the previous proof but now take the pseudosuspension
of the space $X(n, \aleph_0)$ over~$P$ by~$q$,
where $q : X(n, \aleph_0) \to P$ is a quotient map such that
$A=q[X(n, \aleph_0)]$ is the decomposition space of~$X(n, \aleph_0)$
into its components.
\end{proof}
\begin{remark
If one uses Theorem~\ref{thm.2.2} instead of Marde\v{s}i\'{c}'s
Factorization Theorem,
and standard topological reasoning
(see \cite{E}*{proofs of Theorems~5.4.3 and~3.4.2})
one gets the following results.
\begin{proposition}
For every hereditarily indecomposable compact space~$X$ such
that $\dim X = n$ and the weight of~$X$ is equal to~$\tau$, there exists
an inverse system $\mathbf{S}=\{X_{\sigma}, \pi_{\rho}^{\sigma}, \Sigma \}$,
where $|\Sigma|\le \tau$, of metrizable hereditarily indecomposable
compact spaces of dimension~$n$ whose limit is homeomorphic to~$X$.
If $X$~is a continuum, then all~$X_\sigma$ are continua.
\end{proposition}
\begin{proposition}
Every normal $n$-dimensional space~$X$ of weight~$\tau$
that has Property~(KM) has a hereditarily indecomposable
compactification $\tilde{X}$ of dimension~$n$ and of weight~$\tau$.
\end{proposition}
\end{remark}
\begin{remark}
The results of this paper remain valid if in the formulation of Property~(KM)
one replaces closed sets by zero-sets and open sets by cozero-sets.
This implies that in Theorem~\ref{thm.2.1} one can relax the assumption
of normality to complete regularity.
\end{remark}
\begin{bibdiv}
\begin{biblist}
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author={Aarts, J. M.},
title={Wallman-Shanin Compactification},
pages={218\ndash 220},
book={ title={Encyclopedia of general topology},
editor={Hart, Klaas Pieter},
editor={Nagata, Jun-iti},
editor={Vaughan, Jerry E.},
publisher={Elsevier Science Publishers B.V.},
place={Amsterdam},
date={2004},
pages={x+526},
isbn={0-444-50355-2},
review={\MR {2049453 (2005d:54001)}},
}}
\bib{MR1785837}{article}{
author={Bankston, Paul},
title={Some applications of the ultrapower theorem to the theory of
compacta},
note={Papers in honour of Bernhard Banaschewski (Cape Town, 1996)},
journal={Applied Categorical Structures},
volume={8},
date={2000},
number={1-2},
pages={45--66},
issn={0927-2852},
review={\MR{1785837 (2001f:54011)}},
}
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author={Engelking, Ryszard},
title={Theory of dimensions finite and infinite},
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date={1995},
pages={viii+401},
isbn={3-88538-010-2},
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}
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author={Pol, Roman},
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journal={Topology Proceedings},
volume={25},
date={2000},
pages={179--206 (2002)},
issn={0146-4124},
review={\MR{1925683 (2003k:54028)}},
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title={A shorter model theory},
publisher={Cambridge University Press},
place={Cambridge},
date={1997},
pages={x+310},
isbn={0-521-58713-1},
review={\MR{98i:03041}},
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\bib{KM}{article}{
author={Krasinkiewicz, J{\'o}zef},
author={Minc, Piotr},
title={Mappings onto indecomposable continua},
journal={Bulletin de l'Acad\'emie Polonaise des Sciences.
S\'erie des Sciences Math\'ematiques, Astronomiques et Physiques},
volume={25},
date={1977},
pages={675--680},
issn={0001-4117},
review={\MR{0464184 (57 \#4119)}},
}
\bib{Ku1}{book}{
author={Kuratowski, K.},
title={Topology. Vol. I},
publisher={Academic Press},
place={New York},
date={1966},
pages={xx+560},
review={\MR{0217751 (36 \#840)}},
}
\bib{Ku}{book}{
author={Kuratowski, K.},
title={Topology. Vol. II},
publisher={Academic Press},
place={New York},
date={1968},
pages={xiv+608},
review={\MR{0259835 (41 \#4467)}},
}
\bib{M1}{article}{
author={Ma{\'c}kowiak, T.},
title={The condensation of singularities in arc-like continua},
journal={Houston Journal of Mathematics},
volume={11},
date={1985},
pages={535--558},
issn={0362-1588},
review={\MR{837992 (87m:54099)}},
}
\bib{M2}{article}{
author={Ma{\'c}kowiak, T.},
title={A universal hereditarily indecomposable continuum},
journal={Proceedings of the American Mathematical Society},
volume={94},
date={1985},
pages={167--172},
issn={0002-9939},
review={\MR{781076 (86j:54061)}},
}
\bib{vM}{book}{
author={van Mill, Jan},
title={The infinite-dimensional topology of function spaces},
series={North-Holland Mathematical Library},
volume={64},
publisher={North-Holland Publishing Co.},
place={Amsterdam},
date={2001},
pages={xii+630},
isbn={0-444-50557-1},
review={\MR{1851014 (2002h:57031)}},
}
\bib{vdS}{thesis}{
author={van der Steeg, B. J.},
title={Models in Topology},
type={PhD thesis},
date={2003},
institution={TU Delft}
}
\bib{Wallman}{article}{
author={Wallman, Henry},
title={Lattices and topological spaces},
journal={Annals of Mathematics},
volume={39},
date={1938},
pages={112--126},
review={\MR{1503392}}
}
\end{biblist}
\end{bibdiv}
\end{document}
|
0805.2997
|
\section{Flat FLRW with barotropic fluids}
The spatially flat FLRW metric (with scale factor $a$)
\begin{equation*}
\ensuremath{\, \mathrm{d}} s^2 = - \ensuremath{\, \mathrm{d}} t^2 + a^2(t)\left(\ensuremath{\, \mathrm{d}} x^2 + \ensuremath{\, \mathrm{d}} y^2 + \ensuremath{\, \mathrm{d}} z^2\right)
\end{equation*}
yields the following Einstein equations:
\begin{subequations} \label{einst}
\begin{align}
3 H^2 &= \kappa^2 \sum \rho_i\es, \label{et}
\\
2 \dot H &= - \kappa^2 \sum (\rho_i + p_i) \label{etd}\es,
\end{align}
\end{subequations}
where $H \defin \tfrac{\dot a} {a}$ is the Hubble rate, $\rho_i$ and $p_i$ are the energy density and the pressure of the $i$-th fluid, and the gravitational constant is set to 1 ($\kappa\defin\sqrt{8 \pi} $). If the fluids can be modeled as barotropic perfect fluids, then, by definition, their pressures are functions of their corresponding energy densities. Let us denote the negative of the enthalpy by $f$, so that
\begin{equation} \label{barot}
f_i(\rho_i) \defin - (\rho_i + p_i)
\end{equation}
are functions of $\rho_i$ alone. In terms of $f_i$, the equation of state parameters can be written as
\begin{equation*}
w_i = - 1 - \frac{f_i} {\rho_i } \es.
\end{equation*}
Note that the $i$-th fluid corresponds to phantom energy as long as $f_i$ is positive definite. Using \eqrf{barot}, we can remove the pressures from Einstein's equations and from the conservation of energy equations of each fluid, which now read
\begin{subequations}
\begin{align}
3 H &= \frac{\dot {\rho_i} + Q_i} {f_i} \label{cem}\es,\\
\sum Q_i &= 0\es.
\end{align}
\end{subequations}
The $Q_i$ account for the transfer of energy between fluids. As usual, conservation of energy and Einstein's equations are not independent.
Now, we will make the assumption that there is no interaction amongst the fluids, i.e. $Q_i = 0$. This step will be justified by its compatibility with the results: If at the end of the calculations there is only one dominant fluid, phantom energy near the singularity, whose energy density is much larger than the energy densities of all other fluids combined, then we can neglect the interaction between fluids. In this case, we can use \eqrf{cem} to find $\rho_i$ as functions of $a$, and \eqrf{et} to find the evolution of $a$ in time:
\begin{align}
3\log\left( \frac{a} {a_0}\right) &= \int^{\rho_i}_{ {\rho_i}_0 } \frac{\ensuremath{\, \mathrm{d}}
\rho_i} {f_i(\rho_i)} \label{rfa}\es,
\\
\frac{\kappa} {\sqrt{3}}(t - t_0) &= \int^a_{a_0} \frac{\ensuremath{\, \mathrm{d}} a} {a\sqrt{\sum
\rho_i(a)}}\es. \label{tfa}
\end{align}
The subindex zero indicates the value of the quantity at a given time, say today.
From \eqrf{rfa}, we can reason that $\rho_i$ are monotonic functions of $a$ as long as $f_i$ don't cross zero, cross infinity or jump branches ($f_i$ could be multivalued). Analyzing \eqrf{tfa}, we can deduct that $a$ is a monotonic function of $t$ provided that the total energy density $\sum \rho_i$ does not vanish or diverge. We can conclude that $\rho_i$ are monotonic functions of time until $f_i$ or $\sum \rho_i$ vanish or diverge or $f_i$ change branches.
Finite-time future singularities will appear if for some reason the integral in \eqrf{tfa} cannot keep increasing; so that time, in the left hand side of equation of \eqrf{tfa}, wouldn't keep increasing either. A classification of finite-time future singularities was introduced by Nojiri \emph{et al.} \cite{Nojiri2005} and completed by Copeland \emph{et al.} \cite{Copeland2006}. It is summarized in \tbrf{tb:sing}.
If the integrand in \eqrf{tfa} cannot be evaluated, say because the integrand fails to be a real number for $a = a_s$, then we might have a sudden or type-III singularity at $a_s$. On the other hand, if $a$, the upper limit of \eqrf{tfa}, can go to infinity, then we have a Big Rip singularity if the integral converges; and there is no singularity if the integral diverges.
One corollary out of these statements is that a singularity type IV cannot be produced by a single barotropic non-interacting fluid in a realistic model. This case, although mathematically possible, is physically impossible. Baryonic matter will always be present and its contribution to the right hand side of \eqrf{et}, which evolves as $a^{-3}$, does not vanish because $a_s$ remains finite per definition of type-IV singularity. If the energy density of phantom fluid vanishes then the energy density of baryonic matter takes over and drives the evolution of $a$ (without any singularity whatsoever).
\begin{table}
\caption{\label{tb:sing} Classification of finite-time future singularities \cite{Copeland2006,Nojiri2005}. A dash indicates non-specified behavior. The equivalences of behavior, $\rho \sim\abs{\dot{a}}$ and $\abs{p}\sim\abs{\ddot{a}}$, follow from Eqs.~(\ref{einst}). }
\begin{ruledtabular}
\begin{tabular}{l|cccc}
& $a$ & $\rho\sim\abs{\dot{a}}$ & $\abs{p}\sim\abs{\ddot{a}}$ & $\abs{\dddot{a}}$ and higher\\
\hline
I - Big Rip & $\infty$ & $\infty$ & $\infty$ & - \\
III & $a_s$ & $\infty$ & $\infty$ & - \\
II - sudden & $a_s$ & $\rho_s$ & $\infty$ & - \\
\hline
IV & $a_s$ & $0$ & $0$ & $\infty$ \\
V & $\infty$ & $\rho_s$ & $p_s$ & $\infty$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Sudden singularities}
A sudden singularity occurs when the pressure diverges but the energy density and the scale factor remain finite. By \eqrf{einst}, the first derivative of the scale factor also remain finite but the second derivative diverges. Hence, $a$ can be approximated, near the singularity, by
\begin{equation} \label{aaprox}
a(t) \approx a_s - A \, (t_s - t) + B \, (t_s - t)^{1 + \frac{1} {1 + \delta}} +
\mathcal{O}((t_s - t)^{1 + \frac{1} {1 + \delta} + \epsilon} )
\end{equation}
with positive definite both $\delta$ and $\epsilon$. Possible higher order terms in $t_s - t$ have been omitted ($\epsilon > 0$) because we only need to show the divergence of the second derivative while keeping the first derivative finite ($\delta > 0$). It is easy to prove that the behaviors of the total energy density and total pressure near the singularity are given by
\begin{subequations}
\begin{align}
\sum \rho_i &\approx \frac{3\, A^2}{a_s^{\phantom{s}2}\, \kappa ^2} - \frac{6
\,A\, B\, (\delta +2) \,(t_s - t)^{\frac{1}{1 + \delta}}} {a_s^{\phantom{s}2}\,
(\delta + 1)\, \kappa ^2}\es,\label{eq:EneBeh}\text{ and}
\\
\sum p_i &\approx - \frac{2 \,B\, (\delta +2)}{a_s (\delta + 1)^2} \,(t_s - t) ^{
- 1 + \frac{1}{1 + \delta}}\es.
\end{align}
\end{subequations}
Note that the signs in front of $A$ and $B$ in \eqrf{aaprox} have been chosen in such a way that if $A$ and $B$ are both positive then $a$ approaches $a_s$ from below, $\rho$ also approaches $\rho_s$ from below, and $p$ diverges to $-\infty$. If $A$ was negative, then $a'(t_s)$ would be negative. But this cannot happen because of the monotonicity of $a$ --Einstein equations for flat FLRW don't have curvature terms to change the sign of $a'(t)$-- and we know that $a'(t)$ is positive today.
Now, let us analyze the conditions under which the contributions from fluids like dark matter or electromagnetic radiation would not be significant near the singularity. We need the total energy density to be much larger than the dark-matter energy density. This is, the constant term in the behavior of $\sum \rho_i$, \emph{cf.} \eqrf{eq:EneBeh}, must be much larger than ${\rho_m}_0 a_0^{\phantom{0}3}/a_s^{\phantom{s}3}$, where ${\rho_m}_0$ is the energy density of dark matter measured today. Therefore, $a_s$ and $A$ must be big enough to satisfy
\begin{equation*}
A^2a_s \gg \frac{\kappa^2} {3}\,a_0^{\phantom{0}3}\,{\rho_m}_0\,.
\end{equation*}
If $A$ vanished, then this inequality could not be satisfied. This is, only positive definite $A$ is compatible with the assumption of non-interacting fluids.
If $B$ was negative, the fluid causing the singularity would have positive both energy density and pressure. Thus, such fluid would, near the singularity, simply not be phantom energy.
Assuming that only one fluid, dark energy, contributes significantly near the singularity, then \eqrf{barot} for such fluid has the form
\begin{equation} \label{fappAB}
f \approx \sign{B}\frac{(3 \,\abs{A})^\delta(2\, \abs {B} \, (\delta +2))^{1 + \delta}
}{\kappa ^ {2 \delta}\,(\delta +1)^{2 + \delta} \,a_s^{\phantom{s}1 + 2 \delta}}
\,\abs{\frac{3 A^2}{a_s^{\phantom{s}2} \kappa ^2} - \rho}^{ - \delta}.
\end{equation}
Although it has been argued that only positive $A$ and $B$ are physically relevant for phantom-energy driven future singularities, the above formula shows the correct sign dependence should $A$ or $B$ be negative. Keeping the sign dependence allows for easier comparison with other calculations presented in the literature.
Conversely, if phantom energy is modeled by
\begin{equation} \label{fsud}
f = \frac{C} { (\rho_s - \rho) ^\delta} + \mathcal{O}((\rho_s - \rho)^{1 -
\delta})\,,
\end{equation}
with positive $C$, $\rho_s$, and $\delta$, then the evolution is such that the scale
factor near the singularity is given by
\begin{widetext}
\begin{equation}
a(t) \approx a_s \left( 1 - \tau + \frac { \left( \frac {3 \,C \, ( 1 + \delta ) } { \kappa^2} \right)^{ \frac {1} { 1 + \delta } } ( 1 + \delta ) } {2 \,( 2 + \delta ) \, \rho_s} \, \tau^{1 + \frac {1} {1 + \delta} } + \order { \tau^{1 + \frac {1} {1 + \delta } + \epsilon } } \right) \label {atrunc}
\end{equation}
\end{widetext}
where
\begin{equation}
\tau \defin \kappa \sqrt {\frac {\rho_s} {3} } ( t_s - t ) \es, \label {deft}
\end{equation}
and
\begin{equation}
\epsilon = \min\left\{2, \frac{\delta + 3} {\delta + 1}\right\} - 1 - \frac{1} {1 +
\delta}\es.
\end{equation}
Note that \eqrf{atrunc} is of the form of \eqrf{aaprox} and that $\epsilon$ is positive for $\delta > 0$. While Copeland \emph{et al.} showed that the first term of \eqrf{fsud} yields a sudden singularity (see Eq.~(461) in \cite{Copeland2006}), the calculation shown here is more general in that it only analyzes the behavior near the singularity (hence the operator $\mathcal{O}$ and the need to keep track of $\epsilon$). Thus, it encompasses other models, {\it e.g.} model (32) in \cite{Nojiri2005}, that might behave differently far from the singularity. As implied in remarks by Catt{\"o}en and Visser \cite{Cattoen2005}, proving that \eqrf{fappAB} can be obtained from \eqrf{aaprox} is relatively trivial. However, the proof of the converse is rather laborious because one must prove that the dismissed terms in \eqrf{fsud} can also be dismissed in \eqrf{atrunc}. Such proof follows the lines of the calculation in the appendix.
We reach then the following conclusion: a phantom-energy model where barotropic dark energy is the only significant fluid near the singularity will produce a sudden singularity, \eqrf{aaprox}, if and only if its behavior near the singularity has the form of \eqrf{fsud}. The relationship between $(A, B)$ and $ (C, \rho_s) $ can be read off from equations (\ref{aaprox}) and (\ref{atrunc}). One implication of this theorem is that sudden singularities cannot be achieved with a static equation of state parameter, it must evolve according to
\begin{equation} \label{wend}
w \approx - \frac{C} { \rho\, (\rho_s - \rho) ^\delta} = \mathcal{O}(\tau ^{ - 1 + \frac{1} {1 + \delta }})
\end{equation}
near the singularity.
\section{Semiclassical fields}
\begin{table}
\caption{\label{tb:cof} Spin-dependent coefficients in \eqrf{tmn}.}
\begin{ruledtabular}
\begin{tabular}{ccc}
spin & $\alpha$ & $\beta$
\\
\hline
0 & $\frac{1} {2800\pi^2}$ & $\frac{1} {2800\pi^2}$
\\
$ \frac{1} {2}$ & $\frac{3} {2800\pi^2}$& $\frac{11} {5600\pi^2}$
\\
1 & $ - \frac{9} {1400\pi^2}$ & $\frac{31} {1400\pi^2}$
\end{tabular}
\end{ruledtabular}
\end{table}
We will use the semiclassical expression for the vacuum stress-energy of conformally invariant quantized fields in a vacuum state conformally obtained from Minkowski spacetime \cite{Candelas1979}:
\begin{multline} \label{tmn}
\expv{T_{\mu\nu}} = \frac{\alpha} {3}\left( g_{\mu\nu} \,
R^{;\sigma}_{\phantom{;\sigma};\sigma} - R_{;\mu\nu} + R\,R_{\mu\nu} - \frac{1}
{4}g_{\mu\nu}\,R^2\right) +\\ \beta\left( \frac{2} {3}R\, R_{\mu\nu} -
R^{\sigma}_\mu\,R_{\nu\sigma} + \frac{1} {2} g_{\mu\nu}\, R_{\sigma \tau }\,R^{\sigma \tau } - \frac{1} {4}g_{\mu \nu }\,R^2 \right)\,,
\end{multline}
where $R$ is the Ricci scalar, $R_{\mu\nu}$ is the Ricci tensor, and $\alpha$ and $\beta$ are spin-dependent coefficients given in \tbrf{tb:cof}. Because the derivatives of $R$ diverge faster than $R$ or $g$, the first two terms of the coefficient of $\alpha$ are the ones that contribute the most:
\begin{widetext}
\begin{subequations} \label {eq:rhopSudden}
\begin{align}
\rho_a \defin \expv{T_{00}} \vert_{\text {spin} = a} & = \alpha \frac { \kappa^4 \, \rho_s} {3} ( 3 \,C )^{ \frac {1} {1 + \delta }} \, \delta \left( ( 1 + \delta ) \, \tau \right)^{- 2 + \frac {1} {1 + \delta}} \es ,
\\
p_a \defin \expv{T_{11}} \vert_{\text {spin} = a} & = - \alpha \frac {\kappa^4 \, \rho_s} {9} ( 3 \,C )^{\frac {1} {1 + \delta} } \, \delta \,( 1 + 2 \,\delta )\, {a_s}^2 \,\left( ( 1 + \delta ) \, \tau \right)^{- 3 + \frac {1} {1 + \delta}} \es . \label{eq:exppSudden}
\end{align}
\end{subequations}
\end{widetext}
Both expectation values diverge as the singularity approaches, the pressure faster than the energy density. The equation of state parameter of the quantum contributions is then given by
\begin{equation}
w = - {a_s}^2 \frac {1 + 2 \, \delta} {3 \,( 1 + \delta ) } \tau^{ -1} \es .
\end{equation}
It is negative, it doesn't depend on the spin of the field, and it diverges at the singularity faster than \eqrf{wend}. These divergences mean that the approximation breaks down before the singularity occurs. Nevertheless, a qualitative analysis is in order. The behavior of the perturbation is determined by the sign of $\alpha $, which is the only quantity not defined positive in \eqrf{eq:exppSudden}. Because $\alpha$ is positive for both scalar and spinor fields, then $\rho_a >0$, $p_a<0$ and therefore these vacuum states enhance the singularity. This happens because adding these vacuum perturbations is equivalent to adding more phantom energy to the right hand side of \eqrf{einst}. Vector fields, with negative $\alpha$, counter the contributions from dark energy and therefore soften the singularity.
\section{Thermodynamical Considerations}
While the vacuum state of a system described by an exact Lagrangian has no thermal properties, it is not impossible for a vacuum state to become thermal (\emph{e.g.} \cite{Mitter2000, Horwitz1986}). In general (see \cite{Katz1967} for a canonical exposition), small unknown terms in the Hamiltonian cause the thermalization of the system (equation (12.2) leads to equation (13.13) in the reference).
In this section, the vacua are considered as individual subsystems. We admit that their evolution is known only up to some terms of first order in $\hbar$ and that their complete Lagrangian would contain higher order terms in $\hbar$, self-interactions, and interactions with the phantom fluid. Thus, these vacua qualify for thermal calculations. The previous section provide the energy density and pressure of the fields. The pressure is interpreted as the partial pressure of the vacuum state in the mixture of cosmological fluids. We don't need to assume that the process is quasi-static because we are not concerned with the temperature or the entropy of the subsystems; we shall be satisfied with computing the sign of the change of the enthalpy of formation.
In the first law of thermodynamics $\delta U = \delta Q - \delta W$, we can replace
\begin{subequations}
\begin{align}
U & \propto \rho_a \, a^3\,,\\
W & \propto p_a \, a^3 \,. \label{wprop}
\end{align}
\end{subequations}
The approximations for \eqrf{tmn} are valid only for free fields (see \cite{Birrell1984} page 5). Thus, the lack of interaction terms in \eqrf{wprop} is justified. We are interested in the sign of $\delta Q$ because it determines whether the expansion is exothermic or endothermic. As shown in Eqs.~(\ref{eq:rhopSudden}), the pressure diverges faster than the energy density and therefore $\delta Q \approx \delta W$. Hence,
\begin{equation}
\sign {\delta Q} = \sign {3\, p_a\, a^2 \, \delta a + a^3 \, \delta p_a}\,.
\end{equation}
We can now dismiss the first term in the right hand side because it behaves as $\tau^{-3 + \frac {1} {1 + \delta} }$ which is slower than the $\tau^{-4 + \frac {1} {1 + \delta} }$ divergence of $\delta p_a$. Therefore, the sign of the heat flowing \emph{into} the system is the same as the sign of pressure change. The latter is determined by the negative of the sign of $\alpha$ because all of the other quantities in \eqrf{eq:exppSudden} are positive definite. The expansion is then exothermic for both scalar and spinor fields and endothermic for vector fields. Exothermic reactions are spontaneous and thus they enhance the singularity. We conclude again that scalar and spinor vacua enhance the singularity and vector vacua soften it. This is in agreement with the dynamical results of the previous section.
While the models considered in this paper furnish an example of dynamics and thermodynamics coinciding in their predictions, this might not necessarily be true in general. A counterexample would be extremely valuable because it will point to a deficiency of Quantum Field Theory in Curved Spacetimes (QFTCS). This theory is considered complete up to first order in $\hbar$. But if the dynamics and thermodynamics don't coincide, then the predictive power of QFTCS would be limited because it wouldn't be possible to determine the evolution of an entropy-driven system without a Statistical Mechanics theory of gravity (let alone Quantum Gravity).
|
0805.1836
|
\section{Introduction}
Several complex systems have underlying structures that are
described by networks or graphs
\cite{Strogatz,rev-Barabasi}. Recent interest in networks is due to
the discovery that several naturally occurring
networks come under some universal classes and they can
be simulated with simple mathematical models, viz small-world
networks \cite{Watts}, scale-free networks \cite{scalefree} etc.
Several networks in the real world consist of dynamical elements interacting
with each other and the interactions define the links of the network.
Several of these networks have a large number
of degrees of freedom and it is important to understand their dynamical
behavior. Here, we study the synchronization
and cluster formation in networks consisting of interacting
dynamical elements. A general model of coupled dynamical systems on
networks will consist
of the following three elements.
\begin{enumerate}
\item The evolution of uncoupled elements.
\item The nature of couplings.
\item The topology of the network.
\end{enumerate}
Most of the earlier studies of synchronized cluster formation in
coupled chaotic systems have focused on
networks with large number of connections ($\sim N^2$)
\cite{rev-Kaneko}.
In this paper, we consider networks
with number of connections
of the order of $N$. This small number of connections
allows us to study the role that different connections play in
synchronizing different nodes and
the mechanism of synchronized cluster formation. The study reveals two
interesting phenomena. First,
when nodes form synchronized clusters, there can be some nodes which
show an intermittent behaviour between independent evolution and
evolution synchronized with some cluster. Secondly,
the cluster formation can be in two different ways, driven and self-organized
phase synchronization \cite{sarika-REA1}.
The connections or couplings in the self-organized phase synchronized
clusters are mostly of the intra-cluster type while those in the
driven phase synchronized clusters are mostly of
the inter-cluster type.
\section{Coupled dynamical systems and synchronized clusters}
Consider a network of $N$ nodes and $N_c$ connections (or couplings)
between the nodes. Let each node of the network be assigned an $m$-dimensional
dynamical variable ${\bf x}^i, i=1,2,\ldots,N$. A very general dynamical
evolution can be written as
\begin{equation}
\frac{d{\bf x}_i}{dt} = {\bf F}(\{ {\bf x}_i \}).
\end{equation}
In this paper, we consider a separable case and the evolution equation
can be written as,
\begin{equation}
\frac{d{\bf x}_i}{dt} = {\bf f}({\bf x}_i) + \frac{\epsilon}{k_i}
\sum_{j \in \{k_i\}} {\bf g}({\bf x}_j).
\label{evol-cont}
\end{equation}
where $\epsilon$ is the coupling constant, $k_i$ is the degree of node
$i$, and $\{k_i\}$ is the set of nodes connected to the node $i$.
A sort of diffusion version of the evolution equation~(\ref{evol-cont}) is
\begin{equation}
\frac{d{\bf x}_i}{dt} = {\bf f}({\bf x}_i) + \frac{\epsilon}{k_i}
\sum_{j \in \{k_i\}} \left({\bf g}({\bf x}_j)-{\bf g}({\bf
x}_i)\right).
\label{evol-cont-diff}
\end{equation}
Discrete versions of Eqs.~(\ref{evol-cont})
and~(\ref{evol-cont-diff}) are
\begin{equation}
{\bf x}_i(t+1) = {\bf f}({\bf x}_i(t)) + \frac{\epsilon}{k_i}
\sum_{j \in \{k_i\}} {\bf g}({\bf x}_j(t)).
\label{evol-disc}
\end{equation}
and
\begin{equation}
{\bf x}_i(t+1) = {\bf f}({\bf x}_i(t)) + \frac{\epsilon}{k_i}
\sum_{j \in \{k_i\}} \left({\bf g}({\bf x}_j(t))-{\bf g}({\bf
x}_i(t))\right).
\label{evol-disc-diff}
\end{equation}
For the discrete evolution we use logistic or circle maps while for
the continuous case we use Lorenz or R\"ossler systems.
\subsection{Phase synchronization and synchronized clusters}
Synchronization of coupled dynamical systems
\cite{book-syn} is manifested by the appearance of some
relation between the functionals of different dynamical variables.
The exact synchronization corresponds to
the situation where the dynamical variables for different nodes have identical
values. The phase synchronization corresponds to the situation where the
dynamical variables
for different nodes have some definite relation between their phases
\cite{phase1,phase2}. When the number
of connections in the network
is small ($N_C \sim N$) and when the local dynamics of the
nodes (i.e. function $f(x)$) is in the chaotic zone, and we look at
exact synchronization, we find that only
few synchronized clusters with small number of nodes are formed.
However, when we look at phase synchronization, synchronized clusters
with larger number of nodes are obtained. Hence, in our numerical study we
concentrate on phase synchronization.
\section{General properties of synchronized dynamics}
We consider some general properties of synchronized dynamics.
They are valid
for any coupled discrete and continuous dynamical systems.
Also, these
properties are applicable for exact as well as phase or any other type
of synchronization
and are independent of the type of network.
\subsection{Behavior of individual nodes}
As the network evolves, it splits into several synchronized clusters.
Depending on their asymptotic dynamical behaviour the nodes
of the network can be divided into three types. \\
(a) {\it Cluster nodes}: A node of this type synchronizes with other nodes and
forms a synchronized cluster. Once this node enters a synchronized cluster
it remains in that cluster afterwards. \\
(b) {\it Isolated nodes}: A node of this type does not synchronize
with any other node
and remains isolated for all the time. \\
(c) {\it Floating Nodes}: A node of this type keeps on switching intermittently
between an independent evolution and a synchronized evolution
attached to some cluster.
Of particular interest are the floating nodes and we will discuss some
of their properties afterwards.
\subsection{Mechanism of cluster formation}
The study of the relation between the synchronized clusters and the couplings
between the nodes represented by the
adjacency matrix $C$ shows two different
mechanisms of cluster formation \cite{sarika-REA1,pre2}. \\
(i) Self-organized clusters: The nodes of a cluster can be
synchronized because of intra-cluster
couplings. We refer to this as
the self-organized synchronization and
the corresponding synchronized clusters as self-organized clusters. \\
(ii) Driven clusters: The nodes of a cluster can be
synchronized because
of inter-cluster couplings.
Here the nodes of one cluster are driven by those
of the others. We refer to this as the driven
synchronization and the corresponding clusters as driven clusters.
In our numerical studies we have been able to identify ideal clusters
of both the types,
as well as clusters of the mixed type where both ways of
synchronization contribute
to cluster formation.
(Fig.~1 of Ref.~\cite{sarika-REA1} gives examples of ideal as well as
mixed clusters in coupled map networks.)
In general we find
that the scale free
networks and the Caley tree networks lead to better cluster formation than
the other types of networks with the same average connectivity.
Geometrically the two mechanisms of cluster formation can be easily
understood by considering a tree type network. A tree
can be broken into different clusters in different ways. \\
(a) A tree can be broken into two or more disjoint clusters with only
intra-cluster couplings by breaking one
or more connections. Clearly, this splitting is not unique
and will lead to self-organized clusters. Figure~\ref{tree-clus}(a)
shows a tree forming two synchronized clusters of self-organized
type. This situation is
similar to an Ising ferromagnet where domains of up and down spins can
be formed. \\
(b) A tree can
also be divided into two clusters by putting connected nodes into different
clusters. This division is unique and leads to two clusters with only
inter-cluster couplings, i.e. driven clusters. Figure~\ref{tree-clus}(b)
shows a tree forming two synchronized clusters of the driven type.
This situation is
similar to an Ising anti-ferromagnet where two sub-lattices of up and
down spins are formed.\\
(c) Several other ways of splitting a tree are possible. E.g. it is easy
to see that a tree can be broken into three clusters of the driven
type. This is shown in figure~\ref{tree-clus}(c). There is no simple
magnetic analog for this type of cluster formation. It
can be observed close to a period
three orbit. We note that four or more clusters of the driven type are
also possible. As compared to the cases (a) and (b) discussed above
which are commonly observed,
the clusters of case (c) are not so common and are
observed only for some values of the parameters.
\begin{figure}[hb]
\begin{center}
\includegraphics[width=14cm]{R_E_Amritkar_fig1.eps}
\end{center}
\caption{Different ways of cluster formation in a tree structure are
demonstrated. The open, solid and gray circles show nodes belonging to
different clusters. (a) shows two clusters of the self-organized
type, (b) shows two clusters of driven type and (c) shows three
clusters of the driven type.}
\label{tree-clus}
\end{figure}
\section{Linear stability analysis}
A suitable network to study the stability of self-organized
synchronized clusters is the globally coupled network. The stability
of globally coupled maps is well studied in the literature
\cite{GCM-stab1,GCM-stab2,GCM-stab3}.
An ideal example to consider the stability of the driven synchronized
state is a complete bipartite network.
A complete bipartite network consists of two sets of nodes with each
node of one set connected
with all the nodes of the other set and no connection between the
nodes of the same set. Let $N_1$ and $N_2$ be the number of nodes
belonging to
the two sets. We define a bipartite synchronized
state as the one that
has all $N_1$ elements of the first set synchronized
to some value, say ${\bf X}_1(t)$, and
all $N_2$ elements of the second set synchronized
to some other value, say ${\bf X}_2(t)$.
All the eigenvectors and the eigenvalues of the Jacobian matrix for the bipartite synchronized state can be determined explicitly.
The eigenvectors of the type
$(\alpha,\ldots,\alpha,\beta,\ldots,\beta)^T$ determine the
synchronization manifold and this manifold has dimension two.
All other eigenvectors correspond to
the transverse manifold.
Lyapunov exponents corresponding to the transverse eigenvectors
for Eq.~(\ref{evol-disc-diff}) with one dimensional variables and $g(x)=f(x)$
are
\begin{eqnarray}
\lambda_1 &=& \ln|(1-\epsilon)| + \frac{1}{\tau} \lim_{\tau \to\infty}
\sum^\tau_{t=1} \ln |f^{\prime}(X_1)|,
\nonumber \\
\lambda_2 &=& \ln|(1-\epsilon)| + \frac{1}{\tau} \lim_{\tau \to\infty}
\sum^\tau_{t=1} \ln |f^{\prime}(X_2)|,
\label{lya-driven-Nlarge}
\end{eqnarray}
and $\lambda_1$ and $\lambda_2$ are respectively $N_1-1$ and $N_2-1$
fold degenerate \cite{pre2}. Here, $f^{\prime}_(X_1)$ and $f^{\prime}_(X_2)$ are
the derivatives of $f(x)$ at $X_1$ and $X_2$ respectively.
The synchronized state is stable provided the transverse Lyapunov
exponents are negative.
If $f^\prime$ is bounded then from
Eqs.~(\ref{lya-driven-Nlarge}) we see that for $\epsilon$ larger than
some critical value, $\epsilon_b (<1)$, bipartite synchronized state will be
stable. Note that this bipartite synchronized state will be stable even if one
or both the remaining Lyapunov exponents corresponding to the
synchronization manifold are
positive, i.e. the trajectories are chaotic. The linear stability
analysis for other type of couplings and dynamical systems can be done
along similar lines.
\section{Floating nodes}
We had noted earlier that the nodes can be divided into three types,
namely cluster nodes, isolated nodes and floating nodes, depending on
the asymptotic behavior of the nodes.
Here, we discuss some properties of the floating nodes which show an
intermittent behavior between synchronized evolution with some cluster
and an independent evolution.
Let $\tau$ denote the residence time of a floating node in a cluster
(i.e. the continuous time interval that the node is in a cluster).
Figure~\ref{freq-floating} plots the frequency of residence time $\nu(\tau)$
of a floating node as a
function of the residence time $\tau$.
A good straight line fit on log-linear
plot shows an exponential dependence,
\begin{equation}
\nu(\tau) \sim \exp(-\tau/ \tau_r)
\label{exp-dist-floating}
\end{equation}
where
$\tau_r$ is the typical residence time
for a given node. We have also studied the distribution of the time
intervals for which a floating node is not synchronized with a given
cluster. This also shows an exponential distribution.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{R_E_Amritkar_fig2.ps}
\end{center}
\caption{The figure plots the frequency of residence time $\nu(\tau)$
of a floating node in a cluster as a
function of the residence time $\tau$. A good straight line fit on log-linear
plot shows exponential dependence.}
\label{freq-floating}
\end{figure}
Let us now consider the condition for the occurrence of floating nodes.
Consider a floating node in a cluster. The stability of this cluster
is ensured if the transverse Lyapunov exponents are all negative. The
floating node will leave the cluster
provided the conditional Lyapunov exponent for this node,
assuming that the other nodes in the cluster remain synchronized,
changes sign and becomes positive. Thus the fluctuation of the
conditional Lyapunov exponent about zero can be taken as the condition
for the existence of a floating node.
Several natural systems show examples of floating nodes,
e.g. some birds may show intermittent behaviour between free flying and
flying in a flock. An interesting example in physics is that of particles
or molecules in a liquid in
equilibrium with its vapor where the particles
intermittently belong to the liquid and vapor. Under suitable
conditions it is possible to
argue that the residence time of a tagged particle in the liquid phase
should have an exponential distribution \cite{pre2}, i.e. a behavior similar to
that of the floating nodes (Eq.~(\ref{exp-dist-floating})).
\section{Conclusion and Discussion}
We have studied the properties of coupled dynamical elements on
different types of networks. We find that
in the course of time evolution they form synchronized
clusters.
In several cases when synchronized clusters are formed there are some
isolated nodes which do not belong to any cluster. More interestingly
there are some {\it floating} nodes which show an intermittent behavior
between an independent evolution and an evolution synchronized with some
cluster. The residence time spent by a floating node in the synchronized cluster
shows an exponential distribution.
We have identified two mechanisms of cluster formation,
self-organized and driven phase synchronization. For self-organized
clusters intra-cluster couplings dominate while for driven clusters
inter-cluster couplings dominate.
|
0805.3471
|
\section{Introduction}
Heterogeneity often plays an important role in various adsorption
processes and its presence may profoundly modify the adsorption
isotherms and other thermodynamic properties compared to the
homogeneous situation. Consequently, numerous
articles\cite{B04,ASWM02}, reviews \cite{D01,Adamczyk2005} and
monographs \cite{JM88,RE92} have been published that describe
experimental, theoretical and numerical studies in the area. The
disorder may originate from the energetic or structural heterogeneity
of the substrate or from the adsorbate species. Heterogeneity may also
result if the adsorbed molecules are large enough so that multisite
adsorption becomes possible \cite{JWA96}.
In a recent article \cite{TTV07}, we investigated the properties of a
lattice model of adsorption on a disordered substrate that can be
solved exactly (See also Refs.\cite{Oshanin2003,OBB03}). We showed
that there exists an exact mapping to the system without disorder in
the limits of small and infinite activities and we exploited this
result to obtain an approximate, but accurate description of the
disordered system.
For continuous systems, structural disorder may be represented by the
random site model (RSM) in which adsorption sites are uniformly and
randomly distributed on a plane \cite{JWTT93,OT07}. The molecules,
represented by hard spheres, can bind to these immobile sites. Steric
exclusion is expressed by the fact that a site is available for
adsorption only if the nearest occupied site is at least one particle
diameter away. In addition, adsorption energy is assumed equal for
each adsorbed molecule. Therefore, the disorder of this model is
characterized by the dimensionless site density, $\rho_s$, only. The
degree of complexity increases drastically because steric effects,
which usually dominate the adsorption on continuous surfaces, are
modified by the local disordered structure of adsorption sites. This
model may be appropriate for the reversible adsorption of proteins on
disordered substrates\cite{OGMRW03,LJ05}.
Oleyar and Talbot \cite{OT07} examined the
reversible adsorption of hard spheres on the 2D RSM and proposed an
approximate theory for the adsorption isotherms based on a cluster
expansion of the grand canonical partition function. Although
successful at low site densities, the quality of the theory
deteriorates rapidly with increasing site density and fails completely
above a certain density.
In sections II and III we develop a general statistical mechanical
description of the RSM model and we confirm the intuitive result that
in the limit of large site density the system maps to hard spheres
adsorbing on a continuous surface. Then, for finite site density, we
prove that for the one-dimensional system in the limit of infinite
activity, there is a mapping to a hard rod system at a pressure $\beta
P=\rho_s$. We also present an argument supporting the validity of this
result in higher dimensions. Note that when the activity is
``strictly'' infinite, desorption is no longer possible and the
adsorption process is irreversible. In this case the model has an
exact mapping to the Random Sequential Adsorption (RSA) of hard
particles on a continuous surface
\cite{JWTT93}.
In sections IV and V we propose approximate theoretical schemes for
the adsorption isotherms in one and two dimensions that interpolate
between the limits of small and large activities for a given site
density $\rho_s$. Comparison with simulation results shows that these
approaches are a considerable improvement over the cluster expansion.
For completeness we note that in addition to its application to
adsorption, the model is also interesting because of its relationship
to the vertex cover problem\cite{WH00,WH01}. A vertex cover of an
undirected graph is a subset of the vertices of the graph which
contains at least one of the two endpoints of each edge. In the vertex
cover problem one seeks the {\it minimal vertex cover} or the vertex
cover of minimum size of the graph. This is an NP-complete problem
meaning that it is unlikely that there is an efficient algorithm to
solve it. The connection to the adsorption model is made by
associating a vertex with each adsorption site. An edge is present
between any two vertices (or sites) if they are closer than the
adsorbing particle diameter. The minimal vertex cover corresponds to
densest particle packings. Weight and Hartmann\cite{WH00,WH01}
obtained an analytical solution for the densest packing of hard
spheres on random graphs, but the existence of the geometry in
adsorption processes implies that the machinery developed to describe
adsorption on random graphs cannot be used in RSM models.
\section{Statistical Mechanics of the Random Site Model }
The adsorption surface is generated by placing $n_s$ points,
representing adsorption sites, randomly and uniformly on a substrate,
either a line in 1D or a plane in 2D (the boundary conditions are
irrelevant in the large $n_s$ limit). Spheres of diameter $\sigma$ may
bind, centered, on an available adsorption site. A site is available
if the nearest occupied site is at least a distance $\sigma$ away. Two
points are connected, and therefore cannot be simultaneously occupied,
if they are closer than $\sigma$.
The positions of the $n_s$ sites are denoted by ${\bf R}_i$ where $i$ is a
index running from $1$ to $n_s$. The sites are quenched during the
adsorption-desorption process. The adsorbed phase in equilibrium with
a bulk phase containing adsorbate at activity $\lambda$ can be
formally described with the grand canonical partition function:
\begin{equation}\label{eq:pf}
\Xi(\lambda,\{{\bf R}_i\})= 1+\sum_{n=1}^{\infty}\frac{\lambda^n}{n!}
\int\cdots\int d{\bf r}^n\prod_{i>j}(1+f_{ij})\prod_{i=1}^n\eta({\bf r}_i)
\end{equation}
where the microscopic density of sites $\eta({\bf r})$ is given by
\begin{equation}\label{eq:7}
\eta({\bf r})=\sum_{i=1}^{n_s}\delta({\bf r}-{\bf R}_i),
\end{equation}
$\lambda=\exp(\beta\mu)$ is the activity, ${\bf r}_i$ denotes the
position of sphere $i$ and $f_{ij}$ is the Mayer f-function which is
equal to $-1$ if spheres $i$ and $j$ are closer than $\sigma$ and $0$
otherwise. Eq. (\ref{eq:pf}) applies to a particular realization of
the quenched variables $\{R_i\}$. In order to average over disorder,
it is necessary to take the average not of the partition function, but
of the logarithm of the partition function \cite{RST94}. By using
standard rules of diagram theory\cite{HM76}, one obtains that
\begin{align}\label{eq:11}
\overline{\ln(\Xi)}=\sum_{n=1}^{\infty}\frac{\lambda^n}{n!}
\int d{\bf r}^n U_n({\bf r}_1,{\bf r}_2,...,{\bf r}_n)\prod_{i=1}^n\overline{\eta({\bf r}_i)}
\end{align}
where the bar means that the average is taken over
disorder, $U_n({\bf r}_1,{\bf r}_2,...,{\bf r}_n)$
denotes the Ursell function associated with the Mayer f-functions of hard
particles, $\prod_{i>j}(1+f_{ij})$\cite{G76}.
Let us denote the probability of finding adsorption
sites at positions ${\bf R_i},i=1,...,n_s$, as
$P({\bf R}_1,{\bf R}_2,...{\bf R}_s)$. We will assume that the positions are uncorrelated so that
\begin{equation}\label{eq:15}
P({\bf R}_1,{\bf R}_2,...{\bf R}_s)=\prod_{i=1}^{n_s}P({\bf R}_i)
\end{equation}
and will consider a Poissonian distribution of points,
$P({\bf R})=1/A$.
The average of the site density $\overline{\eta({\bf r})}$ is given by
\begin{align}\label{eq:13}
\overline{\eta({\bf r})}&=\int...\int\prod_{i=1}^{n_s}d{\bf R}_i P({\bf R}_1,{\bf R}_2,...{\bf R}_s)\eta({\bf r})\nonumber\\
&=\rho_s
\end{align}
where $\rho_s=n_s/A$ is the site density of the particles in a system
of area $A$ (length in one dimension).
We show in Appendix A that the average of the logarithm of the partition function over disorder can be written as
\begin{equation}
\overline{\ln(\Xi)}= \ln(\Xi^*(z=\lambda\rho_s)) +O(1/\rho_s)
\end{equation}
where
\begin{align}\label{eq:12}
\ln(\Xi^*(z))&=\sum_{n=1}^{\infty}\frac{z^n}{n!}
\int d{\bf r}^n U_n({\bf r}_1,{\bf r}_2,...,{\bf r}_n)
\end{align}
i.e., $\ln(\Xi^*(z))$, is the partition function of hard spheres on a
continuous surface at an activity $z=\lambda\rho_s$. This result
shows that when the site density $\rho_s$ goes to infinity and that
the activity $\lambda$ goes to $0$, with the constraint that the
product $\lambda\rho_s$ remains finite, the Random Site Model maps to
a system of hard particles in continuous space.
The number density of adsorbed molecules
can be computed directly from the partition function:
\begin{equation}
\rho(\lambda,\rho_s)=\frac{z}{A}\left(\frac{\partial\overline{\ln \Xi}}{\partial z}\right)_{n_s}
\end{equation}
with again $z=\lambda\rho_s$.
By using Eq.(\ref{eq:21}) given in the Appendix, one obtains to the second-order in $1/\rho_s$
\begin{equation}
\rho(\lambda,\rho_s)=\rho^*(z)-\frac{\rho^*(z)}{\rho_s+\rho^*(z)}\frac{d\rho^*(z)}{dz}+O(1/\rho_s^2)
\end{equation}
where $\rho^*(z)$ is the number density of the hard sphere model in continuous space.
Since $\rho^*(z)$ is an increasing function of the activity $z$, it is
an upper bound for $\rho(\lambda,\rho_s)$.
The expansion of $\overline{\ln \Xi}$ and $\rho(\lambda,\rho_s)$ in powers of the activity when $\lambda\rightarrow 0$ can also be generated from the above equations in straightforward way.
\section{The limit of large activity}
\subsection{A mapping with the (homogeneous) equilibrium hard sphere model}
The limit of large activity, $\lambda\rightarrow\infty$, is expected
to lead to the maximum density of adsorbed spheres for a given density
$\rho_s$ of adsorption sites. Indeed, this limit combines the presence
of a relaxation mechanism, which is induced by the infinitesimally small but
non-zero desorption process that allows a sampling of hard-sphere
configurations, with the guarantee that no sites left open for
adsorption will stay empty. The one-dimensional model is then amenable
to an exact solution in the limit
$\lambda\rightarrow\infty$. Interestingly, it maps onto an equilibrium
system of hard rods in continuum (1D) space at the same density (which
is of course a function of $\rho_s$)
The adsorbed density $\rho(\rho_s,\lambda\rightarrow\infty)=\rho_{\rm
max}(\rho_s)$ in the 1D case can be obtained exactly with the
following simple probabilistic argument. Assume that a given site is
occupied. Then all sites that lie within a distance $\sigma$ from
this site must be unoccupied. In the the optimally packed system the
next site beyond this must be occupied. The average distance from an
arbitrary point to the first site is $1/\rho_s$ giving the average
distance between two occupied sites as $\sigma+1/\rho_s$. The average
coverage in the maximally occupied system is thus
\begin{equation}\label{eq:rmax}
\rho_{\rm max}(\rho_s)=\frac{\rho_s}{1+\rho_s\sigma}.
\end{equation}
In the following we set $\sigma=1$. When the site density is low, $\rho_s<<1$, one recovers the
independent site approximation (Langmuir model), $\rho_{\rm
max}=\rho_s$. As the site density increases Eq. (\ref{eq:rmax}) shows
that there is a continuous increase of $\rho_{\rm max}$ and that a
closed packed configuration is obtained as $\rho_s$ approaches
infinity
The above argument can be generalized
to describe the correlation functions, or more conveniently in this 1D
system, the gap distribution functions. Specifically let
$F(x)$ denote the
probability density associated with finding a gap of size between $x$
and $x+dx$. In an optimally packed configuration, the probability to
find a gap of length $x$ is related to the probability to find the
first site at a given distance $x$ (say to the right) of an arbitrary
point. For a Poissonian distribution of sites this simply leads to
\begin{equation}\label{eq:fmax}
F_{\rm max}(x;\rho_s)=\rho_s e^{-\rho_s x}.
\end{equation}
The reasoning is easily extended to multi-gap distribution functions,
$F(x_1,x_2),F(x_1,x_2,x_3),...$, where two successive gaps are
neighbors in the sense that they are separated by a single
particle. One then shows that all these higher-order gap functions
factorize in products of one-gap distribution functions, e.g. $ F_{\rm
max}(x_1,x_2) = F_{\rm max}(x_1)F_{\rm max}(x_2)$. The outcome of this
probabilistic argument is that the 1D random site model is equivalent
to an equilibrium system of hard rods on a (continuous) line at the
pressure
\begin{equation}\label{eq:bp}
\beta P=\rho_s.
\end{equation}
Indeed, from the known equation of state of the hard rod fluid
\cite{T36} one has $\beta P=\rho/(1-\rho)$, i.e. $\rho=\beta
P/(1+\beta P)$, which corresponds to Eq. (\ref{eq:rmax}) after
insertion of Eq. (\ref{eq:bp}), and $F(x) = \beta Pe^{-\beta Px}$,
which reduces again to Eq. (\ref{eq:fmax}). The multi-gap distribution
functions are also given by products of 1-gap distribution functions,
which completes the proof of equivalence.
Extension of the above arguments to higher dimensions is far from
straightforward. First, in $d=2$ and higher, one may encounter at high
adsorbed density phase transitions to ordered, crystalline-like,
phases. Second, the simplicity of the reasoning in terms of gaps
characterized only by their length is lost when one leaves the
one-dimensional case. For these reasons, we have not been able to
develop a rigorous demonstration of a mapping between the 2D RSM and
the (homogeneous) hard-disk system at equilibrium at the same density
when the activity $\lambda$ goes to infinity.
To nonetheless make some progress, let us consider the nearest
neighbor radial distribution function $H(r)$ introduced by Torquato
\cite{T95}. $H(r)$ (and also used in the context of irreversible
adsorption models\cite{RTT96,VTT98} is the probability density
associated with finding a nearest neighbor particle center at some
radial distance $r$ from the reference particle center. This somewhat
generalizes the 1-gap distribution function to any spatial
dimension. For a Poissonian distribution of particle centers of
density $\rho_s$ in $d=2$, one finds that
\begin{equation}\label{eq:hpoisson}
H(r) = 2\pi r\rho_s e^{-\pi r^2\rho_s}
\end{equation}
No exact formula exists for $H(r)$ in a hard disk fluid. Using the results
of reference \cite{T95} one can,
however, demonstrate that in the large $r$ limit at an equilibrium
pressure $P$ it is given by
\begin{equation}\label{eq:hlim}
H(r)\sim 2\pi r\beta P e^{-\pi r^2\beta P}
\end{equation}
The reasoning now goes as follows. Consider a maximally occupied
configuration (associated with the limit $\lambda\rightarrow\infty$)
in the 2D RSM with an adsorption site density $\rho_s$ and consider a
given adsorbed particle. All sites within a radial distance $\sigma$
of the central occupied site must be unoccupied. $H(r)$ is associated
with situations such that the nearest adsorbed particle center is at a
radial distance $r$ of the reference one. Therefore, there should be
no adsorption sites in the region delimited by the two circles of
radius $\sigma$ (inside) and $r$ (outside). Actually, this statement
is not quite right: in 2 dimensions, there may be an exclusion effect
due to other particle centers at a distance $\gtrsim r$ of the central
site. To be more rigorous, the outside circle delimiting the region
with no adsorption sites should be (at least) of radius
$r-\sigma$. When $r\rightarrow\infty$, one can neglect the
contribution of width $\sigma$ due either to the disk of radius
$\sigma$ centered on the reference site or to the shell of width
$\sigma$ located between $r-\sigma$ and $r$. When
$r\rightarrow\infty$, the probability that, given that a reference
particle is centered at the origin, a spherical region of radius $r$
is empty of adsorption sites is given by $\exp(-\rho_s\pi r^2)$, so that
asymptotically $H_{\rm max}(r)$ goes as in Eq. (\ref{eq:hpoisson}),
i.e.
\begin{equation}
H_{\rm max}(r)\sim 2\pi r\rho_s e^{-\pi r^2\rho_s}
\end{equation}
Comparison with Eq. (\ref{eq:hlim}) tells us that it is the same
asymptotic behavior as that of an equilibrium hard disk system in the
plane with $\beta P=\rho_s$. This is the same result as found in $d=1$
(see above), except that the density $\rho$ and pressure $P$ are no
longer related by a simple analytical expression.
\begin{figure}[t]
\resizebox{8cm}{!}{\includegraphics{g12m.ps}}
\caption{Simulated adsorption Isotherms. $\rho_s=1,2,4$ top to bottom. The dashed lines
show the predictions of Eq. (\ref{eq:rmax}).}\label{fig:isotherms}
\end{figure}
\subsection{Numerical verification of the suggested mapping}
\begin{figure}[th]
\resizebox{8cm}{!}{\includegraphics{s3a.ps}}
\caption{Gap distribution function $F_{\rm max}(x)$ for configurations of hard rods at infinite activity
on the random site surface with $\rho_s = 0.7,1,2.0, 4.0$ from right to
left, bottom. The dashed lines are the predictions of Eq.(\ref{eq:fmax})
The solid lines correspond to the simulation results.}\label{fig:gapdist1}.
\end{figure}
We confirmed Eq. (\ref{eq:rmax}) in two ways. The first generates
configurations of maximum density directly. A number $n_s$ points are
distributed uniformly and randomly in the unit interval. The first
site is occupied by a rod of length $\sigma=1/ n_s$. Each site to the
right is checked in order until one is found that is at least a
distance $\sigma$ from the occupied one. This site is occupied and the
process iterated until all sites have been accounted for. A number of
averages over different configurations of sites is performed. We also
verified Eq. (\ref{eq:rmax}) by determining the adsorption isotherms for
different values of the site density $\rho_s$ and taking the limit
$\lambda\to\infty$: See Fig. \ref{fig:isotherms}.
For the hard rod fluid at equilibrium the gap distribution function is
given exactly by:
\begin{equation}\label{eq:gdist}
F(\rho,x)=\frac{\rho}{1-\rho}\exp(-\frac{\rho \,x}{1-\rho}).
\end{equation}
The gap distribution function calculated
for the densest configurations of the
random site model is in agreement with the predictions of Eq.
(\ref{eq:gdist}) using a density computed from Eq. (\ref{eq:rmax}),
confirming that the configuration of rods on the
random site surface has the same structure as the equilibrium hard rod
fluid at the same density (see Fig.\ref{fig:gapdist1}). This is consistent with the behavior of a
related lattice model for which we showed that there is an exact
mapping to the system without disorder at infinite
activity\cite{TTV07}.
In order to use Eq.(\ref{eq:bp}) to estimate the maximum coverage of
the RSM in two-dimensions, we combine it with the approximate equation
of state for hard disks on a continuous surface, which was proposed by
Wang\cite{W02}:
\begin{equation}\label{eq:wang}
\frac{\beta P}{\rho}=1+\frac{D}{1-\theta / \theta_0}-(D+2a\theta+4b\theta^2+8c\theta^3+16d\theta^4+32e\theta^5)
\end{equation}
where $\theta=\pi\sigma^2\rho/4$ is the coverage, $\theta_0=0.907..$
is the coverage of a hexagonal close-packed configuration,
$D=4.08768, a=1.25366, b=0.46051, c=0.152797,d=0.04412, e=0.00929.$.
For a given value of the site density, $\rho_s$,
Eq. (\ref{eq:wang}) is solved numerically for $\theta$. It is
convenient to introduce the dimensionless site density
\begin{equation}\label{eq:14}
\alpha = \frac{\pi}{4}\sigma^2\rho_s
\end{equation}
The results, shown in Fig.~\ref{fig:maxcov2d} are in excellent
agreement with the simulation results for the entire range of
$\alpha$. This should be compared with cluster expansion to second
order that gives good predictions only for
$\alpha\lesssim0.3$\cite{OT07}. Note that the approximate equation of
state, Eq.(\ref{eq:wang}), does not include the possible presence of a
phase transition.
\begin{figure}[t]
\resizebox{8cm}{!}{\includegraphics{g17new.ps}}
\caption{Maximum coverage of the RSM as a function
of the dimensionless site density. The symbols show the simulation
results, the dashed line is the cluster approximation to second order
\cite{OT07}, and the solid line shows the predictions of
Eqs. (\ref{eq:bp}) and (\ref{eq:wang}). }\label{fig:maxcov2d}.
\end{figure}
\begin{figure}[t]
\begin{center}
\resizebox{8cm}{!}{\includegraphics{clusters.eps}}
\caption{Lowest order clusters of adsorption sites.
A solid line connects two sites that are closer than $\sigma$ and a
dashed line indicates that two sites are further apart than
$\sigma$. Triplets of type ``3a'' (left) and ``3b'' (right) are shown
left to right in the second row. The expression to the right of the
clusters is the corresponding grand canonical partition function for
the adsorbed particles.}\label{fig:clusters}
\end{center}
\end{figure}
\section{RSM in one dimension}
\subsection{Low site density expansion}
To provide a description of the adsorption isotherms at finite
activity, it is useful to consider an expansion in the density of
adsorption sites. Assuming that the adsorption surface consists of
isolated clusters of sites, the (averaged) logarithm of the partition
function may be expressed as
\begin{equation}
\overline{\ln \Xi }=N_1\ln \Xi_1 +N_2\ln \Xi_2 +N_{3a}\ln \Xi_{a} +N_{3b}\ln \Xi_{3b}+
...,
\end{equation}
where $N_i=x_iN_s$ is the number of clusters of type $i$ and $i\equiv
(n,a)$ with $n$ being the number of sites in the cluster and $a$
characterizing when necessary the subclass of clusters (see
Fig.\ref{fig:clusters}). This simple expansion is possible because
adsorption on a given cluster does not affect any of the others. The
adsorption isotherm is then
\begin{equation}\label{eq:2}
\rho A = \lambda N_1\left(\frac{\partial\ln \Xi_1}{\partial\lambda}\right)+\lambda N_2\left(\frac{\partial\ln \Xi_2}{\partial\lambda}\right)+...,
\end{equation}
which gives
\begin{align}\label{eq:ce}
\rho &= \lambda \rho_s\bigl(x_1\frac{1}{1+\lambda}+\frac{2}{1+2\lambda}x_2\nonumber\\
+&\frac{3+2\lambda}{1+3\lambda+\lambda^2}x_{3a}+\frac{3}{1+3\lambda}x_{3b}+...\bigr).
\end{align}
In one-dimension, exact expressions can be obtained for the
first few clusters following the approach of Quintanilla and Torquato \cite{QT96}:
\begin{align}\label{eq:cex1}
x_1&=\exp(-2\rho_s)\\
\label{eq:cex2}
x_2&=\exp(-2\rho_s)(1-\exp(-\rho_s))\\
\label{eq:cex3a}
x_{3a}&=\exp(-3\rho_s)(\exp(-\rho_s)+\rho_s-1)\\
\label{eq:cex3b}
x_{3b}&=\exp(-2\rho_s)(1-(1+\rho_s)\exp(-\rho_s))
\end{align}
where $x_2$, for example, is the number of clusters with two linked
sites (per adsorption site). At the triplet level we need to
distinguish between two subclasses of clusters shown in Fig. \ref{fig:clusters}.
In type ``3a'' two of the sites can be simultaneously occupied, while in type
``3b'' only one of the sites can be occupied since all are mutually closer
than the particle diameter.
The predictions of Eq.~(\ref{eq:ce}) are compared with the simulation
results in Fig. \ref{fig:expan}. The expansion to third order
provides a good description of the isotherm for $\rho_s=0.2$, but the
quality deteriorates rapidly for larger site density. Eq.~(\ref{eq:ce})
is unable to predict the correct limit when $\lambda \rightarrow
\infty$. This disagreement is more pronounced when $\rho_s$ increases:
for instance, when $\rho_s=1$, the expansion fails completely, because the
predicted density is lower than for $\rho_s=0.5$.
Since the above expansion is limited to small values of $\rho_s$, we
simplify Eq.~(\ref{eq:ce}) by performing a series expansion in
$\rho_s$, giving
\begin{align}\label{eq:ce1}
\rho &= \lambda \rho_s\bigl(\frac{1}{1+\lambda}-\frac{2\lambda}{(1+\lambda)(1+2\lambda)}\rho_s \nonumber\\
+&
\frac{\lambda^2(12\lambda^2+29\lambda+9)}{2(1+\lambda)(1+2\lambda)(1+3\lambda)(1+3\lambda+\lambda^2)}\rho_s^2 +O(\rho_s^3)\bigr)
\end{align}
We note that Eq. (\ref{eq:ce1}) can be expressed as
\begin{align}
\rho(\rho_s,\lambda)&=\sum_{l=1}^3\left(-
\left(\frac{-\lambda}{1+\lambda}\right)^l +F_l(\rho_s,\lambda)\right)\rho_s^l
\end{align}
where $ F_l(\rho_s,\lambda)$ represents the remaining terms in the exact
expansion and, consistent with these terms, has the property that
$F_l(\rho_s,\lambda)\to 0$ when $\lambda\to 0 $ and
$\lambda\to\infty$. This property has been verified order by order for
a random lattice model\cite{TTV07}, and only for the three first orders
for this model. If we partially resum the
series, neglecting the remaining terms, we obtain that
\begin{align}\label{eq:1}
\rho(\rho_s,\lambda)&=\frac{\lambda\rho_s}{1+\lambda(1+\rho_s)}
\end{align}
Although the site density expansion, Eq. (\ref{eq:ce1}), is valid only
for small $\rho_s$, the partial resummation leads to an expression for
$\rho$ that is exact in the limit of very large activities $\lambda$,
whatever the value of $\rho_s$. It thus provides a sensible
approximation based on the site density expansion.
Isotherms given by Eq. (\ref{eq:1}) are plotted in Fig.
\ref{fig:expan} (dotted curves). Although for very low site density
the third order expansion gives a better estimation, the quality of
Eq. (\ref{eq:1}) is significantly better for $\rho_s=0.5$, and it
always gives the saturation density exactly. For larger values of the
site density, $\rho_s>1$, the approximation overestimates the adsorbed
amount for intermediate activities. This is expected since the
neglected terms are non-zero for intermediate bulk activity. Attempts
to obtain a better resummation of the series expansion,
Eq. (\ref{eq:ce1}), were fruitless. Therefore, we follow the approach
that we used in our study of the model of dimer
adsorption\cite{TTV07}, by introducing an effective activity.
\begin{figure}
\resizebox{8cm}{!}{\includegraphics{j4log.ps}}
\caption{Adsorption isotherms versus activity $\lambda$ for several site densities:
$\rho_s=0.5,0.3,0.2$ from top to bottom. The dash-dotted and dotted
lines show the predictions of Eq.~\ref{eq:ce} (exact expansion to
third order in $\rho_s$) and Eq.~\ref{eq:1}, respectively. The solid lines
show the simulation results. }\label{fig:expan}.
\end{figure}
\subsection{Effective activity approach}
Despite the exact mapping found in the infinite activity limit, no simple
reasoning can be used to obtain exact results at finite bulk
activity. We therefore investigated several approximate methods, the
most successful of which is based on an effective activity.
The equation of state of the hard rod fluid on a continuous line is \cite{T36}
\begin{equation}\label{eq:eos1d}
\beta P=\frac{\rho}{1-\rho\sigma}
\end{equation}
According to the results in Section III, the hard rod fluid at a
pressure $\beta P=\rho_s$ has the same density (obtained by inverting
Eq. (\ref{eq:eos1d})) as the densest configuration of hard spheres on
the random site surface, Eq. (\ref{eq:rmax}). We seek a generalization
of this mapping for an arbitrary value of the bulk phase activity,
i.e. an effective activity $\lambda_{\rm eff}$ such that the density of
adsorbed rods in the RSM, $\rho(\rho_s,\lambda)$ is given by
\begin{equation}\label{eq:leff}
\rho(\rho_s,\lambda)=\rho^*(\lambda_{\rm eff}(\lambda,\rho_s)),
\end{equation}
where $\rho^*(\lambda)$ is the density of rods on a continuous substrate at an
activity $\lambda$. This is given exactly by
\begin{equation}\label{eq:rholam}
\rho^*(\lambda)=\frac{L_w(\lambda)}{1+L_w(\lambda)},
\end{equation}
where $L_w(x)$, the Lambert-W function, is the solution of $x=L_w(x)\exp(L_w(x))$.
The inverse relation is
\begin{equation}\label{eq:lam}
\lambda = \frac{\rho}{1-\rho}\exp\bigl(\frac{\rho}{1-\rho}\bigr).
\end{equation}
From the exact result, Eq. (\ref{eq:rmax}), we have that $\lambda_{\rm
eff}(\lambda=\infty)=\rho_s e^{\rho_s}$. Taking Eqs. (\ref{eq:rholam}) as a
mere definition of $\lambda_{\rm eff}(\lambda,\rho_s)$, we have computed
it from the simulated adsorption isotherms. Results are shown in
Figs.~\ref{fig:effa1}-\ref{fig:effa4} and will be used as a reference
for testing the validity of various approximations.
\begin{figure}
\resizebox{7cm}{!}{\includegraphics{lambdaeffrho1.eps}}
\caption{Effective activity $\lambda_{\rm eff}$ as a function of the logarithm of the activity $\ln \lambda$ with $\rho_s=1$.
The solid line shows the simulation result, calculated from the
density by using Eq. (\ref{eq:lam}). The predictions of
Eq.~(\ref{eq:leffa}) combined with Eq.~(\ref{eq:3}) and
Eq.(\ref{eq:10}) are shown as dotted and dashed lines, respectively.
They are almost indistinguishable. The horizontal dashed line shows
the exact limiting value of $\rho_s e^{\rho_s}$.}\label{fig:effa1}.
\end{figure}
\begin{figure}
\resizebox{7cm}{!}{\includegraphics{lambdaeffrho2.eps}}
\caption{Same as Fig. \ref{fig:effa1}, except that $\rho_s=2$. The prediction of
Eq.~(\ref{eq:10}) is now more accurate than that of Eq.~(\ref{eq:3}).}\label{fig:effa2}.
\end{figure}
\begin{figure}
\resizebox{7cm}{!}{\includegraphics{lambdaeffrho4.eps}}
\caption{Same as Fig. \ref{fig:effa2}, except that $\rho_s=4$.
Eq.~(\ref{eq:3}) fails to reproduce the simulation data for intermediate
value of $\ln(\lambda)$ whereas the prediction
of Eq.~(\ref{eq:10}) remains accurate. }\label{fig:effa4}.
\end{figure}
Following the method introduced in the study of the lattice
model\cite{TTV07}, we attempt to construct simple expressions that can
reproduce the numerical results, when available and describe the
isotherms for a wide range of parameters $\rho_s$ and $\lambda$. The
effective activity as a function of the activity $\lambda$ and of the
density site $\rho_s$ can be written as
\begin{equation}\label{eq:leffa}
\frac{1}{\lambda_{\rm eff}}=\frac{f(\rho_s,\lambda)}{\rho_s\lambda}+\frac{1}{\rho_s e^{\rho_s}}
\end{equation}
where $f(\rho_s,\lambda)$ is a function to be determined. We know that at
small $\lambda$, the exact behavior is given by $\lambda_{\rm eff}=\rho_s\lambda$,
which imposes that $f(\rho_s,\lambda)$ goes to $1$. Conversely, when
$\lambda$ is large, the effective activity $\lambda_{\rm eff}$ behaves as
$\rho_s e^{\rho_s}+O(\lambda)$, which leads to the constraint that
$\lim_{\lambda\rightarrow
\infty}f(\rho_s,\lambda)=A(\rho_s)$ where $A(\rho_s)$ is an unknown function of $\rho_s$.
If $A(\rho_s)$ remains close to one, the simplest choice consists of
choosing
\begin{equation}\label{eq:3}
f(\rho_s,\lambda)=1
\end{equation}
for all values of $\lambda$. While this simple choice gives a fair
agreement with the simulated values when $\rho_s=1$ (see Fig.
\ref{fig:effa1}), the situation deteriorates for larger values of
$\rho_s$ (see Figs.~\ref{fig:effa2}-~\ref{fig:effa4}). However,
contrary to the one-dimensional lattice model, we do not know the
exact asymptotic behavior of $f(\rho_s,\lambda)$ for large $\lambda$. In
order to improve the description, we consider a homographic function
of $\lambda$ verifying the exact behavior, when $\lambda\rightarrow 0$ and
$\lambda\rightarrow\infty$,
\begin{equation}\label{eq:4}
f(\rho_s,\lambda)= \frac {A(\rho_s) \lambda+B(\rho_s)}{ \lambda+B(\rho_s),
}
\end{equation}
where $A(\rho_s)$ and $B(\rho_s)$ are unknown functions. Examination of the
series expansion in the limit of infinite bulk activity suggests that
$B(\rho_s)$ should grow as $e^{\rho_s}$ in order to give the right
behavior. In the absence of additional criteria, we have chosen
$A(\rho_s)=\rho_s$ and $B(\rho_s)=\rho_se^{\rho_s}$, which gives
\begin{equation}\label{eq:10}
f(\rho_s,\lambda)= \frac {\rho_s(e^{\rho_s}+\lambda)}{ \lambda+\rho_s e^{\rho_s} }.
\end{equation}
Note that when $\rho_s=1$, this function gives $f(\rho_s,\lambda)=1$. Figs.
\ref{fig:effa1}-~\ref{fig:effa4} show that the quality of the
approximation given by Eq. (\ref{eq:leffa}) and Eq. (\ref{eq:3})
deteriorates with increasing $\rho_s$. This is a consequence of the
fact that the asymptotic approach to the saturated state is not
correctly captured with this simple approximation. A better agreement
with simulations is obtained by using Eq.(\ref{eq:leffa}) and
Eq.(\ref{eq:10}) for $\rho_s=2$ and $\rho_s=4$ (see Figs. \ref{fig:effa2}
and \ref{fig:effa4}).
The purpose of this exercise is ultimately to obtain an approximate,
but accurate, description of the adsorption isotherms. This is done
by substituting Eq. (\ref{eq:leffa}) in Eq. (\ref{eq:rholam}).
Fig.~\ref{fig:leff} shows a comparison of these estimates with the
simulation results. The effective activity approach is a considerable
improvement over the cluster expansion, whose accuracy is restricted
to small density site $\rho_s<0.5$ (c.f. Fig. \ref{fig:expan}) and
allows one to describe adsorption even when $\rho_s>1$.
\begin{figure}[t]
\resizebox{7cm}{!}{\includegraphics{rholambda.eps}}
\caption{Adsorption isotherms versus activity $\lambda$ for several site densities: $\rho_s=1,2,4$,
predicted by the effective activity approach, Eqs. (\ref{eq:lam}) and
(\ref{eq:rholam}) by using Eq. (\ref{eq:3}) (dotted lines) and
combined with Eq. (\ref{eq:10}) (dashed lines). The solid lines show
the simulation results. }\label{fig:leff}.
\end{figure}
\subsection{Structure}
\begin{figure}[th]
\resizebox{8cm}{!}{\includegraphics{s1a.ps}}
\caption{Gap distribution function for configurations of hard rods at finite activity
on the random site surface with $\rho_s = 10$. The dashed lines are the
predictions of Eq. \ref{eq:gdist} with a density given by the
equilibrium isotherm value and the solid lines show the simulation
results. $\lambda=1, 5, 50, 1000$ from left to right in the
bottom part.}\label{fig:gapdist2}.
\end{figure}
We have also examined the structure of the hard-rod configurations at
finite activity. As in the case of the lattice model\cite{TTV07}, we
do not expect an exact mapping to the homogeneous system at the same
density. Nevertheless, as shown in Fig. \ref{fig:gapdist2}, the
distributions computed from the simulation are very well described by
Eq. (\ref{eq:gdist}) with a density equal to the equilibrium isotherm
value.
\section{Two-dimensional model}
We construct approximate isotherms in the same way as
for one-dimensional model, i.e. by introducing an effective
activity as a function of bulk activity.
\begin{figure}[t]
\resizebox{8cm}{!}{\includegraphics{rss2d.ps}}
\caption{Adsorption Isotherms for dimensionless site densities $\alpha=1,2,4$ from bottom to top. The dashed lines
show the predictions of Eq.(\ref{eq:8}) combined with Eq.(\ref{eq:9}).
The solid lines show the simulation results obtained with 20 realizations of $n_s=2000$ sites. }\label{fig:rss2d}.
\end{figure}
We first consider the homogeneous hard-disk system. We determine the
Helmholtz free energy per particle by integration along the isotherms:
\begin{equation}\label{eq:5}
\beta f^{ex}=\int_0^\theta \left(\frac{\beta P}{\rho}-1\right)\frac{d\theta'}{\theta'}.
\end{equation}
The excess chemical potential can then be obtained from
\begin{equation}\label{eq:6}
\beta \mu^{ex}=\beta f^{ex}+\theta\frac{\partial\beta f^{ex}}{\partial \theta}.
\end{equation}
By inserting the approximate equation of state, Eq. (\ref{eq:wang}) in Eq.(\ref{eq:5}), and substituting the result in Eq. (\ref{eq:6}), one obtains for the activity, $\lambda=\exp(\beta \mu)$ the relation
\begin{align}\label{eq:8}
\ln(\lambda)=\ln(4\theta/\pi) -{\frac {96}{5}}\,e{\theta}^{5}-
20\,d{\theta}^{4}-{\frac {32}{3}}\,c{\theta}^{3}-6\,b{\theta}^{2}-4\,a\theta\nonumber\\- D
\ln \left( 1 -\frac{2\,\sqrt {3}}{\pi}\theta \right) +2\,{\frac { D
\sqrt {3}\theta}{\pi -2\,\sqrt {3}\theta}},
\end{align}
where, we recall, $\theta=\pi\sigma^2\rho/4$.
The isotherm for the $2D$ RSM can be calculated by following a
procedure similar to that used in the one-dimensional case. By means
of Eq. (\ref{eq:bp}), one obtains the saturation
coverage when the bulk activity is infinite, $\theta_\infty$. Then, by
inserting this quantity in Eq.~(\ref{eq:8}), the effective activity
$\lambda_{\rm eff}^\infty\equiv \lambda_{\rm eff}(\theta_\infty)$ is obtained.
The counterpart of
Eq. (\ref{eq:leffa}) for the two-dimensional system is now
\begin{equation}\label{eq:9}
\frac{1}{\lambda_{\rm eff}}=\frac{1}{\rho_s\lambda}+\frac{1}{\lambda_{\rm eff}^\infty}
\end{equation}
where, in the absence of more information, we have set $f(\rho_s,\lambda)=1$.
Therefore, at a given bulk activity $\lambda$ and site density $\rho_s$, one obtains an
effective activity by using Eq. (\ref{eq:9}), and by inserting the
effective activity in Eq. (\ref{eq:8}), one deduces the corresponding
coverage in the 2D RSM.
Fig.~\ref{fig:rss2d} compares the simulation results and the
approximate isotherms for different values of the site density. While
the agreement is not perfect, the scheme represents a major
improvement compared to the pertubative approach (third-order density
expansion) for describing situations where the density site is
moderate to high and allows one to develop new approximate isotherm
equations.
\section{Conclusion}
We have presented a theoretical and numerical study of the reversible
adsorption of hard spheres on randomly distributed adsorption
sites. Unlike the case of irreversible adsorption, there is no simple
mapping between the RSM and a system of hard spheres on a homogeneous
surface. We have, nonetheless, been able to obtain some exact results
in various limits: small and large bulk activity and small and large
site density. We have proposed an effective activity approach, which
interpolates between the known behavior, to obtain an approximate
description for intermediate situations. The adsorption isotherms
predicted by this approach are in excellent agreement with simulation
results. In the two-dimensional case, we have not investigated phase
transitions in detail, although there does appear to be a solid-like
phase for sufficiently high site density and activity. It will also
be interesting to investigate reversible adsorption on a substrate
when a second source of disorder is present: distribution of
adsorption site energies. The present approach should apply to this
case as well.
|
0805.2582
|
\section{INTRODUCTION}
\label{chap:intro}
The measurements of Lilly et al. \cite{Lilly} and Du et
al.\cite{Du} reveal strong anisotropic transport properties
of the two-dimensional electron gas (2DEG) for the half-filled
Landau-level system for high Landau-levels (LL) and at very low temperature.
The anisotropic behavior in the transport
properties is consistent with stripe and bubble charge-density-wave
phases which were predicted early on in
Refs.~\onlinecite{Koulakov,Folger,Moessner} by means of Hartree-Fock
calculations of the 2DEG and were confirmed more recently by
numerical studies of systems with up to 12 electrons\cite{Haldane,Yang}.
However, Fradkin et al.\cite{Kivelson} have
challenged this interpretation and suggested that the anisotropic
transport might be due to a possible nematic
phase of the 2DEG in a magnetic field. This idea finds support in
the good comparison between the
results of the temperature dependence of the anisotropy of the
resistivity obtained by means of a Monte Carlo simulation
of the nematic phase\cite{Fradkin} with that which has been
experimentally observed.
In addition, the idea is supported by the experiments of Cooper et
al.\cite{Cooper} where an in-plane magnetic field was applied
in the 2DEG and the results of the experiment were interpreted
on the basis of the presence of a nematic state; further support of
the idea is provided by the fact that
the theoretically estimated transition temperature from an isotropic
to nematic phase\cite{Wexler_Dorsey} is of similar magnitude as the
experimentally determined temperature at which the on-set of the
anisotropic transport occurs.
Rather recently we have presented\cite{FHNC_nematic}
a variational calculation of the nematic state as ground state of the
half-filled Landau-level system in a magnetic field based on an ansatz
ground state wavefunction proposed by Oganesyan et al. \cite{Oganesyan}
which is of the Jastrow-Slater form and is given by the following expression:
\begin{eqnarray}
\Psi \left( {\vec {r}_1 ,\vec {r}_2 ,...,\vec {r}_N } \right)&=&\hat
{P}_0 \prod\limits_{j<k}^N {(z_j -} z_k )^2
e^{-{\sum_{k=1}^N |{z_k } |^2}/4 } \nonumber \\
&\times & \det \left| {\varphi _{\vec {k}} (\vec {r}_i )}
\right|\label{WF}
\end{eqnarray}
where $\hat P_0$ is the projection operator onto the lowest LL,
$\phi_{\vec k}(\vec r_i)$ are two-dimensional (2D) plane-wave states.
Here, $z_j=x_j+iy_j$ is the complex 2D coordinate of the $j$ electron.
This wavefunction is a Jastrow correlated Slater determinant with Jastrow
part similar to the Laughlin state \cite{Laughlin}.
This ground-state wavefunction has the same form as
the form proposed by Rezayi and Read\cite{RR}, however, the
single-particle momenta form an elliptical Fermi sea as opposed to the
circular Fermi sea. There is a broken-symmetry parameter which is
the ratio $\alpha =k_1 /k_2$ of the semi-major $k_1$ and semi-minor
$k_2$ axes of the elliptic Fermi sea. Using this wave function to describe
the nematic state we had carried out a variational study of the half-filled
system using the so-called Fermi-hypernetted-chain (FHNC)
approximation\cite{FHNC_nematic}.
The results of the above mentioned variational calculation indicate
that there is a certain value of the parameter $\lambda$ ($\lambda$ is
proportional to the 2DEG layer thickness\cite{ZDS}) below which the nematic
state is energetically favorable as compared to the isotropic and the
stripe-ordered ground states for the second excited LL. It is interesting
to note that this critical value of $\lambda$ is very close to the value
of $\lambda$ which can be estimated based on the actual
experimental conditions which are applicable for the case of the data by Lilly
et al.\cite{Lilly} and by Du et al.\cite{Du}. However, one of the weak points
of the above described variational study is the fact that the FHNC
approximation is plagued by an unknown-size error and the results
cannot be improved in a controlled manner\cite{Manousakis_Pandharipande}.
Therefore, there is a need to check
the validity of these results and conclusions using the variational
Monte Carlo method and this task is undertaken in the present work.
There is a different variational approach to the problem of a broken
rotational state of the half-filled LL introduced by
Ciftja and Wexler\cite{Ciftja}. They have used the Fermi-hypernetted-chain
(FHNC) approximation to study a broken rotational state of the
half-filled LL where the symmetry-breaking parameter was introduced in
the correlation part of the wavefunction as $(z_i-z_j)^2\rightarrow
(z_i-z_j-\alpha)(z_i-z_j+\alpha)$, and they used the standard
single-particle determinant with a circular Fermi sea.
In this and in the work of Ref.~\onlinecite{FHNC_nematic}, we
considered the unprojected wavefunction of the
nematic state. The advantage of this simplified version is that it
has a Jastrow form with a Slater determinant so it can be applied
directly with FHNC and it allows us to study large-size
systems using the variational Monte Carlo method.
The paper is organized as follows: In the following Section
we discuss the formulation and the procedure; in Sec.~\ref{MC_Results}
we present the results and we compare them with those obtained
for the case of a stripe-ordered state and the isotropic state.
In Sec.~\ref{Conclusions} we summarize the conclusions of the
present calculation.
\section{Method}
\label{Procedure}
We have adopted the toroidal geometry of a square with periodic boundary
conditions. This geometry has the advantage of naturally adapting to
the nematic and isotropic state wavefunction.
There are several steps in applying
the MC approach for this problem. First, as part of the
wavefunction of nematic state we
construct a Slater determinant of plane waves characterized by
momentum vectors which lie inside
an elliptical Fermi sea. Second, since the pseudo-potential
is $ln(r)$, which is a long-range interaction, we need to
to take into
account all periodic image charge interactions. One of the methods to
do this is the Ewald summation technique.
In subsection \ref{Ewald} of the appendix we describe the
Ewald summation technique for the case of toroidal boundary
conditions and the $ln(r)$ interaction.
In the present section we will discuss our implementation of
the MC to study the nematic state.
Given a value of $\alpha$ there are definite values of the number of
particles $N$ which correspond to a closed shell. These definite values
of $N$ are calculated as follows. The occupied states characterized
by $k_x,k_y$ must satisfy the following condition:
\begin{eqnarray}
\bigg(\frac{k_x}{k_1}\bigg)^2+\bigg(\frac{k_y}{k_2}\bigg)^2 \leq 1,
\end{eqnarray}
where $k_1$ and $k_2$ are the major and minor axis of the Fermi sea
and given by:
\begin{eqnarray}
k_1 = \sqrt{\frac{4\pi \rho}{\alpha}},\\ k_2 = \sqrt{4\pi \rho
\alpha},
\end{eqnarray}
where $\rho$ is the uniform particle density of the system. For a
finite system of size $L\times L$, $k_x = n_x\Delta k$ and $k_y =
n_y\Delta k$ where $\Delta k=2\pi/L$ and $n_x,n_y \in Z$. So one can
deduce the conditions for $n_x,n_y$ such that:
\begin{eqnarray}
\frac{\pi}{N}\bigg(\alpha n_x^2+\frac{n_y^2}{\alpha}\bigg) \leq 1.
\label{n_x_n_y}
\end{eqnarray}
For a value of $N$ to be acceptable the number of states, i.e., the
number of pairs $(n_x,n_y)$ satisfying the above inequality should
be equal to $N$. For example,
for $\alpha=1$, $N$ can be $1,5,9,13,21,25,29,37,45,\ldots$; for
$\alpha =2$, they can be $1,3,7,11,15,17,21,\ldots$
\begin{figure}[htp]
\vskip 0.3 in
\begin{tabular}{cc}
\epsfig{file=Figure1a.eps,width=0.8\linewidth,clip=}\\
\epsfig{file=Figure1b.eps,width=0.8\linewidth,clip=}
\end{tabular}
\caption{Occupied states for the nematic state with
$\alpha=2$ (top) and $\alpha=4$ (bottom) for the case of 89 particles.}
\label{K89}
\vskip 0.3 in
\end{figure}
In Figs. \ref{K89}, we present two
examples of closed shell which correspond to $\alpha =2$ and
$4$ for $N=89$. Notice that with anisotropy parameter $\alpha=k_x/k_y > 1$,
the occupied states (i.e those satisfying equation \ref{n_x_n_y}) are
anisotropically distributed along the preferred $k_x$ axis.
In our MC calculation we will use these cases as well
as larger size systems up to 145 particles.
We follow the Metropolis MC scheme for
sampling the wavefunction where the ratio needed between the new and the
old wavefunction is:
\begin{eqnarray}
\bigg \vert \frac{\psi(\vec r_{new})}{\psi_(\vec r_{old})}\bigg \vert
^2= exp(u(\vec r_{new})-u(\vec r_{old})) \bigg\vert\frac{Det(e^{i\vec
k\cdot \vec r_{new}})}{Det(e^{i\vec k\cdot \vec r_{old}})}\bigg
\vert.
\end{eqnarray}
where $u(\vec r)$ is the periodic pseudo-potential which is derived in
the subsection (\ref{Ewald}) of the appendix.
To carry out the calculation of the ratio between the Slater
determinant of the new configuration and the old configuration
efficiently, we use the inverse updating technique developed by
Ceperley et al.\cite{Ceperley}. We found that the number
of MC steps needed for ``thermalization'' is of the order of $10^5$ and we
use of the order of $2\times 10^6$ MC steps to calculate averages of
the distribution function.
The potential energy of the high LL can be expressed \cite{Ciftja} via
the pair distribution function of the LLL using the single mode
approximation discussed in Ref.~\onlinecite{MacDonald}, namely,
\begin{eqnarray}
V_L = \frac{\rho}{2}\int\bigg[g(\vec r)-1\bigg]V_{eff}^L(r)d^2
r\label{V_L}
\end{eqnarray}
where the effective potential $V_{eff}^L(r)$ for Landau level L is the
convolution of the effective interaction\cite{ZDS}
\begin{eqnarray}
V(r) = e^2
/\epsilon\sqrt{r^2+\lambda^2}
\label{lambda}
\end{eqnarray}
with the $L$-order Laguerre
polynomial; namely, it is the Fourier transform of:
\begin{eqnarray}
\tilde{V}^L_{eff}(q) = \frac{2\pi e^2}{\epsilon q} e^{-\lambda q}
\bigg[L_L(q^2/2)\bigg]^2\label{V_eff}
\end{eqnarray}
In the above formula, $\lambda$ is a length scale which characterizes
the confinement of the electron wave function in the direction
perpendicular to the heterojunction \cite{ZDS}.
We use the single mode approximation to calculate the interaction
energy at high LL (equation \ref{V_L}) and we are only interested in obtaining
the pair distribution function $g(\vec r)$. The kinetic energy
advantage of the isotropic phase over the nematic phase is
calculated in the subsection (\ref{Kinetic}) pf the appendix.
The approach can be divided into the following steps:
\begin{itemize}
\item The pair distribution function for the LLL for
different anisotropic parameters $\alpha$ is calculated.
\item The single mode approximation\cite{MacDonald} is used to calculate
the interaction energies at a high LL.
\item The kinetic energy for different anisotropic parameters
is evaluated (see appendix \ref{Kinetic}).
\item We compare total energies of the isotropic and nematic state to
determine at what LL the nematic becomes energetically favorable.
\item The optimum value of $\alpha$ is determined by minimizing the
total energy.
\item The HF results which have been reported so
far\cite{Stripe1,Stripe2,Stripe3} correspond to the case of
$\lambda=0$.
Therefore, we needed to carry out Hartree-Fock calculations following
Refs~\onlinecite{Stripe1,Stripe2,Stripe3}
for the case of the interaction given by Eq.~\ref{lambda}
for $\lambda\ne 0$.
The optimum total energies of the nematic states will be
compared with those of the stripe states at different values of
$\lambda$ for the 2$^{nd}$ excited LL to determine a critical value of
$\lambda$ below which the nematic state maybe energetically favorable.
\item The above mentioned critical value of $\lambda$ is compared
with the value which corresponds to those samples used in the
experiment\cite{Lilly}.
\item A comparison of the MC results to the ones
obtained by FHNC will also be presented.
\end{itemize}
\section{Results}
\label{MC_Results}
The pair distribution function $g(r)$ obtained
using MC integration has important differences when compared to
$g(r)$ obtained by
FHNC\cite{FHNC_nematic} as illustrated in Fig. \ref{g_FHNC_MC}.
Thus, it is important to
obtain the energies of the nematic state at high LL by MC and to
compare them with those obtained by FHNC.
We first compare the interaction energies obtained for
different values of $\alpha > 1$ with the potential energy
of the isotropic state
($\alpha=1$) (Figs. \ref{V_0_MC} and \ref{V_1_MC}).
Notice from Figs. \ref{V_0_MC} and \ref{V_1_MC} that the
potential energy of the isotropic state is lower than the potential
energy of the nematic state for the 1$^{st}$ excited LL and LLL for
all values of the parameter $\lambda$. The
potential energy is calculated with the pseudo-potential obtained
using the Ewald sum as discussed in subsection (\ref{Ewald}) of the
appendix. Essentially the same result is also found with a
pseudo-potential obtained using the Lekner summation
technique\cite{Jensen}. Furthermore, as shown in
appendix \ref{Kinetic}, the kinetic energy of the isotropic state is
below that of the nematic state, and, thus, the total energy of the
isotropic state is always lower than that of the nematic state. Hence,
our MC calculation shows that the isotropic state is
energetically favorable as compared to the nematic state for the LLL and
the 1$^{st}$ excited LL for all values of the parameter $\lambda$.
Also note that the same conclusion was reached
using the FHNC technique\cite{FHNC_nematic} with the same wavefunction.
These findings solidify the
conclusion that for the LLL and the 1$^{st}$ excited LL, the isotropic
state is more stable than the nematic state, which is also in
agreement with the experimental findings of Refs. \onlinecite{Lilly} and
\onlinecite{Du}.
\begin{figure}[htp]
\vskip 0.3 in
\includegraphics[width=3.275 in]{Figure2.eps}
\caption{Comparison of the pair distribution function obtained by FHNC
and MC.}\label{g_FHNC_MC}
\end{figure}
\vskip 0.3 in
\begin{figure}[htp]
\vskip 0.3 in
\includegraphics[width=3.275 in]{Figure3.eps}
\caption{Comparison of the potential energy of the nematic state
calculated for various values of $\alpha \neq 1$ as function of
$\lambda$ with the isotropic state ($\alpha=1$) for
LLL.}\label{V_0_MC}
\vskip 0.3 in
\end{figure}
\begin{figure}[htp]
\vskip 0.3 in
\includegraphics[width=3.275 in]{Figure4.eps}
\caption{Comparison of the potential energy of the nematic state
calculated for various values of $\alpha \neq 1$ as function of
$\lambda$ with the isotropic state ($\alpha=1$) for the 1$^{st}$
excited LL.}\label{V_1_MC}
\vskip 0.3 in
\end{figure}
\begin{figure}[htp]
\vskip 0.3 in
\centerline{\includegraphics[width=3.275 in]{Figure5.eps}}
\caption{Comparison between the potential energy of the nematic state
calculated for various values of the anisotropic parameter $\alpha
\neq 1$ as a function of $\lambda$ and the potential
energy of the isotropic state
($\alpha=1$) at the $2^{nd}$ excited LL}\label{V_2_MC}
\vskip 0.3 in
\end{figure}
\begin{figure}[htp]
\vskip 0.3 in
\includegraphics[width=3.275 in]{Figure6.eps}
\caption{Comparison of total energy of the nematic state calculated
for various values of the anisotropic parameter $\alpha \neq 1$ as
functions of $\lambda$ with the isotropic state ($\alpha=1$) at the
$2^{nd}$ excited LL}\label{E_2_MC}
\vskip 0.3 in
\end{figure}
For the 2$^{nd}$ excited LL, however, the situation changes as
illustrated in Fig. \ref{V_2_MC}.
The conclusion which can be drawn from the comparison of
Fig.~\ref{V_2_MC} is that the interaction energy of the nematic
state is lower than that of the isotropic state for all values of
$\lambda$. However, we need to
compare the total energy of the nematic state with that of the
isotropic state for the 2$^{nd}$ excited LL (Fig. \ref{E_2_MC}).
From Fig. \ref{E_2_MC}, we conclude that the nematic
state is energetically favorable as compared to the isotropic state for
the 2$^{nd}$ excited LL for the range of the parameter $\lambda \leq
0.4$. Note that using FHNC we found\cite{FHNC_nematic} that for
$\lambda \leq 0.6$ the total energy of the
nematic state is lower than the energy of the isotropic state.
In summary, both
FHNC and MC yield similar conclusions about the stability of the
nematic state against the isotropic state for the 2$^{nd}$ excited LL.
In Refs.~\onlinecite{Koulakov,Folger,Moessner}
the stripe-ordered phase was predicted based on HF calculations
and this ordering can also explain the anisotropy
observed in the transport properties of the 2DEG at low
temperature. Therefore, we need to investigate the stability of
the nematic state against the stripe-ordered state as follows. First,
we find the optimum energies of the nematic state with respect to the
anisotropic parameter $\alpha$ for various values of $\lambda$. Next,
we compare these with the optimum energies of the stripe state
obtained by the HF approximation\cite{Stripe1,Stripe2,Stripe3}.
Calculations for the case where
$\lambda=0$ have been carried out in Refs. \onlinecite{Stripe1,Stripe2}
and \onlinecite{Stripe3}. For making a comparison with the optimum
energy of the nematic state at various values of $\lambda$,
we carried out detailed HF calculations for the case where
$\lambda \neq 0$. For the stripe-ordered state, the
optimum energy is obtained by minimizing the energy with respect to
the uniaxial anisotropy parameter $\varepsilon$ defined in
Ref.~\onlinecite{Stripe3}.
Fig. \ref{Ewald_MC_HF} shows the comparison of the optimal
energies obtained by MC for the nematic state with the optimum (with
respect to $\varepsilon$) energy for the stripe state
obtained by HF. Note that, for $\lambda \geq 0.5$, the optimum nematic
state is obtained for $\alpha=1$, i.e., it is the isotropic state.
Furthermore, Fig. \ref{Ewald_MC_HF} demonstrates that the nematic state is
energetically lower than the stripe state for the values of $\lambda
\leq \lambda_c = 0.37$.
\begin{figure}[htp]
\vskip 0.3 in
\includegraphics[width=3.275 in]{Figure7.eps}
\caption{Comparison of the optimal nematic state calculated by MC
using the pseudo-potential using the Ewald sum with the stripe state
calculated by HF as function of $\lambda$.}\label{Ewald_MC_HF}
\vskip 0.3 in
\end{figure}
As discussed earlier the
pseudo-potential can be obtained by using either the Ewald or the
Lekner summation technique\cite{Jensen}.
We have also carried out the same calculation using
the Lekner summation technique and the results obtained
are in good agreement with those obtained using the Ewald summation
method. Thus, we can conclude
that with MC calculation, for $\lambda \leq \lambda_c=0.37$, the
energy of the nematic state is lower than the stripe-ordered state.
\section{Conclusions}
\label{Conclusions}
In Fig.~\ref{Ewald_MC_FHNC} the results for the optimum
total energy of the nematic state obtained with the variational MC
method is compared with that obtained by FHNC in Ref.~\onlinecite{FHNC_nematic}
and with the optimum energy of the stripe-ordered state.
The critical value of $\lambda_c$ we found from FHNC\cite{FHNC_nematic}
is $0.4$ which is close to the value of 0.37 obtained above by MC.
The critical value of $\lambda$ corresponding to the sample used in
experiment \cite{Lilly} which was calculated in
Ref.~\onlinecite{FHNC_nematic}, using the conditions of the experiment
and sample characteristics,
is approximately $0.34$, which can be below the critical value found
above. Thus, both MC and FHNC calculations
indicate that the nematic state might be the state observed
experimentally for the 2DEG at the heterojunction in the samples used
in experiment described in Ref.\onlinecite{Lilly}.
\begin{figure}[htp]
\vskip 0.3 in
\includegraphics[width=3.275 in]{Figure8.eps}
\caption{Comparison of the optimum nematic state obtained from FHNC
and MC with the stripe state obtained from HF}\label{Ewald_MC_FHNC}
\vskip 0.3 in
\end{figure}
There is still a remaining question about the validity of our approximation
to neglect the projection operator in the wavefunction (\ref{WF}).
However, in both FHNC treatments of the problem\cite{FHNC_nematic,Ciftja},
where, in addition to neglecting the projection operator for arguments
presented there, there was a second rather annoying question (and rather
straightforward to answer) of
the validity of the FHNC approximation in evaluating the energy
expectation value. In the present paper the latter question is answered
by employing the Monte Carlo method.
Therefore, we conclude that the present calculations eliminates
the suspicion that the conclusions drawn in Ref.~\onlinecite{FHNC_nematic}
might be due to an artifact of the FHNC approximation employed in
Ref.~\onlinecite{FHNC_nematic}.
\section{Appendix}
\subsection{Ewald summation technique for the logarithmic potential}
\label{Ewald}
The long-range nature of the pseudo-potential $ln(r)$ which appears in
the exponent of the wavefunction of the Jastrow factor in the case of
periodic boundary conditions requires a summation over all periodic
image charges. Specifically, the ``charge'' distribution required to
give rise to a logarithmic interaction is given as:
\begin{eqnarray}
\rho(\vec r) = \sum_{\vec R}\delta(\vec r-\vec R)+\rho_{background}
\end{eqnarray}
The two-dimensional (2D) Poisson equation is given by:
\begin{eqnarray}
\nabla^2\Phi(\vec r) = -2\pi \rho(\vec r)
\end{eqnarray}
and its solution in 2D is the logarithmic interaction.
We need to solve the above equation for a periodic square $L \times L$.
The idea of Ewald summation is to add around each charge an opposite
Gaussian charge distribution of an appropriately chosen width $\mu$,
and, in addition, to subtract the same Gaussian charge
distribution.
Let us split $\rho$ into long-range and short-range portions in the
following manner:
\begin{eqnarray}
\rho(\vec r) = \rho_1(\vec r) +\rho_2(\vec r)\\ \rho_1(\vec
r)=\sum_{\vec R}\frac{e^{-\frac{(\vec r-\vec
R)^2}{\mu^2}}}{\pi\mu^2}+\rho_{background}\\ \rho_2(\vec
r)=\sum_{\vec R}\bigg[\delta(\vec r-\vec R)-\frac{e^{-\frac{(\vec
r-\vec R)^2}{\mu^2}}}{\pi\mu^2}\bigg]
\end{eqnarray}
$\phi_1$, which corresponds to $\rho_1$ is a short-range potential,
and, thus, we can calculate $\phi_1$ in
real space since it converges very quickly. The other combined charge
configuration, i.e., $\rho_2$, consisting of the Gaussian and the
background charge and the corresponding potential is denoted by
$\phi_2$. Since $\phi_2$ is a long-range potential it will be calculated
in Fourier space. The solution to each of the Poisson's equations
for the two charge distributions and the corresponding potential is
straightforward. We note that for our case the ``charge'' of the particle
is $e^2=2m$. We find
\begin{eqnarray}
\phi_1(\vec r)=\frac{4m\pi}{A}\sum_{\vec k\neq
0}\frac{e^{-\mu^2k^2/4}}{k^2}e^{i\vec k\cdot\vec
r}\label{Ewald_1},\\ \phi_2(\vec r)=-m\sum_{\vec
R}Ei\bigg[-\frac{(\vec r-\vec R)^2}{\mu^2}\bigg]\label{Ewald_2}.
\end{eqnarray}
where $\vec k = 2\pi/L \vec n$ with $\vec n\in Z^2$ and
$Ei(t)$ is the Exponential integral function and is defined by:
$Ei(t) = -\int_{-t}^{\infty}\frac{e^{-x}}{x}dx$.
For the Ewald summation, the convergence of (\ref{Ewald_1}) and
(\ref{Ewald_2}) is achieved choosing the width of the Gaussian charge
distribution $\mu=1$, the number of cells for the sum in
(\ref{Ewald_2}) to be 10 and by carrying out the sum in momentum space
in (\ref{Ewald_1}) over 200 k-states.
In order to check the validity of this approach for the case
of our use of toroidal boundary conditions we calculated the
distribution function and the energy for the 1/3 ($m=3$) case
using the expressions (\ref{Ewald_1},\ref{Ewald_2}) and our
results for the energy and distribution function are identical
to the results of Morf and Halperin\cite{Morf} who
used the disk geometry.
\subsection{Evaluation of kinetic energy of the nematic
state}\label{Kinetic}
In this subsection of the appendix, we compute the
kinetic energy difference between the nematic and the
isotropic state. In the single-LL approximation, the kinetic energy is
quenched. In addition, the same is true in the HF treatment of the
stripe, namely, there is no kinetic energy due to any correlation
factors or operators. While this approximation gives a significant
difference between the potential energy of the isotropic state and the
nematic state, it gives no difference between their kinetic energy
which is unacceptable because of the difference in the geometry of the
Fermi sea. We want to estimate this difference. We can start with:
\begin{eqnarray}
(\vec \nabla-\vec A)^2F\Phi\\
= (\vec \nabla-\vec A)^2F\Phi+2\bigg[(\vec \nabla-\vec
A)F\bigg]\nabla\Phi+F\nabla^2\Phi
\end{eqnarray}
The first term in the above equation yields:
\begin{eqnarray}
(\vec \nabla-\vec A)^2F\Phi = \frac{\hbar\omega_c}{2}F\Phi,
\end{eqnarray}
which is common for all states under our consideration so for
simplicity we can drop it. The last term is:
\begin{eqnarray}
F\nabla^2\Phi=F\sum_{k}\frac{\hbar^2k^2}{2m^{\star}}\Phi
\end{eqnarray}
So the contribution of the last term is:
\begin{equation}
\sum_{\vec k \in FFS}\frac{\hbar^2k^2}{2m^{\star}}
\end{equation}
where $\vec k \in FFS$ stands for a summation over all vectors
$\vec k$ in the corresponding filled Fermi sea.
The summation over the circular Fermi sea in the isotropic case
is given by:
\begin{eqnarray}
\frac{1}{N}\sum_{\vec k}\frac{\hbar^2
k^2}{2m^\star}=\frac{\hbar^2k_F^2}{4 m^\star},
\end{eqnarray}
and the in the case of the elliptic Fermi sea in the anisotropic
case the summation is given by
\begin{eqnarray}
\frac{1}{N}\sum_{\vec k}
\frac{\hbar^2k^2}{2m^\star}=\frac{\hbar^2}{4m^{\star}}\frac{k_1^2+k_2^2}{2}
\end{eqnarray}
Using the facts that $k_F^2=k_1\cdot k_2$ and $k_1/k_2=\alpha$, the
kinetic energy difference between the isotropic state and the nematic
state is given as follows:
\begin{eqnarray}
\Delta (KE) = -\frac{\hbar^2k_F^2}{4m^{\star}}\frac{(1-\alpha)^2}{2\alpha}.
\end{eqnarray}
\section{Acknowledgements}
We would like to thank Eduardo Fradkin, Steve Kivelson and Kun Yang
and Lloyd Engel for useful discussions.
|
1208.2544
|
\section{Introduction}
An affine manifold is a differential manifold $M$ with a special atlas of coordinate charts such that the coordinate changes extend to affine automorphisms of $ \mathbb{R}^{n}$. These distinguished charts are called affine charts. Given an affine structure on $M$ is equivalent to have a flat and torsion free linear connection $\nabla $ on $M$. A tensor field on $M$, for example a Poisson bivector, is called polynomial if in affine coordinates its coefficients are polynomial functions. For some results on affine geometry and polynomial tensor fields see ( [1],[2],[3],[4],[5] ). If $M=G$ is a Lie group and $\nabla $ is a left invariant affine structure on $G$ the pair $(G,\nabla)$ is called an\textbf{ affine Lie group}. From the infinitesimal point of view this means that the Lie algebra $ {\cal G}=T_{\epsilon}G$ of $G$ is endowed with a left symmetric product $ab:=L_{a}b=R_{b}a$ compatible with the Lie bracket of ${\cal G}$ in the sense that we have $ab-ba=[a,b]$. A pair $(G,\omega^{+})$ where $\omega^{+}$ is a left invariant symplectic form on $G$ is called a \textbf{symplectic Lie group}. For elements of the theory of symplectic Lie groups see [6]. The authors have described in [7],( see also [8],[9] ), the Lie algebras corresponding to nilpotent symplectic Lie groups in terms of the so-called symplectic double extension for the Lie algebras. For the structure of symplectic Lie groups see [10]. It is well known, see [11], that the formula
\begin{center}
\begin{equation}
\omega(ab,c)=-\omega(b,[a,c])
\end{equation}
\end {center}
with $\omega^{+}_{\epsilon}=\omega$ and $\epsilon$ the unit in $G$, defines a left invariant affine structure on $G$. These two structures are used in [12] to prove the existence of polynomial Poisson tensors on some compact solvmanifolds.
A pair $(G,\alpha^{+})$ where $\alpha^{+}$ is a left invariant contact form on the Lie group $G$, is called a \textbf{contact Lie group.} There are very strong links between the symplectic and the contact filiform Lie groups ([13]).
If a compact nilmanifold $M:=\Gamma\backslash G$ is endowed with a symplectic form $\Omega$ , it is well known that there exists a left invariant symplectic form $\omega^{+}$ on $G$ inducing a symplectic form on $M$ cohomologous to $\Omega$ ([14]).The study of the geometry of compact symplectic nilmanifolds $(M,\Omega)$ with $\Omega$ induced by $\omega^{+}$, called here \textbf{Symplectic Nilmanifolds}, is the other aim of this work. For a description of lattices in simply connected 4-dimensional solvable symplectic Lie groups, see [15].
\section{Lattices in 6-dimensional real 2-nilpotent symplectic Lie groups}
\subsection{ Case with derived Lie group 2-dimensional}
In this subsection ${G}$ is a 6-dimensional real 2-step nilpotent symplectic Lie group with Lie algebra ${\cal G}$ and derived group of dimension two. Denote by ${\cal H}_{1}(\mathbb{A})$ the 3-dimensional Heisenberg Lie $\mathbb{A}$-algebra where $\mathbb{A}$ is the field of real numbers, the field of complex numbers, or the ring $\mathbb{D}$ of dual numbers. The corresponding real Lie algebra ${\cal G}$ is one from ${\cal H}_{1}(\mathbb{C})$, ${\cal H}_{1}(\mathbb{R}){\times}{\cal H}_{1}(\mathbb{R})$ or ${\cal H}_{1}(\mathbb{D})$, as shown below. Each of these algebras is symplectic i.e. have an invertible scalar 2-cocycle (see [13]). In this section we want to study the lattices in $G$.
\subsubsection {Real Lie algebra structures on $\cal G $}
Let $A$ be a 6-dimensional 2-nilpotent real Lie algebra with center $ Z(A)=D(A)$ of dimension two. If $V$ is a vector subspace of $A$ such that $A=V \oplus D(A)$ , the bracket on $A$ is given by an onto linear map $ u: \Lambda^{2}(V)\rightarrow D(A)$. Consequently the transpose map of $ u $ is injective. Hence, given $A$ is given a vector subspace $W$ of dimension two of $\Lambda^{2}(V^{*})$ with maximal support i.e. $ \bigcap_{\eta\in W}Ker(\eta) =0$ (*).
The Pfaffian on $\Lambda^{2}(V^{*})$ induces a quadratic map $ q:W\rightarrow \Lambda^{4}(V^{*})$, $\eta\mapsto Pf\eta:=\frac{1}{2}\eta^{2}$. If $ q $ were null there would be a basis $\{{\eta_{1},\eta_{2}}\}$ of $ W $ such that $\eta_{1}^{2}=\eta_{2}^{2}=\eta_{1}\wedge\eta_{2}= 0$. As $Ker\eta_{i}=P_{i}$ is a plane, $ \eta_{1}\wedge\eta_{2}= 0$ implies that $ P_{1}\bigcap P_{2}=D $ is a line and (*) is not verified. Now, any non zero element of $\Lambda^{4}(V^{*})$ determines, via $q$, a quadratic form on $ W $ we still note $q$. There are three cases for $q$ : 1) $q$ has rank two and is isotropic , 2) $q$ has rank two and is anisotropic, 3) $q$ has rank one.
In the first case there exists a basis $\{{\eta_{1},\eta_{2}}\}$ of $W$ such that $\eta_{1}^{2}=0=\eta_{2}^{2}$ and $ \eta_{1}\wedge\eta_{2} \neq 0$. If $\eta_{1}=f_{1}\wedge g_{1} $ and $\eta_{2}=f_{2}\wedge g_{2} $ then $\{{f_{1},g_{1},f_{2},g_{2}\}}$ is a basis of $V^{*}$. If $\{{e_{1},e_{2},e_{3},e_{4}\}}$ denote its dual basis ,we have $\eta_{1}=e_{1}^{*}\wedge e_{2}^{*}$ and $\eta_{2}=e_{3}^{*}\wedge e_{4}^{*}$. Hence, the Lie algebra $A=V \oplus D(A)$ has two non null brackets $[e_{1},e_{2}]=:e_{5}$ and $[e_{3},e_{4}]=:e_{6}$. Moreover as $D(A)$ is a plane, the Lie algebra $A$ is isomorphic to the direct product ${\cal H}_{1}(\mathbb{R}){\times}{\cal H}_{1}(\mathbb{R})$.
Suppose $q$ anisotropic with rank two. Let $A_{\mathbb{C}}$ , $W_{\mathbb{C}}$ and $q_{\mathbb{C}}$ be the corresponding complexifications. It is clear that $q_{\mathbb{C}}$ is isotropic and there exists $\eta\in W_{\mathbb{C}}$ non zero such that $\eta^{2}=0$. Putting $\eta=\eta_{1}+i\eta_{2}$ with $\eta_{i}\in W $ , we have $\eta_{1}^{2}=\eta_{2}^{2}\neq 0 $ and $ \eta_{1}\wedge\eta_{2}=0$. As $\eta^{2}=0$ then $\eta=f\wedge g $ with $f:=f_{1}+i f_{2}$ and $g=g_{1}+i g_{2}$ where $f_{1},f_{2},g_{1},g_{2}\in V^{*}$. The fact that $ \eta_{1}=Re\eta, \eta_{2}=Im\eta $ implies that
$ \eta_{1}= f_{1}\wedge g_{1}-f_{2}\wedge g_{2}$ , $ \eta_{2}= f_{1}\wedge g_{2}+f_{2}\wedge g_{1}$ and $B^{*} =\{{f_{1},g_{2},f_{2},g_{1}\}}$ is a basis of $V^{*}$. Let $\{{e_{1},e_{2},e_{3},e_{4}\}}$ be the dual basis of $B^{*}$. We have then, $ \eta_{2}=e_{1}^{*}\wedge e_{2}^{*} + e_{3}^{*}\wedge e_{4}^{*}$ ; $ \eta_{1}=e_{1}^{*}\wedge e_{4}^{*} + e_{2}^{*}\wedge e_{3}^{*}$ . Consequently the product in the Lie algebra $A=V \oplus D(A)$ is given by the brackets $ [e_{1}, e_{2}]=[e_{3}, e_{4}]=:e_{5}$ and $ [e_{1}, e_{4}]=[e_{2}, e_{3}]=:e_{6} $. Now consider a basis $\{{f_{1},f_{2},f_{3}\}}$ of the Heisenberg complex Lie algebra of dimension three with $[f_{1},f_{2}]=f_{3}$. The real basis $\{{f_{1},if_{2},if_{1},f_{2},if_{3},f_{3}\}}$ denoted by $\{{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}}$ verifies $ [e_{1}, e_{2}]=[e_{3}, e_{4}]=:e_{5}$ and $ [e_{1}, e_{4}]=[e_{2}, e_{3}]=:e_{6} $ , so $A$ is isomorphic to ${\cal H}_{1}(\mathbb{C})$.
Finally, if $ q $ has rank 1, there is a basis $ \{\eta_{1},\eta_{2}\} $ of $ W $ such that $\eta_{1}^{2}\neq o $ ; $\eta_{2}^{2}=\eta_{1}\wedge\eta_{2}= 0 $. From $\eta_{2}^{2}= o $, it comes, $\eta_{2}=f\wedge g $ and $ \eta_{1}\wedge\eta_{2}= 0 $ implies $\eta_{1}= f \wedge h + g\wedge k $ with $B^ {*} =\{f,g,h,k\}$ a basis of $ V^{*} $. Let $\{{e_{1},e_{2},e_{3},e_{4}}\} $ be the dual basis of $B^{*}$. We have $ \eta_{1}=e_{1}^{*}\wedge e_{3}^{*} + e_{2}^{*}\wedge e_{4}^{*}$ and $\eta_{2}=e_{1}^{*}\wedge e_{2}^{*}$. Then, in terms of $ B $ the bracket in $ A $ is given by $ [e_{1},e_{3}]= [e_{2},e_{4}]=:e_{5}$ and $[e_{1},e_{2}]= :e_{6} $. To show that the Lie algebra $A$ is isomorphic to ${\cal H}(\mathbb{D})$, consider a $\mathbb{R}[\epsilon]$-basis $\{{f_{1},f_{2},f_{3}}\} $ of the last algebra with $[f_{1},f_{2}]=f_{3}$. As $ \epsilon^{2}=0$ in the $\mathbb{R}$-basis $\{e_{i}; 1\leq i\leq 6\}=\{f_{1},f_{2},\epsilon f_{2},-\epsilon f_{1},\epsilon f_{3},f_{3}\} $ we have $[e_{1},e_{2}]=e_{6}$ ; $[e_{1},e_{3}]= e_{5}=[e_{2},e_{4}]$ ; $[e_{3},e_{4}]=0$ and the two real Lie algebras $A$ and ${\cal H}_{1}(\mathbb{D})$ are isomorphic.
\subsubsection {Rational Lie algebra structures on ${\cal G}$ and lattices}
Let $A$ be a rational Lie algebra of dimension 6 such that the real Lie algebra $ A\otimes \mathbb{R} $ obtained by extension of the base field to $\mathbb{R}$ is isomorphic to one of Lie algebras described in 2.1. It is clear that $A$ is 2-step nilpotent and $ Z(A)=D(A)$ has dimension two. The choice of a subspace $V$ of $A$ as above determines a plane $W$ of $\Lambda^{2}(V^{*})$ and a quadratic map $ q:W\rightarrow \Lambda^{4}(V^{*})$.
If the rank of $q$ is 1, there is a basis $ \{\eta_{1},\eta_{2}\} $ of $ W $ such that
$\eta_{1}^{2}\neq o $ ; $\eta_{2}^{2}=\eta_{1}\wedge\eta_{2}= 0 $ and the above calculation shows the existence of a basis of $A$ in which the multiplication table is that of ${\cal H}_{1}(\mathbb{D})$. Hence, the corresponding real 1-connected Lie group $G$ is the unimodular group of matrix
\begin{center}
$\begin{pmatrix}
1 & u & v \\
0 & 1 & w \\
0 & 0 & 1
\end{pmatrix}$
\end{center}
\begin{flushleft}
with $u,v,w \in \mathbb{R}[\epsilon]=:\mathbb{D}$.\\
\end{flushleft}
In this case, any lattice in $G$ is commensurable to the lattice $\Gamma$ whose elements are the matrices in $G$ with $u,v,w\in \mathbb{Z}[\epsilon]$.
If $q$ has rank two, the choice of a basis of $V$ determines a quadratic form on $W$. Moreover, it is cleat that the quadratic form produced for another basis choice, is collinear with the first. In other words, the set of the $q$ maps obtained is in bijection with the $\mathbb{Q}^{*}$- orbits in the set of quadratic forms of rank two on $W$ where the action is given by homotheties.
In every equivalent class there is an unique representative of the form $X^{2}+dY^{2}$ with $d$ the common discriminant of the quadratic forms of the class. It is clear that $d$ can be chosen a squarefree integer.
Hence we have :
If $d< 0$, the real algebra $ A\otimes \mathbb{R} $ is isomorphic to ${\cal H}_{1}(\mathbb{R}){\times}{\cal H}_{1}(\mathbb{R})$ .
If $d > 0$, $ A\otimes \mathbb{R} $ is isomorphic to ${\cal H}_{1}(\mathbb{C})$ .
If $ d\neq -1 $ then ${\cal K }:= \mathbb{Q}(\sqrt {-d} )$ is a field, where $ \mathbb{Q} $ is the field of rational numbers. As a consequence, the extension of the scalar field to ${ \cal K}$ produces the Lie algebra direct product ${{\cal H}}_{1}({\cal K})\times {\cal H }_{1}({\cal K})$. Consider $q_{A}:W\rightarrow \Lambda^{4}(V^{*})$, the form composed of the injection of $W$ in $\Lambda^{2}(V^{*})$ with $\gamma_{2}=Pf$ where $Pf$ is the Pfaffian. Obviously $q_{A}$ is proportional to $X^{2}+dY^{2}$ and its extension to the tensor product $A\otimes \mathbb{Q}(\sqrt {-d})$, relative to $\mathbb{Q}$, becomes isotropic: that is, there exists $ u\in W\otimes{\cal K }$, $u\neq0 $, $\gamma_{2}(u)=0$ and $u$ is decomposable i.e. $u= (f_{1}+\sqrt {-d}f_{2})\wedge (f_{3}+\sqrt {-d}f_{4})$ with $f_{j}\in V^{*}$. As $u=( f_{1}\wedge f_{3}-d f_{2}\wedge f_{4})+\sqrt {-d}( f_{1}\wedge f_{4}+ f_{2}\wedge f_{3})=u_{1}+\sqrt {-d}u_{2}$ where $u_{1}=\frac{u+\overline{u}}{2}$, $u_{2}=\frac{u-\overline{u}}{2 \sqrt {-d}}$ with $\overline{u}$ the conjugate of $u$ .Since $u_{1}$ and $u_{2}$ have rank 4 , $B^{*}=\{f_{1},f_{2},f_{3},f_{4}\}$ is a basis of $V^{*}$ and $\{u_{1},u_{2}\}$ is a basis of $W$ over $\mathbb{Q}$ . At this point we can write the bracket of the Lie algebra $A$ over $\mathbb{Q}$ .Let $ B= \{e_{1},e_{2},e_{3},e_{4}\}$ the dual basis of $B^{*}$. As $ u_{1}=e_{1}^{*}\wedge e_{3}^{*}-d e_{2}^{*}\wedge e_{4}^{*} $ and $u_{2}=e_{1}^{*}\wedge e_{4}^{*}+ e_{2}^{*}\wedge e_{3}^{*} $, the bracket in $A$ is given by
\begin{center}
\begin{equation}
[e_{1},e_{4}]=[e_{2},e_{3}]=e_{5} , [e_{1},e_{3}]=e_{6} , [e_{2},e_{4}]=-d e_{6}
\label{eq:ED1}
\end{equation}
\end{center}
\begin{flushleft}
where $d$ is a non null squarefree integer \\
\end{flushleft}
The formula (2) gives all the $\mathbb{Q}$ -structures of Lie algebra on the real Lie algebras
${\cal H}_{1}(\mathbb{R}){\times}{\cal H}_{1}(\mathbb{R})$ and ${\cal H}_{1}(\mathbb{C})$ if $d>0$ or $d<0$ respectively (see [16], page 47).
We want, in the following, describe a lattice in each class of commensurability.
Notice if $d > 0$ , the corresponding 1-connected Lie group is the 3-dimensional Heisenberg Lie group $G= H_{1}(\mathbb{C})$ and we have the following tower of groups $G= H(\mathbb{C}) \supset G_{d}= H({\cal K})\supset\Gamma_{d}=H({\cal O}_{-d})$ where ${\cal O}_{-d}$ is the ring of integers of ${\cal K }$. In other words:
\begin{center}
${\cal O}_{-d} =\mathbb{Z}(\sqrt {-d})$ if $ d \equiv 1,2 ( mod\ 4)$
${\cal O}_{-d}=\mathbb{Z}[\frac{1+\sqrt{-d}}{2}]$ if $d\equiv 3 ( mod \ 4 )$
\end{center}
\textbf{Remark 1.} Obviously the lattice $\Gamma_{d}:=N_{d}(\mathbb{Z})$ is embedded into $N_{d}(\mathbb{Q})=: H_{1}({\cal K})$ . Also if $d_{1}\neq d_{2}$ are two squarefree positive integers, the lattices $N_{d_{1}}(\mathbb{Z})$ and $N_{d_{2}}(\mathbb{Z})$ are not commensurables.
From the multiplication in the group $G'$ of matrices
\begin{center}
$\begin{pmatrix}
1 & a & c \\
0 & 1 & b \\
0 & 0 & 1
\end{pmatrix}$
\end{center}
\begin{flushleft}
with $a,b,c \in {\cal O}_{-d}$ , we can deduce the product in the lattice $H_{1}({\cal O}_{-d})$. In fact, we have $N_{d}(\mathbb{Q})=\mathbb{Q}^{6}$ and the product in the lattice $N_{d}(\mathbb{Z})$ is induced by the product of $G'$. More precisely, if $ d\equiv 1,2 (mod \ 4)$ putting $a=x_{1}+\varpi x_{2}$ and $b'=y_{3}+\varpi y_{4}$ with $\varpi^{2}=-d $ it is found that the the products in the groups $H_{1}(\mathbb{Z}[\sqrt{-d}])$ and $N_{d}(\mathbb{Z})$ are identical.
\end{flushleft}
In a similar way, the equality $\varpi^{2}=\varpi-\frac{1+d}{4}$ determines the product in the lattice when $d\equiv 3 (mod\ 4 )$.
Let $d\in\mathbb{Z}$ be squarefree . If $d > 0 $ then $\mathbb{Q}( \sqrt{-d})$ is a subfield of $ \mathbb{C}$ whereas if $d<0$ , $\mathbb{Q}( \sqrt{-d})$ is a subfield of $\mathbb{R}\times \mathbb{R}$. In fact the map $\mathbb{Q}[\sqrt{-d}]=\frac{\mathbb{Q}[X]}{(X^{2}+d)}\rightarrow \mathbb{R}\times \mathbb{R}$ , $\overline{\alpha+\beta X }\mapsto (\alpha+\sqrt{-d} \beta,\alpha-\sqrt{-d} \beta)$ is an injective homomorphism of rings. Hence $H_{1}({\cal O}_{-d})\subset H_{1}({\cal K})$ and the last one is included in $H_{1}(\mathbb{C})$ or in $H_{1}(\mathbb{R}\times \mathbb{R})$ according to the sign of $d$.
Recall that a field of the form $\frac{\mathbb{Q}[X]}{(X^{2}+d)}$ where $d$ is an integer and $d\neq 0,-1 $ is called a quadratic field. If $d < 0$ the field is said real, if $d>0$, the field is imaginary.
In the these terms we have shown the following result:
\textbf{Theorem 1.} {a. Each nilpotent real Lie algebras ${\cal H}_{1}(\mathbb{C})$ and ${\cal H}_{1}(\mathbb{R}){\times}{\cal H}_{1}(\mathbb{R})$ has an infinity of $\mathbb{Q}$ -Lie algebra structures corresponding to squarefree integers $d$. Consequently the corresponding simply connected real Lie groups $ H_{1}(\mathbb{C})$ and $H_{1}(\mathbb{R}){\times}H_{1}(\mathbb{R})$ have an infinity of commensurability class of lattices.
b. In $ H_{1}(\mathbb{C})$ each commensurability class contains a lattice isomorphic to $H_{1}({\cal A})$ where ${\cal A}$ is the ring of integers of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. In $H_{1}(\mathbb{R}){\times}H_{1}(\mathbb{R})$ every class of commensurability either contains a lattice isomorphic to $H_{1}({\cal A})$ with ${\cal A}$ the ring of integers of the real quadratic field $\mathbb{Q}(\sqrt{-d})$ if $d < -1$ or $H_{1}(\mathbb Z)\times H_{1}(\mathbb Z)$ if $d=-1$.}
\textbf{Remark 2.} {In terms of a basis $\{e_{i}, 1\leq i\leq 6 \}$ the brackets in the $\mathbb Q $-Lie algebra, as $ d>1 $ below, are given by
\begin{center}
\begin{equation}
[e_{1},e_{2}]=[e_{3},e_{4}]=e_{5} , [e_{1},e_{3}]=e_{6} , [e_{2},e_{4}]= -d e_{6}
\end{equation}
\end{center}
where $d$ is a squarefree integer.
Moreover, the product $\sigma\times \tau $ in the algebraic group $N_{d}(\mathbb Z )$ for $\sigma=(x_{i})$ and $\tau=(y_{i})$ is equal to either
\begin{equation}
\sigma\tau=(x_{1}+y_{1},..,x_{4}+y_{4}, x_{5}+y_{5}+x_{1}y_{3}+dx_{2}y_{4},x_{6}+y_{6}+x_{1}y_{4}+x_{2}y_{3})
\label{eq:ED1}
\end{equation}
or
\begin{equation}
\sigma\tau=(x_{1}+y_{1},..,x_{4}+y_{4}, x_{5}+y_{5}+x_{1}y_{3}+(\frac{d-1}{4})x_{2}y_{4},x_{6}+y_{6}+x_{1}y_{4}+x_{2}y_{3}+x_{2}y_{4}
\end{equation}
\begin{flushleft}
if $d\equiv 2,3 ( mod\ 4 )$ or $d\equiv 1 ( mod\ 4 )$ respectively ).
\end{flushleft}
\subsection {Lattices in the cotangent bundle of the real Heisenberg-Lie group}
Denote by $G:= T^{*} H_{1}(\mathbb{R}) $ the simply connected Lie group whose Lie algebra ${\cal G}$ is the semi-direct product of the real Heisenberg Lie algebra ${\cal H}_{1}$ by the abelian Lie algebra ${\cal H}_{1}^{*}$ via the coadjoint action of ${\cal H}_{1}$. We will describe the lattices of $G$. This group is a quadratic 2-step nilpotent Lie group with bi-invariant metric of type $ (3,3)$. Also it carries a left invariant symplectic form: the corresponding scalar 2-cocycle $\Omega$ is defined by an invertible derivation $\delta=(\lambda,-\lambda^{t}) $ where $\lambda$ is an invertible derivation of ${\cal H}$.
It is clear that the Lie algebra ${\cal G}$ is isomorphic to the vector space $\mathbb{R}^{3}\oplus \Lambda^{2}(\mathbb{R}^{3})$ endowed with the Lie bracket $[(x,u),(y,v)]=(0,x \wedge y)$ and that ${\cal G}$ has a unique rational Lie algebra structure given by this bracket. So, any two lattices in $G$ are commensurable.
It is useful, for our purpose, to identify $G$ and the vector space $\mathbb{R}^{6}$ endowed with the product given for $\sigma=(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})$ and $\tau= (x'_{1},x'_{2},x'_{3},y'_{1},y'_{2},y'_{3})$ as
\begin{equation}
\sigma\tau=(x_{1}+x'_{1}+y_{2}y'_{3},x_{2}+x'_{2}+y_{3}y'_{1},x_{3}+x'_{3}+y_{1}y'_{2},y_{j}+y'_{j})
\end{equation}
where $1\leq j\leq 3$.
The subset $\Gamma_{0}$ of $G$ whose elements have integer components is a lattice of $G$ and any lattice of $G$ is commensurable to $\Gamma_{0}$. Moreover its derived group is equal to its center. We have :
\textbf{Theorem 2.} Up to isomorphism a lattice $\Gamma $ of $ G $, as above , is characterized by three positive integers $d_{1}, d_{2},d_{3}$ such that $1\leq d_{1}|d_{2}|d_{3}$. More precisely, $\Gamma $ is isomorphic to the group
\begin{center}
$\Gamma_{d_{1},d_{2},d_{3}}=\{(a_{i},b_{j});a_{i},b_{j}\in\mathbb{Z},1\leq i,j\leq3\}$
\end{center}
with multiplication given by
\begin{center}
$(a_{i},b_{j})(a'_{i},b'_{j})=(a_{1}+a'_{1}+d_{1}b_{2}b'_{3},a_{2}+a'_{2}+d_{2}b_{3}b'_{1},a_{3}+a'_{3}+d_{3}b_{1}b'_{2},b_{j}+b'_{j})$
\end{center}
\textbf{Proof}. A calculation shows that $\Gamma_{d_{1},d_{2},d_{3}}$ is a group. It is a subgroup of $G$ because the extension of its multiplication to $\mathbb{R}^{6}$ gives a group isomorphic to $G$. Also it is clear that $\Gamma_{d_{1},d_{2},d_{3}}$ is a lattice of $G$ , so commensurable to $\Gamma_{0}$.
In the other hand, the center and the derived group of $\Gamma_{d_{1},d_{2},d_{3}}$ are respectively
\begin{center}
$ Z(\Gamma_{d_{1},d_{2},d_{3}})=\{( (a_{i}),0,0,0);a_{i}\in\mathbb{Z}\}$
$D(\Gamma_{d_{1},d_{2},d_{3}})=\{(d_{1}a_{1},d_{2}a_{2},d_{3}a_{3},0,0,0);a_{i}\in\mathbb{Z}\}$
\end{center}
\begin{flushleft}
Hence, the factor group $Z/D$ is isomorphic to the direct product of three cyclic groups $C_{d_{1}}\times C_{d_{2}}\times C_{d_{3}}$. Consequently different triples of integers, as in the Theorem 2, produce non isomorphic lattices.
\end{flushleft}
The following presentation of the group $\Gamma_{d_{1},d_{2},d_{3}}$ by generators and relations, will be useful for the proof. Denote by $z_{1},z_{2},z_{3},y_{1},y_{2},y_{3}$ the elements of $\Gamma_{d_{1},d_{2},d_{3}}$ with only one non zero component which is equal to $1$ in the place $i$ for $z_{i}$ and $j+3$ for $y_{j}$. The $z_{i}$ generate the center and satisfy the relations:
\begin{center}
$z_{k}z_{l}=z_{l}z_{k}$ and $ y_{j}z_{i}=z_{i}y_{j}$ for any $i,j,k,l\in\{1,2,3\}$
\end{center}
and a direct calculation gives the other relations
\begin{center}
\begin{equation}
y_{2}y_{3}=y_{3}y_{2}z_{1}^{d_{1}} , y_{3}y_{1}=y_{1}y_{3}z_{2}^{d_{2}} ,y_{1}y_{2}=y_{2}y_{1}z_{3}^{d_{3}}.
\end{equation}
\end{center}
Conversely, a group generated by six elements satisfying these relations with the $d_{i}$ as above is isomorphic to $\Gamma_{d_{1},d_{2},d_{3}}$.
Finally we show that any lattice $\Gamma $ of $G$ has such a presentation.
It is clear that a lattice $\Gamma $ of $G$ is a 2-step nilpotent group: its center $Z$ contains its derived group $D$. Also $D$ is a sublattice of the lattice $Z$ of the 3-dimensional real vector space $Z(G)$. Hence $Z/D$ is a group direct product of three cyclic groups $C_{d_{1}},C_{d_{2}}$ and $ C_{d_{3}}$ where the $d_{i}$ are positive integers and
$1\leq d_{1}|d_{2}|d_{3}$ . Then, using multiplicative notation , there is a basis $\{z_{1},z_{2},z_{3}\}$ of the free Abelian group $Z$ such that $\{z_{1}^{{d}_{1}},z_{2}^{{d}_{2}},z_{3}^{{d}_{3}}\}$ is a basis of $D$.
Let $K$ be an arbitrary group and $ \left\langle g,h\right\rangle:=g^{-1}h^{-1}gh$ the commutator of $g$ and $ h\in K $ . It is easy to verify the formula
\begin{center}
$ \left\langle g,h_{1}h_{2}\right\rangle= \left\langle g,h_{2}\right\rangle \left\langle g,h_{1}\right\rangle^{h_{2}}$
\end{center}
where $a^{b}:=b^{-1}ab$.
Applying this formula to $\Gamma $ we get :
\begin{center}
$\left\langle x,yz\right\rangle=\left\langle x,y\right\rangle\left\langle x,z\right\rangle$
\end{center}
because two commutators commute and $D$ is in the center of $\Gamma $.
Now the commutator map
\begin{center}
$\Gamma\times \Gamma\rightarrow D$ , $(x,y)\rightarrow \left\langle x,y\right\rangle $
\end{center}
induces an antisymmetric map $\varphi: \Gamma/Z \times \Gamma/Z\rightarrow D $. But the last formula shows that $\varphi $ is additive in the second variable. Consequently it is bi-additive and then determines an homomorphism of Abelian groups $\overline{\varphi}:\Lambda^{2}( \Gamma/Z )\rightarrow D $. This homomorphism is onto. Moreover , $ \Gamma/Z $ can be identifyed with a lattice of the real space $G/Z(G)$. As $D$, $\Gamma/Z$ and $\Lambda^{2}( \Gamma/Z )$ are free Abelian groups of finite type and rank 3, it follows that $\overline{\varphi}$ is an isomorphism of groups. Denote by $\{u_{1},u_{2},u_{3}\}$
the basis of $\Lambda^{2}( \Gamma/Z )$ such that $\overline{\varphi}(u_{i})=z_{i}^{{d}_{i}}$. As $\mathbb{Z}$ is a principal ring, there exists a basis $\{t_{1},t_{2},t_{3}\}$ of $\Gamma/Z$ such that $u_{1}=t_{2}\wedge t_{3} $, $u_{2}=t_{3}\wedge t_{1} $ and $u_{3}=t_{1}\wedge t_{2} $ . Taking $y_{i}\in\Gamma $ with $\overline {y_{i}}=t_{i}$ for $1\leq i\leq 3$ we have
$ \left\langle y_{2},y_{3}\right\rangle=z_{1}^{{d}_{1}}$....Hence it is clear that $\Gamma $ is generated by $\{y_{1},y_{2},y_{3},z_{1},z_{2},z_{3}\}$ with the relations:
\begin{center}
$y_{2}^{-1}y_{3}^{-1}y_{2}y_{3}=z_{1}^{{d}_{1}}$,
$y_{3}^{-1}y_{1}^{-1}y_{3}y_{1}=z_{2}^{{d}_{2}}$,
$y_{1}^{-1}y_{2}^{-1}y_{2}y_{1}=z_{3}^{{d}_{3}}$.
$z_{i}x=xz_{i}$
\end{center}
\begin{flushleft}
for every $x \in \Gamma $.
\end{flushleft}
So, $\Gamma $ is isomorphic to $\Gamma_{d_{1},d_{2},d_{3}}$.
\hfill $\square$\\
\section{Lattices in some filiform Lie groups}
For an integer $n\geq 2$ let ${\cal L}_{n}(\mathbb{K})=Span_{\mathbb{K}}\{e_{0},e_{1},...,e_{n}\}$ be the filiform Lie algebra over the field ${\mathbb{K}}$ with the only nonzero brackets given by $ [e_{0},e_{i}]=e_{i+1}$ for $i=1,2,...,n-1$. As ${\cal L}_{n}(\mathbb{K})$ has an invertible derivation,its bracket is compatible with a left symmetric product.
\subsection {Rational Lie algebra structures in ${\cal L}_{n}(\mathbb{R})$}
The algebras ${\cal L}_{n}(\mathbb{K})$ are the only filiform Lie algebras having a unique Abelian ideal of codimension $1$. In fact a codimension $1$ ideal $I$ contains the derived ideal $J=Span_{\mathbb{K}}\{e_{2},...,e_{n}\}$ and is determined by a line in the quotient algebra by $J$, so $I=Span_{\mathbb{K}}\{\alpha e_{0}+\beta e_{1},e_{2},...,e_{n}\}$ with $(\alpha,\beta)\neq (0,0)$. If $\alpha\neq 0$, $I$ is not Abelian whereas, if $\alpha=0$, $I=Span_{\mathbb{K}}\{e_{1},...,e_{n}\}$ is the unique Abelian codimension $1$ ideal of ${\cal L}_{n}(\mathbb{K})$. We have :
\textbf{Lemma 1.} Let $k\subset\mathbb{K}$ be a subfield of the field $\mathbb{K}$ and ${\cal G}$ a $k$-Lie algebra such that the Lie algebra ${\cal G}\otimes_{k}\mathbb{K}$ is isomorphic to ${\cal L}_{n}(\mathbb{K})$. Then ${\cal G}$ has a unique Abelian ideal of codimension 1 and is isomorphic to ${\cal L}_{n}(k)$.
\textbf{Proof.} It is clear that $D({\cal G}){\otimes}_{k}\mathbb{K}$ is isomorphic to
$D({\cal L}_{n}(\mathbb{K}))=Span _{\mathbb{K}}\{e_{2},...e_{n}\}$ and hence $ D({\cal G})$ is a $k$-subspace of dimension $n-1$ of $ D({\cal L}_{n}(\mathbb{K}))$.
Consider a $k$-basis $\{f_{2},f_{3},...,f_{n}\}$ of $D({\cal G})$ completed by $\{f_{0},f_{1}\}$ in a $k$-basis $B$ of ${\cal G} $. As $B$ is also a $\mathbb{K}$-basis of ${\cal L}_{n}(\mathbb{K})$ we may write $e_{1}=\alpha f_{0}+\beta f_{1}+\sum_{i\geq 2} \gamma_{i}f_{i}$ with $\alpha,\beta,\gamma_{i}\in\mathbb{K}$.
Also, as $ ad e_{1}|D({\cal L}_{n}(\mathbb{K}))=0$ and $ ad f_{i}|D({\cal L}_{n}(\mathbb{K}))= 0$ for $i\geq 2$ ,
\begin{center}
$(\alpha ad f_{0}+\beta ad f_{1})|D({\cal L}_{n}(\mathbb{K}))=0$.
\end{center}
Now, if $ad f_{0}|D({\cal L}_{n}(\mathbb{K}))=0$ or $ad f_{1}|D({\cal L}_{n}(\mathbb{K}))=0$ then $D({\cal G})\oplus k f_{i}$ is an Abelian ideal of ${\cal G}$ of codimension 1.
Finally, suppose both adjoints are non zero. Using $ \{f_{2},...,f_{n}\} $ as a $\mathbb{K}$-basis of $D({\cal L}_{n}(\mathbb{K}))$, we get two $n-1$ square matrices, with coefficients in $k$, representing the restrictions of $ad f_{0}$ and $ad f_{1}$ to $D({\cal L}_{n}(\mathbb{K})$, which are $\mathbb{K}$-linearly dependant. This implies that $\alpha$ and $\beta$ are $k$ linearly dependent i.e. $\alpha=\lambda\beta$ with $\lambda\in k-\{0\}$. Then we have $e_{1}=\beta (\lambda f_{0}+f_{1})+\sum_{i\geq 2}\gamma_{i}f_{i}$ with $\lambda f_{0}+f_{1}\in{\cal G}$. It is then clear that $Span_{k}\{\lambda f_{0}+f_{1},f_{2},...,f_{n}\}$ is a codimension $1$ ideal of
${\cal G}$ and that ${\cal G}$ is isomorphic to ${\cal L}_{n}(k)$.
\hfill $\square$\\
Denote by $F_{n}$ the simply connected real Lie group with Lie algebra ${\cal L}_{n}(\mathbb{R})$. We have the following consequence of the Lemma 1 (see [16]).
\textbf{Lemma 2.} The Lie group $F_{n}$ has a unique commensurability class of lattices.
\subsection {Isomorphism classes of lattices in $F_{n}$ }
In the following, we want to describe the isomorphism classes of lattices in $F_{n}$. It is clear that our Lie group is isomorphic to the manifold ${\mathbb{R}}^{n}\times {\mathbb{R}}$ endowed with the product
\begin{center}
$(v,t)(v',t')=(v+ (Exptd )(v'),t+t')$
\end{center}
where $d$ is a principal nilpotent endomorphism of the vector space ${\mathbb{R}}^{n}$. The center of the group is then $Z F_{n}=\{(v,0) ; v\in Ker d \}$ and we have the following canonical exact sequence of Lie groups
\begin{center}
$ 1\leftharpoonup Z F_{n}\rightarrow\stackrel {p_{n}}{F_{n}\rightarrow F_{n-1}} \rightarrow 1 $
\end{center}
Obviously, the ascending central sequence of $F_{n}$ is obtained by pullback by
$p_{n}$ of the corresponding sequence of $F_{n-1}$. By induction on $n\geq2 $ , it follows that
\begin{center}
$0\subset C_{1}F_{n}\subset C_{2}F_{n}\subset...\subset C_{r}F_{n}\subset...$
\end{center}
is a sequence of Abelian groups and $ C_{r}F_{n}$ is isomorphic to ${\mathbb{R}}^{r}$ for $r\leq n-1$ and $C_{n}F_{n}=F_{n}$. Moreover as $C^{1}({\cal L}_{n})=C_{n-1}({\cal L}_{n})$ ;
$C^{2}({\cal L}_{n})=C_{n-2}({\cal L}_{n})$ ...$C^{r}({\cal L}_{n})=C_{n-r}({\cal L}_{n})$ , we also know the descending central sequence of $F_{n}$.
It is clear that the set $\Gamma_{0}:={\mathbb{Z}^{n}}\times \mathbb{Z}$ endowed with the (semidirect) product given by
\begin{center}
$(a_{1},...,a_{n},k)(a'_{1},...,a'_{n},k')=((a_{1},...a_{n})+ g_{0}^{k}(a'_{1},...a'_{n}),k+k')$
\end {center}
with
\begin{center}
$g_{0}^{k}(a'_{1},...a'_{n})= (a'_{1},a'_{2}+ka'_{1},a'_{3}+ka'_{2}+\binom{k}{2}a'_{1},...)$
\end {center}
\begin{flushleft}
is a lattice in $F_{n}$ and any lattice in $F_{n}$ is commensurable with $\Gamma_{0}$.\\
\end{flushleft}
Notice that the action of $\mathbb{Z}$ over \textsl{${\mathbb{Z}^{n}}$} is essentially given by the product of the matrix $g_{0}:=I_{n}+E_{2,1}+E_{3,2}+...E_{n,n-1}$ by the column vector $(a'_{1},a'_{2},...,a'_{n})^{T}$. Here $E_{i,j}$ is the n-square matrix with only one non zero coefficient , which is 1 in the i-line and the j-column.
In the following, we will describe the other lattices in $F_{n}$.
Let $\Gamma_{1}$ be a finite index subgroup of $\Gamma_{0}$ and consider the corresponding canonical sequence of groups:
\begin {center}
$\Gamma_{1}\leftharpoonup \Gamma_{0}\rightarrow \mathbb{Z}$
\end{center}
with $\pi:\Gamma_{0}\rightarrow \mathbb{Z}$ ,$\pi(a_{1},...a_{n},k)=k$. It is clear that
$\pi( \Gamma_{1})=d \mathbb{Z}$ where $d$ a strictly positive integer.
Let $\pi'$ be the restriction of $\pi $ to $\Gamma_{1}$. Obviously $L_{1}:=Ker \pi'\subset Ker\pi= {\mathbb{Z}^{n}}=\{(a_{1},...a_{n},0 );a_{i}\in \mathbb{Z} \}$ is a finite index subgroup of ${\mathbb{Z}^{n}}$ and we obtain the following canonical sequence of groups
\begin{center}
$ 1\leftharpoonup L_{1} \leftharpoonup \Gamma_{1}\rightarrow d \mathbb{Z}\rightarrow 1 $
\end{center}
Thus $\Gamma_{1}$ is the group semi-direct product of $L_{1}$ by $\mathbb{Z}$ with the action of $\mathbb{Z}$ over $L_{1}$ given by $g_{1}:=g_{0}^{d}|L_{1}$.
\textbf{ Let $\Gamma $ be a lattice of $F_{n}$} . This means that $\Gamma $ contains a sublattice of $\Gamma_{0}$ and this one is a finite index subgroup in $\Gamma$ and in $\Gamma_{0}$.
Let $H_{n}$ be the maximal Abelian normal subgroup ( of codimension 1 ) of $F_{n}$ : it is clear that ${\mathbb{Z}^{n}}$ and $L_{1}$ are lattices of $H_{n}$. Consider the canonical injection $j:\Gamma\leftharpoonup F_{n}$ and the canonical surjection $p:F_{n}\rightarrow F_{n}/H_{n}$ . Then we have the exact sequence of groups:
\begin{center}
$ 1\leftharpoonup H_{n}\leftharpoonup F_{n}\rightarrow F_{n}/H_{n}\cong \mathbb{R}$
\end{center}
Since $(p\circ j)(\Gamma_{1})$ is a finite index subgroup of $(p\circ j)(\Gamma)$ , the last one is a lattice of $\mathbb{R}$.
\textbf{Remark 3.} Consider the Lie group $G:=\mathbb{R}^{3}\oplus \Lambda^{2}\mathbb{R}^{3}$ , its lattice $\Gamma=\mathbb{Z}^{3}\oplus \Lambda^{2}\mathbb{Z}^{3}$ and $H:=P\oplus \mathbb{R}^{3}$ where $P$ is any plane of $\mathbb{R}^{3}$. The factor group $G/H$ is isomorphic to $\mathbb{R}$ but the image of $\Gamma$ by the canonical projection of $G$ on $G/H$ may not be a lattice of $G/H$. It suffices to take $P=Span \{e_{1},e_{2}+\alpha e_{3}\}$ where $\{e_{1},e_{2},e_{3}\}$ is a set of generators for $\mathbb{Z}^{3}$ with $\alpha$ irrational .
It is clear that \textbf {$L:=Ker (p\circ j)$ } is a subgroup of $H_{n}$ which contains $L_{1}$ as a finite index subgroup. Hence $L$ is a lattice of $H_{n}$ and we have the following split exact sequence of groups:
\begin{center}
$1\leftharpoonup L\rightarrow \Gamma \rightarrow \mathbb{Z}\rightarrow 1$
\end{center}
\textbf{Theorem 3.} Every lattice $\Gamma$ of the Lie group $F_{n}$ is isomorphic to a group $\Gamma_{g(1)}$ semi-direct product of the normal Abelian group $L=Ker (p\circ j)$ by the group $ \mathbb{Z}$, via the action of the last given by a unipotent matrix $g(1)=I_{n}+\displaystyle\sum_{k> l} a_{kl}E_{k,l}$ in an ordered basis $(e_{1},e_{2},...e_{n})$ of $L$.
Since the map $\mathbb{Z}\rightarrow Aut(L)$, $ p\mapsto g(p)=g(1)^{p}$, is a homomorphism of groups, the lattices in Theorem 3 are only determined by $g(1)$.
\textbf{Example 1}. If $ dim F_{n}=3 $ then $ g(1)=\begin{pmatrix}
\ 1 & 0 \\
\ a & 1
\end{pmatrix}$.
Here the isomorphism classes of lattices of $F_{n}$ are parametrized by the $|a|$ , $a\in \mathbb{Z}$.
Let $\Gamma=L\times \mathbb{Z}$, $\Gamma'=L'\times \mathbb{Z}$ be two lattices in $F_{n}$ and $\phi:\Gamma\rightarrow \Gamma'$ an isomorphism of lattices . From Theorem 3, it is clear that $\phi=\varphi\times id $ where $\varphi:L\rightarrow L'$ is an isomorphism of groups.
As $ (v,p)(w,q)=(v+g(1)^{p}w,p+q)$ and $(\varphi(v),p)(\varphi(w),q)=(\varphi(v)+g'(1)^{p}\varphi(w),p+q)$, we have, as $\varphi$ is an isomorphism of groups:
\begin{center}
$(\varphi(v)+g'(1)^{p}\varphi(w),p+q)=(\varphi(v)+ \varphi(g(1)^{p}w),p+q)$
\end{center}
\begin{flushleft}
and, in particular, $g'(1)\varphi(w)=\varphi(g(1)w)$,i.e.
\end{flushleft}
\begin{center}
$\varphi^{-1}g'(1)\varphi=g(1)$
\end{center}
Conversely, this condition determines an isomorphism between $\Gamma$ and $\Gamma'$. Moreover, this condition implies $ \varphi^{-1}(g'(1)-I_{L'})\varphi=g(1)-I_{L}$ and more generally,
\begin{center}
$(g'(1)-I_{L'})^{k}\circ\varphi=\varphi\circ(g(1)-I_{L})^{k}$
\end{center}
for any $k\in\mathbb{N}$.
Notice that if the lattices associated to $g(1)$ and $g'(1)$ are isomorphic, these matrices are conjugated in $GL(n,\mathbb{Z})$ ,via a triangular matrix, with $\epsilon_{i}\in \{1,-1 \}$ in the diagonal.
Denote by $L_{k}$ the kernel of $(g(1)-I_{L})^{k}$ ; idem for $L'_{k}$. We have $\varphi(L_{k})=L'_{k}$. In conclusion it results:
\textbf{Proposition 1.} Two lattices $L\times_{g(1)}\mathbb{Z}$ and $L'\times_{g'(1)}\mathbb{Z}$ in $F_{n}$ are isomorphic if an only if , in adapted basis to the above associated flags in $L$ and $L'$, we have $ g(1)= \varphi^{-1}\circ g'(1)\circ\varphi $ with $\varphi$ in the semi-direct product $T^{-}_{n}(\mathbb{Z})\times Diag( \epsilon_{i})$ where $T^{-}_{n}(\mathbb{Z})$ is the group of invertible lower triangular matrices with $1$ in the diagonal and coefficients in $\mathbb{Z}$.
\textbf{Remark 4.} Denote by $\{\Gamma_{g}\}$ the set of lattices in $F_{n}$; from the above study we have:
1. $(g(1)-I_{n})^{k}\neq 0 $ for $k$ integer, $ 0\leq k\leq n-1$ or equivalently $ g(1)=Exp(d')$ with $d'$ a $n$-nilpotent rational matrix of maximal nilpotent index.
2. Let $N(T^{-}_{n}(\mathbb{Z}))$ be the normalizer of $T^{-}_{n}(\mathbb{Z})$ in $GL(n,\mathbb{Z})$. The group $N(T^{-}_{n}(\mathbb{Z}))$ acts over the set $\{g(1)\}$ and the orbits of this action are in bijection with the set of lattices $\{\Gamma_{g}\}$.
3. Consider the correspondence
\begin{center}
$\theta:\{\Gamma_{g}\}\rightarrow ({\mathbb N}^{*})^{n-1}$ , $\Gamma_{g}\mapsto (|a_{2,1}|, |a_{3,2}|,...,|a_{n,n-1}|)$
\end{center}
This correspondence is in fact a map. The group $N(T^{-}_{n}(\mathbb{Z}))$ is generated by the matrices $U_{ij}=I_{n}+E_{i,j}$ with $i>j $ and the matrices $ Diag (\epsilon_{1},\epsilon_{2},...,\epsilon_{n})$. Via the conjugate action
\begin{center}
$Diag (\epsilon_{1},\epsilon_{2},...,\epsilon_{n})g(1)Diag (\epsilon_{1},\epsilon_{2},...,\epsilon_{n})$
\end{center}
the coefficient $a_{ij}$ becomes $\epsilon_{i}\epsilon_{j}a_{ij}$ and to
the set $\{a_{2,1},a_{3,2},...,a_{n,n-1}\}$ corresponds the set $\{\epsilon_{1}\epsilon_{2}a_{2,1},\epsilon_{2}\epsilon_{3}a_{3,2},...,\epsilon_{n-1}\epsilon_{n}a_{n,n-1}\}$. Hence the $|a_{i,j}|$ are not changed by the action of the matrices $Diag (\epsilon_{1},\epsilon_{2},...,\epsilon_{n})$.
\textbf{Example 2}. If $n=3$, $ g(1)=
\begin{pmatrix}
1 & 0 & 0 \\
a & 1 & 0 \\
c & b & 1
\end{pmatrix}$
with $a,b,c \in \mathbb{Z}$
and $\varphi=
\begin{pmatrix}
1 & 0 & 0 \\
u & 1 & 0 \\
0 & v & 1
\end{pmatrix}$
with $u,v \in \mathbb{Z}$, because $\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
w & 0 & 1
\end{pmatrix}$ ,
with $w \in \mathbb{Z}$, is an element of $Z(T^{-}_{3}(\mathbb{Z}))$. Then $\varphi^{-1}=
\begin{pmatrix}
1 & 0 & 0 \\
-u & 1 & 0 \\
uv & -v & 1
\end{pmatrix}$ , $g(1)\varphi=
\begin{pmatrix}
1 & 0 & 0 \\
u+a & 1 & 0 \\
c+bu & b+v & 1
\end{pmatrix}$, and $\varphi^{-1}g(1)\varphi=
\begin{pmatrix}
1 & 0 & 0 \\
a & 1 & 0 \\
c+bu-av & b & 1
\end{pmatrix}$.
A simple observation of the coefficient $c':= c+bu-av $ permits the choice of $u$ and $v$ so that $0\leq c'< gcd (a,b)\leq Min(a,b)$. Hence if $ gcd (a,b)=1$ there is only one lattice, whereas if $gcd (a,b)= a>0$ there are $a$ lattices.
As a consequence, two lattices corresponding to $c'_{1}$ and $c'_{2}$ are isomorphic if and only if $c'_{1}+ (bu-av)=c'_{2}$.
We can prove :
\textbf{Proposition 2.} For any point $p\in({\mathbb N}^{*})^{n-1}$ and $\theta$ as in Remark 4, the set $\theta^{-1}(p)$ is finite. More precisely:
Given $g(1)=\begin{pmatrix}
\ 1 & 0 \\
(a_{ij}) & 1
\end{pmatrix}$ as in Theorem 3, there exists $\varphi\in T^{-}_{n}(\mathbb{Z})$ such that $\varphi^{-1}g(1)\varphi=
\begin{pmatrix}
\ 1 & 0 \\
(a'_{ij}) & 1
\end{pmatrix}$ with $ 0\leq a'_{i,j}<a'_{j+1,j}$ for $i>j+1$.
\textbf{Proof.} Denote , in the proof, by $g$ the matrix $g(1)$ and by $g(e_{i})$ the $i$-column of $g$. First of all, by conjugation, we can suppose the coefficients of the first subdiagonal of $g(1)=\begin{pmatrix}
\ 1 & 0 \\
(a_{ij}) & 1
\end{pmatrix}$ strictly positive. Order the elements $a_{i,j}$ as follows:
\begin{center}
$a_{3,1},a_{4,2},...,a_{n,n-2},a_{4,1},a_{5,2},...,a_{n,n-3},a_{5,1},...,a_{n,1}$
\end{center}
A Euclidean division gives $ a_{3,1}=qa_{2,1}+a'_{3,1}$ with $a'_{3,1}\in[0,a_{2,1}-1]$ and the first three columns becomes $ g(e_{1})=e_{1}+a_{2,1}e_{1}+(qa_{2,1}+a'_{3,1})e_{3}+...$ ,
$ g(e_{2})=e_{2}+a_{3,2}+...$ and $ g(e_{3})=e_{3}+...$.Take as a new basis $\{e'_{i};0\leq i\leq n\}$ with $e'_{2}:=e_{2}+qe_{3}$ and $e'_{i}:=e_{i}$ if $i\neq 2$. We have then
$g(e'_{1})=e'_{1}+a_{2,1}e'_{2}+a'_{3,1}e'_{3}+a_{4,1}e'_{4}+...$ ; $g(e'_{2})=e'_{2}+a_{3,2}e'_{3}+...$. After an iteration the Euclidean division for $a_{4,2}$ and $a_{3,2}$ etc,we can hence suppose that $a_{3,1}\in[0,a_{2,1}-1],...,a_{n,n-2}\in[0,a_{n-1,n-2}-1],...,a_{i,j}\in[0,a_{j+1,j}-1]$.
The next step concerns $a_{i+1,j+1}$ if $i<n$ or $a_{n-j+2,1}$ if $i=n$. In the first case, if we make the Euclidean division $a_{i+1,j+1}=qa_{j+2,j+1}+a'_{i+1,j+1}$, the $j+1$-column of $g$ can be writen
\begin{center} $g(e_{j+1})=e_{j+1}+a_{j+2,j+1}e_{j+2}+...a_{i,j+1}e_{i}+(qa_{j+2,j+1}+a'_{i+1,j+1})e_{i+1}+...$
\end{center}
Consider the new basis $B'=\{e'_{i} ; 1\leq i\leq n\}$ with $e'_{j+2}:=e_{j+2}+qe_{i+1}$ and $e'_{i}:=e_{i}$ for the other indices. In the basis $B'$ the only coefficient changed in the $j+1$-column $g(e'_{j+1})$ is in the $i+1$ line where $a'_{i+1,j+1}$ replaces $a_{i+1,j+1}$. In the other columns the coefficients which change are after $a_{i+1,j+1}$ in the order adopted for the matrix terms.
The second case is treated as for $a_{3,1}$ and the proof is over.
\hfill $\square$\\
The lattices in $F_{n}$ with $a_{i+2,i+1}=1$ for every $i$, are all isomorphic and a basis change, as in the proof, reduces to the case where all $a_{i,j}$ with $i>j+1$ are zero : this lattice is $N(\mathbb{Z})$.
For this lattice $\Gamma_{0}$, we have $C_{i}\Gamma_{0}=C^{n-i}\Gamma_{0}$ for $0<i<n$, as one can easily verify using the presentation of $\Gamma_{0}$ :
\begin{center}
$\Gamma_{0}=<y_{1},...,y_{n},z>$ with $y_{i}y_{j}=y_{j}y_{i}$ for every $i,j$
$zy_{i}=y_{i}y_{i+1}z$ for $i<n$ and $zy_{n}=y_{n}z$.
\end{center}
For a general lattice $\Gamma$, the group $C^{n-i}\Gamma$ is contained in $C_{i}\Gamma$ for $0<i<n$ and the factor groups $C_{i}\Gamma/C^{n-i}\Gamma$ are finite Abelian groups. Their cardinalities can be computed in terms of the coefficients $a_{j+1,j}$. These assertions are obtained using the following presentation of $\Gamma$ as semidirect product:
\begin{center}
$\Gamma=<y_{1},...,y_{n},z>$ with $y_{i}y_{j}=y_{j}y_{i}$ for every $i,j$ ; $zy_{n}=y_{n}z$ and for $i<n$, $zy_{i}=y_{i}z_{i}z$ , where $z_{i}=y_{i+1}^{a_{i+1,i}}y_{i+2}^{a_{i+2,i}}...y_{n}^{a_{n,i}}$
\end{center}
Notice that if $C_{i}\Gamma=C^{n-i}\Gamma$ for all $i$, the lattice $\Gamma$ is isomorphic to $\Gamma_{0}$. In fact if $C^{1}\Gamma=C_{n-1}\Gamma$, we already have $\Gamma\cong\Gamma_{0}$. Moreover, two nonisomorphic lattices corresponding to matrices with the same first subdiagonal, may have isomorphic factor groups.
\textbf{Example 3}. Let $n=3$ and consider the lattice $\Gamma$ with $g(1)$ as in \textbf{Example 2} and $0\leq c< gcd (a,b)\leq Min(a,b)$. The factor group $C_{1}\Gamma/C^{2}\Gamma$ is isomorphic to $C_{ab}$ independently of $c$. The other factor group
$C_{2}\Gamma/C^{1}\Gamma$ is the quotient of $\mathbb{Z}^{2}$ by the lattice generated by $(a,c)$ and $(0,b)$. Moreover, it is easy to see that if $\delta=gcd(a,b,c)$, this factor group is $C_{\delta}\times C_{ab\delta^{-1}}$. This shows that two nonisomorphic lattices may have isomorphic factor groups. For instance take $a=6, b=9, c_{1}=1$ for $\Gamma_{1}$ and $a=6, b=9, c_{2}=2$ for $\Gamma_{2}$; since $\delta=1$ for both of them, $C_{2}\Gamma/C^{1}\Gamma$ is cyclic of order $54$ for both lattices.
\section { Some Geometry of the corresponding affine compact nilmanifolds}
The Lie groups considered in section 2 are symplectic Lie groups (see [13]) and hence their left invariant symplectic form $ \Omega^{+}$ defines a left invariant affine structure $\nabla$ on the group by the formula (1). Since these groups are unimodular, this connection is geodesically complete ([6]). Consequently the quotient manifold $M=\Gamma\backslash G $, where $\Gamma$ is a lattice in $ G $, is a compact manifold endowed with a symplectic form and a complete flat and torsion free linear connection $\nabla$. In fact, as these Lie groups are quadratic ( i.e. are endowed with a bi-invariant semi-Riemannian metric ) and the connection $\nabla$ on $G$ is the Levi-Civita connection of a flat left invariant semi-Riemannian metric ([7]), then $M$ carries also a flat semi-Riemannian metric.
In the other hand, a direct calculation shows that the $2n$-dimensional Lie algebra ${\cal L}_{2n-1}(\mathbb{R})$, with bracket $ [e_{0},e_{i}]=e_{i+1}$ for $i=1,2,...,2n-2$ , has a scalar non degenerate cocycle given by $\Omega=\sum_{i=0}^{n-1}(-1)^{i}e_{i}^{*}\wedge e_{2n-i-1}^{*}$. Hence the groups $F_{2n-1}$ are symplectic Lie groups. Moreover as any Lie algebra ${\cal L}_{k}$ has a non singular derivation $D$ its Lie bracket is compatible with the left symmetric product $ab=L_{a}b$ where $L_{a}=D^{-1}ad_{a}D$. Recall also that if $F_{n}$ is odd dimensional its carries a left invariant contact form ([13]).
In short, in the following, we deal with \textbf{ affine compact and geodesically complete nilmanifolds}.The Euler characteristic of a such manifold is always zero (Kostant and Sullivan).
Let us now study \textbf{some special diffeomorphisms of the manifolds $M=\Gamma\backslash G $}. As the groups are nilpotent and simply connected, an automorphism of the Lie group $ G $, preserving a lattice $\Gamma$, is fully determined by its restriction to $\Gamma$ (see [16]). Obviously any automorphism of $G$ stabilizing $\Gamma$, determines a diffeomorphism of $M$.
Consider the Heisenberg Lie group $G:=H_{1}(\mathbb{K})$ where $\mathbb{K}$ is $\mathbb{C}$, $\mathbb{R}\times \mathbb{R}$ or $\mathbb{D}$. The map $\phi:G\rightarrow G$ where the image by $\phi:=\phi_{\alpha,\beta,\gamma}$ of the matrix
\begin{center}
$\begin{pmatrix}
1 & x & z \\
0 & 1 & y \\
0 & 0 & 1
\end{pmatrix}$ is the matrix
$\begin{pmatrix}
1 & \alpha x & \gamma z \\
0 & 1 & \beta y \\
0 & 0 & 1
\end{pmatrix}$
\end{center}
\begin{flushleft}is an automorphism of $G$ if and only if $ \gamma=\alpha\beta$ with $\alpha,\beta,\gamma$
invertibles in $\mathbb{K}$
\end{flushleft}
Let $K$ a quadratic number field and $\Gamma$ the lattice of $G$ where $x,y,z$ are integers of $K$. The map $\phi$ will give an automorphism of $G$, preserving $\Gamma $, if and only if $\alpha,\beta,\gamma$ are invertible elements of the ring ${\cal O}_{K}$ of integers in $K$. Recall that if $K$ is complex, the group of units of ${\cal O}_{K}$ is finite: $C_{2}, C_{4}$ or $C_{6}$. When $K$ is real, the group of units is infinite and direct product of $C_{2}$ by $\mathbb{Z}$, cyclic group generated by a so-called fundamental unit $\epsilon$. This means that any unit of ${\cal O}_{K}$ can be written as $\epsilon^{n}$ or $-\epsilon^{n}$with $n$ in $\mathbb{Z}$. For example, for $K$ the quadratic fields $\mathbb{Q}(\sqrt{k})$ with $k=2,3,5$ the corresponding $\epsilon$ is respectively $1+\sqrt{2}, 2+\sqrt{3}$ and $\frac{1+\sqrt{5}}{2}$.
Let $f$ be an arbitrary automorphism of the Lie group $G$. Obviously $f|Z(G)$ is an automorphism of its center $Z(G)\equiv \mathbb{R}^{2}$ and its characteristic polynomial has two roots. For $f=\phi_{\alpha,\beta,\gamma}$ the map $f|Z(G)$ corresponds to the map $ K\otimes_{\mathbb{Q}}\mathbb{R}\rightarrow K\otimes_{\mathbb{Q}}\mathbb{R}$, $z\otimes\lambda\mapsto \gamma z\otimes\lambda $ and the eigenvalues are the values of $\gamma$ for the two embeddings of $K$ in $\mathbb{R}$ that is $\gamma(\sqrt{d})$ and $\gamma(-\sqrt{d})$.
In the other hand, $f$ induces an automorphism $\overline{f}$ of the 4-dimensional Abelian Lie group $G/Z(G)$ and hence has four characteric roots. In the case $f=\phi_{\alpha,\beta,\gamma}$, the map $\overline{f}:K^{2}\otimes_{\mathbb{Q}}\mathbb{R}\rightarrow K^{2}\otimes_{\mathbb{Q}}\mathbb{R}$ is the extension to $\mathbb{R}$ of the correspondence $(x,y)\mapsto(\alpha x,\beta y)$ and the eigenvalues of $\overline{f}$ are the images of $\alpha$ and $\beta$ by the real embeddings of $K$.
If $\alpha=\epsilon^{n}$ and $\beta=\epsilon^{m}$ then $f=\phi_{\alpha,\beta,\gamma}$, six eigenvalues. Moreover if $nm(n+m)\neq 0$, three of these eigenvalues have modulus $>1$ and the others modulus $<1$ and consequently $f$ induces an \textbf{Anosov diffeomorphism} of the space of right cosets $\Gamma\backslash G $.
In summary , if $U$ denotes the group of units of the ring ${\cal O}_{-d}$ and $Aut_{\Gamma}G$ is the group of automorphisms of $G$ preserving $\Gamma$ we have:
\textbf{Lemma 3.} The map $(\alpha,\beta)\mapsto \phi_{\alpha,\beta,\alpha\beta} $ is an injective homomorphism of groups from $U\times U$ into $Aut_\Gamma G$. In the complex case , $U$ is finite and the eigenvalues of $\phi_{\alpha,\beta,\alpha\beta} $ have modulus 1. In the real case , $U$ is infinite and eigenvalues have inverse modules by pairs.
\textbf{Automorphisms of the Lie group $G:= T^{*}H_{1}(\mathbb{R})$}. In fact, it is easy to describe all the automorphisms of the Lie group $G$. We will only look for the automorphisms of the lattice $\Gamma:=\Gamma_{1,1,1}$ using its presentation
\begin{center}
$y_{2}y_{3}=y_{3}y_{2}z_{1}$ , $y_{3}y_{1}=y_{1}y_{3}z_{2}$ , $y_{1}y_{2}=y_{2}y_{1}z_{3}$
\end{center}
given in section 2. Consider three elements $y'_{1},y'_{2},y'_{3}$ in $\Gamma$ such that their canonical images in $\Gamma/Z\Gamma$ generate a lattice of $G/ZG$. This means that we can write
\begin{center}
\begin{equation}
y'_{1}=y_{1}^{a_{1}}y_{2}^{a_{2}}y_{3}^{a_{3}}z'_{1} ,
y'_{2}=y_{1}^{b_{1}}y_{2}^{b_{2}}y_{3}^{b_{3}}z'_{2} ,
y'_{3}=y_{1}^{c_{1}}y_{2}^{c_{2}}y_{3}^{c_{3}}z'_{3}
\end{equation}
\end{center}
where $a_{i},b_{j},c_{k}$ are integers. The elements $y'_{1},y'_{2},y'_{3}$ generate $\Gamma$ if and only if the matrix
\begin{center}
$\begin{pmatrix}
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2} \\
a_{3} & b_{3} & c_{3}
\end{pmatrix}$
\end{center}
is in $GL(3,\mathbb{Z})$ and the $z'_{i}$ are arbitrary elements of $Z(\Gamma)$. The commutators $<y'_{2},y'_{3}>, <y'_{3},y'_{1}> $ and $<y'_{1},y'_{2}> $ are in $Z(\Gamma)$ and generate it. Thus there is an unique automorphism of $\Gamma$ such that the image of $y_{i}$ is $y'_{i}$ for $i=1,2,3 $.
Hence, using the presentation of $\Gamma:=\Gamma_{1,1,1}$ given in the proof of Theorem 2, we have :
\textbf{Lemma 4.} For $y'_{i}$, with $i=1,2,3$, in $\Gamma:=\Gamma_{1,1,1}$ as in (8), $z'_{i} \in Z(\Gamma)$ and the above matrix in $GL(3,\mathbb{Z})$ , there exists a unique automorphism $\phi$ of $\Gamma$ such that $\phi(y_{i})=y'_{i}$.
Let $f$ be an automorphism of $\Gamma:=\Gamma_{1,1,1}$. Denote by $f|Z(\Gamma)$ its restriction to the center of $\Gamma$ and by $\overline{f}$ the canonical map induced by $f$ on $ \Gamma/Z(\Gamma)$. Since $Z(\Gamma)$ and $ \Gamma/Z(\Gamma)$ are free $\mathbb{Z}$- modules of rank $3$ , we have associated to $f$ two invertible matrices $ A$ and $B$ with entries in $\mathbb{Z}$. Taking basis $ (z_{1}, z_{2},z_{3})$ and $(\overline{y_{1}},\overline{y_{2}},\overline{y_{3}})$ of these groups such that $ <y_{2},y_{3}>=z_{1} , , $, from the relations between $y_{i}$ and $z_{j}$ we have $A=\Lambda^{2}B$. As $det B =1$ or $det B=-1$, it follows $A= (det B) B^{-1}$. Hence, the characteristic polynomials of $B$ and $A$ are respectively given by
\begin{center}
$ p_{B}(X)=X^{3}-(TrB)X^{2}+(Tr \Lambda^{2}B)X-detB $
$q_{A}(X)=X^{3}-(Tr \Lambda^{2}B)X^{2}+((detB)^{-1}TrB)X-1$
\end{center}
Consequently we obtain :
\textbf{Lemma 5.} Let $f$ be an automorphism of the lattice $\Gamma:=\Gamma_{1,1,1}$ of the group $G:= T^{*}H_{1}(\mathbb{R})$. The six eigenvalues associated to the automorphisms $f|Z(\Gamma)$ and the induced one on $\Gamma/Z(\Gamma)$ , are algebraic integers. The product of the eigenvalues of $B$ is $1$ or $-1$ whereas the product of the eigenvalues of $A$ is always $1$ and up to the sign they are the inverses of the preceeding ones. If none have modulus unity, half have modulus bigger than one and the other modulus smaller than one. This happens if and only if no root of unity appears as eigenvalue.
\textbf{ Example 4}. Consider the matrix of $GL(3,\mathbb{Z})$,
\begin{center}
$\begin{pmatrix}
1 & 5 & 2 \\
2 & -1 & -1 \\
3 & 2 & 0
\end{pmatrix}$
\end{center}
\begin{flushleft}
Its characteristic polynomial is $ X^{3}-15X-1$ which has no root of modulus $1$. Hence the automorphism of $G:= T^{*}H_{1}(\mathbb{R})$ associated to this matrix determines an \textbf { Anosov diffeomorphism } of the manifold $\Gamma\backslash G$.
\end{flushleft}
\textbf{Automorphisms of the Lie group $G:=F_{n}$}.
Let $\Gamma_{0}:=\mathbb{Z}^{n}\times \mathbb Z $ be the lattice of $G:=F_{n}$ , described in the subsection 3.2. An automorphism $ f $ of $\Gamma_{0}$ leaves invariant its central descending sequence. In terms of the generators $ y_{1}, y_{2},...y_{n},z $ this means that the Abelian subgroup $C^{1}\Gamma_{0}=<y_{2},...,y_{n}>$ and the flag $ <y_{i},y_{i+1},...,y_{n}>$ are invariant by $f$. Hence $f$ acts on the subgroups of $\Gamma_{0}$ containing $C^{1}\Gamma_{0}$. But all Abelian such subgroups are contained in
${\cal{M}}:= < y_{1},...,y_{n}>$ ; consequently $\cal{M}$ is invariant by $f$. So the restriction of $f$ to $\cal{M}$, in the $\mathbb{Z}$-basis $(y_{1},...,y_{n})$, is given by a lower triangular
matrix with integers entries and diagonal terms $1$ or $-1$. Also, $f$ induces an automorphism of $\Gamma_{0}/\cal{M}$ and hence the image of $z$ by $f$ is $zm$ or $z^{-1}m$ where $m$ is any element of $\cal{M}$.
Notice , in particular, that the eigenvalues of $f$ are $1$ or $-1$. Putting $f(y_{i})=y_{i}^{\epsilon_{i}}$ with $\epsilon_{i}\in\{1,-1\}$ and $f(z)=zm$ and applying $f$ to the relation $zy_{i}=y_{i}y_{i+1}z$ we get $zmy_{i}^{\epsilon_{i}}=y_{i}^{\epsilon_{i}}y_{i+1}^{\epsilon_{i+1}}zm $. As $my_{i}=y_{i}m $ then $zy_{i}^{\epsilon_{i}}=y_{i}^{\epsilon_{i}}y_{i^+1}^{\epsilon_{i+1}}z$ i.e. $\epsilon_{i}=1$ implies $\epsilon_{i+1}=1$. In a similar way $\epsilon_{i}=-1$ implies $\epsilon_{i+1}=-1$.
\textbf{Toral Affine Symplectic actions }.
By definition, the action $L^{G}$ on a symplectic Lie group $(G,\Omega^{+})$ given by $L^{G}_{\sigma}: \tau\mapsto \sigma\tau$, is symplectic. So, the right invariant vector fields $x^{-}$ with $x\in {\cal G}$ on $G$ are symplectic ( or locally Hamiltonian ). If $G$ is simply connected, these vector fields are Hamiltonian , that is, the action $L^{G}$ is Hamiltonian. In fact , in this case, a direct calculation ( see [10])shows that the map $Q
:exp x\mapsto \sum_{k=1}^{\infty} \frac{1}{k!}( ad^{*}_{x})^{k-1}.\Omega(x,.)$ is an equivariant moment map for $L^{G}$ and the action $\rho_{G}$ of $G$ on ${\cal G}^{*}$ given by $\sigma\mu:=Q(\sigma)+Ad^{*}(\sigma)(\mu)$. In particular we have :
\begin{center}
$i(x^{-})\Omega^{+}=-dQ_{x}$ , where $Q_{x}(expy)=<Q(expy),x>$ with $x,y \in {\cal G}$.
\end{center}
As $Q(\sigma\tau)= Q(\sigma)+Ad_{\sigma}^{*}(Q(\tau))$ for any $\sigma,\tau$ in $G$ we will say ([10]) that $Q$ is moment cocycle. Moreover since , $\sigma=exp x\mapsto (Q(\sigma),Ad^{*}(\sigma))$ is a representation of $G$ by affine transformations of $ {\cal G}^{*}$ and $Q$ is a diffeomorphism , $Q$ is a developing map of the affine structure on $M:=\Gamma\backslash G$ with $\Gamma$ any lattice of $G$ and $\nabla$ is geodesically complete. Notice that $Q(\epsilon)=0$, where $\epsilon$ stands for the unit of $G$ , and $h:\gamma\mapsto h(\gamma)=(\gamma,Q(\gamma))$ is the holonomy representation of the affine structure of $M$. We have then:
\begin{center}
$h(\gamma)\circ Q = Q\circ \gamma$ for any $\gamma\in\Gamma$
\end{center}
So, $M=\Gamma\backslash G$ can be identified to the manifold $h(\Gamma)\backslash{\cal G}^{*} $, with $\Gamma$ acting properly and freely on ${\cal G}^{*}$ via $h$. The tangent bundle $TM$ identifies with the fiber product
\begin{center}
$E_{h}:=G\times_{h}{\cal G}^{*}=(G\times {\cal G}^{*})/\Pi_{1}(M)$
\end{center}
\begin{flushleft}
where $\Pi_{1}(M)=\Gamma $ acts diagonally by deck transformations on the factor $G$ and via $h$ on the factor ${\cal G}^{*}$.
\end{flushleft}
The developing map $Q$ defines an isomorphism of vector bundles of $TM$ with $E_{h}$. The flat connection $\nabla$ on $TM$ arise from the representation $h$ in the standard way. For the structure of symplectic Lie groups the reader can refer to [10]. Recall that $M$ is endowed with a Lagrangian foliation (see [17]).
Let $ (u_{1},u_{2},...,u_{n})$ a basis of $ {\cal G}$ and $p:G\rightarrow M$ the canonical projection. It is clear that $p$ is a symplectic and affine covering map. The global system of affine coordinates $Q_{u_{i}}(\sigma):=Q(\sigma)(u_{i})$ for $i=1,2,...n$ on $G$ determines a local system of affine coordinates $(x_{1},x_{2},...,x_{n})$ on $M$ given by $\Omega^{*}(X_{i},.)=dx_{i}$, where $ \Omega^{*}$ is the symplectic form on $M$ and $X_{i}$ is a local vector field on $M$ verifying $p_{*}(u_{i}^{-})=X_{i}$. Using the coordinates $x_{i}$, Boucetta and the second author have shown in [12] the following facts:
1) The Poisson bracket corresponding to $\Omega^{*}$ is polynomial of degree 1.
2) The symplectic form $\Omega^{*}$ is polynomial of degree at most $n-1$ where $n$ is the dimension of $M$.
3) The volume form $(\Omega^{*})^{n}/n!$ is parallel relative to $\nabla$.
4) Any solution of the classical Yang-Baxter equation $ {\cal G}$ gives arise to determines a polynomial Poisson tensor on $M$.
In the following $G$ is nilpotent and simply connected and $\Gamma$ is a lattice in $G$. Notice that the action $\Psi$ of $Z(G)$ on $M$, deduced of $L^{Z(G)}$, is affine and symplectic. As the stabilizer of any point in $M$, under $\Psi$, is $Z(G)\cap\Gamma $, the torus $ T^{*}:=Z(G)/Z(G)\cap \Gamma $ acts freely on $M$. Consequently, using a Corollary of the slice Theorem of Koszul (see [18], page 180 ), $M/T^{*}$ is a manifold and $M$ is a principal $T^{*}$ -bundle over $M/T^{*}$. Obviously the canonical projection of $M$ on $M/T^{*}$ is an affine map.
In the other hand any closed subgroup $H$ of $G$ acts naturally, via the action $L^{H}$, by symplectomorphisms of $G$ and the map $Q_{H}=i'\circ Q$ , with $i'$ the transpose of the canonical injection $i$ of the Lie algebra of $H$ in ${\cal G}$, is a $\rho_{H}$ equivariant invariant moment map relative to $L^{H}$ . Notice that any value of $Q_{H}$ is a regular value.
Recall now that the bracket of two symplectic vector fields is Hamiltonian and suppose $G$ non Abelian. Consequently if we take $H:=Z(G)\bigcap D(G)$ , the action $L_{T'}$ of the torus $T':= H/H\bigcap Z(\Gamma)$ on $M$ becomes Hamiltonian. More precisely if $X,Y$ are symplectic vector fields on $M$ then $[X,Y]$ is Hamiltonian with Hamiltonian function $ \Omega^{*}(Y,X)$. Hence we have (see [18]):
\textbf{Lemma 6.} The action of the torus $T':= H/H\bigcap Z(\Gamma)$ on $M=\Gamma\backslash G$ induced by $L^{H}$ is Hamiltonian if $G$ is nilpotent and non Abelian. In particular if $G=F_{n}$ there is a Hamiltonian action of a circle on $M$ and, following Duistermaat-Heckman, the stationary-phase approximation is in this case exact.
The determination of the Duistermaat-Heckman function in the Lemma could be interesting ([18],page 70).
Denote by $Q_{T'}$ the corresponding moment map in the Lemma 6. According to the convexity theorem of Atiyah and Guillemin-Sternberg ([8], page 59 ), the image of $Q_{T'}$ is the convex hull of the image of the $T'$ fixed point set of $M$. Moreover, in the case where dim $Z(G)\bigcap D(G)$ is the half of dim $ M $ the image of the moment map is necessarily a Delzant polytope ([19],page 75)) because the action $\Psi_{T}$ is effective. This is the case for example if $G:= T^{*}H_{1}(\mathbb{R}) $.
\textbf{Theorem 4.} Let $(G,\Omega^{+})$ be a symplectic nilpotent non Abelian and simply connected Lie group and let $\tau\in Z(G)\bigcap D(G)\bigcap\Gamma$ with $\tau\neq\epsilon$, $\tau=expz$ and $H:=exp(\mathbb{R}z)$. Then the action $L_{T}$ of the cercle $T:=H/exp(\mathbb{Z}z )$ on $M$ given by $\widehat{\tau_{1}}.\overline{\sigma}:=\overline{\tau_{1}\sigma}$, deduced of $L^{H}$, is Hamiltonian and the corresponding moment map $Q_{T}$ is $\rho_{T}$ - equivariant. Moreover
$N:= T\backslash M$ is an affine manifold, $T$ acts on $M$ by (local) translations and $M$ is an affine $T$-principal bundle over $N$. Finally, the origin of $(\mathbb{R}z)^{*}$ is a regular value of $Q_{T}$, the set level $Q_{T}^{-1}(o)$ is an affine hypersurface of $M$ and
$R:=T\backslash Q_{T}^{-1}(o)$ is a reduced (affine) symplectic nilmanifold of $M$ by means of $T$. In fact we have the following canonical sequences of affine manifolds and affine maps :
\begin{center}
$Q_{T}^{-1}(0)\leftharpoonup M\rightarrow (Lie(T))^{*}$
\end{center}
\begin{center}
$T\leftharpoonup Q_{T}^{-1}(0)\rightarrow R $
\end{center}
\textbf{Proof.} First of all notice that $H$ is a central subgroup, $H\bigcap \Gamma$ is a lattice in $H$ and $T$ is a circle. As above, the action of $T$ on $M$ is symplectic and Hamiltonian. Moreover, as $ p $ is a local symplectomorphism, if $ z^{*} $ is the symplectic vector field on $M$, we have $p_{*}(z^{-})= z^{*}$. Consequently the regular value $0$ of $Q_{H}$ is also a regular value of $Q_{T}$. Hence $Q_{T}^{-1}(0)$ is a hypersurface of $M$. A simple verification shows that the action of $T$ on $M$ is free. Using a $T$-invariant Riemannian metric on $M$ and the slice theorem of Koszul, it follows that $N:= T\backslash M$ is a manifold and $M$ is a $T$-principal bundle over $N$. The fact that $H$ acts on $G$ by translations ( see [10] ) implies that $T$ acts (in local coordinates ) on $M$ in a similar way. As a consequence, the canonical maps involved in the $T$-principal bundle are affine maps. Finally,the moment map $Q_{T}$ is $\rho_{T}$-equivariant and the stabilizer group $T_{0}$ , by the action $\rho_{H}$, acts on $Q_{T}^{-1}(0)$ freely and properly. Hence, using the procedure described by Marsden and Weinstein in [20], there exists a unique symplectic form $\Omega^{*}_{0}$ on $R$ with the property $\pi_{0}^{*} \Omega^{*}_{0}=i^{*}_{0} \Omega^{*}$ where $\pi_{0}:Q_{T}^{-1}(0)\rightarrow R$ is the canonical projection and $i_{0}:Q_{T}^{-1}(0)\rightarrow M$ is the canonical inclusion.
\hfill $\square$\\
\textbf {Fundamental group of the reduced manifold.} Given a $2n$-dimensional compact symplectic nilmanifold $( \Gamma\backslash G,\Omega^{*})$, it is natural to ask if we can, in general, easily describe the fundamental group of a $2n-2$ symplectic reduced manifold $N$ of $M$ obtained , as shown by Theorem 4, in terms of $\Gamma $. The response to this question is in general negative, as shown the following analysis.
Recall some facts about the structure of simply connected nilpotent symplectic Lie groups. Denote by $H^{\bot}$ the connected component of the unit in $Q_{H}^{-1}(0)$. Then $H^{\bot}$ is a closed subgroup of $G$ which is $ \Omega^{*}$-orthogonal to $H$ and $K:=H\backslash H^{\bot}$ is a reduced symplectic Lie group. Moreover we have the exact canonical sequence of affine Lie groups
\begin{center}
$ \{\epsilon\}\leftharpoonup H\leftharpoonup H^{\bot}\rightarrow H\backslash H^{\bot}\rightarrow \{\epsilon\}$
\end{center}
and the canonical affine principal fiber bundle
\begin{center}
$ \{\epsilon\}\leftharpoonup H^{\bot}\leftharpoonup G\rightarrow H^{\bot}\backslash G$
\end{center}
\begin{flushleft}
where the projection can be identified with the moment map $Q_{H}$ (see [10]).
\end{flushleft}
Let $\Gamma$ be a lattice in $G$. If $H^{\bot}$ admits lattices and the behaviour of $\Gamma$ relatively to $H$ and $H^{\bot}$ is nice i.e. $H\bigcap \Gamma$ and $H^{\bot}\bigcap\Gamma$ are lattices in $H$ and $H^{\bot}$, we obtain a compact symplectic reduced manifold $R'$ of $M$, with fundamental group $\Gamma':=(\Gamma\bigcap H)\backslash \Gamma\bigcap H^{\bot}$, by means of the compact affine manifold $(H^{\bot}\bigcap\Gamma)\backslash H^{\bot}$ or the symplectic Lie group $H\backslash H^{\bot} $.
Notice that the manifolds involved in the last sequences are covering spaces of the manifolds involved in the sequences described in the Theorem 4.
\textbf {Example 5.} Let $G$ be the Lie group given by the manifold $\mathbb{R}^{6}$ endowed with the product.
\begin{center}
$ (x_{i},y_{j})(x'_{i},y'_{j})= (x_{i}+x'_{i},y_{j}+y'_{j}+x_{k}x'_{l})$
\end{center}
where $\{j,k,l\}$ is a cyclic permutation of $\{1,2,3\}$. The 1-parameter subgroups of $G$ correspond to the maps
\begin{center}
$\varphi:t\rightarrow (tx_{i},ty_{j}+ \frac{t(t-1)}{2}x_{k}x_{l})$
\end{center}
and then
\begin{center}
$\frac{d\varphi}{dt}(0)=(x_{1},x_{2},x_{3},y_{1}-\frac{x_{2}x_{3}}{2}, y_{2}-\frac{x_{3}x_{1}}{2},y_{3}-\frac{x_{1}x_{2}}{2})$
\end{center}
and
$\varphi(1)= (x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})$.
Consequently the exponential map $ exp: {\cal G}\rightarrow G$ is given by
\begin{center}
$(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})\rightarrow (a_{1},a_{2},a_{3},b_{1}+\frac{a_{2}a_{3}}{2},b_{2}+\frac{a_{3}a_{1}}{2},b_{3}+b_{1}+\frac{a_{1}a_{2}}{2})$
\end{center}
\begin{flushleft}
whereas the logarithm map is described by
\end{flushleft}
\begin{center}
$ln: G\rightarrow {\cal G}, (x_{i},y_{j})\longmapsto (x_{i},y_{j}-\frac{y_{k}x_{l}}{2})$
\end{center}
Let us look for the scalar 2-cocycle $\Omega$ on ${\cal G}$. As $D({\cal G})=Z({\cal G})$ then $\Omega$ is totally isotropic on $D({\cal G})$. Using the natural basis $\{e_{i},f_{j}\}$ of ${\cal G}$ we have the condition $\Sigma^{i=3}_{i=1}\Omega(e_{i},f_{i})=0$. Up to a 2-coboundary, the matrix of $\Omega$ can be written
\begin{center}
$\begin{pmatrix}
0& -B^{t} \\
B & -0 \\
\end{pmatrix}$
\end{center}
where $B$ is a square matrix of size 3 and trace 0, invertible if $\Omega$ is non degenerate.
Take for $\Gamma$ the lattice of $G$ with $x$ and $y$ integers. Any 1-dimensional central subgroup $H$ of $G$ which meets $\Gamma$ contains an element $z=(0,0,0,y_{1},y_{2},y_{3})$ where the $y_{i}$ are coprime integers. Using an automorphism of $G$ leaving $\Gamma
$ fixed, $H$ becomes $H=\{(0,0,0,0,0,t), t\in \mathbb{R}\}$. Putting $B=(b_{ij})$ we have then, $I=\mathbb{R}f_{3}$ and $I^{\bot}=V\oplus Z({\cal G})$ with
\begin{center}
$V=\{(a_{1},a_{2},a_{3}); a_{1}b_{31}+a_{2}b_{32}+a_{3}b_{33}=0\}$.
\end{center}
\begin{flushleft}
Hence $ exp( I^{\bot})=\{(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}) ; \Sigma^{i=3}_{i=1}x_{i}b_{3i}=0\}$
\end{flushleft}
We want to describe $\Gamma':=\Gamma\bigcap exp( I^{\bot})$. If the $b_{3,i}$ are $\mathbb{Q}$-linearly independent, the $x_{i}$ are zero and $\Gamma'=Z(\Gamma)$.
If the $\mathbb{Q}$-vector space $W$ generated by the set $\{b_{3i}; i=1,2,3 \}$ has dimension 2, we have $(x_{1},x_{2},x_{3})= d(t_{1},t_{2},t_{3})$ with $d$ and $t_{i}$ in $\mathbb{Q}$. So , $\Gamma'=\{(nu_{1},nu_{2},nu_{3},v_{1},v_{2},v_{3}) ; n,v_{i}\in \mathbb{Z}\}$ where $(u_{1},u_{2},u_{3})$ is an unimodular element of $\mathbb{Z}^{3}$. Finally, if $W$ is 1-dimensional, the relation $ \Sigma^{i=3}_{i=1}x_{i}b_{3i}=0$ tells that there exist a non null linear form $f$ on $ \mathbb{Z}^{3}$ such that
\begin{center}
$ \Gamma'=\{(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}) ; x_{i},y_{j} \in \mathbb{Z}, f(x_{1},x_{1},x_{1})=0\}$
\end{center}
\textbf {In short $ \Gamma'$ is a lattice in the Lie group $H^{\bot}$ only in this last case.}
\textbf{Proposition 3}. Let $G=F_{2m-1}$ , $H=Z(G)$ and $\Gamma=L\times_{g(1)}\mathbb{Z}$ as in Proposition 1. Then $H^{\bot}$ do not depend of the left invariant symplectic form and $\Gamma\bigcap H$ and $\Gamma\bigcap H^{\bot}$ are lattices in $H$ and $H^{\bot}$ respectively. In fact, $H^{\bot}$is the integral subgroup of $G$ associated to the unique Abelian maximal ideal of ${\cal G}$ and $\Gamma_{1}:=L/Ker (d-Id)$ is a lattice of the Abelian Lie group $K:=H\backslash H^{\bot}$. Also, the torus $R:=\Gamma_{1}\backslash K $ is a reduced manifold of the symplectic nilmanifold $M:=\Gamma\backslash G$.
\textbf{Proof.} Let $\{e_{0},e_{1},...,e_{2m-1}\}$ be an adapted basis of ${\cal G}$ in the sense of Vergne i.e. $ad_{e_{0}}(e_{i})=e_{i+1}$ for $0 < i< 2m-1$ . A two scalar cocycle $\omega $ on ${\cal G}$ verifies $\omega (e_{i},e_{j})=0$ whenever $i+j$ is even or bigger than $2m-1$. It follows that
$ Z(\cal G)^{\bot}$ contains $V= Span \{e_{1},...,e_{2m-1}\}$ for every $\omega $. If $\omega $ is nondegenerate, $ Z(\cal G)^{\bot}$ has dimension $2m-1$, so is equal to $V$. This proves the first claim. The description of lattices of $F_{2m-1}$ in Proposition 1, implies the assertions concerning $\Gamma_{1}$.
In the other hand, the group $K$ is a symplectic Lie group, reduction of the symplectic Lie group $G$ by the Lie group $H$. Also, there exists a natural embedding, denoted in the following by $i$, of the torus $N:=(\Gamma\bigcap H^{\bot})\backslash H^{\bot}$ into the manifold $M$ and the natural action of the cercle $C:=(\Gamma\bigcap H)\backslash H$ on $N$ determines the compact manifold $R:=C\backslash N $. Moreover, as this action is free (and proper) a Corollary of the slice Koszul Theorem (see [18]) produces a $C$-principal fiber bundle
\begin{center}
$ C\rightarrow N \stackrel{\pi}{\rightarrow} R (*)$
\end{center}
Notice that the canonical action of $H$ on $H^{\bot}$ preserves the restriction of the symplectic form $\omega^{+}$ to $H^{\bot}$ and this restriction induces a left invariant symplectic form $\omega_{K}$.
Also, the canonical action $\phi$ of $C$ on $K$ is symplectic and $J_{K}(\overline{exp x)}=\sum((ad_{x}^{*})^{k-1}.\omega(x,.))$ , with $x$ in the Lie algebra of $H^{\bot}$, is a moment map for $\phi$. Finally, we have directly ( or using the procedure Marsden-Weinstein ) a symplectic form $\Omega_{R}$ on $R$ such that $\pi^{*}(\Omega_{R})=i^{*}(\Omega)$. In conclusion, the torus $R$ is a symplectic manifold of dimension $2n-2$ reduction of the $2n$ symplectic nilmanifold $M$.
Remark that that the objects and the maps involved in fibration (*) are affine maps. This also the case for
\begin{center}
$N \stackrel{i}{\rightarrow}M\stackrel{J_{K}}{\rightarrow}J_{K}^{-1}(\nu) (**)$
\end{center}
\begin{flushleft}
where $\nu $ is a regular value of $J_{K}$.
\end{flushleft}
Hence we can say, as in [10], that the symplectic manifold $M$ is a symplectic double extension of the symplectic manifold $R$.\hfill $\square$\\
\section {Symplectic Heisenberg Lie groups}
In this section $\mathbb{A}$ is a local associative and commutative $l$- dimensional real algebra and ${\cal H}_{k}(\mathbb{A})$ is the corresponding $l(2k+1)$-dimensional Heisenberg Lie algebra defined by $\mathbb{A}$. We will give a necessary and sufficient condition for the existence of a scalar nondegenerate $2$-cocycle on ${\cal H}_{1}(\mathbb{A})$. More precisely, if we decompose $\mathbb{A}$ as $\mathbb{A}=\mathbb{R}\oplus\mathbb{N}$, where $\mathbb{N}$ is the maximal ideal of $\mathbb{A}$ and we denote by $\mathbb{S}$ the annihilator of $\mathbb{N}$ or socle of $\mathbb{A}$, we have :
\textbf{Theorem 5.} The real Lie algebra ${\cal H}_{1}(\mathbb{A})$ is a symplectic Lie algebra if and only if $\mathbb{A}$ is even dimensional and $d=dim \mathbb{S}\leq 2$. Moreover for $k\geq 2$ the algebra ${\cal H}_{k}(\mathbb{A})$ is not symplectic.
\textbf{Proof.}The schema of the proof is as follows: if $d \geq 3$, we show that any scalar $2$-cocycle is degenerate ; if $ d=1$ then $\mathbb{A}$ is a Frobenius algebra and we describe a nondegenerate scalar $2$-cocycle. The case $ d=2$ requires a more detailed study.
It is clear that ${\cal H}_{1}(\mathbb{A})={\cal H}_{1}(\mathbb{R})\otimes \mathbb{A}= e\otimes \mathbb{A}\oplus f\otimes \mathbb{A}\oplus g\otimes \mathbb{A}$ with $[e,f]=g$.
Suppose $d=1$ and let $\mu $ be a linear form on $\mathbb{A}$ non zero on $\mathbb{S}$ so that the bilinear form $ \varphi (x,y)=\mu (xy) $ satisfying $ \varphi( xy,z)= \varphi (x,yz)$ is nondegenerate. A direct calculation shows that the formula
\begin{center}
$\omega (e\otimes a + f\otimes b+ g\otimes c,e\otimes a' + f\otimes b'+ g\otimes c')= \mu (ac'-a'c)+ \omega'(b,b')$
\end{center}
defines a non degenerate $2$-cocycle on ${\cal H}_{1}(\mathbb{A})$ if and only if $l$ is even and $\omega'(b,b')$ is a nondegenerate alternating bilinear form.
If $d\geq 3$, a simple calculation shows that $\mathbb{N}e+\mathbb{N}f+\mathbb{A}g $ is orthogonal to
$\mathbb{S}g$ relatively to any scalar $2$-cocycle of ${\cal H}_{1}(\mathbb{A})$. Consequently any $2$-cocycle is degenerate.
Now, suppose $d=2$ and recall that an alternate bilinear form $\omega$ on ${\cal H}_{1}(\mathbb{A})$ is a $2$-cocycle if and only if $\mathbb{A}g $ is totally isotropic for $\omega$ and we have $\omega(ae,bg)=\omega(abe,g) $ ; $\omega(af,bg)=\omega(abf,g)$ for any a,b in $\mathbb{A}$.
Let $f_{1}, f_{2}$ two linear forms on $\mathbb{A}$ such that their restrictions to $\mathbb{S}$ are independent and $\varphi_{1}, \varphi_{2}$ the associatives bilinear forms given by $ \varphi_{i}(a,b)=f_{i}(ab)$ for $i=1,2$.
Define a bilinear form $\omega$ on ${\cal H}_{1}(\mathbb{A})$ putting: 1) $\omega(cg,c'g)=0$ , 2) $\omega(ae,bg)= \varphi_{1}(a,b)$ , 3) $\omega (af,bg)= \varphi_{2}(a,b) $, 4) The restriction of $\omega$ to $\mathbb{A}e\oplus \mathbb{A}f $ is any alternate bilinear form. In fact $\omega$ is a $2$-cocycle. Moreover we will show that the linear map $\mathbb{A}g\rightarrow {\cal H}_{1}(\mathbb{A})^{*}$, given by $ cg\rightarrow \omega(cg,.)$ is injective. Consider $xg$ in the kernel of this map, we have $\varphi_{1}(a,x)=\varphi_{2}(a',x)=0$ for any $a, a'$ in $\mathbb{A}$ , so $f_{1}(ax)= f_{2}(a'x)=0$. Let us show that $x=0$. If $x\neq 0$, consider $I$ a minimal nonzero ideal of $\mathbb{A}$ contained in $\mathbb{A}x$. We have
$\mathbb{N}I=\{0\}$, because $\mathbb{N}I$ is an ideal and $\mathbb{N}I=I$ would imply $I=\{0\}$. So, $I$ is contained in $\mathbb{S}$ and there exists $a\in \mathbb{A}$ such that $0\neq ax \in \mathbb{S}$. Then $f_{1}(ax)= f_{2}(ax)$ and as the restrictions to $\mathbb{S}$ of $f_{1}$ and $f_{2}$ are independent,it follows $ax=0$, which is impossible.
Consequently, there exists a subspace $E$ of $\mathbb{A}e\oplus \mathbb{A}f$ in duality with $\mathbb{A}g$ by $\omega$. Denote by $F$ a supplementary subspace of $E$ in $\mathbb{A}e\oplus \mathbb{A}f$. According to condition 4) above, we can choose the restriction of $\omega$ to $\mathbb{A}e\oplus \mathbb{A}f$ verifying $\omega(E,E)=\omega(E,F)=0$ and $\omega|F\times F$, any nondegenerate alternate form (notice that $ dim F=dim \mathbb{A}$ is even by hypothesis). This choice determines a nondegenerate $2$-cocycle of ${\cal H}_{1}(\mathbb{A})$.
Finally consider the case $k\geq 2$. Obviously, ${\cal H}_{k}(\mathbb{A})={\cal H}_{k}(\mathbb{R})\otimes \mathbb{A}$ and ${\cal H}_{k}(\mathbb{A})=\mathbb{A}e_{1}\oplus...\oplus\mathbb{A}e_{k}\oplus\mathbb{A}f_{1}\oplus...\oplus\mathbb{A}f_{k}\oplus\mathbb{A}g$ with $[e_{i},f_{i}]=g$. If $\omega$ is a $2$-cocycle of
${\cal H}_{k}(\mathbb{A})$, we have for any $a\in \mathbb{A}$:
\begin{center}
$\delta\omega(ae_{1},f_{1},e_{i})=0=\delta\omega(ae_{1},f_{1},f_{i})$ for $i \geq2$
\end{center}
and so $\omega(ag,e_{i})=\omega(ag,f_{i})=0$.
As $\delta\omega(ae_{2},f_{2},e_{1})=0=\delta\omega(ae_{2},f_{2},f_{1})$, we also have $\omega(ag,e_{1})=\omega(ag,f_{1})=0$ and
$\delta\omega(ae_{1},f_{1},bg)=0$ implies $\omega(ag,bg)=0$. In summary, $\mathbb{A}g$ is in the kernel of $\omega$. Hence $\omega$ is degenerate. \hfill $\square$\\
\textbf{Remark 5.} It is clear in the proof of Theorem 5 that ${\cal H}_{1}(\mathbb{A})$ is symplectic if $\mathbb{A}$ is any Frobenius algebra non necessary a local algebra.
\textbf{Example 6.} Let $\mathbb{A}:=\mathbb{R}[x,y]/(x^{3}=y^{2}=xy=0) $ , so this algebra can be decomposed as $\mathbb{A}=\mathbb{R}\oplus \mathbb{R}x \oplus \mathbb{R}x^{2}\oplus \mathbb{R}y$ and $\mathbb{S}=\mathbb{R}x^{2}\oplus \mathbb{R}y$ is its socle. Following step by step the proof of the Theorem 5 we can find a scalar nondegenerate cocycle on ${\cal H}_{1}(\mathbb{A})$.
\textbf{Example 7.} Let $\mathbb{A}=\mathbb{R}^{2k}\oplus \mathbb{C}^{h}$ and ${\cal K }:= \mathbb{Q}[X]/(P)$ where $P$ is a polynomial having $2k$ real roots and $2h$ non real roots . Then $ H_{1}({\cal O}_{\cal K})$ is a lattice in the real simply connected Lie group of Lie algebra ${\cal H}_{1}(\mathbb{A})$.
\textbf{Example 8.} Consider the real algebra $\mathbb{A}_{\cal Q}=\mathbb{R}\oplus V\oplus\mathbb{R}$, with $V$ a $2d$- dimensional vector space and product given by
\begin{center}
$(\lambda,v,\mu)(\lambda',v',\mu')=(\lambda\lambda',\lambda v'+\lambda' v,\Phi_{\cal\ Q}(v,v')+\lambda\mu'+\lambda'\mu)$
\end{center}
\begin{flushleft}
where $\Phi_{\cal\ Q}$ is the bilinear symmetric form associated to a non degenerate quadratic form ${\cal Q}: V \rightarrow \mathbb{R}$. The subset $\{(0,v, \mu); v\in V, \mu\in \mathbb{R}\}$ is the maximal ideal of $\mathbb{A}_{\cal Q}$. It is also a Frobenius algebra via the linear form $f(\lambda,v,\mu)=\mu$. Take $ \overline{q}:\mathbb{Q}^{2d}\rightarrow \mathbb{Q}$ a rational quadratic form with $sign( \overline{q})=sign ({\cal Q})$. Then, the quadratic real extension $ {\overline{q}}$ is isometric to ${\cal Q}$. Finally, if we denote by $\mathbb{A}_{\overline{q}}$ the the corresponding algebra associated to $ \overline{q}$ as above, the real Lie algebras ${\cal H}_{1}(\mathbb{A}_{\overline{q}})\otimes_{\mathbb{Q}} \mathbb{R}$ and ${\cal H}_{1}(\mathbb{A}_{\cal Q})$ are isomorphic. This defines a lattice of the real simply connected Lie group of which the Lie algebra is ${\cal H}_{1}(\mathbb{A}_{\cal Q})$.
\end{flushleft}
As two algebras $\mathbb{A}_{\overline{q}}$ and $\mathbb{A}_{\overline{q'}}$ are isomorphic if and only if $ \overline{q}$ and $ \overline{q'}$ are collinear, the commensurability classes of lattices in the group $H_{1}(\mathbb{A}_{\cal Q})$ are in bijection with the classes of collinear rational quadratic forms of the same index than ${\cal Q}$.
\section{Some symplectic solvmanifolds with symplectic or semi-Riemannian affine structure}
First of all recall that the left invariant connection defined by the formula (1) is not symplectic if $G$ is non Abelian. Nevertheless in [15], we have studied lattices in 4-dimensional Lie symplectic Lie groups and showed that these groups admit a complete left invariant symplectic affine structure. Here we give a simpler proof of this result, in a more general framework.
\textbf{Theorem 6.} Let $(G,\omega^{+})$ be a symplectic Lie group having an Abelian normal closed codimension one subgroup. Then $(G,\omega^{+})$ admits a complete left invariant symplectic affine structure. Consequently any homogeneous space $K\backslash G$ inherits a complete flat and torsion free symplectic connection. In particular this is the case if $K$ is a (uniform) lattice in $G$.
\textbf{Proof.} The corresponding Lie bracket on ${\cal G}=I\oplus \mathbb{R}e$, where $I$ is an Abelian ideal, is given by $ [x,e]=u(x)$ for $x\in I$ and $u\in End(I)$. Consider the linear map $L:{\cal G}\rightarrow End(\cal G)$ defined by $L_{x}=0$ for $x\in I$ and $L_{e}=ad_{e}$. It is straightforward to check that $L_{a}b-L_{b}a=[a,b]$ and $L_{[a,b]}=[L_{a},L_{b}]$ for all $a$ and $b$ in ${\cal G}$. So $ab:=L_{a}b$ is a left symmetric product compatible with the bracket of ${\cal G}$. Moreover for any scalar $2$-cocycle $\omega$ of the Lie algebra ${\cal G}$ we have the formula
\begin{center}
$\omega(L_{a}b,c)+ \omega( b,\omega L_{a}c)=0$.
\end{center}
Hence for $a^{+}$ and $b^{+}$, left invariant vector fields in $G$, the formula $\nabla_{a^+}b^{+}=(ab)^{+}$ , where $\omega$ is nondegenerate, determines a left invariant connection verifying the required conditions. \hfill $\square$\\
Some symplectic Lie groups or their dual or their doubles corresponding to some solutions of classical Yang-Baxter equations are endowed with other left invariant geometric structures for example flat semi-Riemannian metrics, complex structures, polynomial Poisson structures etc (see [12],[21],[22],[23]).
If the (simply connected ) symplectic Lie group $(G,\omega^{+})$ is quadratic i.e. has a bi-invariant metric given by a non degenerate bilinear form $k$ on ${\cal G}$, the affine structure defined by (1) is the Levi-Civita connection corresponding to the left invariant semi-Riemannian metric given by the bilinear form $<x,y>:=k(Dx,Dy)$ with $\omega(x,y)=k(Dx,y)$. In this case, the Lie algebra ${\cal G}$ is nilpotent because $D$ is an invertible derivation. Consequently any nilmanifold $\Gamma\backslash G$ is endowed with a flat and complete semi-Riemannian metric. For more details ( see [7],[21]).
Notice that the inverse $r:=\omega^{-1}$ is an invertible solution of the classical Yang-Baxter equation on ${\cal G}$. In [12] it is shown that the Poisson tensors $r^{+}$ and $r^{-}$ are polynomials. In fact the Poisson-Lie tensor $\Pi(r)=r^{+}-r^{-}$ is polynomial of degree at most 2 (see [22]). Moreover, any double Lie group $D(G)$ of $(G,\Pi(r))$ is a Manin Lie group endowed with an affine left invariant structure $\nabla$ and a left invariant complex structure $J$ such that $\nabla J=0$ (see [22] ). In these terms and as consequence of these results we have:
\textbf{Proposition 4. } Let $(G,\omega^{+})$ be a real simply connected nilpotent symplectic Lie group and $D(G)$ the double simply connected Lie group of $(G,\Pi(r))$. Then any lattice $\Gamma$ in $G$ determines naturally a class of lattices in $D(G)$ and if $\Gamma'$ is a lattice in $D(G)$ the nilmanifold $\Gamma'\backslash D(G)$ is endowed with an homogeneous semi-Riemannian structure, two transversal Lagrangians foliations, a flat complete connection $\nabla$ and a complex structure $J$ such that $\nabla J=0$.
\textbf{Proof.} Let $T^{*}G$ the cotangent bundle of $G$ endowed with the Lie structure semidirect produit of $G$ by ${\cal G}$ via the coadjoint action and recall that the map $\theta: D( {\cal G},\Pi(r))\rightarrow t^{*}\cal G $ with $\theta(\alpha,x):=(\alpha,r(\alpha)+x)$ is an isomorphism of Manin algebras ( see for example [22]) for any solution $r$ of the classical Yang-Baxter equation on $r$.
We will prove that $\Gamma$ determines lattices in $D(G)$. Now, the $\mathbb{Z}$-span ${\cal L}$ of the set $ exp_{G}^{-1}(\Gamma)$ is a lattice in the additive group ${\cal G}$. Consider a basis $\mathbb{B}$ of ${\cal G}$ contained in ${\cal L}$. It is clear that the set $\mathbb{B}':=(\mathbb{B}^{*}\times{0})\times ({0}\times\mathbb{B})$ where $\mathbb{B}^{*}$ is the dual basis of $\mathbb{B}$ is a basis of the Lie algebra $t^{*}\cal G={\cal G}^{*}\times{\cal G}$ with rational constants structure. Take now the $\mathbb{Q}$-vector space generated by $\mathbb{B}'$, then if $L$ is any lattice of maximal rank of $t^{*}\cal G$ contained in this $\mathbb{Q}$ space, the group generated by $exp_{t^{*}\cal G}(L)$ is a lattice in $D(G)$. \hfill $\square$\\
\textbf{Acknowledgements.} We are grateful to the referee for helpful comments.
|
1208.2433
|
\section{Introduction}
\label{sec:introduction}
The aim of this paper is to develop a category-theoretical framework to unify various generalizations of the {\em Frobenius--Schur theorem}. We first recall the Frobenius--Schur theorem for compact groups. Let $G$ be a compact group, and let $V$ be a finite-dimensional continuous representation of $G$ with character $\chi_V$. The {\em $n$-th Frobenius--Schur indicator} (or {\em FS indicator}, for short) of $V$ is defined and denoted by
\begin{equation}
\label{eq:FS-formula-cpt}
\nu_n(V) = \int_G \chi_V(g^n) d \mu(g),
\end{equation}
where $\mu$ is the normalized Haar measure on $G$. The Frobenius--Schur theorem states that the value of the second FS indicator $\nu_2(V)$ has the following meaning:
\begin{theorem*}[Frobenius--Schur theorem]
If $V$ is irreducible, then we have
\begin{equation}
\label{eq:FS-ind-RCH}
\nu_2(V) = \begin{cases}
+1 & \text{if $V$ is real}, \\
\phantom{+} 0 & \text{if $V$ is complex}, \\
-1 & \text{if $V$ is quaternionic}.
\end{cases}
\end{equation}
Moreover, the following statements are equivalent: \\
\indent {\rm (1)} $\nu_2(V) \ne 0$. \\
\indent {\rm (2)} $V$ is isomorphic to its dual representation. \\
\indent {\rm (3)} There exists a non-degenerate $G$-invariant bilinear form $b: V \times V \to \mathbb{C}$. \\
If one of the above equivalent statements holds, then such a bilinear form $b$ is unique up to scalar multiples and satisfies $b(w, v) = \nu_2(V) \cdot b(v, w)$ for all $v, w \in V$. In other words, $b$ is symmetric if $\nu_2(V) = +1$ and is skew-symmetric if $\nu_2(V) = -1$.
\end{theorem*}
For $n \ge 3$, the representation-theoretic meaning of the $n$-th FS indicator is less obvious than the second one and there is no such theorem involving the $n$-th FS indicator. Hence, the second FS indicator could be of special interest. Unless otherwise noted, we simply call $\nu_2$ the {\em FS indicator} and refer to $\nu_n$ for $n \ge 3$ as {\em higher FS indicators}.
Generalizing those for compact groups, the FS indicator and higher ones have been defined for various algebraic objects, including (quasi-)Hopf algebras, tensor categories and conformal field theories; see \cite{MR1808131,MR2104908,MR2095575,MR1436801,MR1657800,MR2029790,MR2313527,MR2381536,MR2366965}. Among others, the theory of Ng and Schauenburg \cite{MR2313527,MR2381536,MR2366965} is especially important since it gives a unified category-theoretical understanding of all of \cite{MR1808131,MR2104908,MR2095575,MR1436801,MR1657800,MR2029790}. For the case of a semisimple (quasi-)Hopf algebra, a generalization of the Frobenius--Schur theorem is also formulated and proved; see \cite{MR1808131,MR2104908,MR2095575}. These results have many applications in Hopf algebras and tensor categories; see \cite{MR2725181,MR1942273,MR1919158,MR2213320,KMN09,MR2196640,MR2774703,KenichiShimizu:2012}.
On the other hand, there are several generalizations in other directions. For example, the earlier result of Linchenko and Montgomery \cite{MR1808131} can be thought, in fact, as a generalization of the Frobenius--Schur theorem for a finite-dimensional semisimple algebra with an anti-algebra involution. Based on their result, Hanaki and Terada \cite{TeradaJunya:2006-03} proved a generalization of the Frobenius--Schur theorem for association schemes and gave some applications to association schemes. Doi \cite{MR2674691} reconstructed the results of \cite{MR1808131} with an emphasis on the use of the theory of symmetric algebras. Recently, Geck \cite{2011arXiv1110.5672G} proved a result similar to Doi and gave some applications to finite Coxeter groups.
Unlike Hopf algebras, the representation categories of such algebras do not have a natural structure of a monoidal category and therefore we cannot understand these results in the framework of Ng and Schauenburg. Our first question is:
\begin{question}
\label{Q:motivation-1}
Is there a good category-theoretical framework to understand the FS indicator and the Frobenius--Schur theorem for such algebras?
\end{question}
The second question is about the twisted versions of some of the above. Given an automorphism $\tau$ of a finite group (or a semisimple Hopf algebra), the $n$-th $\tau$-twisted FS indicator $\nu_n^\tau$ is defined by twisting the definition of $\nu_n$ by $\tau$ and the twisted version of the Frobenius--Schur theorem is also formulated and proved; see \cite{MR1078503,MR2068079,MR2879228}. Our second question is:
\begin{question}
\label{Q:motivation-2}
If there is an answer to Question~\ref{Q:motivation-1}, then what is its twisted version?
\end{question}
In this paper, we give answers to these questions. Following the approaches of \cite{MR1657800,MR2029790,MR2313527}, we see that the duality is what we really need to define the FS indicator. This observation leads us to the notion of a {\em category with duality}, which has been well-studied in the theory of Witt groups \cite{MR2181829,MR2520968}. As an answer to Question~\ref{Q:motivation-1}, we introduce and study the FS indicator for categories with duality. Considering a suitable category and suitable duality, we can recover various generalizations of the Frobenius--Schur theorem for compact groups. We also introduce a method to ``twist'' the given duality on a category. This gives an answer to Question~\ref{Q:motivation-2}; in fact, the twisted FS indicator can be understood as the ``untwisted'' FS indicator with respect to the twisted duality.
\subsection{Organization and Summary of Results}
The present paper is organized as follows: In Section~\ref{sec:categ-with-dual}, following Mac Lane \cite{MR1712872}, we recall some basic results on adjoint functors and then introduce a category with duality in terms of adjunctions. By generalizing the definitions of \cite{MR1657800,MR2029790,MR2313527}, we define the FS indicator of an object of a category with duality over a field $k$ (Definition~\ref{def:FS-ind}). We also introduce a general method to ``twist'' the given duality by an adjunction. Then the twisted FS indicator is defined to be the ``untwisted'' FS indicator with respect to the twisted duality.
Pivotal Hopf algebras are introduced as a class of Hopf algebras whose representation category is a pivotal monoidal category; see, e.g., \cite{KMN09}. Motivated by this notion, in Section~\ref{sec:piv-alg}, a {\em pivotal algebra} is defined to be a triple $(A, S, g)$ consisting of an algebra $A$, an anti-algebra map $S: A \to A$ and an invertible element $g \in A$ satisfying certain conditions (Definition~\ref{def:piv-alg}). The representation category of a pivotal algebra is not monoidal in general but has duality. Therefore the FS indicator of an $A$-module is defined in the way of Section~\ref{sec:categ-with-dual}. In Section~\ref{sec:piv-alg}, we study the FS indicator for pivotal algebras and prove some fundamental properties of them.
From our point of view, \eqref{eq:FS-formula-cpt} is not the definition but a formula to compute the FS indicator. It is natural to ask when such a formula exists. In Section~\ref{sec:piv-alg}, we also give a formula for a separable pivotal algebra (Theorem~\ref{thm:piv-FS-ind-ch}); if a pivotal algebra $A = (A, S, g)$ has a separability idempotent $E$, then
\begin{equation}
\label{eq:FS-formula-intro}
\nu(V) = \sum_i \chi_V \Big( S(E'_i) g E''_i \Big)
\quad \Big( E = \sum_i E_i' \otimes E_i'' \Big)
\end{equation}
for all finite-dimensional left $A$-module $V$, where $\chi_V$ is the character of $V$. The relation between this formula and the results of \cite{MR1808131,MR2674691,2011arXiv1110.5672G} is discussed. By specializing \eqref{eq:FS-formula-intro}, we obtain a formula for group-like algebras (Example \ref{ex:FS-ind-ch-GLalg}) and for finite-dimensional weak Hopf $C^*$-algebras (Example \ref{ex:FS-ind-ch-WHA}). We also obtain the formula of Mason and Ng \cite{MR2104908} for finite-dimensional semisimple quasi-Hopf algebras and its twisted version (\S\ref{subsec:quasi-hopf-algebras}).
In Section~\ref{sec:coalgebras}, we introduce a {\em copivotal coalgebra} as the dual notion of pivotal algebras. Each result of Section~\ref{sec:piv-alg} has an analogue in the case of copivotal coalgebras. A crucial difference from the case of algebras is that there are infinite-dimensional coseparable coalgebras. For example, the Hopf algebra $R(G)$ of continuous representative functions on a compact group $G$ is coseparable with coseparability idempotent given by the Haar measure on $G$. The Formula~\eqref{eq:FS-formula-cpt} is obtained by applying the coalgebraic version of~\eqref{eq:FS-formula-intro} to $R(G)$.
In Section~\ref{sec:quantum-sl_2}, we apply our results to the quantum coordinate algebra $\mathcal{O}_q(SL_2)$ and the quantized universal enveloping algebra $U_q(\mathfrak{sl}_2)$. We first determine the FS indicator of all simple right $\mathcal{O}_q(SL_2)$-comodules. In a similar way, we also determine the twisted FS indicator with respect to an involution of $\mathcal{O}_q(SL_2)$ corresponding to the group homomorphism
\begin{equation*}
SL_2(k) \to SL_2(k), \quad
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} a & - b \\ - c & d \end{pmatrix}
\end{equation*}
in the classical limit $q \to 1$. Similar results for $U_q(\mathfrak{sl}_2)$ are also given by using the Hopf pairing between $\mathcal{O}_q(SL_2)$ and $U_q(\mathfrak{sl}_2)$.
\begin{remark}
To answer Question~\ref{Q:motivation-1}, we need to work in a ``non-monoidal'' setting. Since, as we have remarked, the Frobenius--Schur theory has some good applications even in non-monoidal settings, the FS indicator for categories with duality could be interesting. However, to define the higher (twisted) FS indicators, a monoidal structure seems to be necessary. At least, there is a reason why we cannot define higher FS indicators for categories with duality; see Remarks \ref{rem:transposition-and-E-map} and \ref{rem:FS-higher}.
It is interesting to construct a twisted version of \cite{MR2313527,MR2381536,MR2366965}. One of the referees kindly pointed out to the author that in May 2012, Daniel Sage gave a talk on a category-theoretic definition of the higher twisted FS indicators for Hopf algebras at the Lie Theory Workshop held at University of Southern California. Independently, after the submission of the first version of this paper, the author obtained a description of the higher twisted FS indicator for Hopf algebras by using the crossed product monoidal category. In this paper, we also mention higher twisted FS indicators for pivotal monoidal categories but leave the details for future work; see \S\ref{subsec:group-action-pivotal}.
\end{remark}
\begin{remark}
Linchenko and Montgomery \cite{MR1808131} established a relation between the FS indicator and invariant bilinear forms on an irreducible representation. Unlike the case of compact groups, a relation between the FS indicator and ``reality'' of representations is not known in the case of Hopf algebras; in fact, as remarked in \cite{MR1808131}, the reality of a representation of a Hopf algebra is not defined since, in general, a Hopf algebra does not have a good basis like the group elements of the group algebra. In a forthcoming paper, we will introduce the notions of real, complex and quaternionic representations of a Hopf $*$-algebra and Formulate \eqref{eq:FS-ind-RCH} in a Hopf algebraic context. We will also provide an exact quantum analog of the Frobenius--Schur theorem for compact quantum groups.
\end{remark}
\subsection{Notation}
Given a category $\mathcal{C}$ and $X, Y \in \mathcal{C}$, we denote by $\Hom_\mathcal{C}(X, Y)$ the set of all morphisms from $X$ to $Y$. $\mathcal{C}^{\op}$ means the opposite category of $\mathcal{C}$. An object $X \in \mathcal{C}$ is often written as $X^{\op}$ when it is regarded an object of $\mathcal{C}^{\op}$. A similar notation is used for morphisms. A functor $F: \mathcal{C} \to \mathcal{D}$ is denoted by $F^{\op}$ if it is regarded as a functor $\mathcal{C}^{\op} \to \mathcal{D}^{\op}$.
Throughout, we work over a fixed field $k$ whose characteristic is not two. By an algebra, we mean a unital associative algebra over $k$. Given a vector space $V$ (over $k$), we denote by $V^\vee = \Hom_k(V, k)$ the dual space of $V$. For $f \in V^\vee$ and $v \in V$, we often write $f(v)$ as $\langle f, v \rangle$. Unless otherwise noted, the unadorned tensor symbol $\otimes$ means the tensor product over $k$. Given $t \in V^{\otimes n}$, we often write $t$ as
\begin{equation*}
t = t^{1} \otimes t^{2} \otimes \dotsb \otimes t^{n} \in V \otimes V \otimes \dotsb \otimes V
\end{equation*}
The comultiplication and the counit of a coalgebra are denoted by $\Delta$ and $\varepsilon$, respectively. For an element $c$ of a coalgebra, we use Sweedler's notation
\begin{equation*}
\Delta(c) = c_{(1)} \otimes c_{(2)}, \quad
\Delta(c_{(1)}) \otimes c_{(2)} = c_{(1)} \otimes c_{(2)} \otimes c_{(3)} = c_{(1)} \otimes \Delta(c_{(2)}),
\ \dotsc
\end{equation*}
\section{Categories with Duality}
\label{sec:categ-with-dual}
\subsection{Adjunctions}
Following Mac Lane \cite{MR1712872}, we recall basic results on adjunctions. Let $\mathcal{C}$ and $\mathcal{D}$ be categories. An {\em adjunction} from $\mathcal{C}$ to $\mathcal{D}$ is a triple $(F, G, \Phi)$ consisting of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ and a natural bijection $\Phi_{X,Y}: \Hom_\mathcal{D}(F(X), Y) \to \Hom_\mathcal{C}(X, G(Y))$ ($X \in \mathcal{C}$, $Y \in \mathcal{D}$).
Given an adjunction $(F, G, \Phi)$ from $\mathcal{C}$ to $\mathcal{D}$, the {\em unit} $\eta: \id_{\mathcal{C}} \to G F$ and the {\em counit} $\varepsilon: F G \to \id_{\mathcal{D}}$ of the adjunction $(F, G, \Phi)$ are defined by
$\eta_X^{} = \Phi_{X,F(X)}^{}(\id_{F(X)}^{})$ and $\varepsilon_Y^{} = \Phi_{G(Y),Y}^{-1} (\id_{G(Y)}^{})$ for $X \in \mathcal{C}$ and $Y \in \mathcal{D}$, respectively. They satisfy the counit-unit equations
\begin{equation}
\label{eq:adjunction-2}
\varepsilon_{F(X)} \circ F(\eta_{X}^{}) = \id_{F(X)} \text{\quad and \quad}
G(\varepsilon_{Y}) \circ \eta_{G(Y)}^{} = \id_{G(Y)}
\end{equation}
for all $X \in \mathcal{C}$ and $Y \in \mathcal{D}$. By using $\eta$, the natural bijection $\Phi$ is expressed as
\begin{equation}
\label{eq:adjunction-3}
\Phi_{X,Y}(f) = G(f) \circ \eta_{X}
\quad (f \in \Hom_\mathcal{C}(F(X), Y)
\end{equation}
Similarly, by using $\varepsilon$, the inverse of $\Phi$ is expressed as
\begin{equation}
\label{eq:adjunction-4}
\Phi_{X,Y}^{-1}(g) = \varepsilon_{Y} \circ F(g)
\quad (g \in \Hom_\mathcal{D}(X, G(Y))
\end{equation}
Note that $\circ$ at the right-hand side stands for the composition in $\mathcal{D}$. We will deal with the case where $\mathcal{D} = \mathcal{C}^{\op}$, the opposite category of $\mathcal{C}$.
Each adjunction is determined by its unit and counit; indeed, let $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ be functors, and let $\eta: \id_{\mathcal{C}} \to G F$ and $\varepsilon: F G \to \id_{\mathcal{D}}$ be natural transformations satisfying \eqref{eq:adjunction-2}. If we define $\Phi$ by \eqref{eq:adjunction-3}, then the triple $(F, G, \Phi)$ is an adjunction from $\mathcal{C}$ to $\mathcal{D}$ whose unit and counit are $\eta$ and $\varepsilon$. From this reason, we abuse terminology and refer to such a quadruple $(F, G, \eta, \varepsilon)$ as an adjunction from $\mathcal{C}$ to $\mathcal{D}$.
\subsection{Categories with Duality}
\label{subsec:cat-with-duality}
The following terminologies are taken from Balmer \cite{MR2181829} and Calm\'es-Hornbostel \cite{MR2520968}.
\begin{definition}
\label{def:cat-with-duality}
A {\em category with duality} is a triple $\mathcal{C} = (\mathcal{C}, (-)^\vee, j)$ consisting of a category $\mathcal{C}$, a contravariant functor $(-)^\vee: \mathcal{C} \to \mathcal{C}$ and a natural transformation $j: \id_\mathcal{C} \to (-)^{\vee \vee}$ satisfying
\begin{equation}
\label{eq:duality-1}
(j_X)^\vee \circ j_{X^\vee} = \id_{X^\vee}
\end{equation}
for all $X \in \mathcal{C}$. If, moreover, $j$ is a natural isomorphism, then we say that $\mathcal{C}$ is a {\em category with strong duality}, or, simply, $\mathcal{C}$ is {\em strong}.
\end{definition}
Let $\mathcal{C}$ be a category with duality. We call the functor $(-)^\vee: \mathcal{C} \to \mathcal{C}$ the {\em duality functor} of $\mathcal{C}$. A pivotal monoidal category is an example of categories with duality; see \cite[Appendix]{MR2095575}. Thus we call the natural transformation $j: \id_\mathcal{C} \to (-)^{\vee\vee}$ the {\em pivotal morphism} of $\mathcal{C}$.
\begin{example}
\label{ex:cat-w-dual-invol-Hopf}
Let $H$ be a Hopf algebra with antipode $S$. If $H$ is {\em involutory}, {\em i.e.}, $S^2 = \id_H$, then the category $\Mod(H)$ of left $H$-modules is a category with duality; the duality functor is given by taking the dual $H$-module and the pivotal morphism is given by the canonical map
\begin{equation}
\label{eq:k-Vec-piv-mor}
\iota_V: V \to V^{\vee\vee} = \Hom_k(\Hom_k(V, k),k),
\quad \langle \iota(v), f \rangle = \langle f, v \rangle
\quad (v \in V, f \in V^\vee)
\end{equation}
The full subcategory $\fdMod(H)$ of $\Mod(H)$ of finite-dimensional left $H$-modules is a category with strong duality since $\iota_V$ is an isomorphism if (and only if) $\dim_k V < \infty$.
\end{example}
Let $D$ denote the duality functor of $\mathcal{C}$ regarded as a covariant functor from $\mathcal{C}$ to $\mathcal{C}^{\op}$. Definition \ref{def:cat-with-duality} says that the quadruple $(D, D^{\op}, j, j^{\op}): \mathcal{C} \to \mathcal{C}^{\op}$ is an adjunction. Hence we obtain a natural bijection
\begin{equation}
\label{eq:transposition-1}
\begin{aligned}
\Trans_{X,Y}:
\Hom_\mathcal{C}(X, Y^\vee)
& = \Hom_{\mathcal{C}^{\op}}(D(Y), X^{\op}) \\
\longrightarrow & \Hom_\mathcal{C}(Y, D^{\op}(X^{\op}))
= \Hom_\mathcal{C}(Y, X^\vee)
\quad (X, Y \in \mathcal{C})
\end{aligned}
\end{equation}
which we call the {\em transposition map}. By~\eqref{eq:adjunction-3}, $\Trans_{X,Y}$ is expressed as
\begin{equation}
\label{eq:transposition-2}
\Trans_{X,Y}(f) = f^\vee \circ j_Y
\quad (f \in \Hom_\mathcal{C}(X, Y^\vee))
\end{equation}
By \eqref{eq:adjunction-4}, we have $\Trans_{X,Y}^{-1}(g) = g^\vee \circ j_X = \Trans_{Y,X}(g)$ for $g \in \Hom_\mathcal{C}(Y, X^\vee)$. Hence we have
\begin{equation}
\label{eq:transposition-3}
\Trans_{Y,X} \circ \Trans_{X,Y} = \id_{\Hom_\mathcal{C}(X,Y^\vee)}^{}
\end{equation}
Note that $j$ is not necessarily an isomorphism. By understanding a category with duality as a kind of adjunction, we obtain the following characterization of categories with strong duality.
\begin{lemma}
\label{lem:duality-strong}
For a category $\mathcal{C}$ with duality, the following are equivalent: \\
\indent {\rm (1)} $\mathcal{C}$ is a category with strong duality. \\
\indent {\rm (2)} The duality functor $(-)^\vee: \mathcal{C} \to \mathcal{C}^{\op}$ is an equivalence.
\end{lemma}
\begin{proof}
Let, in general, $(F, G, \eta, \varepsilon)$ be an adjunction between some categories. Then $F$ is fully faithful if and only if $\eta$ is an isomorphism \cite[IV.3]{MR1712872}. Now we apply this result to the above quadruple $(D, D^{\op}, j, j^{\op})$ as follows: If $\mathcal{C}$ is strong, then $D$ is fully faithful. Since $X \cong X^{\vee\vee} = D(X^\vee)$ ($X \in \mathcal{C}$), $D$ is essentially surjective. Hence, $D$ is an equivalence. The converse is clear, since an equivalence of categories is fully faithful.
\end{proof}
Following \cite{MR2520968}, we introduce duality preserving functors and related notions:
\begin{definition}
Let $\mathcal{C}$ and $\mathcal{D}$ be categories with duality. A {\em duality preserving functor} from $\mathcal{C}$ to $\mathcal{D}$ is a pair $(F, \xi)$ consisting of a functor $F: \mathcal{C} \to \mathcal{D}$ and a natural transformation $\xi: F(X^\vee) \to F(X)^\vee$ ($X \in \mathcal{C}$) making
\begin{equation}
\label{eq:duality-2}
\begin{CD}
F(X) @>{F(j_X)}>> F(X^{\vee\vee}) \\
@V{j_{F(X)}}VV @VV{\xi_{X^\vee}}V \\
F(X)^{\vee\vee} @>>{\xi_X^{\vee}}> F(X^\vee)^\vee
\end{CD}
\end{equation}
commute for all $X \in \mathcal{C}$. If, moreover, $\xi$ is an isomorphism, then $(F, \xi)$ is said to be a {\em strong}. If $\xi$ is the identity, then $(F, \xi)$ is said to be {\em strict}.
Now let $(F, \xi), (G, \zeta): \mathcal{C} \to \mathcal{D}$ be such functors. A {\em morphism of duality preserving functors} from $(F, \xi)$ to $(G, \zeta)$ is a natural transformation $h: F \to G$ making
\begin{equation*}
\begin{CD}
F(X^\vee) @>{\xi_X}>> F(X)^\vee \\
@V{h_{X^\vee}}VV @AA{h_X^\vee}A \\
G(X^\vee) @>>{\zeta_X}> G(X)^\vee
\end{CD}
\end{equation*}
commute for all $X \in \mathcal{C}$.
\end{definition}
If $(F, \xi): \mathcal{C} \to \mathcal{D}$ and $(G, \zeta): \mathcal{D} \to \mathcal{E}$ are duality preserving functors between categories with duality, then the composition $G \circ F: \mathcal{C} \to \mathcal{E}$ becomes a duality preserving functor with
\begin{equation*}
\begin{CD}
G(F(X^\vee)) @>{G(\xi_X)}>> G(F(X)^\vee) @>{\zeta_{F(X)}}>> G(F(X))^\vee
@. \quad (X \in \mathcal{C})
\end{CD}
\end{equation*}
One can check that categories with duality form a 2-category; 1-arrows are duality preserving functors and 2-arrows are morphisms of duality preserving functors. Hence we can define an {\em isomorphism} and an {\em equivalence} of categories with duality in the usual way.
Given a duality preserving functor $(F, \xi): \mathcal{C} \to \mathcal{D}$, we define
\begin{equation*}
\tilde{F}_{X,Y}: \Hom_\mathcal{C}(X, Y^\vee) \to \Hom_\mathcal{D}(F(X), F(Y)^\vee)
\quad (X, Y \in \mathcal{C})
\end{equation*}
by $\tilde{F}_{X,Y}(f) = \xi_Y \circ F(f)$ for $f: X \to Y^\vee$. $\tilde{F}$ is compatible with the transposition map in the sense that the diagram
\begin{equation}
\label{eq:transposition-4}
\begin{CD}
\Hom_\mathcal{C}(X, Y^\vee) @>{\tilde{F}_{X,Y}}>>
\Hom_\mathcal{D}(F(X), F(Y)^\vee) \\
@V{\Trans_{X,Y}}VV @VV{\Trans_{F(X),F(Y)}}V \\
\Hom_\mathcal{C}(Y, X^\vee) @>>{\tilde{F}_{X,Y}}>
\Hom_\mathcal{D}(F(Y), F(X)^\vee)
\end{CD}
\end{equation}
commutes for all $X, Y \in \mathcal{C}$. Indeed, we have
\begin{align*}
& \Trans_{F(X),F(Y)} (\tilde{F}_{X,Y}(f))
= (\xi_Y \circ F(f))^\vee \circ j_{F(Y)}
= F(f)^\vee \circ \xi_Y^\vee \circ j_{F(Y)} \\
& \qquad = F(f)^\vee \circ \xi_{X^\vee} \circ {F(j_X)}
= \xi_Y \circ F(f^\vee) \circ F(j_X)
= \tilde{F}_{X,Y} (\Trans_{X,Y}(f))
\end{align*}
Now suppose that $\mathcal{C}$ is a category with strong duality. Then:
\begin{lemma}
\label{lem:duality-op}
$\mathcal{C}^{\op}$ is a category with duality with the same duality functor as $\mathcal{C}$ and pivotal morphism $(j^{-1})^{\op}$. The duality functor on $\mathcal{C}$ is an equivalence of categories with duality between $\mathcal{C}$ and $\mathcal{C}^{\op}$.
\end{lemma}
Hence, from~\eqref{eq:transposition-4} with $(F, \xi) = ((-)^\vee, \id_{(-)^\vee}): \mathcal{C}^{\op} \to \mathcal{C}$, we see that
\begin{equation}
\label{eq:transposition-5}
\begin{CD}
\Hom_{\mathcal{C}}(X^\vee, Y) @=
\Hom_{\mathcal{C}^{\op}}(Y^{\op}, (X^\vee)^{\op}) @>{(-)^\vee}>>
\Hom_{\mathcal{C}}(Y^\vee, X^{\vee\vee}) \\
@V{\Trans_{X,Y}^{\op}}VV @. @VV{\Trans_{Y^\vee, X^\vee}}V \\
\Hom_{\mathcal{C}}(Y^\vee, X) @=
\Hom_{\mathcal{C}^{\op}}(X^{\op}, (Y^\vee)^{\op}) @>>{(-)^\vee}>
\Hom_{\mathcal{C}}(X^\vee, Y^{\vee\vee})
\end{CD}
\end{equation}
commutes for all $X, Y \in \mathcal{C}$, where $\Trans_{X,Y}^{\op}$ is the transposition map for $\mathcal{C}^{\op}$ regarded as a map $\Hom_{\mathcal{C}}(X^\vee, Y) \to \Hom_{\mathcal{C}}(Y^\vee, X)$. Explicitly, it is given by $\Trans_{X,Y}^{\op}(f) = j_X^{-1} \circ f^\vee$.
\begin{remark}
\label{rem:transposition-and-E-map}
If $\mathcal{C}$ is a pivotal monoidal category, then there is a natural bijection
\begin{equation*}
\Hom_\mathcal{C}(X^{\vee}, Y) \cong \Hom_\mathcal{C}(\mathbf{1}, X \otimes Y)
\quad (X, Y \in \mathcal{C}),
\end{equation*}
where $\otimes$ is the tensor product of $\mathcal{C}$ and $\mathbf{1} \in \mathcal{C}$ is the unit object. The diagram
\begin{equation*}
\begin{CD}
\Hom_\mathcal{C}(X^{\vee},X) @>{\cong}>> \Hom_\mathcal{C}(\mathbf{1}, X \otimes X) \\
@V{\Trans^{\op}_{X,X}}VV @VV{E_X^{(2)}}V \\
\Hom_\mathcal{C}(X^{\vee},X) @>>{\cong}> \Hom_\mathcal{C}(\mathbf{1}, X \otimes X)
\end{CD}
\end{equation*}
commutes, where the horizontal arrows are the canonical bijections and $E_X^{(2)}$ is the map used in \cite{MR2381536} to define the FS indicator.
\end{remark}
\subsection{Frobenius--Schur Indicator}
Recall that a category $\mathcal{C}$ is said to be {\em $k$-linear} if each hom-set is a vector space over $k$ and the composition of morphisms is $k$-bilinear. A functor $F: \mathcal{C} \to \mathcal{D}$ between $k$-linear categories is said to be {\em $k$-linear} if the map $\Hom_\mathcal{C}(X, Y) \to \Hom_\mathcal{D}(F(X), F(Y))$, $f \mapsto F(f)$ is $k$-linear for all $X, Y \in \mathcal{C}$. Note that $\mathcal{C}^{\op}$ is $k$-linear if $\mathcal{C}$ is. Thus the $k$-linearity of a contravariant functor makes sense.
\begin{definition}
By a {\em category with duality over $k$}, we mean a $k$-linear category with duality whose duality functor is $k$-linear.
\end{definition}
For simplicity, in this section, we always assume that a category $\mathcal{C}$ with duality over $k$ satisfies the following finiteness condition:
\begin{equation}
\label{eq:assumption-fin-dim}
\dim_k \Hom_\mathcal{C}(X, Y) < \infty \quad \text{for all $X, Y \in \mathcal{C}$}
\end{equation}
\begin{definition}
\label{def:FS-ind}
Let $\mathcal{C}$ be a category with duality over $k$. The {\em Frobenius--Schur indicator} (or FS indicator, for short) of $X \in \mathcal{C}$ is defined and denoted by $\nu(X) = \Trace(\Trans_{X,X})$, where $\Trace$ means the trace of a linear map.
\end{definition}
The following is a list of basic properties of the FS indicator:
\begin{proposition}
\label{prop:FS-ind-basic}
Let $\mathcal{C}$ be a category with duality over $k$ and let $X \in \mathcal{C}$.
{\rm (a)} $\nu(X)$ depends on the isomorphism class of $X \in \mathcal{C}$.
{\rm (b)} $\nu(X) = \dim_k B^+_\mathcal{C}(X) - \dim_k B_\mathcal{C}^-(X)$, where
\begin{equation*}
B_\mathcal{C}^{\pm}(X) = \{ b: X \to X^\vee \mid \Trans_{X,X}(b) = \pm b \}.
\end{equation*}
{\rm (c)} Let $X_1, X_2 \in \mathcal{C}$. If their biproduct $X_1 \oplus X_2$ exists, then we have
\begin{equation*}
\nu(X_1 \oplus X_2) = \nu(X_1) + \nu(X_2).
\end{equation*}
\end{proposition}
\begin{proof}
(a) Let $p: X \to Y$ be an isomorphism in $\mathcal{C}$. Then
\begin{equation*}
\Hom_\mathcal{C}(p,p^\vee): \Hom_\mathcal{C}(Y, Y^\vee) \to \Hom_\mathcal{C}(X, X^\vee),
\quad f \mapsto p^\vee \circ f \circ p
\end{equation*}
is an isomorphism. By the naturality of the transposition map, the diagram
\begin{equation*}
\begin{CD}
\Hom_\mathcal{C}(Y, Y^\vee) @>{\Trans_{Y,Y}}>> \Hom_\mathcal{C}(Y, Y^\vee) \\
@V{\Hom_\mathcal{C}(p,p^\vee)}VV @VV{\Hom_\mathcal{C}(p,p^\vee)}V \\
\Hom_\mathcal{C}(X, X^\vee) @>>{\Trans_{X,X}}> \Hom_\mathcal{C}(X, X^\vee)
\end{CD}
\end{equation*}
commutes. Hence, we have $\nu(X) = \Trace(\Trans_{X,X}) = \Trace(\Trans_{Y,Y}) = \nu(Y)$.
(b) The result follows from~\eqref{eq:transposition-3} and the fact that the trace of an operator is the sum of its eigenvalues.
(c) For $a = 1, 2$, let $i_a: X_a \to X_1 \oplus X_2$ and $p_a: X_1 \oplus X_2 \to X_a$ be the inclusion and the projection, respectively. For $a, b, c, d = 1, 2$, we set
\begin{equation*}
\Trans_{a b}^{c d}
= p_{c d} \circ \Trans_{X_1 \oplus X_2, X_1 \oplus X_2} \circ i_{a b}:
\Hom_\mathcal{C}(X_a, X_b^\vee) \to \Hom_\mathcal{C}(X_c, X_d^\vee)
\end{equation*}
where $i_{a b} = \Hom_\mathcal{C}(p_a^{}, p_b^\vee)$ and $p_{c d} = \Hom_\mathcal{C}(i_c^{}, i_d^\vee)$. By linear algebra, we have
\begin{equation}
\label{eq:FS-ind-additive-proof-1}
\Trace(\Trans_{X_1 \oplus X_2, X_1 \oplus X_2})
= \Trace(\Trans_{1 1}^{1 1}) + \Trace(\Trans_{1 2}^{1 2}) + \Trace(\Trans_{2 1}^{2 1}) + \Trace(\Trans_{2 2}^{2 2})
\end{equation}
Now, by the naturality of the transposition map, we compute
\begin{align*}
\Trans_{a b}^{a b}
& = \Hom_\mathcal{C}(i_a, i_b^\vee) \circ \Trans_{X_1 \oplus X_2, X_1 \oplus X_2}
\circ \Hom_\mathcal{C}(p_a^{}, p_b^\vee) \\
& = \Hom_\mathcal{C}(i_a, i_b^\vee)
\circ \Hom_\mathcal{C}(p_b^{}, p_a^\vee) \circ \Trans_{X_a, X_b} \\
& = \Hom_{\mathcal{C}}(p_b^{} \circ i_a^{}, p_a^\vee \circ i_b^\vee) \circ \Trans_{X_a, X_b}
\end{align*}
Hence $\Trans_{a b}^{a b}$ is equal to $\Trans_{X_a, X_a}$ if $a = b$ and is zero if otherwise. Combining this result with~\eqref{eq:FS-ind-additive-proof-1}, we obtain $\nu(X_1 \oplus X_2) = \nu(X_1) + \nu(X_2)$.
\end{proof}
The FS indicator is an invariant of categories with duality over $k$. Indeed, the commutativity of \eqref{eq:transposition-4} yields the following proposition:
\begin{proposition}
\label{prop:FS-ind-inv}
Let $(F, \xi): \mathcal{C} \to \mathcal{D}$ be a strong duality preserving functor. If $F$ is $k$-linear and fully faithful, then we have $\nu(F(X)) = \nu(X)$ for all $X \in \mathcal{C}$.
\end{proposition}
Similarly, we obtain the following proposition from~\eqref{eq:transposition-5}:
\begin{proposition}
\label{prop:FS-ind-dual}
Suppose that $\mathcal{C}$ is a category with strong duality over $k$. Then, for all $X \in \mathcal{C}$, we have $\nu(X^\vee) = \Trace(\Trans^\op_{X,X}) = \nu(X^{\op})$.
\end{proposition}
Let $\mathcal{A}$ be a $k$-linear Abelian category. Recall that a nonzero object of $\mathcal{A}$ is said to be {\em simple} if it has no proper subobjects. We say that a simple object $V \in \mathcal{A}$ is {\em absolutely simple} if $\End_{\mathcal{A}}(V) \cong k$. Note that the opposite category $\mathcal{A}^{\op}$ is also $k$-linear and Abelian. It is easy to see that an object of $\mathcal{A}$ is (absolutely) simple if and only if it is (absolutely) simple as an object of $\mathcal{A}^{\op}$.
\begin{proposition}
\label{prop:FS-ind-abs-simple}
Let $\mathcal{C}$ be an Abelian category with strong duality over $k$, and let $X \in \mathcal{C}$. \\
\indent {\rm (a)} If $X$ is a finite biproduct of simple objects, then $\nu(X) = \nu(X^{\vee})$. \\
\indent {\rm (b)} If $X$ is absolutely simple, then $\nu(X) \in \{ 0, \pm 1 \}$. $\nu(X) \ne 0$ if and only if $X$ is self-dual, that is, $X$ is isomorphic to $X^{\vee}$.
\end{proposition}
\begin{proof}
(a) We first claim that if $V \in \mathcal{C}$ is simple, then $\nu(V) = \nu(V^{\vee})$. Let $V \in \mathcal{C}$ be a simple object. Since $(-)^\vee: \mathcal{C} \to \mathcal{C}^{\op}$ is an equivalence, $V^\vee$ is simple as an object of $\mathcal{C}^{\op}$ and hence it is simple as an object of $\mathcal{C}$. If $V$ is isomorphic to $V^{\vee}$, then our claim is obvious. Otherwise, we have
\begin{equation*}
\Hom_\mathcal{C}(V, V^{\vee}) = 0 \text{\quad and \quad}
\Hom_\mathcal{C}(V^\vee, V^{\vee\vee}) \cong \Hom_\mathcal{C}(V^{\vee}, V) = 0
\end{equation*}
by Schur's lemma. Therefore $\nu(V) = 0 = \nu(V^{\vee})$ follows.
Now write $X$ as $X = V_1 \oplus \dotsb \oplus V_m$ for some simple objects $V_1, \dotsc, V_m \in \mathcal{C}$. By the above arguments and the additivity of the FS indicator, we have
\begin{equation*}
\nu(X^{\vee}) = \nu(V_1^{\vee}) + \dotsb + \nu(V_m^{\vee}) = \nu(V_1) + \dotsb + \nu(V_m) = \nu(X)
\end{equation*}
(b) Suppose that $X \in \mathcal{C}$ is absolutely simple. If $X$ is isomorphic to $X^{\vee}$, then
\begin{equation*}
\dim_k \Hom_\mathcal{C}(X, X^{\vee}) = \dim_k \End_{\mathcal{C}}(X) = 1
\end{equation*}
and hence $\nu(X)$ is either $+1$ or $-1$ by Proposition~\ref{prop:FS-ind-basic} (b). Otherwise, $\nu(X) = 0$ as we have seen in the proof of (a).
\end{proof}
\begin{remark}
\label{rem:FS-higher}
If $\mathcal{C}$ is a pivotal monoidal category over $k$, then the $n$-th FS indicator $\nu_n(X)$ of $X \in \mathcal{C}$ is defined for each integer $n \ge 2$; see \cite{MR2381536}. The commutativity of \eqref{eq:transposition-4} implies $\nu_2(X) = \nu(X^\vee)$ (see also Remark~\ref{rem:transposition-and-E-map}). However, in view of Proposition~\ref{prop:FS-ind-abs-simple}, $\nu(X) = \nu_2(X)$ always holds in the case where $\mathcal{C}$ is strong, Abelian, and semisimple. We prefer our Definition~\ref{def:FS-ind} since it is more convenient when we discuss the relation between the FS indicator and invariant bilinear forms.
One would like to define higher FS indicators for an object of a category with duality over $k$ by extending that for an object of a pivotal monoidal category over $k$. This is impossible because of the following example: For a group $G$, we denote by $\Vect^G_\fdim$ the $k$-linear pivotal monoidal category of finite-dimensional $G$-graded vector spaces over $k$. The $n$-th FS indicator of $V = \bigoplus_{x \in G} V_x \in \Vect^G_\fdim$ is given by
\begin{equation}
\label{eq:FS-ind-G-gr}
\nu_n(V) = \sum_{x \in G[n]} \dim_k(V_x),
\text{\quad where $G[n] = \{ x \in G \mid x^n = 1 \}$}
\end{equation}
Now we put
\begin{equation*}
\mathcal{C} = \Vect_\fdim^{\mathbb{Z}_4 \times \mathbb{Z}_4}
\text{\quad and \quad}
\mathcal{D} = \Vect_\fdim^{\mathbb{Z}_2 \times \mathbb{Z}_8}
\end{equation*}
There exists a bijection $f: \mathbb{Z}_4 \times \mathbb{Z}_4 \to \mathbb{Z}_2 \times \mathbb{Z}_8$ such that $f(x^{-1}) = f(x)^{-1}$ for all $x \in \mathbb{Z}_4 \times \mathbb{Z}_4$. $f$ induces an equivalence $\mathcal{C} \approx \mathcal{D}$ of categories with duality over $k$. If we could define the $n$-th FS indicator for categories with duality over $k$, then there would exist at least one equivalence $F: \mathcal{C} \to \mathcal{D}$ such that $\nu_n(F(X)) = \nu_n(X)$ for all $X \in \mathcal{C}$ and $n \ge 2$. However, by \eqref{eq:FS-ind-G-gr}, there is no such equivalence.
\end{remark}
\subsection{Separable Functors}
\label{subsec:sep-functor}
Let $H$ be an involutory Hopf algebra, e.g., the group algebra of a group $G$. We consider the category $\mathcal{C} = \fdMod(H)$ of Example~\ref{ex:cat-w-dual-invol-Hopf}. The FS indicator $\nu(V)$ of $V \in \mathcal{C}$ is interpreted as follows: Let $\Bil_H(V)$ denote the set of all $H$-invariant bilinear forms on $V$. The transposition map induces
\begin{equation*}
\Sigma_V: \Bil_H(V) \to \Bil_H(V),
\quad \Sigma_V(b)(v, w) = b(w, v)
\quad (b \in \Bil_H(V), v, w \in V)
\end{equation*}
via the canonical isomorphism $\Bil_H(V) \cong \Hom_H(V, V^\vee)$. Now let $\Bil_H^{\pm}(V)$ denote the eigenspace of $\Sigma_V$ with eigenvalue $\pm 1$. Then we have
\begin{equation*}
\nu(V) = \Trace(\Trans_{V,V}) = \Trace(\Sigma_V) = \dim_k \Bil_H^+(V) - \dim_k \Bil_H^-(V)
\end{equation*}
Hence, from our definition, the relation between $\nu(V)$ and $H$-invariant bilinear forms on $V$ is clear. On the other hand, it is not obvious that $\nu(V)$ is expressed by a formula like \eqref{eq:FS-formula-cpt}. It should be emphasized that, from our point of view, \eqref{eq:FS-formula-cpt} is not the definition of $\nu(V)$ but rather a formula to compute $\nu(V)$. We note that a similar point of view is effectively used to derive a formula of the FS indicator for semisimple finite-dimensional quasi-Hopf algebras in \cite{MR2095575}.
A key notion to derive~\eqref{eq:FS-formula-cpt} is a {\em separable functor} \cite{MR1926102}; a functor $U: \mathcal{C} \to \mathcal{V}$ is said to be {\em separable} if there exists a natural transformation
\begin{equation*}
\Pi_{X,Y}: \Hom_\mathcal{V}(U(X), U(Y)) \to \Hom_\mathcal{C}(X, Y) \quad (X, Y \in \mathcal{C})
\end{equation*}
such that $\Pi_{X,Y}(U(f)) = f$ for all $f \in \Hom_\mathcal{C}(X, Y)$. Such a natural transformation $\Pi$ is called a {\em section} of $U$. Suppose that $\mathcal{C}$ and $\mathcal{V}$ be $k$-linear. We say that a section $\Pi$ of $U$ is {\em $k$-linear} if $\Pi_{X,Y}$ is $k$-linear for all $X, Y \in \mathcal{C}$.
Now let $\mathcal{C}$ be a category with duality over $k$, and let $\mathcal{V}$ be a $k$-linear category satisfying \eqref{eq:assumption-fin-dim}. A $k$-linear functor $U: \mathcal{C} \to \mathcal{V}$ induces a $k$-linear map
\begin{equation*}
U_{X,Y}: \Hom_\mathcal{C}(X, Y) \to \Hom_\mathcal{V}(U(X), U(Y))
\quad f \mapsto U(f)
\quad (X, Y \in \mathcal{C})
\end{equation*}
If $U$ has a $k$-linear section $\Pi$, we define a linear map
\begin{equation}
\label{eq:transposition-sep}
\widetilde{\Trans}_{X,Y}: \Hom_\mathcal{V}(U(X), U(Y^\vee)) \to \Hom_\mathcal{V}(U(Y), U(X^\vee))
\quad (X, Y \in \mathcal{C})
\end{equation}
so that the diagram
\begin{equation*}
\begin{CD}
\Hom_\mathcal{V}(U(X), U(Y^\vee)) @>{\widetilde{\Trans}_{X,Y}}>> \Hom_\mathcal{V}(U(Y), U(X^\vee)) \\
@V{\Pi_{X,Y^\vee}}VV @AA{U_{Y,X^\vee}}A \\
\Hom_\mathcal{C}(X, Y^\vee) @>>{\Trans_{X,Y}}> \Hom_\mathcal{C}(Y, X^\vee)
\end{CD}
\end{equation*}
commutes. By using the well-known identity $\Trace(A B) = \Trace(B A)$, we have
\begin{equation}
\label{eq:FS-ind-sep}
\begin{aligned}
\Trace(\widetilde{\Trans}_{X,X})
& = \Trace(U_{X,X^\vee} \circ \Trans_{X,X} \circ \Pi_{X, X^\vee}) \\
& = \Trace(\Pi_{X, X^\vee} \circ U_{X,X^\vee} \circ \Trans_{X,X})
= \Trace(\Trans_{X,X})
= \nu(X)
\end{aligned}
\end{equation}
Before we explain how \eqref{eq:FS-formula-cpt} is derived from \eqref{eq:FS-ind-sep}, we recall the following lemma in linear algebra: Let $f: V \to W$ and $g: W \to V$ be linear maps between finite-dimensional vector spaces. We define
\begin{equation*}
T: V \otimes W \to V \otimes W,
\quad T(v \otimes w) = g(w) \otimes f(v).
\quad (v \in V, w \in W)
\end{equation*}
\begin{lemma}
\label{lem:trace-1}
$\Trace(T) = \Trace(f g)$.
\end{lemma}
The dual of this lemma is also useful: Let $B(V, W)$ be the set of bilinear maps $V \times W \to k$ and consider the map $T^\vee: B(V, W) \to B(V, W)$, $T^\vee(b)(v, w) = b(g(w), f(v))$. Since $T^\vee$ is the dual map of $T$ under the identification $B(V, W) \cong (V \otimes W)^\vee$, we have $\Trace(T^\vee) = \Trace(f g)$.
\begin{proof}
Let $\{ v_i \}$ and $\{ w_j \}$ be a basis of $V$ and $W$, and let $\{ v^i \}$ and $\{ w^j \}$ be the dual basis to $\{ v_i \}$ and $\{ w_j \}$, respectively. Then we have
\begin{align*}
\Trace(T)
= \sum_{i, j} \langle v^i, g(w_j) \rangle \langle w^j, f(v_i) \rangle
= \sum_j \langle w^j, f(g(w_j)) \rangle = \Trace(f g)
\end{align*}
Here, the first and the last equality follow from $\Trace(f) = \sum_i \langle v^i, f(v_i) \rangle$ and the second follows from $x = \sum_i \langle v^i, x \rangle v_i$.
\end{proof}
Now, for simplicity, we assume $k$ to be an algebraically closed field of characteristic zero. Let $H$ be a finite-dimensional semisimple Hopf algebra over $k$, e.g., the group algebra of a finite group $G$. Then, by the theorem of Larson and Radford \cite{MR957441}, $H$ is involutory. Let $\Vect_{\fdim}(k)$ denote the category of finite-dimensional vector spaces over $k$. The forgetful functor $\fdMod(H) \to \Vect_\fdim(k)$ is a separable functor with $k$-linear section
\begin{gather*}
\Pi_{V,W}: \Hom_k(V, W) \to \Hom_H(V, W)
\quad (V, W \in \Mod(H)) \\
\Pi_{V,W}(f)(v) = S(\Lambda_{(1)}) f(\Lambda_{(2)} v)
\quad (f \in \Hom_k(V, W), v \in V)
\end{gather*}
where $\Lambda \in H$ is the {\em Haar integral} ({\em i.e.}, the two-sided integral such that $\varepsilon(\Lambda) = 1$). Let $\Bil(V)$ denote the set of all bilinear forms on $V$. Instead of the map \eqref{eq:transposition-sep}, we prefer to consider the map
\begin{equation*}
\widetilde{\Sigma}_V: \Bil(V) \to \Bil(V),
\quad \widetilde{\Sigma}_V(b)(v, w) = b(\Lambda_{(1)} v, \Lambda_{(2)} w),
\quad (b \in \Bil(V), v, w \in V)
\end{equation*}
which makes the diagram
\begin{equation*}
\begin{CD}
\Bil(V) & \ \cong \ & \Hom_k(V, V^\vee) @>{\Pi_{V,V^\vee}}>> \Hom_H(V, V^\vee) & \ \cong \ & \Bil_H(V) \\
@V{\widetilde{\Sigma}_V}VV @V{\widetilde{\Trans}_{V,V}}VV @VV{\Trans_{V,V}}V @VV{\Sigma_V}V \\
\Bil(V) & \ \cong \ & \Hom_k(V, V^\vee) @<{}<{\text{inclusion}}< \Hom_H(V, V^\vee) & \ \cong \ & \Bil_H(V)
\end{CD}
\end{equation*}
commutes; see \S\ref{sec:piv-alg}, especially Theorem~\ref{thm:piv-FS-ind-ch}, for the details. Let $\rho: H \to \End_k(V)$ be the algebra map corresponding to the action $H \otimes V \to V$. By Lemma~\ref{lem:trace-1}, we have
\begin{equation*}
\nu(V) = \Trace(\widetilde{\Sigma}_{V})
= \Trace \Big( \rho(\Lambda_{(1)}) \circ \rho(\Lambda_{(2)}) \Big)
= \chi_V(\Lambda_{(1)} \Lambda_{(2)})
\end{equation*}
where $\chi_V = \Trace \circ \rho$. This is the FS indicator for $H$ introduced by Linchenko and Montgomery \cite{MR1808131}. In the case where $H = k G$, we have $\nu(V) = |G|^{-1} \sum_{g \in G} \chi_V(g^2)$ since $\Lambda = |G|^{-1} \sum_{g \in G} g$. Formula~\eqref{eq:FS-formula-cpt} for compact groups is obtained in a similar way; see \S\ref{sec:coalgebras} for details.
\subsection{Twisted Duality}
\label{subsec:tw-FS-ind}
To deal with some twisted versions of the Frobenius--Schur theorem, we introduce a {\em twisting adjunction} of a category with duality and give a method to twist the original duality functor by using a twisting adjunction. Our method can be thought as a generalization of the arguments in \cite[\S4]{MR2879228}.
Let $\mathcal{C}$ be a category with duality. Suppose that we are given an adjunction $(F, G, \eta, \varepsilon): \mathcal{C} \to \mathcal{C}$ and a natural transformation $\xi_X: F(X^\vee) \to G(X)^\vee$. Define $\zeta_X$ by
\begin{equation}
\label{eq:duality-trans-1}
\begin{CD}
\zeta_X: G(X^\vee) @>{j_{G(X^\vee)}}>> G(X^\vee)^{\vee\vee}
@>{\xi^\vee_{X^\vee}}>> F(X^{\vee\vee})^\vee
@>{F(j_X)^\vee}>> F(X)^\vee
\end{CD}
\end{equation}
$\zeta_X$ is a natural transformation making the following diagrams commute:
\begin{equation}
\label{eq:duality-trans-2}
\begin{CD}
F(X) @>{F(j_X)}>> F(X^{\vee\vee})\phantom{,} \\
@V{j_{F(X)}}VV @VV{\xi_{X^\vee}}V \\
F(X)^{\vee\vee} @>>{\zeta_X^\vee}> G(X^\vee)^\vee,
\end{CD}
\qquad \qquad
\begin{CD}
G(X) @>{G(j_X)}>> G(X^{\vee\vee})\phantom{.} \\
@V{j_{G(X)}}VV @VV{\zeta_{X^\vee}}V \\
G(X)^{\vee\vee} @>>{\xi_X^\vee}> F(X^\vee)^\vee
\end{CD}
\end{equation}
Indeed, the commutativity of the first diagram is checked as follows:
\begin{align*}
\zeta_X^\vee \circ j_{F(X)}
= j_{G(X^\vee)}^\vee \circ \xi_{X^\vee}^{\vee\vee} \circ F(j_{X})^{\vee\vee} \circ j_{F(X)}
= j_{G(X^\vee)}^\vee \circ j_{G(X^\vee)^\vee} \circ \xi_{X^\vee} \circ F(j_{X})
= \xi_{X^\vee} \circ F(j_{X})
\end{align*}
The commutativity of the second one is checked in a similar way as follows:
\begin{align*}
\zeta_{X^\vee} \circ G(j_X)
& = F(j_{X^\vee})^\vee \circ \xi_{X^{\vee\vee}}^\vee \circ j_{G(X^{\vee\vee})} \circ G(j_X) \\
& = F(j_{X^\vee})^\vee \circ \xi_{X^{\vee\vee}}^\vee \circ G(j_X)^{\vee\vee} \circ j_{G(X)}
= F(j_{X^\vee})^\vee \circ F(j_X^\vee)^{\vee} \circ \xi_{X}^\vee \circ j_{G(X)}
= \xi_{X}^\vee \circ j_{G(X)}
\end{align*}
Now we define the {\em twisted duality functor} by $X^\sharp = G(X^\vee)$. The problem is when $\mathcal{C}$ is a category with duality with this new duality functor $(-)^\sharp$.
\begin{lemma}
\label{lem:twisting-adjunction}
Define $\omega: \id_\mathcal{C} \to (-)^\sharp \circ (-)^\sharp$ by
\begin{equation}
\label{eq:twisted-piv}
\begin{CD}
\omega_X: X @>{\eta_X}>> GF(X) @>{GF(j)}>> GF(X^{\vee\vee})
@>{G(\xi_{X^\vee})}>> G(G(X^\vee)^\vee) = X^{\sharp\sharp}
\end{CD}
\end{equation}
The triple $(\mathcal{C}, (-)^\sharp, \omega)$ is a category with duality if
\begin{equation}
\label{eq:duality-trans-3}
\begin{CD}
\Big( FG(X^\vee)
@>{F(\zeta_X)}>> F(F(X)^\vee)
@>{\xi_{F(X)}}>> (GF(X))^\vee
@>{\eta_X^\vee}>> X^\vee
\Big) = \varepsilon_{X^\vee}
\end{CD}
\end{equation}
holds for all $X \in \mathcal{C}$. If, moreover, $\mathcal{C}$ is strong, then the following are equivalent: \\
\indent {\rm (1)} $F$ is an equivalence. \\
\indent {\rm (2)} $G$ is an equivalence. \\
\indent {\rm (3)} $(\mathcal{C}, (-)^\sharp, \omega)$ is strong.
\end{lemma}
\begin{proof}
Let $X \in \mathcal{C}$. By~\eqref{eq:duality-trans-1}, $\omega_X$ can be expressed in two ways as follows:
\begin{equation*}
\omega_X = G(\xi_{X^\vee} \circ F(j_{X})) \circ \eta_X = G(\zeta_{X} \circ j_{F(X)}) \circ \eta_X
\end{equation*}
For each $X \in \mathcal{C}$, we compute
\begin{align*}
\omega_X^\sharp \circ \omega_{X^\sharp}
& = G(\omega_X^\vee) \circ \omega_{G(X^\vee)}
= G(\omega_X^\vee) \circ G\Big( \xi_{G(X^\vee)^\vee} \circ F(j_{G(X^\vee)}) \Big) \circ \eta_{G(X^\vee)} \\
& = G \Big( \eta_X^\vee \circ G(\zeta_X^\vee \circ j_{F(X)})^\vee
\circ \xi_{G(X^\vee)^\vee} \circ F(j_{G(X^\vee)}) \Big) \circ \eta_{G(X^\vee)} \\
& = G \Big(\eta_X^\vee \circ \xi_{F(X)} \circ F(j_{F(X)}^\vee \circ j_{F(X)^\vee} \circ \zeta_X) \Big) \circ \eta_{G(X^\vee)} \\
& = G \Big(\eta_X^\vee \circ \xi_{F(X)} \circ F(\zeta_X) \Big) \circ \eta_{G(X^\vee)}
= G (\varepsilon_{X^\vee}) \circ \eta_{G(X^\vee)} = \id_{G(X^\vee)} = \id_{X^\sharp}
\end{align*}
Here, the fourth equality follows from the naturality of $\xi$, the fifth from~\eqref{eq:duality-1}, the sixth from the Assumption~\eqref{eq:duality-trans-3}, and the seventh from \eqref{eq:adjunction-2}. Now we have shown that the triple $(\mathcal{C}, (-)^\sharp, \omega_X)$ is a category with duality.
It is easy to prove the rest of the statement; $(1)\Leftrightarrow(2)$ follows from basic properties of adjunctions. To show $(2)\Leftrightarrow(3)$, recall Lemma~\ref{lem:duality-strong}.
\end{proof}
In view of Lemma~\ref{lem:twisting-adjunction}, we call a quintuple $\mathbf{t} = (F, G, \eta, \varepsilon, \xi)$ satisfying \eqref{eq:duality-trans-3} a {\em twisting adjunction} for $\mathcal{C}$. Given such a quintuple $\mathbf{t}$, we denote by $\mathcal{C}^{\mathbf{t}}$ the triple $(\mathcal{C}, (-)^\sharp, \omega)$ constructed in Lemma \ref{lem:twisting-adjunction}. Now we introduce an involution of a category with duality:
\begin{definition}
\label{def:duality-involution}
An {\em involution} of $\mathcal{C}$ is a triple $\mathbf{t} = (F, \xi, \eta)$ such that $(F, \xi)$ is a strong duality preserving functor on $\mathcal{C}$ and $\eta$ is an isomorphism
\begin{equation}
\label{eq:cat-w-dual-invol-1}
\eta: (\id_\mathcal{C}, \id_{(-)^\vee}) \to (F, \xi) \circ (F, \xi)
\end{equation}
of duality preserving functors satisfying
\begin{equation}
\label{eq:cat-w-dual-invol-2}
\eta_{F(X)} = F(\eta_X)
\end{equation}
for all $X \in \mathcal{C}$. We say that $\mathbf{t}$ is {\em strict} if $\xi$ and $\eta$ are identities.
\end{definition}
Note that \eqref{eq:cat-w-dual-invol-1} is an isomorphism of such functors if and only if
\begin{equation}
\label{eq:cat-w-dual-invol-3}
\eta_X^\vee \circ \xi_{F(X)} \circ F(\xi_X) \circ \eta_{X^\vee} = \id_{X^\vee}
\end{equation}
holds for all $X \in \mathcal{C}$.
An involution $(F, \xi, \eta)$ of $\mathcal{C}$ is a special type of twisting adjunction; indeed, by~\eqref{eq:cat-w-dual-invol-2}, the quadruple $(F, F, \eta, \eta^{-1})$ is an adjunction. By the definition of duality preserving functors, the natural transformation \eqref{eq:duality-trans-1} is given by
\begin{align*}
\zeta_X
& = F(j_{X})^\vee \circ \xi_{X^{\vee}}^\vee \circ j_{F(X^{\vee})}
= F(j_{X})^\vee \circ \xi_{X^{\vee\vee}} \circ F(j_{X^{\vee}})
= \xi_{X^{\vee\vee}} \circ F(j_X^\vee) \circ F(j_{X^{\vee}}) = \xi_X
\end{align*}
Since the counit is $\eta^{-1}$, \eqref{eq:duality-trans-3} is equivalent to \eqref{eq:cat-w-dual-invol-3}. Hence $(F, F, \eta, \eta^{-1}, \xi)$ is a twisting adjunction for $\mathcal{C}$. From now, we identify an involution of $\mathcal{C}$ with the corresponding twisting adjunction.
Suppose that $\mathcal{C}$ is a category with duality over $k$. Let $\mathbf{t} = (F, G, \dotsc)$ be a twisting adjunction for $\mathcal{C}$. $\mathbf{t}$ is said to be {\em $k$-linear} if the functor $F: \mathcal{C} \to \mathcal{C}$ is $k$-linear. If this is the case, $G$ is also $k$-linear as the right adjoint of $F$ (see, e.g., \cite[IV.1]{MR1712872}) and hence $\mathcal{C}^\mathbf{t}$ is a category with duality over $k$.
\begin{definition}
\label{def:tw-FS-ind}
Let $\mathcal{C}$ be a category with duality over $k$, and let $\mathbf{t}$ be a $k$-linear twisting adjunction for $\mathcal{C}$. The ($\mathbf{t}$-){\em twisted FS indicator} $\nu^\mathbf{t}(X)$ of $X \in \mathcal{C}$ is defined by
\begin{equation*}
\nu^\mathbf{t}(X) = \nu(X^\mathbf{t})
\end{equation*}
where $X^\mathbf{t} \in \mathcal{C}^{\mathbf{t}}$ is the object $X$ regarded as an object of $\mathcal{C}^\mathbf{t}$.
\end{definition}
To study the twisted FS indicator, it is useful to introduce the twisted transposition map. Let $\mathcal{C}$ be a category with duality (not necessarily over $k$). Given a twisting adjunction $\mathbf{t} = (F, G, \eta, \varepsilon, \xi)$ for $\mathcal{C}$, the ($\mathbf{t}$-){\em twisted transposition map}
\begin{equation}
\label{eq:tw-trans-1}
\Trans_{X,Y}^\mathbf{t}: \Hom_\mathcal{C}(F(X), Y^\vee) \to \Hom_\mathcal{C}(F(Y), X^\vee)
\quad (X, Y \in \mathcal{C})
\end{equation}
is defined for $f: F(X) \to Y^\vee$ by
\begin{equation*}
\begin{CD}
\Trans_{X,Y}^\mathbf{t}(f): F(Y) @>{F(f^\vee \circ j_Y)}>> F(F(X)^\vee)
@>{\xi_{F(X)}}>> (GF(X))^\vee @>{\eta_X^\vee}>> X^\vee
\end{CD}
\end{equation*}
For a while, let $\Trans^\sharp_{X,Y}: \Hom_\mathcal{C}(X, Y^\sharp) \to \Hom_\mathcal{C}(Y, X^\sharp)$ denote the transposition map for $\mathcal{C}^\mathbf{t}$. One can easily check that the diagram
\begin{equation}
\label{eq:tw-trans-2}
\begin{CD}
\Hom_\mathcal{C}(F(X), Y^\vee) @>{\cong}>> \Hom_\mathcal{C}(X, G(Y^\vee))
& \ = \ & \Hom_\mathcal{C}(X, Y^\sharp) \\
@V{\Trans_{X,Y}^\mathbf{t}}VV @. @VV{\Trans_{X,Y}^\sharp}V \\
\Hom_\mathcal{C}(F(Y), X^\vee) @>>{\cong}> \Hom_\mathcal{C}(Y, G(X^\vee))
& \ = \ & \Hom_\mathcal{C}(Y, X^\sharp)
\end{CD}
\end{equation}
commutes for all $X, Y \in \mathcal{C}$, where the horizontal arrows are the natural bijection given by~\eqref{eq:adjunction-3}. Hence, under the condition of Definition~\ref{def:tw-FS-ind}, we have
\begin{equation*}
\nu^\mathbf{t}(X) = \Trace(\Trans^\sharp_{X,X}) = \Trace(\Trans_{X,X}^\mathbf{t})
\end{equation*}
The properties of the twisted FS indicator can be obtained as follows: Apply previous results to $\mathcal{C}^\mathbf{t}$ and then interpret the results in terms of $\mathcal{C}$ and $\mathbf{t}$ by using the commutative Diagram~\eqref{eq:tw-trans-2}. Following this scheme, a twisted version of Proposition~\ref{prop:FS-ind-abs-simple} is established as follows:
\begin{proposition}
\label{prop:tw-FS-ind-abs-simple}
Let $\mathcal{C}$ be an Abelian category with strong duality over $k$, let $X \in \mathcal{C}$, and let $\mathbf{t} = (F, G, \dotsc)$ be a $k$-linear twisting adjunction for $\mathcal{C}$ such that $F$ is an equivalence of categories. Then:
{\rm (a)} If $X$ is a finite biproduct of simple objects, then $\nu^\mathbf{t}(F(X)) = \nu^\mathbf{t}(X^{\vee})$.
{\rm (b)} If $X$ is absolutely simple, then $\nu^\mathbf{t}(X) \in \{ 0, \pm 1 \}$. $\nu^\mathbf{t}(X) \ne 0$ if and only if $F(X) \cong X^\vee$.
\end{proposition}
Now let $H$ be a finite-dimensional semisimple Hopf algebra over an algebraically closed field $k$ of characteristic zero. We give two examples of $k$-linear twisting adjunctions for $\fdMod(H)$.
\begin{example}
\label{ex:tw-adj-1}
An automorphism $\tau$ of $H$ induces a strict monoidal autoequivalence on $\fdMod(H)$. If $\tau^2 = \id_H$, then it gives rise to a strict involution of $\fdMod(H)$, which we denote by the same symbol $\tau$. The study of the $\tau$-twisted FS indicator leads us to the results of Sage and Vega \cite{MR2879228}; see \S\ref{sec:piv-alg}.
\end{example}
\begin{example}
Let $L \in \fdMod(H)$ be a left $H$-module such that $h_{(1)} \ell \otimes h_{(2)} = h_{(2)} \ell \otimes h_{(1)}$ holds for all $h \in H$ and $\ell \in L$. This condition implies that, for each $X \in \fdMod(H)$, the map
\begin{equation*}
\mathrm{flip}: L \otimes X \to X \otimes L,
\quad \ell \otimes x \mapsto x \otimes \ell
\quad (\ell \in L, x \in X)
\end{equation*}
is an isomorphism of left $H$-modules. Fix a basis $\{ \ell_i \}$ of $L$ and define
\begin{gather*}
\eta_X: X \to L \otimes L^\vee \otimes X,
\quad \eta_X(x) = \sum \ell_i \otimes \ell_i^\vee \otimes x \\
\varepsilon_X: L^\vee \otimes L \otimes X \to X,
\quad \varepsilon_X(\ell_i^\vee \otimes \ell_j \otimes x) = \delta_{i j} x
\end{gather*}
for $x \in X$, where $\{ \ell_i^\vee \}$ is the dual basis of $\{ \ell_i \}$. The quadruple $(F = L^\vee \otimes (-), G = L \otimes (-), \eta, \varepsilon)$ is an adjunction on $\fdMod(H)$. Moreover, if we define $\xi_X$ by
\begin{equation*}
\begin{CD}
\xi_X: L^\vee \otimes X^\vee @>{\cong}>> (X \otimes L)^\vee @>{\mathrm{flip}^\vee}>> (L \otimes X)^\vee
\end{CD}
\end{equation*}
then the quintuple $\mathbf{t}(L): = (F, G, \eta, \varepsilon, \xi)$ is a twisting adjunction for $\fdMod(H)$. Since, in general, $F$ and $G$ are not monoidal, $\mathbf{t}(L)$ is of different type of twisting adjunctions from Example~\ref{ex:tw-adj-1}.
Following the above notation, we write $X^\sharp$ for $G(X^\vee)$. To interpret the $\mathbf{t}(L)$-twisted FS indicator $\nu(V; L) := \nu^{\mathbf{t}(L)}(V)$, we recall that there is an isomorphism
\begin{equation*}
\Hom_H(X, Y^\sharp) = \Hom_H(X, L \otimes Y^\vee) \cong \Hom_H(X \otimes Y, L)
\end{equation*}
natural in $X, Y \in \fdMod(H)$. The transposition map for $\fdMod(H)^{\mathbf{t}(L)}$ induces
\begin{equation*}
\Sigma_V^L: \Hom_H(V \otimes V, L) \to \Hom_H(V \otimes V, L),
\quad \Sigma_V^L(b)(v, w) = b(w, v)
\end{equation*}
via the above isomorphism. Hence we have $\nu(V; L) = \dim_k B_H^+(V; L) - \dim_k B_H^-(V; L)$, where $B_H^{\pm}(V; L)$ is the eigenspace of $\Sigma_V^L$ with eigenvalue $\pm 1$.
Let $B(V; L)$ denote the set of all bilinear maps $V \times V \to L$. To express $\nu(V; L)$ by using the characters of $V$ and $L$, we use the map
\begin{equation*}
\widetilde{\Sigma}_V^L: B(V; L) \to B(V; L),
\quad \widetilde{\Sigma}_V^L(b)(v, w)
= S(\Lambda_{(1)}) \cdot b(\Lambda_{(2)} w, \Lambda_{(3)} v)
\end{equation*}
which makes the following diagrams commute:
\begin{equation*}
\begin{CD}
B(V; L) & \ \cong \ & \Hom_k(V \otimes V, L) @>{\Pi_{V \otimes V, L}}>>
\Hom_H(V \otimes V, L) & \ \cong \ & \Hom_H(V, V^\sharp)\phantom{.} \\
@V{\widetilde{\Sigma}_V^L}VV @. @VV{\Sigma_{V}^{L}}V @VV{\Trans_{V,V}^\sharp}V \\
B(V; L) & \ \cong \ & \Hom_k(V \otimes V, L) @<{}<{\text{inclusion}}<
\Hom_H(V \otimes V, L) & \ \cong \ & \Hom_H(V, V^\sharp)
\end{CD}
\end{equation*}
Now, let, in general, $f: A \to B$, $g: B \to A$ and $h: M \to M$ be linear maps between finite-dimensional vector spaces. Then, in a similar way as Lemma~\ref{lem:trace-1}, one can show that the trace of
\begin{equation*}
T: \Hom_k(A \otimes B, M) \to \Hom_k(A \otimes B, M),
\quad T(\mu)(a \otimes b) = h \mu(g(b) \otimes f(a))
\end{equation*}
is given by $\Trace(T) = \Trace(h) \Trace(f g)$. By using this formula, we have
\begin{equation*}
\nu(V; L) = \Trace(\widetilde{\Sigma}_{V}^{L}) = \chi_L(S(\Lambda_{(1)})) \chi_V(\Lambda_{(2)} \Lambda_{(3)})
\end{equation*}
\end{example}
If $\dim_k L = 1$, then $L \otimes (-)$ is an equivalence and hence Proposition~\ref{prop:tw-FS-ind-abs-simple} can be applied to the above example. By the above arguments, we now obtain in the following another type of twisted version of the Frobenius--Schur theorem for semisimple Hopf algebras.
\begin{theorem}
\label{thm:FS-ind-Hopf-tw-2}
Let $\alpha: H \to k$ be an algebra map such that $\alpha(h_{(1)}) h_{(2)} = \alpha(h_{(2)}) h_{(1)}$ holds for all $h \in H$ (or, equivalently, let $\alpha$ be a central grouplike element of the dual Hopf algebra $H^\vee$), and let $L$ be the left $H$-module corresponding to $\alpha$. Then, for all simple module $V \in \fdMod(H)$, we have
\begin{equation*}
\nu(V; L) = \alpha(S(\Lambda_{(1)})) \chi_V(\Lambda_{(2)} \Lambda_{(3)}) \in \{ 0, \pm 1 \}
\end{equation*}
Moreover, for a simple module $V \in \fdMod(H)$, the following statements are equivalent: \\
\indent {\rm (1)} $\nu(V; L) \ne 0$. \\
\indent {\rm (2)} $V \cong L \otimes V^\vee$. \\
\indent {\rm (3)} There exists a non-degenerate bilinear form $b: V \otimes V \to k$ satisfying
\begin{equation*}
b(h_{(1)} v, h_{(2)} w) = \alpha(h) b(v, w)
\text{\quad for all $h \in H$ and $v, w \in V$}
\end{equation*}
If one of the above equivalent statements holds, then such a bilinear form $b$ is unique up to scalar multiples and satisfies $b(w, v) = \nu(V; L) b(v, w)$ for all $v, w \in V$.
\end{theorem}
For the case where $H = k G$ is the group algebra of a finite group $G$, the above theorem has been obtained by Mizukawa \cite[Theorem~3.5]{MR2787651}.
\subsection{Group Action on a Pivotal Monoidal Categories}
\label{subsec:group-action-pivotal}
We have concentrated on studying generalizations of the second FS indicator. Here we briefly explain how to define the higher twisted FS indicators for $k$-linear pivotal monoidal categories by generalizing those for semisimple Hopf algebras due to Sage and Vega \cite{MR2879228}. As we have remarked in Section 1, the details are left for future work.
For a set $S$, we denote by $\underline{S}$ the category whose objects are the elements of $S$ and whose morphisms are the identity morphisms. If $G$ is a group, then $\underline{G}$ is a strict monoidal category with tensor product given by $x \otimes y = x y$ ($x, y \in G$).
Let $\mathcal{C}$ be a $k$-linear pivotal monoidal category with pivotal structure $j$. We denote by $\underline{\mathrm{Aut}}_\mathrm{piv}(\mathcal{C})$ the category of $k$-linear monoidal autoequivalences of $\mathcal{C}$ that {\em preserve the pivotal structure} in the sense of \cite{MR2381536}. This is a strict monoidal category with respect to the composition of monoidal functors. By an {\em action} of $G$ on $\mathcal{C}$, we mean a strong monoidal functor
\begin{equation*}
\underline{G} \to \underline{\mathrm{Aut}}_{\mathrm{piv}}(\mathcal{C}), \quad g \mapsto F_g \quad (g \in G)
\end{equation*}
Note that, by definition, there are natural isomorphisms $\id_\mathcal{C} \cong F_1$ and $F_{x} \circ F_{y} \cong F_{x y}$ of monoidal functors. We say that an action $\underline{G} \to \underline{\mathrm{Aut}}_{\mathrm{piv}}(\mathcal{C})$ is {\em strict} if it is strict as a monoidal functor and, moreover, $F_g: \mathcal{C} \to \mathcal{C}$ is strict as a monoidal functor for all $g \in G$.
Now suppose that an action of $G$ on $\mathcal{C}$ is given. The {\em crossed product} $\mathcal{C} \rtimes G$ is a monoidal category defined as follows: As a $k$-linear category, $\mathcal{C} \rtimes G = \bigoplus_{g \in G} \mathcal{C} \rtimes g$, where $\mathcal{C} \rtimes g = \mathcal{C}$ is a copy of $\mathcal{C}$. Given an object $X \in \mathcal{C}$, we denote by $(X, g)$ the object $X$ regarded as an object of $\mathcal{C} \rtimes g \subset \mathcal{C} \rtimes G$. The tensor product of $\mathcal{C} \rtimes G$ is given by
\begin{equation*}
(X, g) \otimes (Y, g') = (X \otimes F_g(Y), g g')
\end{equation*}
see \cite{MR1815142} and \cite{MR2480712} for details. We now claim:
\begin{lemma}
\label{lem:crossed-pivotal}
$\mathcal{C} \rtimes G$ is rigid. The dual object of $(X, g) \in \mathcal{C} \rtimes g$ is given by
\begin{equation*}
(X, g)^\vee = (F_{g^{-1}}(X^\vee), g^{-1})
\end{equation*}
Moreover, $\mathcal{C} \rtimes G$ is a pivotal monoidal category with pivotal structure given by
\begin{equation*}
\begin{CD}
(X, g) = X @>{j_X}>> X^{\vee\vee} \cong F_g^{} F_{g^{-1}} (X^{\vee\vee})
@>{F_g^{}(\xi_{g^{-1};X^{\vee}})}>> F_{g}^{}(F_{g^{-1}}^{} (X^\vee)^\vee)
= (X, g)^{\vee\vee}
\end{CD}
\end{equation*}
where $\xi_{g^{-1};V}: F_{g^{-1}}(V^\vee) \to F_{g^{-1}}(V)^{\vee}$ is the duality transform \cite[\S1]{MR2381536} of $F_{g^{-1}}: \mathcal{C} \to \mathcal{C}$.
\end{lemma}
In the most important case for us where the action of $G$ is strict, this lemma is easy to prove. The proof for general cases is tedious and omitted for brevity.
\begin{definition}
Let $\mathcal{C}$ be a $k$-linear pivotal monoidal category satisfying~\eqref{eq:assumption-fin-dim} and suppose that an action of $G$ on $\mathcal{C}$ is given. For a positive integer $n$ and an element $g \in G$, we define the {\em $g$-twisted $n$-th FS indicator $\nu_n^g(V)$} of $V \in \mathcal{C}$ by $\nu_n^g(V) = \nu_n((V, g))$, where $\nu_n$ in the right-hand side stands for the $n$-th FS indicator of Ng and Schauenburg \cite{MR2381536} for the $k$-linear pivotal monoidal category $\mathcal{C} \rtimes G$.
\end{definition}
For a pair of positive integers $(n, r)$, the $(n, r)$-th FS indicator $\nu_{n,r}$ is also defined in \cite{MR2381536}. It is clear how to define the {\em $g$-twisted $(n, r)$-th FS indicator $\nu_{n,r}^g$} of $V \in \mathcal{C}$.
We explain that our definition agrees with that of Sage and Vega \cite{MR2879228}. For simplicity, we treat all monoidal categories as if they were strict. Note that, as an object of $\mathcal{C}$, we have
\begin{equation*}
(V, g)^{\otimes n} =
V \otimes F_g^{}(V) \otimes F_g^2(V) \otimes \dotsb \otimes F_g^{n - 1} (V)
\quad (:= \widetilde{V^{\otimes n}})
\end{equation*}
Now let $H$ be a finite-dimensional semisimple Hopf algebra over an algebraically closed field $k$ of characteristic zero. Then the group $G = \mathrm{Aut}_{\text{Hopf}}(H)^{\op}$ naturally acts on $\mathcal{C} = \fdMod(H)$. If $g: H \to H$ is an automorphism such that $g^n = \id_H$, then the map
\begin{equation*}
E_{(V, g)}^{(n)}: \Hom_{\mathcal{C} \rtimes G}(\mathbf{1}_{\mathcal{C} \rtimes G}, (V, g)^{\otimes n})
\to \Hom_{\mathcal{C} \rtimes G}(\mathbf{1}_{\mathcal{C} \rtimes G}, (V, g)^{\otimes n})
\end{equation*}
used to define the $n$-th FS indicator in \cite{MR2381536} coincides with the map
\begin{equation*}
\alpha: (\widetilde{V^{\otimes n}})^H \to (\widetilde{V^{\otimes n}})^H, \quad
\sum_{i_1, \dotsc, i_n} v_{i_1}^1 \otimes v_{i_2}^2 \dotsb \otimes v_{i_n}^n \mapsto
\sum_{i_1, \dotsc, i_n} v_{i_1}^2 \otimes \dotsb \otimes v_{i_n}^n \otimes v_{i_1}^1
\end{equation*}
under the canonical identification $(\widetilde{V^{\otimes n}})^H = \Hom_{\mathcal{C} \rtimes G}(\mathbf{1}_{\mathcal{C} \rtimes G}, (V, g)^{\otimes n})$. Since the twisted FS indicator of \cite{MR2879228} is equal to the trace of the map $\alpha$ \cite[Theorem~3.5]{MR2879228}, our definition agrees with that \cite{MR2879228} in the case where both are defined.
Recall that a pivotal monoidal category is a category with duality. If $G = \langle a \mid a^2 = 1 \rangle$ acts on a $k$-linear pivotal monoidal category $\mathcal{C}$, then the functor $F_a: \mathcal{C} \to \mathcal{C}$ is naturally an involution of $\mathcal{C}$ in the sense of Definition~\ref{def:duality-involution}. Let $\mathbf{t}$ denote this involution. By using the crossed product $\mathcal{C} \rtimes G$, the category $\mathcal{C}^\mathbf{t}$ constructed in Lemma~\ref{lem:twisting-adjunction} can be described as follows:
\begin{proposition}
$\mathcal{C}^{\mathbf{t}} \to \mathcal{C} \rtimes G$, $X \mapsto (X, a)$ is a $k$-linear fully faithful duality preserving functor.
\end{proposition}
This is clear from Lemma~\ref{lem:crossed-pivotal} and the definition of $\mathcal{C}^\mathbf{t}$.
\section{Pivotal Algebras}
\label{sec:piv-alg}
\subsection{Pivotal Algebras}
In this section, we introduce and study a class of algebras such that its representation category is a category with strong duality. We first recall that $\fdMod(H)$ is a pivotal monoidal category if $H$ is a pivotal Hopf algebra \cite{KMN09}. Note that the monoidal structure of $\fdMod(H)$ is defined by using the comultiplication of $H$. Since we do not need a monoidal structure, it seems to be a good way to consider ``pivotal Hopf algebras with no comultiplication''. This is the notion of {\em pivotal algebras}, which is formally defined as follows:
\begin{definition}
\label{def:piv-alg}
A {\em pivotal algebra} is a triple $(A, S, g)$ consisting of an algebra $A$, an anti-algebra map $S: A \to A$, and an invertible element $g \in A$ satisfying
$S(g) = g^{-1}$ and $S^2(a) = g a g^{-1}$ for all $a \in A$.
\end{definition}
Let $A = (A, S, g)$ be a pivotal algebra. We denote by $\Mod(A)$ the category of left $A$-modules and by $\fdMod(A)$ its full subcategory of finite-dimensional modules. Given $V \in \Mod(A)$, we can make its dual space $V^\vee$ into a left $A$-module by
\begin{equation}
\label{eq:dual-module}
\langle a f, v \rangle := \langle f, S(a)v \rangle \quad (a \in A, f \in V^\vee, v \in V)
\end{equation}
The assignment $V \mapsto V^\vee$ extends to a contravariant endofunctor on $\Mod(A)$. Now, for each $V \in \Mod(A)$, we define $j_V: V \to V^{\vee\vee}$ by
\begin{equation}
\label{eq:piv-mor-alg}
\langle j_V(v), f \rangle = \langle f, g v \rangle
\quad (v \in V, f \in V^\vee)
\end{equation}
The following computation shows that $j_V: V \to V^{\vee\vee}$ is $A$-linear:
\begin{gather*}
\langle a \cdot j_V(v), f \rangle
= \langle S(a) f, g v \rangle
= \langle f, S^2(a) g v \rangle
= \langle f, g a v \rangle
= \langle j_V(a v), f \rangle
\end{gather*}
It is obvious that $j_V$ is natural in $V \in \Mod(A)$. Now we verify~\eqref{eq:duality-1} as follows:
\begin{equation*}
\langle (j_V^{})^\vee j_{V^\vee}^{}(f), v \rangle
= \langle j_{V^\vee}^{}(f), j_V^{}(v) \rangle
= \langle g f, g v \rangle
= \langle f, S(g) g v \rangle
= \langle f, v \rangle
\end{equation*}
Note that $j_V$ is an isomorphism if and only if $\dim_k V < \infty$. We conclude:
\begin{proposition}
\label{sec:piv-alg-cat-dual}
Let $A$ be a pivotal algebra. Then $\Mod(A) = (\Mod(A), (-)^\vee, j)$ is an Abelian category with duality over $k$. The full subcategory $\fdMod(A)$ is an Abelian category with strong duality over $k$.
\end{proposition}
Let $A$ and $B$ be algebras. Given an algebra map $f: A \to B$, we can make each left $B$-module into a left $A$-module by defining $a \cdot v = f(a) v$ ($a \in A, v \in V$). We denote by $f^\natural(V)$ the left $A$-module obtained in this way from $V$. The assignment $V \mapsto f^\natural(V)$ extends to a functor
\begin{equation}
\label{eq:pull-back-func}
f^\natural: \Mod(B) \to \Mod(A)
\end{equation}
By restriction, we also obtain a functor
\begin{equation}
\label{eq:pull-back-func-fd}
f^\natural|_\fdim: \fdMod(B) \to \fdMod(A)
\end{equation}
It is easy to see that these functors are $k$-linear, exact and faithful. If, moreover, $f$ is surjective, then they are full.
Suppose that $A = (A, S, g)$ and $B = (B, S', g')$ are pivotal algebras. A {\em morphism of pivotal algebras} from $A$ to $B$ is an algebra map $f: A \to B$ satisfying $f(g) = g'$ and $S'(f(a)) = f(S(a))$ for all $a \in A$. If $f$ is such a morphism, then the Functors \eqref{eq:pull-back-func} and \eqref{eq:pull-back-func-fd} are strict duality preserving functors.
An {\em involution} of $A$ is a morphism $\tau: A \to A$ of pivotal algebras such that $\tau^2 = \id_A$. Such a $\tau$ gives rise to a strict involution of $\Mod(A)$, which is usually denoted by the same symbol $\tau$. The proof of the following proposition is straightforward and omitted.
\begin{proposition}
Let $A = (A, S, g)$ be a pivotal algebra, and let $\tau$ be an involution of $A$. Put $S^\tau = S \circ \tau \, (=\tau \circ S)$. Then: \\
\indent {\rm (a)} The triple $A^\tau = (A, S^\tau, g)$ is a pivotal algebra. \\
\indent {\rm (b)} $\id_{\Mod(A)}$ is a strict duality preserving functor $\Mod(A^\tau) \to \Mod(A)^\tau$.
\end{proposition}
This implies that the $\tau$-twisted FS indicator of $V \in \fdMod(A)$ is equal to the untwisted FS indicator of $V$ regarded as a left $A^\tau$-module. Thus, in principle, the theory of the $\tau$-twisted FS indicator reduces to that of the untwisted FS indicator of $A^\tau$, which is again a pivotal algebra.
\subsection{FS Indicator for Pivotal Algebras}
Let $A = (A, S, g)$ be a pivotal algebra, and let $V \in \fdMod(A)$. Since $\fdMod(A)$ is a category with duality over $k$ satisfying \eqref{eq:assumption-fin-dim}, we can define $\nu(V)$ in the way of Section \ref{sec:categ-with-dual}.
The FS indicator $\nu(V)$ is interpreted as follows: Let $\Bil(V)$ be the set of all bilinear forms on $V$. Recall that there is a canonical isomorphism
\begin{equation}
\label{eq:canonical-map}
B_V: \Hom_k(V, V^\vee) \to \Bil(V), \quad
B_V(f)(v, w) = \langle f(v), w \rangle
\end{equation}
Let $\Bil_A(V)$ be the subset of $\Bil(V)$ consisting of those $b \in \Bil(V)$ such that
\begin{equation}
\label{eq:S-adjoint-bilin}
b(a v, w) = b(v, S(a)w)
\quad (a \in A, v, w \in V)
\end{equation}
The set $\Bil_A(V)$ is in fact the image of $\Hom_A(V,V^\vee) \subset \Hom_k(V, V^\vee)$ under the canonical isomorphism \eqref{eq:canonical-map}. Now we define $\Sigma_V: \Bil_A(V) \to \Bil_A(V)$ so that
\begin{equation*}
\begin{CD}
\Hom_A(V, V^\vee) @>{B_V}>> \Bil_{A}(V) \\
@V{\Trans_{V,V}}VV @VV{\Sigma_V}V \\
\Hom_A(V, V^\vee) @>>{B_V}> \Bil_{A}(V)
\end{CD}
\end{equation*}
commutes. If $b = B_V(f)$ for some $f \in \Hom_A(V, V^\vee)$, then we have
\begin{equation}
\label{eq:transposition-bilin-1}
\begin{aligned}
\Sigma_V(b)(v, w)
= B_V(f^\vee j_V)(v, w)
= \langle f^\vee j_V(v), w \rangle
= \langle f(w), g v \rangle
= b(w, g v)
\end{aligned}
\end{equation}
for all $v, w \in V$. In view of \eqref{eq:transposition-bilin-1}, we set
\begin{equation*}
\Bil_A^{\pm}(V) = \{ b \in \Bil_A(V) \mid \text{$b(w, g v) = \pm b(v, w)$ for all $v, w \in V$} \}
\end{equation*}
Then, as a counterpart of Proposition~\ref{prop:FS-ind-basic} (b), we have a formula
\begin{equation}
\label{eq:FS-ind-bilin}
\nu(V) = \dim_k \Bil_A^+(V) - \dim_k \Bil_A^-(V)
\end{equation}
Rephrasing the results of Section 2 by using these notations, we immediately obtain the following theorem:
\begin{theorem}
\label{thm:piv-FS}
If $V \in \fdMod(A)$ is absolutely simple, then we have $\nu(V) \in \{ 0, \pm 1 \}$.
Moreover, the following are equivalent: \\
\indent {\rm (1)} $\nu(V) \ne 0$. \\
\indent {\rm (2)} $V$ is isomorphic to $V^\vee$ as a left $A$-module. \\
\indent {\rm (3)} There exists a non-degenerate bilinear form $b$ on $V$ satisfying~\eqref{eq:S-adjoint-bilin}. \\
If one of the above statements holds, then such a bilinear form $b$ is unique up to scalar multiples and satisfies $b(w, g v) = \nu(V) \cdot b(v, w)$ for all $v, w \in V$.
\end{theorem}
We denote by $\reg_A$ the left regular representation of $A$. If $A$ is finite-dimensional, then the FS indicator of $\reg_A$ is defined. In the case where $A = k G$ is the group algebra of a finite group $G$, there is a well-known formula $\nu(\reg_{k G}) = \# \{ x \in G \mid x^2 = 1 \}$. This formula is generalized to finite-dimensional pivotal algebras as follows:
\begin{theorem}
\label{thm:piv-Tr-S}
Let $A = (A, S, g)$ be a finite-dimensional pivotal algebra. \\
\indent {\rm (a)} $\nu(\reg_A) = \Trace(Q)$, where $Q: A \to A$, $Q(a) = S(a)g$. \\
\indent {\rm (b)} If $A$ is Frobenius, then $\reg_A \cong \reg_A^\vee$ as left $A$-modules. Hence, $\nu(\reg_A) = \nu(\reg_A^\vee)$.
\end{theorem}
The part (b) is motivated by Remark~\ref{rem:FS-higher}; as we have mentioned, our definition of the FS indicator is different from that of \cite{MR2381536}. Therefore, if, for example, $A$ is a finite-dimensional pivotal Hopf algebra, then there are two definitions of the FS indicator of the regular representation of $A$. Nevertheless they are equal since a finite-dimensional Hopf algebra is Frobenius.
\begin{proof}
(a) Recall that there is an isomorphism $\Phi: \Hom_A(\reg_A, \reg_A^\vee) \to \reg_A^\vee$ given by $\Phi(f) = f(1)$. For $f \in \Hom_A(\reg_A, \reg_A^\vee)$ and $a \in A$, we have
\begin{equation*}
\langle \Phi \Trans_{A,A}(f), a \rangle
= \langle (f^\vee j_V)(1), a \rangle
= \langle j_V(1), f(a) \rangle
= \langle f(a), g \rangle
\end{equation*}
Recalling that $f: \reg_A \to \reg_A^\vee$ is $A$-linear, we compute
\begin{equation*}
\langle f(a), g \rangle
= \langle f(a \cdot 1), g \rangle
= \langle a \cdot f(1), g \rangle
= \langle \Phi(f), S(a) g \rangle,
= \langle Q^\vee \Phi(f), a \rangle
\end{equation*}
Hence we have $\Trans_{A,A} = \Phi^{-1} \circ Q^\vee \circ \Phi^{}$ and therefore
\begin{equation*}
\nu(A) = \Trace(\Trans_{A,A}) = \Trace(Q^\vee) = \Trace(Q)
\end{equation*}
(b) Suppose that $A$ is Frobenius. By definition, there exists $\phi \in A^\vee$ such that the bilinear map $A \times A \to k$, $(a, b)\mapsto \phi(a b)$ ($a, b \in A$) is non-degenerate. By using $\phi$, we define a linear map $f: \reg_A \to \reg_A^\vee$ by $\langle f(a), b \rangle = \phi(S(a) b)$ ($a, b \in A$). It is obvious that $f$ is bijective. For $a, b, c \in A$, we compute
\begin{equation*}
\langle f(a b), c \rangle
= \phi(S(a b) c)
= \phi(S(b) S(a) c)
= \langle f(b), S(a) c \rangle
= \langle a \cdot f(b), c \rangle
\end{equation*}
Thus $f$ is $A$-linear and therefore $\reg_A \cong \reg_A^\vee$ as left $A$-modules.
\end{proof}
The following Theorem~\ref{thm:piv-Tr-Sv} is motivated by the {\em trace-like invariant} of Hopf algebras studied in \cite{MR2724230} and \cite{MR2804686}. Given $V \in \Mod(A)$, we denote by $\rho_V: A \to \End_k(V)$ the algebra map induced by the action of $A$. Let $I_V := \Ker(\rho_V)$ denote the annihilator of $V$. By the definition of the dual module $V^\vee$, we have $I_{V^\vee} = S(I_V)$. Hence, if $V$ is self-dual, then $\mathrm{Im}(\rho_V) \to \mathrm{Im}(\rho_V)$, $\rho_V(a) \mapsto \rho_V(S(a))$ ($a \in A$) is well-defined. We also note that if $V \in \fdMod(A)$ is absolutely simple, then:
\begin{equation}
\label{eq:abs-simp-surj}
\text{The algebra map $\rho_V: A \to \End_k(V)$ is surjective}
\end{equation}
\begin{theorem}
\label{thm:piv-Tr-Sv}
Let $V \in \fdMod(A)$ be an absolutely simple module. If $V$ is self-dual, then, by the above arguments, the map
\begin{equation*}
S_V: \End_k(V) \to \End_k(V), \quad \rho_V(a) \mapsto \rho_V(S(a)) \quad (a \in A)
\end{equation*}
is well-defined. By using $S_V$, we also define
\begin{equation*}
Q_V: \End_k(V) \to \End_k(V), \quad Q_V(f) = S_V(f) \circ \rho_V(g) \quad (f \in \End_k(V))
\end{equation*}
Then we have:
\begin{equation*}
{\rm (a)} \Trace(S_V) = \nu(V) \cdot \chi_V^{}(g), \qquad \qquad
{\rm (b)} \Trace(Q_V) = \nu(V) \cdot \dim_k(V)
\end{equation*}
\end{theorem}
\begin{proof}
(a) This can be proved in the same way as \cite[Proposition 4.5]{KMN09}. Here we give another proof: Fix an isomorphism $p: V \to V^\vee$ of left $A$-modules and define $q: V \to V^\vee$ by $q = p^\vee \iota$, where $\iota = \iota_V$ is the canonical isomorphism \eqref{eq:k-Vec-piv-mor}. Our first claim is
\begin{equation*}
S_V(f) = q^{-1} f^\vee q \quad (f \in \End_k(V))
\end{equation*}
Indeed, \eqref{eq:abs-simp-surj}, there exists $a\ in A$ such that $f = \rho_V(a)$ for some $a \in A$. Hence, we compute
\begin{align*}
f^\vee q = (p \rho_V(a))^\vee \iota
= (\rho_V(S a)^\vee p)^\vee \iota
= p^\vee \rho_V(S a)^{\vee\vee} \iota
= q \rho_V(S a) = q S_V(f)
\end{align*}
Next, we determine the map $V \otimes V^\vee \to V \otimes V^\vee$ induced by $S_V$ via
\begin{equation*}
V \otimes V^\vee \to \End_k(V), \quad v \otimes \lambda \mapsto (x \mapsto \lambda(x) v)
\quad (\lambda \in V^\vee, v, x \in V)
\end{equation*}
If $f \in \End_k(V)$ is the element corresponding to $v \otimes \lambda \in V \otimes V^\vee$, then we have
\begin{equation*}
\langle f^\vee q(x), y \rangle
= \langle p^\vee \iota(x), f(y) \rangle
= \langle p(v), x \rangle \, \lambda(y)
\quad (x, y \in V)
\end{equation*}
and therefore $S_V(f)(x) = \langle p(v), x \rangle q^{-1}(\lambda)$. This means that $S_V(f)$ corresponds to the element $q^{-1}(\lambda) \otimes p(v) \in V \otimes V^\vee$ via the above isomorphism.
By the above observation, we have that the trace of $S_V$ is equal to that of
\begin{equation*}
V \otimes V^\vee \to V \otimes V^\vee,
\quad v \otimes \lambda \mapsto q^{-1}(\lambda) \otimes p(v)
\quad (v \in V, \lambda \in V^\vee)
\end{equation*}
Applying Lemma~\ref{lem:trace-1}, we have $\Trace(S_V) = \Trace(q^{-1} p)$. Now we recall the definition of the transposition map and compute $q = p^\vee \iota = p^\vee j_V \rho_V(g)^{-1} = \Trans_{V,V}(p) \rho_V(g)^{-1} = \nu(V) \cdot p \, \rho_V(g)^{-1}$. Hence, we conclude $\Trace(S_V) = \Trace(q^{-1}p) = \nu(V) \Trace(\rho_V(g)) = \nu(V) \chi_V(g)$.
(b) The triple $E = (\End_k(V), S_V, \rho_V(g))$ is a pivotal algebra and $\rho_V: A \to E$ is a morphism of pivotal algebras. Let $V_0$ denote the vector space $V$ regarded as a left $E$-module. By Proposition~\ref{prop:FS-ind-inv}, the functor $\rho_V^\natural: \Mod(E) \to \Mod(A)$ preserves the FS indicator. Since $V = \rho_V^\natural(V_0)$, we have $\nu(V_0) = \nu(V)$.
Now let $d = \dim_k(V)$. Then we have $\reg_E \cong V_0^{\oplus d}$ as left $E$-modules and therefore $\nu(E) = \nu(V_0) d = \nu(V) d$ by Proposition~\ref{prop:FS-ind-basic}. On the other hand, $\nu(E) = \Trace(Q_V)$ by Proposition~\ref{thm:piv-Tr-S}. Thus $\Trace(Q_V) = \nu(V) d$ follows.
\end{proof}
The following is a generalization of \cite[Theorem~8.8 (iii)]{MR2104908}.
\begin{corollary}
\label{cor:Tr-S}
Suppose that $k$ is algebraically closed and that $A = (A, S, g)$ is a finite-dimensional semisimple pivotal algebra. Let $\{ V_i \}_{i = 1, \dotsc, n}$ be a complete set of representatives of the isomorphism classes of simple left $A$-modules. Then
\begin{equation*}
\Trace(S) = \sum_{i = 1}^n \nu(V_i) \chi_{i}(g)
\end{equation*}
where $\chi_i = \chi_{V_i}^{}$ is the character of $V_i$.
\end{corollary}
\begin{proof}
Put $I = \{ 1, \dotsc, n \}$. For $i \in I$, let $\rho_i: A \to \End_k(V_i)$ denote the action of $A$ on $V_i$. By the Artin--Wedderburn theorem, we have an isomorphism
\begin{equation*}
A \to \End_k(V_1) \oplus \dotsb \oplus \End_k(V_n),
\quad a \mapsto (\rho_1(a), \dotsc, \rho_n(a))
\end{equation*}
of algebras. $S: A \to A$ induces an anti-algebra map
\begin{equation*}
\tilde{S}: \End_k(V_1) \oplus \dotsb \oplus \End_k(V_n) \to \End_k(V_1) \oplus \dotsb \oplus \End_k(V_n)
\end{equation*}
via the isomorphism. For each $i \in I$, we have $\tilde{S}(\End_k(V_i)) \subset \End_k(V_{i^*})$, where $i^* \in I$ is the element such that $V_i^\vee \cong V_{i^*}^{}$. Hence we obtain
\begin{equation*}
\Trace(S) = \Trace(\tilde{S})
= \sum_{i \in I, i^* = i} \Trace(\widetilde{S}|_{\End_k(V_i)})
= \sum_{i \in I, i^* = i} \nu(V_i) \chi_i(g)
\end{equation*}
by Theorem~\ref{thm:piv-Tr-Sv}. The sum in the right-hand side is equal to $\sum_{i = 1}^n \nu(V_i) \chi_i(g)$ since $\nu(V_i) = 0$ unless $i = i^*$. The proof is done.
\end{proof}
\subsection{Separable Pivotal Algebras}
Recall that an algebra $A$ is said to be {\em separable} if it has a {\em separability idempotent}, {\em i.e.}, an element $E \in A \otimes A$ such that $E^{1} E^{2} = 1$ and $a E^{1} \otimes E^{2} = E^{1} \otimes E^{2} a$ for all $a \in A$. If such an element exists, then the forgetful functor $\Mod(A) \to \Vect_\fdim$ is separable with section $\Pi_{V,W}: \Hom_k(V, W) \to \Hom_A(V, W)$ ($V, W \in \Mod(A)$) given by
\begin{equation*}
\Pi_{V,W}(f)(v) = E^1 f(E^2 v)
\quad (f \in \Hom_k(V, W))
\end{equation*}
Hence, if a pivotal algebra $A = (A, S, g)$ is separable (as an algebra), then we can apply the arguments of \S\ref{subsec:sep-functor}. This is a rationale for the following theorem:
\begin{theorem}
\label{thm:piv-FS-ind-ch}
Let $A = (A, S, g)$ be a separable pivotal algebra with separability idempotent $E \in A \otimes A$. Then, for all $V \in \fdMod(A)$, we have
\begin{equation*}
\nu(V) = \chi_V(S(E^1) g E^2)
\end{equation*}
\end{theorem}
\begin{proof}
Define $\widetilde{\Sigma}_V: \Bil(V) \to \Bil(V)$ so that the following diagrams commute:
\begin{equation*}
\begin{CD}
\Bil(V) @>{B_V^{-1}}>> \Hom_k(V, V^\vee) @>{\Pi_{V,V^\vee}}>> \Hom_A(V, V^\vee) @>{B_V^{}}>> \Bil_A(V)\phantom{.} \\
@V{\widetilde{\Sigma}_V}VV @V{\widetilde{\Trans}_{V,V}}VV @VV{\Trans_{V,V}}V @VV{\Sigma_V}V \\
\Bil(V) @<<{B_V^{}}< \Hom_k(V, V^\vee) @<<{\text{inclusion}}< \Hom_A(V, V^\vee) @<<{B_V^{-1}}< \Bil_A(V)
\end{CD}
\end{equation*}
By the arguments in \S\ref{subsec:sep-functor}, $\nu(V)$ is equal to $\Trace(\widetilde{\Trans}_{V,V})$. However, to make the computation easier, we prefer to compute $\Trace(\widetilde{\Sigma}_V)$, which is also equal to $\nu(V)$.
Let $\Pi'_V: \Bil(V) \to \Bil_A(V)$ be the composition of the arrows of the first row of the above diagram. If $b = B_V(f)$ for some $f \in \Hom_k(V, V^\vee)$, we have
\begin{gather*}
\Pi'_V(b)(v, w)
= \Big \langle \Pi_{V,V^\vee}(f)(v), w \Big \rangle
= \Big \langle f (E^2v), S(E^1)w \Big \rangle
= b \Big( E^2v, S(E^1)w \Big)
\end{gather*}
Hence, by~\eqref{eq:transposition-bilin-1}, we have
\begin{equation*}
\widetilde{\Sigma}_V(b)(v, w)
= \Sigma_V \Big( \Pi'_V(b) \Big)(v, w)
= b \Big( E^2 w, S(E^1) g v \Big)
\end{equation*}
Applying Lemma~\ref{lem:trace-1}, we obtain $\nu(V) = \chi_V(S(E^1) g E^2)$.
\end{proof}
We discuss the relation between Theorem~\ref{thm:piv-FS-ind-ch} and the results of \cite{MR1808131,MR2674691,2011arXiv1110.5672G}. Let $A = (A, S, g)$ be a pivotal algebra such that the algebra $A$ is symmetric with trace form $\phi: A \to k$. By definition, the map
\begin{equation}
\label{eq:sym-alg-trace}
A \times A \to k,
\quad (a, b) \mapsto \phi(a b)
\quad (a, b \in A)
\end{equation}
is a non-degenerate bilinear symmetric form. Fix a basis $\{ b_i \}_{i \in I}$ of $A$ and let $\{ b_i^\vee \}_{i \in I}$ be the dual basis of $\{ b_i \}$ with respect to~\eqref{eq:sym-alg-trace}. As remarked in \cite{MR2674691}, we have
\begin{equation*}
\sum_{i \in I} a b_i \otimes b_i^\vee = \sum_{i \in I} b_i \otimes b_i^\vee a
\end{equation*}
for all $a \in A$. Hence $v_A = \sum_{i \in I} b_i b_i^\vee \in A$ is a central element, called the {\em volume} of $(A, \phi)$.
Now we suppose that the base field $k$ is of characteristic zero and $A$ is split semisimple over $k$. Then, as Doi showed in \cite{MR2674691}, the volume $v_A$ is invertible and hence $E = \sum_{i \in I} b_i \otimes b_i^\vee v_A^{-1} \in A \otimes A$ is a separability idempotent of $A$. By Theorem~\ref{thm:piv-FS-ind-ch}, the FS indicator of $V \in \Rep(A)$ is given by
\begin{equation*}
\nu(V) = \sum_{i \in I} \chi_V(S(b_i) g b_i^\vee v_A^{-1})
\end{equation*}
If $V$ is simple, then, by Schur's lemma, $v_A$ acts on $V$ as $\chi_V(v_A) \dim_k(V)^{-1} \cdot \id_V$. Hence, we have
\begin{equation}
\label{eq:FS-symm-alg-1}
\nu(V) = \frac{\dim_k(V)}{\chi_V(v_A)} \sum_{i \in I} \chi_V(S(b_i) g b_i^\vee)
\end{equation}
The {\em Schur element} of a simple module $V \in \Rep(A)$ is given by $c_V = \chi_V(v_A) \dim_k(V)^{-2}$ (see Remark 1.6 of \cite{MR2674691}). By using the Schur element, $\nu(V)$ is expressed as
\begin{equation}
\label{eq:FS-symm-alg-2}
\nu(V) = \frac{1}{c_V \dim_k(V)} \sum_{i \in I} \chi_V(S(b_i) g b_i^\vee)
\end{equation}
Letting $g = 1$, we recover the results of \cite{MR1808131,MR2674691}. Geck \cite{2011arXiv1110.5672G} assumed that $g = 1$ and $A$ has a basis $\{ b_i \}_{i \in I}$ such that $b_i^\vee = S(b_i)$. If this is the case, then we have
\begin{equation*}
\nu(V) = \frac{1}{c_V \dim_k(V)} \sum_{i \in I} \chi_V(b_i^2)
\end{equation*}
\begin{example}[Group-like algebras]
\label{ex:FS-ind-ch-GLalg}
As a generalization of the group algebra of a finite group and the adjacency algebra of an association scheme, Doi \cite{MR2051736,MR2674691} introduced a {\em group-like algebra}; it is defined to be a quadruple $(A, \varepsilon, \mathcal{B}, *)$ consisting of a finite-dimensional algebra $A$, an algebra map $\varepsilon: A \to k$, a basis $\mathcal{B} = \{ b_i \}_{i \in I}$ of $A$ indexed by a set $I$, and an involutive map $*: I \to I$, $i \mapsto i^*$ satisfying the following conditions: \\
\indent (G0) There is a special element $0 \in I$ such that $b_0 = 1$ is the unit of $A$. \\
\indent (G1) $\varepsilon(b_i) = \varepsilon(b_{i^*}) \ne 0$ for all $i \in I$. \\
\indent (G2) $p_{i j}^k = p_{j^* i^*}^{k^*}$ for all $i, j, k \in I$, where $p_{i j}^k$ is given by $b_i \cdot b_j = \sum_{k \in I} p_{i j}^k b_k$ ($i, j, k \in I$). \\
\indent (G3) $p_{i j}^0 = \delta_{i j^*} \varepsilon(b_i)$ for all $i \in I$. \\
Now let $A = (A, \varepsilon, \mathcal{B}, *)$ be a group-like algebra. Define a linear map $S: A \to A$ by $S(b_i) = b_{i^*}$ for $i \in I$. One can check that the triple $A = (A, S, 1)$ is a pivotal algebra. By an {\em involution} of $A$, we mean an involutive map $\tau: I \to I$ satisfying
\begin{equation*}
\tau(i^*) = \tau(i)^* \text{\quad and \quad} p_{\tau(i),\tau(j)}^{\tau(k)} = p_{i j}^k
\end{equation*}
for all $i, j, k \in I$. Such a map gives rise to an involution of the pivotal algebra $(A, S, 1)$.
In what follows, we compute the $\tau$-twisted FS indicator $\nu^\tau(V)$ of a simple module $V \in \Rep(A)$ under the assumption that the base field is $\mathbb{C}$ and $\varepsilon(b_i) > 0$ for all $i \in I$. Then, by the results of Doi \cite{MR2674691}, $A$ is semisimple. Note that $A$ is a symmetric algebra with trace form given by $\phi(b_i) = \delta_{i 0}$ and the dual basis of $\{ b_i \}$ with respect to \eqref{eq:sym-alg-trace} is given by $b_i^\vee = \varepsilon(b_i)^{-1} b_{i^*}$. Applying \eqref{eq:FS-symm-alg-1} and \eqref{eq:FS-symm-alg-2} to $A^\tau = (A, S \circ \tau, 1)$, we obtain the following formula:
\begin{equation*}
\nu^\tau(V)
= \frac{\dim_\mathbb{C}(V)}{\chi_V(v_A)} \sum_{i \in I} \frac{1}{\varepsilon(b_i)} \chi_V^{}(b_{\tau(i)} b_i)
= \frac{1}{c_V \dim_\mathbb{C}(V)} \sum_{i \in I} \frac{1}{\varepsilon(b_i)} \chi_V^{}(b_{\tau(i)} b_i)
\end{equation*}
In particular, applying this formula to the adjacency algebra of an association scheme, we recover the formula of Hanaki and Terada \cite{TeradaJunya:2006-03}.
To obtain the twisted Frobenius--Schur theorem for this class of algebras, combine the above formula with Theorem~\ref{thm:piv-FS}; we then have $\nu^\tau(V) \in \{ 0, \pm 1 \}$. Moreover, $\nu^\tau(V) \ne 0$ if and only if there exists a non-degenerate bilinear form $\beta$ on $V$ such that $\beta(b_{\tau(i)} v, w) = \beta(v, b_{i^*} w)$ for all $i \in I$ and $v, w \in V$. Such a bilinear form $\beta$ is symmetric if $\nu^\tau(V) = +1$ and skew-symmetric if $\nu^\tau(V) = -1$.
\end{example}
\begin{example}[Weak Hopf $C^*$-algebras]
\label{ex:FS-ind-ch-WHA}
We assume that the base field is $\mathbb{C}$. A {\em weak Hopf algebra} is an algebra $H$ which is a coalgebra at the same time such that there exists a special map $S: H \to H$ called the antipode; see \cite{MR1726707} and \cite{MR1793595} for the precise definition. We note that the antipode of a weak Hopf algebra is known to be an anti-algebra map.
Let $H$ be a finite-dimensional weak Hopf $C^*$-algebra; see \cite[\S4]{MR1726707} for the precise definition. There exists an element $g \in H$, called the {\em canonical grouplike element} \cite[\S4]{MR1726707}, satisfying $S(g) = g^{-1}$, $S^2(x) = g x g^{-1}$ for all $x \in H$, and some other good properties. In particular, the triple $(H, S, g)$ is a pivotal algebra and therefore the FS indicator $\nu(V)$ is defined for each $V \in \fdMod(H)$.
We can express $\nu(V)$ by using the {\em Haar integral} \cite[\S3]{MR1726707}; if $\Lambda \in H$ is the Haar integral in $H$, then $E = S(\Lambda_{(1)}) \otimes \Lambda_{(2)}$ is a separability idempotent of $H$ ({\em cf}. the proof of Theorem 3.13 of \cite{MR1726707}). Applying Theorem~\ref{thm:piv-FS-ind-ch}, we have
\begin{equation}
\label{eq:FS-formula-WHA}
\nu(V) = \chi_V(S^2(\Lambda_{(1)}) g \Lambda_{(2)}) = \chi_V(g \Lambda_{(1)} \Lambda_{(2)})
\end{equation}
Combining the above formula with Theorem~\ref{thm:piv-FS}, we obtain the Frobenius--Schur theorem for semisimple weak Hopf algebras. We finally give some remarks concerning this example:
(1) Takahiro Hayashi (in private communication with the author) has proved \eqref{eq:FS-formula-WHA} and analogous formulas of the higher FS indicators for weak Hopf algebras in the case where $S^2 = \id_H$.
(2) The formula of Linchenko and Montgomery \cite{MR1808131} is the case where $H$ is an ordinary Hopf algebra. If this is the case, then the $C^*$-condition for $H$ is not needed since we have $S^2 = \id_H$ by the theorem of Larson and Radford \cite{MR957441}. It is not known whether every semisimple weak Hopf algebra $H$ has a grouplike element $g$ such that $S^2(h) = g h g^{-1}$ for all $h \in H$. An affirmative answer to this question proves Conjecture 2.8 of \cite{MR2183279}, which states that every fusion category admits a pivotal structure.
(3) We do not know whether our Formula \eqref{eq:FS-formula-WHA} is equivalent to \cite[(4.3)]{MR1657800} or \cite[(3.70)]{MR1793595}. In \cite{MR2104908,MR1657800,MR1793595}, formulas are proved by finding a central element $e$ such that $\nu(V) = \chi_V(e)$ for all $V$. On the other hand, the element $S(E^1) g E^2$ of our Theorem~\ref{thm:piv-FS-ind-ch} is not central in general. In \S\ref{subsec:quasi-hopf-algebras}, we give a formula of the FS indicator for quasi-Hopf algebras and its twisted version. For the above reason, it is not straightforward to derive the formula of Mason and Ng \cite{MR2104908} from our formula.
(4) By an involution of $H$, we mean an involutive algebra map $\tau: H \to H$ which is also a coalgebra map. Such a map $\tau$ is in fact an involution of the pivotal algebra $(H, S, g)$ and the $\tau$-twisted FS indicator of $V \in \Rep(H)$ is given by $\nu^\tau(V) = \chi_V(g \tau(\Lambda_{(1)}) \Lambda_{(2)})$. We omit the details since these results can be proved in a similar way as the case of quasi-Hopf algebras; see \S\ref{subsec:quasi-hopf-algebras}.
\end{example}
\begin{example}[Twisting by $L \otimes (-)$]
By using separable pivotal algebras, Theorem~\ref{thm:FS-ind-Hopf-tw-2} can be proved as follows: Let $H$, $\alpha$ and $L$ be as in that theorem and define $T_\alpha: H \to H$ by $T_\alpha(h) = \alpha(h_{(1)}) S(h_{(2)})$ ($h \in H$). It is easy to see that the triple $H' = (H, T_\alpha, 1)$ is a pivotal algebra and the identity functor is a strict duality preserving functor between $\fdMod(H)^{\mathbf{t}(L)}$ and $\fdMod(H')$. Hence, by Proposition~\ref{prop:FS-ind-inv} and Theorem~\ref{thm:piv-FS-ind-ch}, we have
\begin{equation*}
\nu(V; L) = \chi_V(T_\alpha S(\Lambda_{(1)}) \Lambda_{(2)})
= \alpha(S(\Lambda_{(1)})) \chi_V(\Lambda_{(2)} \Lambda_{(3)})
\end{equation*}
for all $V \in \fdMod(H)$. The meaning of $\nu(V; L)$ is obtained by applying Theorem~\ref{thm:piv-FS} to $H'$.
\end{example}
\subsection{Quasi-Hopf Algebras}
\label{subsec:quasi-hopf-algebras}
We derive a formula of Mason and Ng \cite{MR2104908} and its twisted version from our results.
Recall that a {\em quasi-Hopf algebra} \cite{MR1047964} is a data $H = (H, \Delta, \varepsilon, \Phi, S, \alpha, \beta)$ consisting of an algebra $H$, algebra maps $\Delta: H \to H \otimes H$ and $\varepsilon: H \to k$, an anti-algebra automorphism $S: H \to H$, elements $\alpha, \beta \in H$ and an invertible element $\Phi \in H^{\otimes 3}$ with inverse $\overline{\Phi}$ satisfying numerous conditions. Let $H_i = (H_i, \Delta_i, \varepsilon_i, \Phi_i, S_i, \alpha_i, \beta_i)$ be quasi-Hopf algebras $(i = 1, 2$). A {\em morphism of quasi-Hopf algebras} from $H_1$ to $H_2$ is an algebra map $f: H_1 \to H_2$ satisfying
\begin{gather*}
\Delta_2 f = (f \otimes f)\Delta_1, \quad
\varepsilon_2 f = \varepsilon_1, \quad
\Phi_2 = (f \otimes f \otimes f)(\Phi_1) \\
S_2 f = f S_1, \quad
\alpha_2 = f(\alpha_1), \quad
\beta_2 = f(\beta_2)
\end{gather*}
Hence, by an {\em involution} of a quasi-Hopf algebra $H$, we shall mean a morphism $\tau: H \to H$ of quasi-Hopf algebras such that $\tau^2 = \id_H$.
If $H$ is a quasi-Hopf algebra, then $\fdMod(H)$ is a rigid monoidal category. Given $V \in \fdMod(H)$, we denote by $e_V: V^\vee \otimes V \to k$ and $c_V: k \to V \otimes V^\vee$ the evaluation and the coevaluation, respectively. Here we need to recall that the dual module of $V$ is defined by the same way as~\eqref{eq:dual-module} and the maps $e_V$ and $c_V$ are given by
\begin{equation*}
e_V(\lambda \otimes v) = \langle \lambda, \alpha v \rangle
\quad (\lambda \in V^\vee, v \in V); \quad
c_V(1) = \sum_{i = 1}^n \beta v_i \otimes v^i
\end{equation*}
where $\{ v_i \}_{i = 1, \dotsc, n}$ is a basis of $V$ and $\{ v^i \}$ is the dual basis.
We need additional assumptions on $H$ so that $H$ is a pivotal algebra. In what follows, we suppose that $k$ is an algebraically closed field of characteristic zero and $H$ is a finite-dimensional semisimple quasi-Hopf algebra. Then $\fdMod(H)$ is a fusion category \cite{MR2183279} such that each its object has an integral Frobenius--Perron dimension. Therefore, by the results of \cite{MR2183279}, $\fdMod(H)$ has a {\em canonical pivotal structure}, {\em i.e.}, an isomorphism $j: \id_{\fdMod(H)} \to (-)^{\vee\vee}$ of $k$-linear monoidal functors such that, for all $V \in \fdMod(H)$, the composition
\begin{equation*}
\begin{CD}
k @>{c_V}>> V \otimes V^\vee @>{j_V \otimes \id_{V^\vee}}>> V^{\vee\vee} \otimes V^\vee @>{e_{V^\vee}}>> k
\end{CD}
\end{equation*}
maps $1 \in k$ to $\dim_k(V) \in k$. Now let $g \in H$ be the image of $1 \in H$ under
\begin{equation*}
\begin{CD}
H @>{j_H}>> H^{\vee\vee} @>{\iota_H^{-1}}>> H
\end{CD}
\end{equation*}
where $\iota_H$ is the Isomorphism~\eqref{eq:k-Vec-piv-mor}. We call $g$ the {\em canonical pivotal element} of $H$. By definition, $g$ is invertible and satisfies $S(g) g = 1$ and $S^2(h) = g a g^{-1}$ for all $a \in H$; see \cite{MR2104908} and \cite{MR2095575} for details. Hence $(H, S, g)$ is a pivotal algebra. Now we remark:
\begin{lemma}
\label{lem:can-piv-1}
Let $f: H_1 \to H_2$ be an isomorphism between finite-dimensional semisimple quasi-Hopf algebras. Then we have $f(g_1) = g_2$, where $g_i \in H_i$ is the canonical pivotal element of $H_i$.
\end{lemma}
\begin{proof}
$f$ induces a functor $f^\natural: \fdMod(H_1) \to \fdMod(H_2)$. By the definition of morphisms of quasi-Hopf algebras, the functor $f^\natural$ is a $k$-linear strict monoidal equivalence. The result follows from the fact that such a functor preserves the canonical pivotal structure \cite[Corollary~6.2]{MR2381536}.
\end{proof}
From this lemma, we see that an involution $\tau$ of the quasi-Hopf algebra $H$ is an involution of the pivotal algebra $(H, S, g)$. As we have observed in \S\ref{sec:piv-alg-cat-dual}, $\tau$ gives rise to an involution of $\fdMod(H)$ and hence the $\tau$-twisted FS indicator $\nu^\tau(V)$ is defined for $V \in \fdMod(H)$. By Theorem~\ref{thm:piv-FS} applied to $A = (H, S \circ \tau, g)$, we have the following property of $\nu^\tau$:
\begin{theorem}
Let $V \in \fdMod(H)$ be a simple module. Then $\nu^\tau(V) \in \{ 0, \pm 1 \}$ and the following statements are equivalent: \\
\indent {\rm (1)} $\nu^\tau(V) \ne 0$. \\
\indent {\rm (2)} $\tau^\natural(V)$ is isomorphic to the dual module $V^\vee$ as a $H$-module. \\
\indent {\rm (3)} There exists a non-degenerate bilinear form $b$ on $V$ satisfying \\
\begin{equation*}
b(\tau(h) v, w) = b(v, S(h) w) \text{\quad for all $v, w \in V$}
\end{equation*}
If one of the above statements holds, then such a bilinear form $b$ is unique up to scalar multiples and satisfies $b(w, g v) = \nu^\tau(V) \cdot b(v, w)$ for all $v, w \in V$.
\end{theorem}
Next we express the number $\nu^\tau(V)$ by using the character of $V$. To that end, it is sufficient to find a separability idempotent of $H$. Let $\Lambda \in H$ be the Haar integral of $H$ (see \cite{1999math4164H} and \cite{MR1615781}). We set
\begin{align*}
p_L & = \Phi^2 S^{-1}(\Phi^1 \beta) \otimes \Phi^{3}, &
q_L & = S(\overline{\Phi}{}^1) \alpha \overline{\Phi}{}^2 \otimes \overline{\Phi}{}^{3} \\
p_R & = \overline{\Phi}{}^1 \otimes \overline{\Phi}{}^2 \beta S(\overline{\Phi}{}^1), &
q_R & = \Phi^1 \otimes S^{-1}(\alpha \Phi^{3}) \Phi^2
\end{align*}
and fix $p \in \{ p_L, p_R \}$ and $q \in \{ q_L, q_R \}$. Following \cite[Lemma~3.1]{MR2104908}, we have
\begin{align}
\label{eq:q-Hopf-integ-1}
\Lambda_{(1)} p^1 a \otimes \Lambda_{(2)} p^2
& = \Lambda_{(1)} p^1 \otimes \Lambda_{(2)} p^2 S(a) \\
\label{eq:q-Hopf-integ-2}
S(a) q^1 \Lambda_{(1)} \otimes q^2 \Lambda_{(2)}
& = q^1 \Lambda_{(1)} \otimes a q^2 \Lambda_{(2)}
\end{align}
for all $a \in A$. From these identities, we see that both $S(\Lambda_{(1)}p^1) \otimes \alpha \Lambda_{(2)} p^2$ and $q^1 \Lambda_{(1)} \beta \otimes S(q^2 \Lambda_{(2)})$ are separability idempotents. Applying Theorem~\ref{thm:piv-FS-ind-ch} to $(H, S \tau, g)$, we have
\begin{equation*}
\nu^\tau(V)
= \chi_V \Big( S \tau S(\Lambda_{(1)}p^1) g \alpha \Lambda_{(2)} p^2 \Big)
= \chi_V \Big( S \tau (q^1 \Lambda_{(1)} \beta) g S(q^2 \Lambda_{(2)}) \Big)
\end{equation*}
for all $V \in \fdMod(H)$. Hence, by using the former expression, we compute
\begin{align*}
\nu^\tau(V)
& = \chi_V(S^2(\tau(\Lambda_{(1)}p^1)) g \cdot \alpha \Lambda_{(2)} p^2)
= \chi_V(g \cdot \tau(\Lambda_{(1)}p^1) \cdot \alpha \Lambda_{(2)} p^2) \\
& = \chi_V(g \cdot \tau(\Lambda_{(1)}p^1 \tau(\alpha)) \cdot \Lambda_{(2)} p^2)
\mathop{=}^{\eqref{eq:q-Hopf-integ-1}} \chi_V(g \cdot \tau(\Lambda_{(1)}p^1) \Lambda_{(2)} p^2 \cdot S\tau(\alpha))) \\
& = \chi_V(S\tau(\alpha) g \cdot \tau(\Lambda_{(1)}p^1) \Lambda_{(2)} p^2)
\end{align*}
Note that the formula of Mason and Ng in \cite{MR2104908} does not involve $g$. To exclude $g$ from the above formula of $\nu^\tau(V)$, we require:
\begin{lemma}
\label{lem:can-piv-2}
Fix $p \in \{ p_L, p_R \}$ and $q \in \{ q_L, q_R \}$. Then we have
\begin{equation}
\label{eq:q-Hopf-integ-3}
g^{-1} S(\beta) = S(\Lambda_{(1)} p^1) \Lambda_{(2)} p^2, \quad
S(\alpha)g = S(q^2\Lambda_{(2)}) q^1\Lambda_{(1)}
\end{equation}
\end{lemma}
\begin{proof}
The first identity is proved in \cite{MR2095575} (where our $g$ appears as $g^{-1}$) and the second can be proved in a similar way. For the sake of completeness, we give a detailed proof of the second identity.
Let $V$ be a simple $H$-module and set $c = c_V(1)$. The map
\begin{equation*}
e_1: V \otimes V^\vee \to k,
\quad e_2(v \otimes f) = \langle q^2\Lambda_{(2)} f, q^1 \Lambda_{(1)} v \rangle
\quad (v \in V, f \in V^\vee)
\end{equation*}
is an $H$-linear map such that $e_1(c) = \dim_k(V)$. On the other hand, by the definition of the canonical pivotal structure, we see that
\begin{equation*}
\begin{CD}
e_2: V \otimes V^\vee @>{j_V \otimes \id_{V^\vee}}>> V^{\vee\vee} \otimes V^\vee @>{e_V}>> k
\end{CD}
\end{equation*}
has the same property. Since $\Hom_H(V \otimes V^\vee, k) \cong \Hom_H(V, V) \cong k$, we have $e_1 = e_2$. This implies that $\langle f, S(\alpha)g v \rangle = \langle f, S(q^2 \Lambda_{(2)}) t^1 \Lambda_{(1)} v \rangle$ holds for all $f \in V^\vee$ and $v \in V$. In conclusion, $S(\alpha)g = S(q^2\Lambda_{(2)}) q^1\Lambda_{(1)}$ holds on each simple module $V$. Since $H$ is semisimple, the identity holds in $H$.
\end{proof}
By Lemmas~\ref{lem:can-piv-1} and~\ref{lem:can-piv-2}, we have $S\tau(\alpha) g = \tau(S(\alpha)g) = S(\tau(q^2\Lambda'_{(2)})) \cdot \tau(q^1 \Lambda'_{(1)})$, where $\Lambda' = \Lambda$ is a copy of $\Lambda$. Hence we compute:
\begin{align*}
\nu^\tau(V)
& = \chi_V(S(\tau(q^2\Lambda'_{(2)})) \cdot \tau(q^1 \Lambda'_{(1)})
\cdot \tau(\Lambda_{(1)}p^1) \Lambda_{(2)} p^2) \\
& = \chi_V(\tau(q^1 \Lambda'_{(1)} \Lambda_{(1)}p^1)
\cdot \Lambda_{(2)} p^2 S(\tau(q^2\Lambda'_{(2)}))) \\
& = \chi_V(\tau(q^1 \Lambda'_{(1)} \Lambda_{(1)}p^1 \tau(q^2\Lambda'_{(2)})) \cdot \Lambda_{(2)} p^2) \\
& = \chi_V(\tau(q^1 \Lambda'_{(1)} \Lambda_{(1)}p^1)
q^2\Lambda'_{(2)} \Lambda_{(2)} p^2)
\end{align*}
Since $\Delta: H \to H \otimes H$ is an algebra map, we have $\Lambda_{(1)}' \Lambda_{(1)} \otimes \Lambda_{(2)}' \Lambda_{(2)} = \Delta(\Lambda' \Lambda) = \varepsilon(\Lambda') \Delta(\Lambda) = \Delta(\Lambda)$. Hence, we finally obtain $\nu^\tau(V) = \chi_V (\tau(q^1 \Lambda_{(1)}p^1) q^2 \Lambda_{(2)} p^2)$. Letting $\tau = \id_H$, we recover the results of Mason and Ng \cite{MR2104908}. Assuming $H$ to be a Hopf algebra, we recover the results of Sage and Vega \cite{MR2879228}.
\section{Coalgebras}
\label{sec:coalgebras}
\subsection{Copivotal Coalgebras}
In this section, we introduce the dual notion of pivotal algebras and study the Frobenius--Schur theory for them. For the reader's convenience, we briefly recall some basic results on coalgebras.
Given a coalgebra $C$, we denote by $\Com(C)$ the category of {\em right} $C$-comodules and by $\fdCom(C)$ its full subcategory of finite-dimensional objects. We express the coaction of $V \in \Com(C)$ as
\begin{equation*}
\rho_V: V \to V \otimes C,
\quad v \mapsto v_{(0)} \otimes v_{(1)}
\quad (v \in V)
\end{equation*}
The {\em convolution product} of $\lambda, \mu \in C^\vee$ is defined by $\langle \lambda \star \mu, c \rangle = \langle \lambda, c_{(1)} \rangle \langle \mu, c_{(2)} \rangle$ for all $c \in C$. $C^\vee$ is an algebra, called the {\em dual algebra}, with multiplication $\star$ and unit $\varepsilon$. The algebra $C^\vee$ acts from the left on each $V \in \Com(C)$ by
$\text{$\rightharpoonup$}: C^\vee \otimes V \to V$, $\lambda \rightharpoonup v = v_{(0)} \langle \lambda, v_{(1)} \rangle$ ($\lambda \in C^\vee, v \in V$). This defines a $k$-linear fully faithful functors $\Com(C) \to \Mod(C^\vee)$ and
\begin{equation}
\label{eq:com-to-mod}
\fdCom(C) \to \fdMod(C^\vee)
\end{equation}
which are not equivalences in general. If $C$ is finite-dimensional, then these functors are isomorphisms of categories. See, e.g., \cite{MR1786197} for details.
Fix a basis $\{ v_i \}_{i = 1, \dotsc, n}$ of $V \in \fdCom(C)$. Then we can define $c_{i j} \in C$ by $\rho_V(v_j) = \sum_{i = 1}^n v_i \otimes c_{i j}$ ($j = 1, \dotsc, n$). The matrix $(c_{i j})$ is called the {\em matrix corepresentation} of $V$ with respect to the basis $\{ v_i \}$. By the definition of comodules, we have
\begin{equation}
\label{eq:matrix-corep}
\Delta(c_{i j}) = \sum_{s = 1}^n c_{i s} \otimes c_{s j},
\quad \varepsilon(c_{i j}) = \delta_{i j}
\end{equation}
for all $i, j = 1, \dotsc, n$. Hence $C_V = \mathrm{span}_k \{ c_{i j} \mid i, j = 1, \dotsc, n \}$ is a subcoalgebra of $C$. We call $C_V$ the {\em coefficient subcoalgebra} of $C$. $C_V$ has a special element $t_V = \sum_i c_{i i}$, called the {\em character} of $V$. If we regard $V$ as a left $C^\vee$-module via \eqref{eq:com-to-mod} and denote its character by $\chi_V$, then we have
\begin{equation}
\label{eq:coalg-character}
\chi_V(\lambda)
= \langle \lambda, c_{11} \rangle + \dotsb + \langle \lambda, c_{n n} \rangle
= \lambda(t_V)
\end{equation}
for all $\lambda \in C^\vee$.
Now let $V$ be a finite-dimensional vector space with basis $\{ v_i \}_{i = 1, \dotsc, n}$ and let $\{ v^i \}$ denote the dual basis. Then $\coEnd(V) = V^\vee \otimes V$ has a basis $e_{i j} = v^i \otimes v_j$ ($i, j = 1, \dotsc, n$) and turns into a coalgebra with $\Delta(e_{i j}) = \sum_{s = 1}^n e_{i s} \otimes e_{s j}$, $\varepsilon(e_{i j}) = \delta_{i j}$. $\coEnd(V)$ coacts on $V$ from the right by
\begin{equation*}
V \to V \otimes \coEnd(V),
\quad v_j \mapsto \sum_{j = 1}^n v_i \otimes e_{i j}
\quad (j = 1, \dotsc, n)
\end{equation*}
Suppose that $C$ coacts on $V$. Let $(c_{i j})$ be the matrix corepresentation of $V$ with respect to $\{ v_i \}$. By \eqref{eq:matrix-corep}, the linear map $\phi: \coEnd(V) \to C$, $\phi(e_{i j}) = c_{i j}$ is a coalgebra map. Conversely, if a coalgebra map $\phi: \coEnd(V) \to C$ is given, $V$ is a right $C$-comodule by
\begin{equation*}
\rho: V \to V \otimes C,
\quad v_j \mapsto \sum_{i = 1}^n v_i \otimes \phi(e_{i j})
\quad (j = 1, \dotsc, n)
\end{equation*}
These constructions give a bijection between the set of linear maps $\rho: V \to V \otimes C$ making $V$ into a right $C$-comodule and the set of coalgebra maps $\phi: \coEnd(V) \to C$.
Suppose that $V \in \fdCom(C)$ is absolutely simple. As the dual of~\eqref{eq:abs-simp-surj}, we have that the corresponding coalgebra map $\phi: \coEnd(V) \to C$ is injective. Let $(c_{i j})$ be the matrix corepresentation of $V$ with respect to some basis of $V$. The injectivity of $\phi$ implies that the set $\{ c_{i j} \}$ is linearly independent.
Now we introduce {\em copivotal coalgebras} as the dual notion of pivotal algebras:
\begin{definition}
A {\em copivotal coalgebra} is a triple $(C, S, \gamma)$ consisting of a coalgebra $C$, an anti-coalgebra map $S: C \to C$ and a linear map $\gamma: C \to k$ satisfying
\begin{equation*}
S^2(c) = \langle \gamma, c_{(1)} \rangle c_{(2)} \langle \overline{\gamma}, c_{(3)} \rangle
\text{\quad and \quad} \overline{\gamma} = \gamma \circ S
\end{equation*}
for all $c \in C$, where $\overline{\gamma}: C \to k$ is the inverse of $\gamma$ with respect to $\star$.
\end{definition}
Let $C = (C, S, \gamma)$ is a copivotal coalgebra. If $V \in \fdCom(C)$, then we can make $V^\vee$ into a right $C$-comodule as follows: First fix a basis $\{ v_i \}$ of $V$ and let $(c_{i j})$ be the matrix corepresentation of $V$ with respect to the basis $\{ v_i \}$. Then we define the coaction of $C$ on $V^\vee$ by
\begin{equation}
\label{eq:dual-comodule}
\rho_{V^\vee}: V^\vee \to V^\vee \otimes C,
\quad \rho_{V^\vee}(v^j) = \sum_{i = 1}^n v^i \otimes S(c_{j i})
\quad (i = 1, \dotsc, n)
\end{equation}
where $\{ v^i \}$ is the dual basis of $\{ v_i \}$. This coaction does not depend on the choice of the basis and has the following characterization, which is rather useful than the above explicit formula:
\begin{equation*}
\langle f_{(0)}, v \rangle f_{(1)} = \langle f, v_{(0)} \rangle S(v_{(1)})
\quad (f \in V^\vee, v \in V)
\end{equation*}
For each $V \in \fdCom(C)$, we define $j_V: V \to V^{\vee\vee}$ by
\begin{equation*}
\langle j_V(v), f \rangle = \langle f, \gamma \rightharpoonup v \rangle
\quad (= \langle f, v_{(0)} \rangle \langle \gamma, v_{(1)} \rangle)
\quad (f \in V^\vee, v \in V)
\end{equation*}
In a similar way as Proposition~\ref{sec:piv-alg-cat-dual}, we prove:
\begin{proposition}
$\fdMod(C)$ is a category with strong duality over $k$.
\end{proposition}
The triple $C^\vee = (C^\vee, S^\vee, \gamma)$ is a pivotal algebra, which we call the {\em dual pivotal algebra} of $C$. Let $V \in \fdMod(C)$. For all $\lambda \in C^\vee$, $f \in V^\vee$ and $v \in V$, we have
\begin{equation*}
\langle \lambda \rightharpoonup f, v \rangle
= \langle f_{(0)}, v \rangle \langle \lambda, f_{(1)} \rangle
= \langle f, v_{(0)} \rangle \langle \lambda, S(v_{(1)}) \rangle
= \langle f, S^\vee(\lambda) \rightharpoonup v \rangle
\end{equation*}
This implies that the Functor \eqref{eq:com-to-mod} is in fact a strict duality preserving functor. In what follows, we often regard $\fdCom(C)$ as a full subcategory of $\fdMod(C^\vee)$.
\subsection{FS Indicator for Copivotal Coalgebras}
Let $C = (C, S, \gamma)$ be a copivotal coalgebra, and let $V \in \fdCom(V)$. We denote by $\Bil(V)$ the set of all bilinear forms on $V$ and by $\Bil_C(V)$ its subset consisting of those $b \in \Bil(V)$ satisfying
\begin{equation}
\label{eq:S-adjoint-bilin-coalg}
b(v_{(0)}, w) v_{(1)} = b(v, w_{(0)}) S(w_{(1)}) \text{\quad for all $v, w \in V$}
\end{equation}
$\Bil_C(V)$ is the image of $\Hom_C(V, V^\vee) \subset \Hom_k(V, V^\vee)$ under the canonical Isomorphism \eqref{eq:canonical-map}. Define $\Sigma_V: \Bil_C(V) \to \Bil_C(V)$ in the same way as before. Then, for all $b \in \Bil_C(V), v, w \in V$, we have
\begin{equation*}
\Sigma_V(b)(v, w)
= b(w, \gamma \rightharpoonup v)
\end{equation*}
Now let $\Bil_{C}^\pm(V)$ be the eigenspace of $\Sigma_V$ with eigenvalue $\pm 1$:
\begin{equation*}
\Bil_{C}^{\pm}(V) = \{ b \in \Bil_{C}(V) \mid
\text{$b(w, \gamma \rightharpoonup v) = \pm b(v, w)$ for all $v, w \in V$} \}
\end{equation*}
Then we have $\nu(V) = \dim_k \Bil_C^+(V) - \dim_k \Bil_C^-(V)$ as a counterpart of Proposition~\ref{prop:FS-ind-basic} (b). Now we immediately obtain the following coalgebraic version of Theorem \ref{thm:piv-FS}:
\begin{theorem}
\label{thm:copiv-FS}
If $V \in \fdCom(C)$ is absolutely simple, then we have $\nu(V) \in \{ 0, \pm 1 \}$. Moreover, the following are equivalent: \\
\indent {\rm (1)} $\nu(V) \ne 0$. \\
\indent {\rm (2)} $V$ is isomorphic to $V^\vee$ as a right $C$-comodule. \\
\indent {\rm (3)} There exists a non-degenerate bilinear form $b$ on $V$ satisfying~\eqref{eq:S-adjoint-bilin-coalg}. \\
If one of the above statements holds, then such a bilinear form $b$ is unique up to scalar multiples and satisfies $b(w, \gamma \rightharpoonup v) = \nu(V) \cdot b(v, w)$ for all $v, w \in V$.
\end{theorem}
We prove several statements concerning the FS indicator of $V \in \fdCom(C)$. The proof will be done by reducing to the case of pivotal algebras in the following way: First fix a subcoalgebra $D \subset C$ satisfying
\begin{equation}
\label{eq:copiv-subcoalg}
\dim_k(D) < \infty, \quad C_V \subset D \text{\quad and \quad} S(D) \subset D
\end{equation}
Note that such a subcoalgebra $D$ always exists. Indeed, by~\eqref{eq:dual-comodule}, we have $C_{X^\vee} = S(C_X)$ for all $X \in \fdCom(C)$. If $V$ is a subcomodule of $X$, then $C_V$ is a subcoalgebra of $C_X$. Therefore, since $X = V \oplus V^\vee$ is self-dual and has $V$ as a subcomodule, $D = C_{V \oplus V^\vee}$ satisfies \eqref{eq:copiv-subcoalg}.
It is obvious that the triple $D = (D, S|_D, \gamma|_D)$ is a copivotal coalgebra and hence $D^\vee$ is a pivotal algebra. As we remarked in the above, the functor
\begin{equation*}
\begin{CD}
F_D: \fdMod(D^\vee) @>{\cong}>{\eqref{eq:com-to-mod}}> \fdCom(D) @>{\text{inclusion}}>> \fdCom(C)
\end{CD}
\end{equation*}
is a $k$-linear fully faithful strict duality preserving functor. Hence, by Proposition~\ref{prop:FS-ind-inv}, $F_D$ preserves the FS indicator. By {\em regarding $V$ as a left $D^\vee$-module}, we mean taking $V_0 \in \fdMod(D^\vee)$ such that $F_D(V_0) = V$ and then identifying $V_0$ with $V$.
Now we prove an analogue of Theorem~\ref{thm:piv-Tr-S}. Let $\reg_C$ denote the coalgebra $C$ regarded as a right $C$-comodule by the comultiplication.
\begin{theorem}
\label{thm:copiv-Tr-S}
Let $C = (C, S, \gamma)$ be a finite-dimensional copivotal coalgebra. \\
{\rm (a)} $\nu(\reg_C^\vee) = \Trace(Q)$, where $Q: C \to C$, $c \mapsto S(c_{(1)}) \langle \gamma, c_{(2)} \rangle$. \\
{\rm (b)} If $C$ is co-Frobenius, then $\reg_C \cong \reg_C^\vee$ as right $C$-comodules. Hence, $\nu(\reg_C) = \nu(\reg_C^\vee)$.
\end{theorem}
\begin{proof}
(a) Write $A = C^\vee$ and regard the right $C$-comodule $\reg_C^\vee$ as a left $A$-module via~\eqref{eq:com-to-mod}. To avoid confusion, we denote by $\rightharpoonup$ the action of $A$ on $\reg_A$ and by $\rightharpoondown$ that on $\reg_C^\vee$. Since the coaction of $\mu \in \reg_C^\vee$ is characterized as $\langle \mu_{(0)}, c \rangle \mu_{(1)} = \langle \mu, c_{(1)} \rangle S(c_{(2)})$, the action $\rightharpoondown: A \times \reg_C^\vee \to \reg_C^\vee$ is given by
\begin{equation*}
\langle f \rightharpoondown \mu, c \rangle
= \langle f, S(c_{(2)}) \rangle \langle \mu, c_{(1)} \rangle
\quad (f \in A, \mu \in \reg_C^\vee, c \in C)
\end{equation*}
Consider the map $S^\vee: \reg_A \to \reg_C^\vee$. For $\lambda \in A$, $f \in \reg_A$ and $c \in C$, we have
\begin{gather*}
\langle S^\vee(\lambda \rightharpoonup f), c \rangle
= \langle \lambda \star f, S(c) \rangle
= \langle \lambda, S(c_{(2)}) \rangle \langle f, S(c_{(1)}) \rangle \\
= \langle \lambda, S(c_{(2)}) \rangle \langle S^\vee(f), c_{(1)} \rangle
= \langle \lambda \rightharpoondown S^\vee(f), c \rangle
\end{gather*}
and therefore $S^\vee: \reg_A \to \reg_C^\vee$ is an isomorphism of $A$-modules. Applying Theorem~\ref{thm:piv-Tr-S} to $A = C^\vee$, we see that $\nu(\reg_C^\vee) = \nu(\reg_A)$ is equal to the trace of the map $Q': A \to A$, $\lambda \mapsto S^\vee(\lambda) \star \gamma$ ($\lambda \in C^\vee$). Since $\langle Q'(\lambda), c \rangle = \langle \lambda, S(c_{(1)}) \rangle \langle \gamma, c_{(2)} \rangle = \langle \lambda, Q(c) \rangle$, $Q'$ is the dual map of $Q$. Hence, we have $\nu(\reg_C^\vee) = \Trace(Q') = \Trace(Q)$.
(b) If $C$ is co-Frobenius, then $A$ is Frobenius. Thus, by Theorem~\ref{thm:piv-Tr-S}, there is an isomorphism $\varphi: \reg_A \to \reg_A^\vee$ of $A$-modules. In the proof of (1), we see that $S^\vee: \reg_A \to \reg_C^\vee$ is an isomorphism of $A$-modules. Regarding them as isomorphisms in the category $\fdCom(C)$, we obtain an isomorphism
\begin{equation*}
\begin{CD}
\reg_C @>{j}>> \reg_C^{\vee\vee} @>{S^{\vee\vee}}>> \reg_A^\vee @>{\varphi}>> \reg_A @>{S^\vee}>> \reg_C^\vee
\end{CD}
\end{equation*}
of right $C$-comodules.
\end{proof}
The following is a coalgebraic version of Theorem~\ref{thm:piv-Tr-Sv}.
\begin{theorem}
\label{thm:copiv-Tr-Sv}
Let $V \in \fdCom(C)$ be an absolutely simple comodule and suppose that $V$ is self-dual. Then, by \eqref{eq:dual-comodule}, the map
\begin{equation*}
S_V: C_V \to C_V, \quad S_V(c) = S(c) \quad (c \in C)
\end{equation*}
is well-defined. We also define
\begin{equation*}
Q_V: C_V \to C_V, \quad Q_V(c) = S(\gamma \rightharpoonup c)
\quad (= S(c_{(1)}) \gamma(c_{(2)})) \quad (c \in C)
\end{equation*}
Then we have:
\begin{equation*}
\mathrm{(1)} \Trace(S_V) = \nu(V) \cdot \gamma(t_V) \qquad
\mathrm{(2)} \Trace(Q_V) = \nu(V) \cdot \dim_k(V)
\end{equation*}
\end{theorem}
\begin{proof}
(1) We regard $V$ as a left $(C_V)^\vee$-module and denote its character by $\chi_V$. Applying Theorem~\ref{thm:piv-Tr-Sv} to the dual pivotal algebra $(C_V)^\vee$ and by using \eqref{eq:coalg-character}, we have $\Trace(S_V) = \nu(V) \cdot \chi_V(\gamma) = \nu(V) \cdot \gamma(t_V)$.
(2) We regard $C_V$ as a right $C_V$-comodule. Since $C_V \cong \coEnd(V)$ is co-Frobenius, by Theorem~\ref{thm:copiv-Tr-S}, we have $\nu(C_V) = \Trace(Q_V)$. Let $d = \dim_k(V)$. Since $C_V \cong V^{\oplus d}$ as a right $C$-comodule, we have $\nu(C_V) = \nu(V) d$. Hence, $\Trace(Q_V) = \nu(V) d$.
\end{proof}
Applying Corollary~\ref{cor:Tr-S} to the dual pivotal algebra of $C$, we have:
\begin{corollary}
\label{cor:Tr-S-copiv}
Suppose that $k$ is algebraically closed and that $C = (C, S, \gamma)$ is a finite-dimensional cosemisimple copivotal coalgebra. Let $\{ V_i \}_{i = 1, \dotsc, n}$ be a complete set of representatives of the isomorphism classes of simple right $C$-comodules. Then
\begin{equation*}
\Trace(S) = \sum_{i = 1}^n \nu(V_i) \gamma(t_i)
\end{equation*}
where $t_i = t_{V_i}$ is the character of $V_i$.
\end{corollary}
\subsection{Coseparable Copivotal Coalgebras}
A coalgebra $C$ is said to be {\em coseparable} if it has a {\em coseparability idempotent}, {\em i.e.}, a bilinear form $\lambda: C \times C \to k$ satisfying $c_{(1)} \lambda(c_{(2)}, d) = \lambda(c, d_{(1)}) d_{(2)}$ and $\lambda(c_{(1)}, c_{(2)}) = \varepsilon(c)$ for all $c, d \in C$. If such a form exists, then the forgetful functor $\Com(C) \to \Vect(k)$ is separable with section $\Pi_{V,W}: \Hom_k(V, W) \to \Hom_C(V, W)$ given by
\begin{equation*}
\begin{CD}
\Pi_{V,W}(f):
V @>{\rho_V}>> V \otimes C
@>{f \otimes \id_V}>> W \otimes C
@>{\rho_W}>> W \otimes C \otimes C
@>{\id_W \otimes \lambda}>> W
\end{CD}
\end{equation*}
for $f \in \Hom_k(V, W)$. The following theorem can be proved by the arguments of \S\ref{subsec:sep-functor}. Nevertheless, to avoid notational difficulties, we do not use $\Pi$ and prove the theorem by reducing to Theorem \ref{thm:piv-FS-ind-ch}.
\begin{theorem}
\label{thm:copiv-FS-ind-ch}
If $C = (C, S, \gamma)$ is a coseparable copivotal coalgebra with coseparability idempotent $\lambda$, then, for all $V \in \fdCom(C)$, we have
\begin{equation*}
\nu(V)
= \lambda(S(\gamma \rightharpoonup t_{V(1)}), t_{V(2)})
\quad \Big( = \lambda(S(t_{V(1)}), t_{V(3)}) \gamma(t_{V(2)}) \Big)
\end{equation*}
\end{theorem}
\begin{proof}
Fix a subcoalgebra $D$ of $C$ satisfying~\eqref{eq:copiv-subcoalg}. $D$ is coseparable with $\lambda_D = \lambda|_{D \times D}$. Since $D$ is finite-dimensional, there exist finite number of linear maps $\lambda_i', \lambda_i'': D \to k$ such that
\begin{equation*}
\lambda_D(x, y)
= \sum_{i} \lambda_i'(x) \lambda_i''(y)
\end{equation*}
for all $x, y \in D$. It is easy to see that $E = \sum_{i} \lambda_i' \otimes \lambda_i''$ is a separability idempotent for the dual pivotal algebra $D^\vee$. Now we regard $V$ as a left $D^\vee$-module and denote its character by $\chi_V$. Applying Theorem \ref{thm:piv-FS-ind-ch} to $D$, we obtain
\begin{equation*}
\nu(V) = \sum_i \chi_V (S_X^\vee(\lambda_i') \star \gamma \star \lambda_i'')
\end{equation*}
Now the desired formula is obtained by using~\eqref{eq:coalg-character}.
\end{proof}
A {\em copivotal Hopf algebra} is a Hopf algebra $H = (H, \Delta, \varepsilon, S)$ equipped with an algebra map $\gamma: H \to k$ satisfying $S^2(x) = \langle \gamma, x_{(1)} \rangle x_{(2)} \langle \gamma, S(x_{(3)}) \rangle$ for all $x \in H$. Since $\gamma$ is an algebra map, $\gamma \circ S$ is the inverse of $\gamma$ with respect to the convolution product. Therefore a copivotal Hopf algebra is a copivotal coalgebra.
A {\em Haar functional} of a Hopf algebra $H$ is a linear map $\lambda: H \to k$ satisfying
$\langle \lambda, 1 \rangle = 1$ and
$\langle \lambda, x_{(1)} \rangle x_{(2)} = \varepsilon(x) 1 = x_{(1)} \langle \lambda, x_{(2)} \rangle$
for all $x \in H$. If $\lambda$ is a Haar functional of $H$, then the map
\begin{equation*}
\tilde{\lambda}: H \times H \to k,
\quad \tilde{\lambda}(x, y) = \langle \lambda, S(x) y \rangle
\quad (x, y \in H)
\end{equation*}
is a coseparability idempotent of the coalgebra $H$. Note that we have
\begin{equation*}
S^2(x_{(1)}) \langle \gamma, x_{(2)} \rangle
= \langle \gamma, x_{(1)} \rangle x_{(2)} \langle \gamma, S(x_{(3)}) \rangle \langle \gamma, x_{(4)} \rangle
= \langle \gamma, x_{(1)} \rangle x_{(2)}
\end{equation*}
for all $x \in H$. The following corollary is a direct consequence of Theorem~\ref{thm:copiv-FS-ind-ch}:
\begin{corollary}
\label{cor:FS-ind-Haar-int}
Regard a copivotal Hopf algebra $H = (H, \Delta, \varepsilon, S; \gamma)$ as a copivotal coalgebra. If there exists a Haar functional $\lambda: H \to k$ on $H$, then we have
\begin{equation*}
\nu(V) = \langle \gamma, t_{V(1)} \rangle \langle \lambda, t_{V(2)} t_{V(3)} \rangle
\end{equation*}
for all $V \in \fdCom(H)$.
\end{corollary}
We shall explain how can we obtain~\eqref{eq:FS-formula-cpt} from Corollary~\ref{cor:FS-ind-Haar-int}.
\begin{example}
We work over $\mathbb{C}$. Let $G$ be a compact group. A function $f: G \to \mathbb{C}$ is said to be {\em representative} if there exist finite number of functions $f_i, g_i: G \to \mathbb{C}$ such that $f(x y) = \sum_i f_i(x) g_i(y)$ for all $x, y \in G$. We denote by $R(G)$ the algebra of continuous representative functions on $G$. $R(G)$ is in fact a Hopf algebra; the comultiplication, the counit and the antipode are given by
\begin{equation*}
f_{(1)}(x) f_{(2)}(y)= f(x y),
\quad \varepsilon(f) = f(1),
\quad S(f)(x) = f(x^{-1})
\end{equation*}
for $f \in R(G)$, $x, y \in G$. Define $\lambda: R(G) \to \mathbb{C}$ by $\lambda(f) = \int_G f(x) d \mu(x)$, where $\mu$ is the normalized Haar measure on $G$. We see that $\lambda$ is a Haar functional of $R(G)$ (in fact, this is the origin of this term).
The group $G$ acts continuously from the left on each $V \in \fdCom(R(G))$ by $x \cdot v = v_{(1)}(x) \cdot v_{(0)}$ ($x \in G, v \in V$). Conversely, if $V$ is a finite-dimensional continuous representation of $G$, then $R(G)$ coacts from the right on $V$. If we fix a basis $\{ v_i \}_{i = 1, \dotsc, n}$ of $V$, the coaction of $R(G)$ is described as follows: Define $f_{i j}: G \to \mathbb{C}$ by
\begin{equation}
\label{eq:FS-ind-cpt-grp-1}
x \cdot v_i = \sum_{i = 1}^n f_{i j}(x) v_j \quad (x \in G, i = 1, \dotsc, n)
\end{equation}
Then each $f_{i j}$ is an element of $R(G)$. The coaction of $R(G)$ on $V$ is defined by
\begin{equation*}
V \to V \otimes R(G), \quad v_i \mapsto \sum_{j = 1}^n v_j \otimes f_{i j} \quad (i = 1, \dotsc, n)
\end{equation*}
These correspondences give an isomorphism of categories with duality over $\mathbb{C}$ between $\fdCom(R(G))$ and the category of continuous representations of $G$.
Now let $V$ be a continuous representation of $G$ with character $\chi_V$. Regarding $V$ as a right $R(G)$-comodule via the above category isomorphism, we obtain $\nu(V) = \lambda(t_{V(1)} t_{V(2)})$ by Corollary~\ref{cor:FS-ind-Haar-int}. To compute this value, we fix a basis $\{ v_i \}_{i = 1, \dotsc, n}$ of $V$ and define $f_{i j}$ by \eqref{eq:FS-ind-cpt-grp-1}. Then $t_V = f_{1 1} + \dotsb + f_{n n}$. Hence, by~\eqref{eq:matrix-corep}, we compute
\begin{align}
\label{eq:FS-ind-cpt-grp-2}
\nu(V)
= \lambda(t_{V(1)} t_{V(2)})
= \int_G^{} \left( \sum_{i, j = 1}^n f_{i j}(x)f_{ji}(x) \right) d \mu(x)
\end{align}
Since the action of $x \in G$ is represented by $\rho(x) = (f_{i j}(x))_{i, j = 1, \dotsc, n}$, we have
\begin{equation*}
\chi_V(x^2)
= \Trace \Big( \rho(x)^2 \Big)
= \sum_{i, j = 1}^n f_{i j}(x)f_{ji}(x)
\end{equation*}
Substituting this to~\eqref{eq:FS-ind-cpt-grp-2}, we obtain~\eqref{eq:FS-formula-cpt}.
\end{example}
\section{Quantum $SL_2$}
\label{sec:quantum-sl_2}
\subsection{The Hopf Algebra $\mathcal{O}_q(SL_2)$}
In this section, we give some applications of our results to the quantum coordinate algebra $\mathcal{O}_q(SL_2)$ and the quantized universal enveloping algebra $U_q(\mathfrak{sl}_2)$. For details on these Hopf algebras, we refer the reader to \cite{MR1321145} and \cite{MR1492989}.
Throughout, the base field $k$ is assumed to be an algebraically closed field of characteristic zero. $q \in k$ denotes a fixed non-zero parameter that is not a root of unity. We use the following standard notations:
\begin{equation*}
[n]_q = \frac{q^n - q^{-n}}{q - q^{-1}},
\quad [n]_q! = [n]_q \cdot [n - 1]_q!
\quad (n \ge 1),
\quad [0]_q! = 1
\end{equation*}
for $n \in \mathbb{N}_0 = \{ 0, 1, 2, \dotsc \}$.
The quantum coordinate algebra $\mathcal{O}_q(SL_2)$ is a Hopf algebra defined as follows: As an algebra, it is generated by $a$, $b$, $c$ and $d$ with relations
\begin{gather*}
a b = q b a, \quad a c = q c a, \quad b d = q d b, \quad c d = q d c, \quad b c = c b \\
a d - q b c = 1 = d a - q^{-1} b c
\end{gather*}
The comultiplication $\Delta$ and the counit $\varepsilon$ are defined by
\begin{gather*}
\Delta(a) = a \otimes a + b \otimes c, \quad \Delta(b) = a \otimes b + b \otimes d,
\quad \varepsilon(a) = 1, \quad \varepsilon(b) = 0 \\
\Delta(c) = c \otimes a + d \otimes c, \quad \Delta(d) = c \otimes b + d \otimes d,
\quad \varepsilon(c) = 0, \quad \varepsilon(d) = 1
\end{gather*}
and the antipode $S$ is given by
\begin{equation*}
S(a) = d, \quad S(b) = -q^{-1} b, \quad S(c) = - q c, \quad S(d) = a
\end{equation*}
We define an algebra map $\gamma: \mathcal{O}_q(SL_2) \to k$ by
\begin{equation*}
\gamma(a) = q^{-1}, \quad \gamma(b) = \gamma(c) = 0, \quad \gamma(d) = q
\end{equation*}
One can check that $\mathcal{O}_q(SL_2)$ is a copivotal Hopf algebra with $\gamma$. In what follows, we determine the FS indicator of simple $\mathcal{O}_q(SL_2)$-comodules.
For each $\ell \in \frac{1}{2}\mathbb{N}_0$, we put $I_\ell = \{ - \ell, - \ell + 1, \dotsc, \ell - 1, \ell \}$ and
\begin{equation*}
X_\ell = \mathrm{span}_k \{ a^{\ell - i} b^{\ell + i} \mid i \in I_\ell \} \subset \mathcal{O}_q(SL_2)
\end{equation*}
$X_\ell$ is a right coideal and hence it is a right $\mathcal{O}_q(SL_2)$-comodule. It is known that each $X_\ell$ is simple and $\{ X_\ell \mid \ell \in \frac{1}{2}\mathbb{N}_0 \}$ is a complete set of representatives of the isomorphism classes of simple right $\mathcal{O}_q(SL_2)$-comodules. This implies, in particular, that $X_\ell$ is self-dual.
In this section, we first prove the following result:
\begin{theorem}
\label{thm:FS-ind-Oq}
$\nu(X_\ell) = (-1)^{2 \ell}$.
\end{theorem}
By Theorem~\ref{thm:copiv-FS}, this result reads as follows: For each $\ell \in \frac{1}{2}\mathbb{N}_0$, there exists a non-degenerate bilinear form $\beta$ on $X_\ell$ satisfying $\beta(x_{(0)}, y) x_{(1)} = \beta(x, y_{(0)}) S(y_{(1)})$ and $b(y, \gamma \rightharpoonup x) = (-1)^{2\ell} \cdot b(x, y)$ for all $x, y \in X_\ell$.
To prove Theorem~\ref{thm:FS-ind-Oq}, we need a matrix corepresentation of $X_\ell$. For each $i \in I_\ell$, we fix a square root $\mu_i \in k$ of $[\ell + i]_{q^{-2}}!$ and take
\begin{equation*}
x_i^{(\ell)} = \left[ \begin{array}{c} 2 \ell \\ \ell + i \end{array} \right]^{1/2}_{q^{-2}}
a^{\ell - i} b^{\ell + i} \quad (i \in I_\ell), \text{\quad where \quad}
\left[ \begin{array}{c} 2 \ell \\ \ell + i \end{array} \right]^{1/2}_{q^{-2}}
:= \frac{\mu_{\ell}}{\mu_{i} \cdot \mu_{-i}}
\end{equation*}
as a basis of $X_\ell$. Define $c_{i j}^{(\ell)}$ by $\Delta(x_j^{(\ell)}) = \sum x_j^{(\ell)} \otimes c_{i j}^{(\ell)}$. The matrix $(c_{i j}^{(\ell)})$ has been explicitly determined and well-studied in relation to unitary representations of a real form of $\mathcal{O}_q(SL_2)$; see, e.g., \cite[\S4]{MR1492989}. Following {\em loc. cit.}, we have
\begin{equation*}
c_{i j}^{(\ell)} = \begin{cases}
N_{i j \ell}^{+} \cdot a^{-i-j} c^{i-j} \cdot p_{\ell + j} (\zeta; q^{-2(i - j)}, q^{2(i + j)} | q^{-2})
& (i + j \le 0, i \ge j) \\
N_{j i \ell}^{+} \cdot a^{-i-j} b^{j-i} \cdot p_{\ell + i} (\zeta; q^{-2(j - i)}, q^{2(i + j)} | q^{-2})
& (i + j \le 0, i \le j) \\
N_{j i \ell}^{-} \cdot p_{\ell - i} (\zeta; q^{-2(i - j)}, q^{2(i + j)} | q^{-2}) \cdot c^{i - j} d^{i + j}
& (i + j \ge 0, i \ge j) \\
N_{i j \ell}^{-} \cdot p_{\ell - j} (\zeta; q^{-2(i - j)}, q^{2(i + j)} | q^{-2}) \cdot b^{i - j} d^{i + j}
& (i + j \ge 0, i \le j) \\
\end{cases}
\end{equation*}
where $\zeta = - q b c$,
\begin{equation*}
N_{i j \ell}^+ = \frac{q^{-(\ell + j)(j - i)}}{[i - j]_{q^{-2}}!}
\cdot \frac{\mu_{+i} \, \mu_{- j}}{\mu_{+j} \, \mu_{-i}}, \quad
N_{i j \ell}^- = \frac{q^{(\ell - j)(j - i)}}{[j - i]_{q^{-2}}!}
\cdot \frac{\mu_{-i} \, \mu_{+j}}{\mu_{-j} \, \mu_{+i}}
\quad (= N_{-i,-j, \ell}^+)
\end{equation*}
and $p_m$ is the {\em little $q$-Jacobi polynomial} \cite[\S2]{MR1492989}. We omit the definition of $p_m$; in what follows, we need only the fact that $p_m(\zeta; q_1, q_2 | q_3)$ ($q_i \in k$) is a polynomial of $\zeta$. Note that, since $S(\zeta) = \zeta$, we have
\begin{equation}
\label{eq:q-Jacobi-1}
S(p_m(\zeta; q_1, q_2 | q_3)) = p_m(S(\zeta); q_1, q_2 | q_3) = p_m(\zeta; q_1, q_2 | q_3)
\end{equation}
Since $f \mapsto (\gamma \rightharpoonup f)$ ($f \in \mathcal{O}_q(SL_2)$) is an algebra map, we also have
\begin{equation}
\label{eq:q-Jacobi-2}
\gamma \rightharpoonup p_m(\zeta; q_1, q_2 | q_3) = p_m(\gamma \rightharpoonup \zeta; q_1, q_2 | q_3) = p_m(\zeta; q_1, q_2 | q_3)
\end{equation}
\begin{proof}[Proof of Theorem~\ref{thm:FS-ind-Oq}]
Let $C_\ell$ be the coefficient subcoalgebra of $X_\ell$. Define
\begin{equation*}
Q_\ell: C_\ell \to C_\ell
\quad Q_\ell(f) = S(\gamma \rightharpoonup f)
\quad (f \in C_\ell)
\end{equation*}
$Q_\ell$ is well-defined since $X_\ell$ is self-dual. By Theorem~\ref{thm:copiv-Tr-Sv}, we have
\begin{equation*}
\nu(X_\ell) = \frac{\Trace(Q_\ell)}{\dim_k(X_\ell)} = \frac{\Trace(Q_\ell)}{2 \ell + 1}
\end{equation*}
By~\eqref{eq:q-Jacobi-1}, \eqref{eq:q-Jacobi-2} and the above description of $c_{i j}^{(\ell)}$, we have
\begin{equation*}
Q_\ell(c_{i j}^{(\ell)})
= S(\gamma \rightharpoonup c_{i j}^{(\ell)})
= \text{(constant)} \times S(c_{i j}^{(\ell)})
= \text{(constant)} \times c_{-j, -i}^{(\ell)}
\end{equation*}
for all $i, j \in I_\ell$. Recall that $\{ c_{i j}^{(\ell)} \}$ is a basis of $C_\ell$ since $X_\ell$ is simple. The above computation means that $Q_\ell$ is represented by a generalized permutation matrix with respect to this basis.
Note that $(i, j) = (-j, -i)$ if and only if $j = -i$. If $i \ge 0$, then
\begin{align*}
Q_\ell(c_{i, -i}^{(\ell)})
& = q^{-2i} \cdot S \Big( N_{i,-i,\ell}^{+} \cdot c^{2i} \cdot p_{\ell - i} (\zeta; q^{-4i}, 1 | q^{-2}) \Big) \\
& = q^{-2i} \cdot N_{i,-i,\ell}^{+} \cdot p_{\ell - i} (\zeta; q^{-4i}, 1 | q^{-2}) \cdot (-q)^{2 i}
= (-1)^{2i} \cdot c_{i, -i}^{(\ell)}
\end{align*}
Since $\ell - i \in \mathbb{Z}$, we have $(-1)^{2 i} = (-1)^{2 \ell}$. In a similar way, we also have $Q_\ell(c_{i,-i}^{(\ell)}) = (-1)^{2 \ell}$ for $i < 0$. Hence we obtain $\Trace(Q_\ell) = (-1)^{2 \ell} \cdot (2 \ell + 1)$.
\end{proof}
$\mathcal{O}_q(SL_2)$ has a Hopf algebra automorphism $\tau$ given by
\begin{equation*}
\tau(a) = a, \quad \tau(b) = - b, \quad \tau(c) = - c, \quad \tau(d) = d
\end{equation*}
$\tau$ is an involution such that $\gamma \circ \tau = \gamma$ and hence the $\tau$-twisted FS indicator $\nu^\tau(X)$ is defined for each $X \in \fdCom(\mathcal{O}_q(SL_2))$. Replacing $S$ in the proof of Theorem~\ref{thm:FS-ind-Oq} with $S \circ \tau$, we have the following theorem:
\begin{theorem}
\label{thm:tw-FS-ind-Oq}
$\nu^\tau(X_\ell) = +1$.
\end{theorem}
By Theorem~\ref{thm:copiv-FS}, this result reads as follows: For each $\ell \in \frac{1}{2}\mathbb{N}_0$, there exists a non-degenerate bilinear form $\beta$ on $X_\ell$ satisfying $\beta(x_{(0)}, y) \tau(x_{(1)}) = \beta(x, y_{(0)}) S(y_{(1)})$ and $\beta(w, \gamma \rightharpoonup v) = \beta(v, w)$ for all $v, w \in X_\ell$.
\begin{remark}
\label{rem:FS-ind-Oq}
The character $t_\ell$ of $X_\ell$ is given by
\begin{equation*}
t_\ell = \sum_{i \in I_\ell} c_{i i}^{(\ell)}
= \sum_{i \in I_\ell, i \ge 0} a^{2 i} p_{\ell + i}(\zeta; 1, q^{-4i} | q^{-2})
+ \sum_{i \in I_\ell, i > 0} p_{\ell + i}(\zeta; 1, q^{4i} | q^{-2}) d^{2 i}
\end{equation*}
One can prove Theorems~\ref{thm:FS-ind-Oq} and \ref{thm:tw-FS-ind-Oq} by Theorem~\ref{thm:copiv-FS-ind-ch} and its corollary (see \cite[\S4]{MR1492989} for a description of the Haar functional on $\mathcal{O}_q(SL_2)$). However, the computation will become more difficult than the above proof.
\end{remark}
\subsection{The Hopf Algebra $U_q(\mathfrak{sl}_2)$}
The quantized enveloping algebra $U_q(\mathfrak{sl}_2)$ is a Hopf algebra defined as follows: As an algebra, it is generated by $E$, $F$, $K$ and $K^{-1}$ with relations $K K^{-1} = 1 = K^{-1} K$,
\begin{gather*}
K E K^{-1} = q^2 E,
\quad K F K^{-1} = q^{-2} F
\text{\quad and \quad} E F - F E = \frac{K - K^{-1}}{q - q^{-1}}
\end{gather*}
The comultiplication $\Delta$, the counit $\varepsilon$ and the antipode $S$ are given by
\begin{gather*}
\Delta(K) = K \otimes K,
\quad \Delta(E) = E \otimes K + 1 \otimes E,
\quad \Delta(F) = F \otimes 1 + K^{-1} \otimes F \\
S(K) = K^{-1}, \ S(E) = - E K^{-1}, \ S(F) = - K F,
\ \varepsilon(K) = 1,
\ \varepsilon(E) = \varepsilon(F) = 0
\end{gather*}
We have $S^2(u) = K u K^{-1}$ for all $u \in U_q(\mathfrak{sl}_2)$. Hence the Hopf algebra $U_q(\mathfrak{sl}_2)$ is pivotal with pivotal grouplike element $K$.
For each $\ell \in \frac{1}{2} \mathbb{N}_0$, we define a left $U_q(\mathfrak{sl}_2)$-module $V_\ell$ as follows: As a vector space, it has a basis $\{ v_i \}_{i \in I_\ell}$. The action of $U_q(\mathfrak{sl}_2)$ on $V_\ell$ is defined by
\begin{equation*}
K \cdot v_i = q^{2i} v_i,
\ E \cdot v_i = [\ell - i + 1]_q v_{i - 1},
\ F \cdot v_i = [\ell + i + 1]_q v_{i + 1}
\ (i \in I_\ell)
\end{equation*}
where $v_{\ell+1} = v_{-(\ell+1)} = 0$.
There is a unique Hopf pairing $\langle -, - \rangle: U_q(\mathfrak{sl}_2) \times \mathcal{O}_q(SL_2) \to k$ such that
\begin{gather*}
\langle K, a \rangle = q^{-1},
\quad \langle K, d \rangle = q,
\quad \langle E, c \rangle = 1,
\quad \langle F, b \rangle = 1 \\
\langle K, b \rangle = \langle K, c \rangle
= \langle E, a \rangle = \langle E, b \rangle = \langle E, d \rangle
= \langle F, a \rangle = \langle F, c \rangle = \langle F, d \rangle = 0
\end{gather*}
see \cite[\S4]{MR1492989} and \cite[V.7]{MR1321145}. This pairing induces an algebra map $\varphi: U_q(\mathfrak{sl}_2) \to \mathcal{O}_q(SL_2)^\vee$. Since $\varphi(K) = \gamma$, $\varphi$ is in fact a morphism of pivotal algebras. Hence we obtain a $k$-linear duality preserving functor
\begin{equation*}
\begin{CD}
\Phi: \fdCom\Big( \mathcal{O}_q(SL_2) \Big)
@>>{\eqref{eq:com-to-mod}}>
\fdMod\Big( \mathcal{O}_q(SL_2)^\vee \Big)
@>{\varphi^\natural}>>
\fdMod\Big(U_q(\mathfrak{sl}_2) \Big)
\end{CD}
\end{equation*}
One has $\Phi(X_\ell) \cong V_\ell$. In particular, $\Phi$ maps simple objects to simple objects. Since $\mathcal{O}_q(SL_2)$ is cosemisimple, the functor $\Phi$ is fully faithful and therefore $\Phi$ preserves the FS indicator. Hence, by Theorem~\ref{thm:FS-ind-Oq}, we have:
\begin{theorem}
$\nu(V_\ell) = (-1)^{2 \ell}$.
\end{theorem}
By Theorem~\ref{thm:piv-FS-ind-ch}, this result reads as follows: For each $\ell \in \frac{1}{2} \mathbb{N}_0$, there exists a non-degenerate bilinear form $\beta$ on $V_\ell$ satisfying $\beta(u v, w) = \beta(v, S(u) w)$ and $\beta(w, K v) = (-1)^{2 \ell} \cdot \beta(v, w)$ for all $u \in U_q(\mathfrak{sl}_2)$ and $v, w \in V_\ell$.
$U_q(\mathfrak{sl}_2)$ has a Hopf algebra automorphism $\tau$ defined by $\tau(E) = -E$, $\tau(F) = -F$, $\tau(K) = K$. It is obvious that $\tau$ is an involution of the pivotal algebra $U_q(\mathfrak{sl}_2)$. Since $\langle \tau(u), f \rangle = \langle u, \tau(f) \rangle$ for all $u \in U_q(\mathfrak{sl}_2)$ and $f \in \mathcal{O}_q(SL_2)$, $\Phi$ also preserves the $\tau$-twisted FS indicator. Therefore we have:
\begin{theorem}
$\nu^\tau(V_\ell) = +1$.
\end{theorem}
This result reads as follows: For each $\ell \in \frac{1}{2} \mathbb{N}_0$, there exists a non-degenerate bilinear form $\beta$ on $V_\ell$ satisfying $\beta(\tau(u) v, w) = b(v, S(u)w)$ and $\beta(w, K v) = b(v, w)$ for all $u \in U_q(\mathfrak{sl}_2)$ and $v, w \in V_\ell$.
\section{Conclusions}
As we have briefly reviewed in Section~\ref{sec:introduction}, the celebrated theorem of Frobenius and Schur has several generalizations. To give a category-theoretical understanding of these generalizations, in Section~\ref{sec:categ-with-dual} we have introduced the FS indicator for categories with duality over a field $k$; if $\mathcal{C}$ is a category with duality over $k$, then a linear map $\Trans_{X,Y}: \Hom_\mathcal{C}(X, Y^\vee) \to \Hom_\mathcal{C}(Y, X^\vee)$, $f \mapsto f^\vee \circ j$ is defined for each $X, Y \in \mathcal{C}$. We call $\Trans_{X,Y}$ the transposition map. The FS indicator $\nu(X)$ of $X \in \mathcal{C}$ is defined to be the trace of $\Trans_{X,X}: \Hom_\mathcal{C}(X, X^\vee) \to \Hom_\mathcal{C}(X, X^\vee)$. We have also introduced a general method to twist the given duality by an adjunction, which is a category-theoretical counterpart of several twisted versions of the Frobenius--Schur theorem.
In Section~\ref{sec:piv-alg}, we have introduced the notion of a pivotal algebra. The representation category of a pivotal algebra has duality and therefore the FS indicator is defined for each of its representation. We have given a representation-theoretic interpretation of the FS indicator and a formula of the FS indicator for separable pivotal algebras. These results yield the Frobenius--Schur-type theorems for Hopf algebras, quasi-Hopf algebras, weak Hopf $C^*$-algebras and Doi's group-like algebras. The notion of pivotal algebras is useful to deal with the twisted FS indicator; as a demonstration, we have constructed the twisted Frobenius--Schur theory for quasi-Hopf algebras.
In Section~\ref{sec:coalgebras}, we have introduced the notion of a copivotal coalgebra as the dual notion of a pivotal algebra and gave results for copivotal coalgebras analogous to pivotal algebras. In particular, we have given a representation-theoretic interpretation of the FS indicator and a formula of the FS indicator for coseparable copivotal coalgebras.
In Section~\ref{sec:quantum-sl_2}, we have applied our results to the quantum coordinate ring $\mathcal{O}_q(SL_2)$ and the quantum enveloping algebra $U_q(\mathfrak{sl}_2)$. For each $\ell \in \frac{1}{2}\mathbb{N}_0$, $\mathcal{O}_q(SL_2)$ has a unique simple right comodule $X_\ell$ of dimension $2 \ell$. We have proved $\nu(X_\ell) = (-1)^{2 \ell}$ and analogous results for the twisted case and $U_q(\mathfrak{sl}_2)$ case. As we have remarked, the Haar functional on $\mathcal{O}_q(SL_2)$ is not used in our proof. We expect that the FS indicator for general $\mathcal{O}_q(G)$ will be determined by using the Haar functional.
\section*{Acknowledgments}
The author would like to thank the referees for careful reading the manuscript. The author is supported by Grant-in-Aid for JSPS Fellows (24$\cdot$3606).
|
1208.2185
|
\section{Preliminaries}
Throughout we consider unitary associative algebras over a field $K$ of characteristic 0. Let $K\langle T\rangle$ be the free associative algebra freely generated over $K$ by the set $T=\{t_1,t_2, \dots\}$. If $A$ is a PI algebra we denote by $I=T(A)\subseteq K\langle T\rangle$ its T-ideal, that is the ideal of all identities of $A$. The algebra $U(A)=K\langle T\rangle/I$ is the relatively free (or generic) algebra in the variety of algebras defined by $A$. When $T$ is a finite set, say $T=\{t_1,\dots,t_k\}$ one obtains the relatively free (or generic) algebra of $A$ of rank $k$ and denotes it by $U_k(A)$. We shall use the same letters $t_i$ for the generators of $K\langle T\rangle$ and for their images under the canonical projection $K\langle T\rangle\to K\langle T\rangle/T(A) = U(A)$.
We recall the construction of the free supercommutative algebra $K[X;Y]$. Let $K\langle X\cup Y\rangle$ be the free associative algebra freely generated by the set $X\cup Y$ where $X\cap Y=\emptyset$. This algebra is 2-graded in a natural way assuming the variables in $X$ of degree 0, and those in $Y$ of degree 1. Let $I$ be the ideal generated by all $ab-(-1)^{|a|\cdot|b|}ba$ where $a$ and $b$ run over the homogeneous elements in $K\langle X\cup Y\rangle$, and $|a|$ is the $\mathbb{Z}_2$-degree of the homogeneous element $a$, and put $K[X;Y]= K\langle X\cup Y\rangle/I$. Then $K[X;Y]\cong K[X]\otimes_K E(Y)$, here $E(Y)$ is the Grassmann algebra on the vector space with a basis $Y$, see for more detail \cite{bereleca}. If a 2-graded algebra $A=A_0\oplus A_1$ satisfies $ab-(-1)^{|a|\cdot|b|}ba=0$ for all homogeneous $a$ and $b$ then it is called supercommutative. Clearly the Grassmann algebra is supercommutative.
Take $X=\{x_{ij}^r\}$, $Y=\{y_{ij}^r\}$ where $1\le i,j\le n$, $r=1$, 2, \dots; here $r$ is an upper index, not an exponent. One defines
the matrices $A_r=(x_{ij}^r)$, $B_r=(x_{ij}^r+y_{ij}^r)$, $C_r=(z_{ij}^r)$ where $z=x$ whenever $1\le i,j\le a$ or $a+1\le i,j\le a+b$, and $z=y$ for all remaining possibilities for $i$ and $j$. Suppose $a+b=n$, and consider the following subalgebras of $M_n(K[X;Y])$. The first is generated by the generic matrices $A_r$, $K[A_r\mid r\ge 1]$. It is isomorphic to the relatively free (or universal) algebra $U(M_n(K))$ of $M_n(K)$. In
\cite[Theorem 2]{bereleca} it was proved that $U(M_n(E))\cong K[B_r\mid r\ge 1]$, also $U(M_{ab}(E))\cong K[C_r\mid r\ge 1]$. The relatively
free algebras of finite rank $k$, denoted by $U_k$, can be obtained by letting $r=1$, \dots, $k$, that is by taking the first $k$ matrices.
The algebra $K\langle T\rangle$ is multigraded, counting the degree of its monomials in each variable.
We work over a field $K$ of characteristic 0 therefore every T-ideal is generated by its multilinear elements, see for example \cite[Section 4.2]{drenskybook}.
The polynomial identities of $M_{11}(E)$ were described by Popov in characteristic 0, see the main theorem of \cite{popov}. The theorem reads that the polynomials
\begin{equation}
\label{basispim11}
[[t_1,t_2]^2,t_1], \qquad [[t_1,t_2],[t_3,t_4],t_5]
\end{equation}
generate the T-ideal of $E\otimes E$, and of $M_{11}(E)$ as well.
Here $[a,b]=ab-ba$ is the usual commutator of $a$ and $b$. The higher commutators without inner brackets will be considered left normed that is $[a,b,c] = [[a,b],c]$, and so on.
Let $L(T)$ be the free Lie algebra on the free generators $T$. If one substitutes the usual product in an associative algebra $A$ by the bracket $[a,b] = ab-ba$ one gets a Lie algebra denoted by $A^-$. It is well known that $L(T)$ is the subalgebra of $K\langle T\rangle^-$ generated by $T$. Moreover $K\langle T\rangle$ is the universal enveloping algebra of $L(T)$. Choose an ordered basis of $L(T)$ consisting of $T$ and left normed commutators. Suppose further that if $u$ and $v$ are elements of the basis then $\deg u<\deg v$ implies $u<v$, in this way the free generators in $T$ precede all remaining basis elements. Then a basis of $K\langle T\rangle$ is given by 1 and all elements $t_1^{n_1}\cdots t_k^{n_k} u_1\cdots u_m$ where $n_i\ge 0$, and $u_i$ are commutators, $u_1\le\cdots \le u_m$. Let $B(T)$ be the subalgebra of $K\langle T\rangle$ generated by 1 and all commutators of degree at least two. Thus $B(T)$ is spanned by 1 and the products of commutators. The elements of $B(T)$ are called proper polynomials. As we work with unitary algebras it is well known that every T-ideal $I$ is generated by its proper elements, see for example \cite[Section 4.3]{drenskybook}.
\section{The generic algebra of $M_{11}(E)$ in two generators}
In the paper \cite{pktcm} we studied the generic algebra of $M_{11}(E)$ in two generators, $F=K[C_1,C_2]$ where $C_1=\begin{pmatrix}
x_1&y_1\\ y_1'& x_1'\end{pmatrix}$, $C_2=\begin{pmatrix} x_2&y_2\\ y_2'& x_2'\end{pmatrix}$. The entries of $C_1$ and $C_2$ lie in the free supercommutative algebra $K[X;Y]$, $X=\{x_1,x_2,x_1', x_2'\}$, $Y=\{y_1,y_2, y_1', y_2'\}$. Here we recall some results of \cite{pktcm} that we need.
The algebra $K[X;Y]$ is 2-graded in a natural way. It can be given a $\mathbb{Z}$-grading by counting the degree in the variables of $Y$ only: $K[X;Y] = \oplus_{n\in\mathbb{Z}}
K[X;Y]^{(n)}$ where $K[X;Y]^{(n)}$ is the span of all monomials of degree $n$ in the variables from $Y$. Obviously $K[X;Y]^{(n)}=0$ unless $0\le n\le 4$. Also
\begin{eqnarray*}
K[X;Y]_0 &=& K[X;Y]^{(0)} + K[X;Y]^{(2)} + K[X;Y]^{(4)}; \\
K[X;Y]_1 &=& K[X;Y]^{(1)} + K[X;Y]^{(3)}.
\end{eqnarray*}
We set $B_0=\{1\}$, $B_1=\{y_1,y_2,y_1', y_2'\}$, $B_2=\{y_1y_2, y_1y_1', y_1y_2', y_2y_1', y_2y_2', y_1'y_2'\}$,
$B_3=\{y_1y_2y_1', y_1y_2y_2', y_1y_1'y_2', y_2y_1'y_2'\}$, $B_4=\{y_1y_2y_1'y_2'\}$. Then for $n=0$, 1, 2, 3, 4, $K[X;Y]^{(n)}$ is a free module over $K[X]$, with a basis $B_n$. As a consequence $K[X;Y]$ is a free module over $K[X]$ with a basis $B=B_0\cup B_1\cup B_2\cup B_3\cup B_4$. Also the ideals in $K[X;Y]$ are $K[X]$-submodules.
The following elements were introduced in \cite{pktcm}.
\begin{eqnarray*}
h_1&=& y_1y_2y_1'y_2';\\
h_2 &=& y_1y_2 (y_1'(x_2'-x_2) - y_2'(x_1'-x_1)); \\
h_3 &=& y_1'y_2' (y_1(x_2'-x_2) - y_2(x_1'-x_1)); \\
h_4 &=& (y_1'(x_2'-x_2) - y_2'(x_1'-x_1))
(y_1(x_2'-x_2) - y_2(x_1'-x_1)).
\end{eqnarray*}
It is immediate to check they satisfy the following relations in $K[X;Y]$.
\begin{eqnarray*}
&&h_1y_1 = h_1y_1' = h_1y_2 = h_1y_2' = 0; \quad h_2y_1 = h_2y_2 = h_3y_1' = h_3y_2' = 0;\\
&&h_2y_1' = h_3y_1 = (x_1'-x_1)h_1; \quad h_2y_2' = h_3y_2 = (x_2'-x_2) h_1; \\
&&h_4y_1 = (x_1'-x_1)h_2; \quad h_4y_2 = (x_2'-x_2) h_2; \\
&&h_4y_1' = -(x_1'-x_1) h_3; \quad h_4y_2' = -(x_2'-x_2)h_3.
\end{eqnarray*}
We shall also need the elements
\begin{eqnarray*}
&&q_n(x_1,x_1') = \sum\nolimits_{i=0}^nx_1^i x_1'^{n-i}; \quad
Q_n(x_2,x_2') = q_n(x_2,x_2');\\
&&r_n(x_1, x_1') = \sum\nolimits_{i=0}^{n-1} (n-i)x_1^{n-1-i}x_1'^i; \quad
R_n(x_2,x_2') = r_n(x_2,x_2');\\
&&s_n(x_1,x_1') = r_n(x_1', x_1); \quad
S_n(x_2,x_2') = s_n(x_2,x_2').
\end{eqnarray*}
One verifies by an obvious induction that
\begin{eqnarray*}
&&r_n=q_{n-1}+x_1r_{n-1}; \quad s_n=q_{n-1} + x_1's_{n-1}; \quad s_n+r_n = (n+1) q_{n-1}; \\
&&q_n=x_1^n + x_1' q_{n-1} = x_1'{}^n + x_1q_{n-1};\quad (x_1'-x_1)q_{n-1} = x_1'{}^n - x_1^n;\\
&& x_1^nx_1'{}^m - x_1^mx_1'{}^n = (x_1'-x_1)(q_nq_{m-1} - q_mq_{n-1}).
\end{eqnarray*}
We compute directly that for every $m$ and $n$ we have
\begin{eqnarray*}
C_1^n &=&\begin{pmatrix}
x_1^n+y_1y_1'r_{n-1}& y_1q_{n-1}\\
y_1'q_{n-1}& x_1'{}^n-y_1y_1's_{n-1}
\end{pmatrix}; \\
C_2^m &=& \begin{pmatrix}
x_2^m + y_2 y_2' R_{m-1} & y_2 Q_{m-1}\\
y_2' Q_{m-1} & x_2'{}^m - y_2y_2' S_{m-1}
\end{pmatrix}.
\end{eqnarray*}
In this way the product $C_1^nC_2^m$ equals
\begin{equation}
\label{explicitproduct}
C_1^nC_2^m = \begin{pmatrix}
x_1^nx_2^m+a+d & y_1x_2'{}^mq_{n-1} + y_2x_1^n Q_{m-1} + c\\
y_1'x_2^m q_{n-1} + y_2'x_1'{}^nQ_{m-1} + c' & x_1'{}^n x_2'{}^m + a'+d'
\end{pmatrix}
\end{equation}
where $a$, $a'\in K[X;Y]^{(2)}$, $d$, $d'\in K[X;Y]^{(4)}$, and $c$, $c'\in K[X;Y]^{(3)}$.
Let $A=\begin{pmatrix} a&b\\ c&d\end{pmatrix}\in F\cong K[C_1,C_2]$ be central. It was shown in \cite{pktcm} that \[
d-a\in J= Ann ((x_2'-x_2)y_1 - (x_1'-x_1)y_2)\cap
Ann ((x_2'-x_2)y_1' - (x_1'-x_1)y_2').
\]
By \cite[Proposition 5]{pktcm} the $K[X]$-module $J$ is spanned by $\{h_1,h_2,h_3,h_4\}$. Corollary 6 from \cite{pktcm} states that the matrix $A$ commutes with $C_1$ and $C_2$ if and only if $b=f_4 h_2$, $c = -f_4 h_3$, $d = a+f_1h_1+f_4h_4$ for some $f_1$, $f_4\in K[X]$. Hence $A$ is central if and only if
$
A= aI + f_1\begin{pmatrix} 0&0\\ 0&h_1\end{pmatrix} + f_4\begin{pmatrix} 0&h_2\\ -h_3& h_4\end{pmatrix}$.
Following the notation used in \cite{pktcm} we define matrices in $F$:
\[
A_0=\begin{pmatrix} h_1&0\\ 0&h_1\end{pmatrix}; \quad
A_1=\begin{pmatrix} 0&0\\ 0&h_1\end{pmatrix}; \quad
A_2=\begin{pmatrix} 0&h_2\\ -h_3&h_4\end{pmatrix}; \quad
A_3=\begin{pmatrix} h_4&0\\ 0&h_4\end{pmatrix}.
\]
An element $a\in F$ is \textsl{strongly central} if it is central, and also for each $b\in F$ the element $ab$ is central in $F$.
One checks (\cite[Lemma 7]{pktcm}) that for arbitrary $\alpha_i\in K[X]$ the elements $\alpha_0A_0 + \alpha_1A_1 + \alpha_2A_2 + \alpha_3A_3$ are strongly central in $F$.
We shall need a couple of technical statements from \cite{pktcm}. Take a left normed commutator $f(t_1,t_2) = [t_1,t_2,t_{i_3}, \ldots, t_{i_k}]$, $i_j=1$, 2, such that $\deg_{t_1} f=n$, $\deg_{t_2} f=m$, $n+m=k$. Then Lemma 8 from \cite{pktcm} states that $f(C_1,C_2) = (x_1'-x_1)^{n-1} (x_2'-x_2)^{m-1} A(k)$ where
\begin{eqnarray*}
A(k) &=& \begin{pmatrix} F(k) & y_1(x_2'-x_2) - y_2 (x_1'-x_1)\\
(-1)^k (y_2'(x_1'-x_1) - y_1'(x_2'-x_2))& F(k) \end{pmatrix},\\
F(k) &=& \frac{(y_1(x_2'-x_2) - y_2(x_1'-x_1)) y_i' + (-1)^k y_i (y_2'(x_1'-x_1) - y_1'(x_2'-x_2))}{x_i'-x_i} .
\end{eqnarray*}
In the expression for $F(k)$ the index $i$ stands for $i_k$. The formula for $f(C_1,C_2)$ yields that if $f_1$, $f_2$ are non-zero commutators in $F$
then $f_1(C_1,C_2)f_2(C_1,C_2)$ is strongly central in $F$, see \cite[Lemma 9]{pktcm}.
\section{Some identities for the algebra $F$}
At the beginning of the previous section we denotted by $F$ the $K$-algebra generated by the matrices $C_1$ and $C_2$, $F=K[C_1, C_2]$.
\begin{lemma}
\label{3comm}
Let $f_1(t_1,t_2)$, $f_2(t_1,t_2)$, $f_3(t_1,t_2)$ be three commutators and put $g=f_1f_2f_3$. Then $g(C_1,C_2)$ vanishes on $F$.
\end{lemma}
\textit{Proof}\textrm{. }
Write the $f_i$ as linear combinations of left-normed commutators. It suffices to consider the case when all $f_i$ are left-normed. By \cite[Lemma 9]{pktcm} the product $f_1(C_1,C_2) f_2(C_1,C_2)$ is, in turn, a combination of the matrices $A_0$, $A_2$, $A_3$ defined above. Now the entries of the matrices $A_i$ are either zeros or, up to a sign, some of the elements $h_j$, $1\le j\le 4$. Looking at the above expression for $f_3(C_1,C_2)$ we see that its entries vanish when multiplied by $h_j$.
\hfill$\diamondsuit$\medskip
\begin{corollary}
\label{iffproper}
Let $f(t_1,t_2) = t_1^nt_2^m u_1^{k_1}\cdots u_r^{k_r}$ where the $u_i$ are left-normed commutators in $t_1$ and $t_2$. Then $f(C_1,C_2)=0$ in $F$ if and only if $k_1+\cdots+k_r\ge 3$.
\end{corollary}
\textit{Proof}\textrm{. } Lemma~\ref{3comm} implies the ``if" part. Suppose $k_1+\cdots+k_r\le 2$. According to the proof of \cite[Lemma 9]{pktcm} the product of two
left-normed commutators, evaluated on $C_1$ and $C_2$ in $F$, is a linear combination of the matrices $A_0$, $A_2$ and $A_3$, the latter with coefficient $\pm 1$,
hence it cannot be 0 in $F$. Also the matrices $C_1$ and $C_2$ are not zero divisors in $F$ according to \cite[Lemma 2]{pktcm}. \hfill$\diamondsuit$\medskip
We make use of the following two equalities that are valid in every associative algebra. They are well known and their proofs consist of an easy induction. We separate them into a lemma for further reference.
\begin{lemma}
\label{equalities}
Let $z$, $a$, $b\in K\langle T\rangle$, and $n\ge 1$ then
\[
[z^n,b] = \sum_{i=0}^{n-1} z^{n-i-1} [z,b] z^i; \qquad
[a,b]z^n = \sum_{i=0}^n {n\choose i} z^i [a,b,\underbrace{z, \ldots,z}_{n-i}].
\]
\end{lemma}
\begin{lemma}
\label{leftright}
Let $u$ be a left-normed commutator in $K\langle T\rangle$ and let $v\in K\langle T\rangle$. Then $uv = \sum_i v_i u_i$ where $u_i$ are left-normed commutators, $v_i\in K\langle T\rangle$, and $\deg u_i\ge \deg u$ for all $i$.
\end{lemma}
\textit{Proof}\textrm{. }
It suffices to consider $v=t_1$, then $ut_1 = [u,t_1] + t_1u$. Then $[u,t_1]$ is left-normed and $\deg [u,t_1]>\deg u$.
\hfill$\diamondsuit$\medskip
\begin{proposition}
\label{1-2}
The polynomials $[[t_1,t_2][t_3,t_4],t_5]$ and $[t_1,t_2][t_3,t_4][t_5,t_6]$ are identities for the algebra $F$.
\end{proposition}
\textit{Proof}\textrm{. }
Both polynomials are multilinear therefore it is enough to evaluate them on a spanning set of the algebra $F$. The algebra $F$ is spanned by elements of the type $C_1^nC_2^m u_1\cdots u_k$ where $n$, $m\ge 0$ and $u_i$ are left-normed commutators. Moreover if $k\ge 2$ then $u_1\cdots u_k$ is strongly central hence $C_1^nC_2^m u_1\cdots u_k$ will be central. But central elements vanish the commutators hence we consider substitutions by elements of the types $C_1^nC_2^m$ and $C_1^nC_2^m u$ only where $u$ is a left-normed commutator.
According to Lemmas~\ref{equalities}, \ref{leftright} one has $[C_1^nC_2^m, C_1^pC_2^q] = \sum_i w_i u_i$ where the $u_i$ are left-normed commutators, analogously for $[C_1^nC_2^mu, C_1^pC_2^q]$ and also for $[C_1^nC_2^m u, C_1^pC_2^q v]$. But then $[t_1,t_2][t_3,t_4]$ becomes $\sum_i w_iu_iv_i$ where $u_i$ and $v_i$ are left-normed commutators. The latter sum is (strongly) central and thus the first polynomial is an identity for $F$. The same procedure applied to the second polynomial yields a combination of products of three commutators which is 0 in $F$ according to Lemma~\ref{3comm}.
\hfill$\diamondsuit$\medskip
\begin{corollary}
\label{2times3}
The identity $[t_1,t_2,t_5][t_3,t_4] + [t_1,t_2][t_3,t_4,t_5]=0$ holds in $F$.
\end{corollary}
\textit{Proof}\textrm{. }
It is another form of the first identity of Proposition~\ref{1-2}.
\hfill$\diamondsuit$\medskip
\noindent \textbf{Remark} The polynomial $[t_1,t_2][t_3,t_4]$ is a central polynomial for $U_2(M_{11}(E))$, and in particular $[t_1,t_2]^2$ is central as
well. On the other hand the latter polynomial is \textsl{not} central for $M_{11}(E)$. It is well known that for the matrix algebra $M_n(K)$ a polynomial
$f(t_1,\ldots,t_k)$ is central if and only if $f(A_1,\ldots,A_k)$ lies in the centre of the generic algebra $U_k(M_n(K))$ generated by $A_1$, \dots, $A_k$,
see \cite[Proposition 1.2, p. 171]{Procesi}. This is a sharp difference in the behaviour of the generic algebras for $M_n(K)$ and for $M_{ab}(E)$.
\medskip
\begin{proposition}
\label{s_4}
The standard polynomial $s_4= \sum (-1)^\sigma t_{\sigma(1)} t_{\sigma(2)} t_{\sigma(3)} t_{\sigma(4)}$ is an identity for $F$. Here $\sigma$ runs over the permutations of the symmetric group $S_4$, and $(-1)^\sigma$ stands for the sign of $\sigma$.
\end{proposition}
\textit{Proof}\textrm{. }
We write $s_4 = [t_1,t_2]\circ [t_3,t_4] - [t_1,t_3]\circ [t_2,t_4] + [t_1,t_4]\circ [t_2,t_3]$ where $a\circ b=ab+ba$.
As in Proposition~\ref{1-2} we shall substitute the variables by elements of the types $C_1^nC_2^m$ and $C_1^nC_2^m u$, $u$ a left-normed commutator.
First suppose $t_1=v_1u$, $t_i=v_i$ where $u$ is a left-normed commutator, and $v_i\in F$ are arbitrary. Then $[v_1u,v_2] = v_1[u,v_2] + [v_1,v_2]u$. The product of three commutators vanishes in $F$ hence
\[
[v_1u,v_2]\circ [v_3,v_4] = ([v_1,v_2]u) \circ [v_3,v_4]+ (v_1[u,v_2])\circ [v_3,v_4] = (v_1[u,v_2])\circ [v_3,v_4].
\]
Simple manipulations show that
\begin{eqnarray*}
(v_1[u,v_2])\circ [v_3,v_4] &=&
v_1[u,v_2][v_3,v_4] +[v_3,v_4]v_1[u,v_2]\\
&=& - v_1u[v_3,v_4,v_2] + v_1[v_3,v_4][u,v_2] + [v_3,v_4,v_1][u,v_2]\\
&=& - v_1u[v_3,v_4,v_2] - v_1[v_3,v_4,v_2]u - [v_3,v_4,v_1,v_2]u.
\end{eqnarray*}
Then one obtains by the Jacobi identity
\begin{eqnarray*}
s_4(v_1u,v_2,v_3,v_4) &=&
-v_1u([v_3,v_4,v_2] - [v_2,v_4,v_4] + [v_2,v_3,v_4])\\
&&- v_1([v_3,v_4,v_2] - [v_2,v_4,v_4] + [v_2,v_3,v_4])u\\
&&-([v_3,v_4,v_1,v_2] - [v_2,v_4,v_1,v_3] + [v_2,v_3,v_1,v_4])u\\
&=& -([v_3,v_4,v_1,v_2] - [v_2,v_4,v_1,v_3] + [v_2,v_3,v_1,v_4])u.
\end{eqnarray*}
The latter sum equals, once again by Jacobi,
\[
([[v_1,v_2],[v_4,v_3]] + [[v_1,v_3],[v_2,v_4]] + [[v_1,v_4],[v_3,v_2]])u
\]
and this vanishes as a combination of products of three commutators.
Now we consider the substitution of $t_i$ by $C_1^{n_i}C_2^{m_i}$, $1\le i\le 4$. First one defines on $K[X;Y]$ an automorphism $'$ of order two by
$x_i\mapsto x_i'$, $y_i\mapsto y_i'$, and then extending to the whole supercommutative algebra. It is easy to see that for every $D=(d_{ij})\in F$ it holds
$d_{22}=d_{11}'$ and $d_{21} = d_{12}'$. This notation agrees also with the formula for the product $C_1^n C_2^m$ in (\ref{explicitproduct}). Suppose
$D_i=C_1^{n_i}C_2^{m_i}$. Hence we may consider $D_i =
\begin{pmatrix} a_i&b_i\\ b_i' & a_i' \end{pmatrix}$, for some $a_i\in K[X;Y]_0$ and $b_i\in K[X;Y]_1$.
Put $D=(d_{ij})=s_4(D_1,D_2,D_3,D_4)\in F$; according to the above it suffices to prove that $d_{11}=d_{21}=0$.
Direct computation shows that the $(1,1)$-entry of $[D_1,D_2]\circ [D_3,D_4]$ is
\begin{eqnarray*}
&&2(b_1b_2'+b_1'b_2)(b_3b_4'+b_3'b_4)\\
&+& (b_1(a_2'-a_2)-b_2(a_1'-a_1))(b_4'(a_3'-a_3)-b_3'(a_4'-a_4))\\
&+&(b_3(a_4'-a_4)-b_4(a_3'-a_3))(b_2'(a_1'-a_1)-b_1'(a_2'-a_2)).
\end{eqnarray*}
Now writing down the analogous expressions for the remaining two summands of $s_4$ as above, and summing up we have
\begin{eqnarray*}
d_{11}&=& 2(b_1b_2'+b_1'b_2)(b_3b_4'+b_3'b_4)-2(b_1b_3'+b_1'b_3)(b_2b_4'+b_2'b_4)\\
&+& 2(b_1b_4'+b_1'b_4)(b_2b_3'+b_2'b_3)\\
&+&(b_1(a_2'-a_2)-b_2(a_1'-a_1))(b_4'(a_3'-a_3)-b_3'(a_4'-a_4))\\
&+&(b_3(a_4'-a_4)-b_4(a_3'-a_3))(b_2'(a_1'-a_1)-b_1'(a_2'-a_2))\\
&-&(b_1(a_3'-a_3)-b_3(a_1'-a_1))(b_4'(a_2'-a_2)-b_2'(a_4'-a_4))\\
&-&(b_2(a_4'-a_4)-b_4(a_2'-a_2))(b_3'(a_1'-a_1)-b_1'(a_3'-a_3))\\
&+&(b_1(a_4'-a_4)-b_4(a_1'-a_1))(b_3'(a_2'-a_2)-b_2'(a_3'-a_3))\\
&+&(b_2(a_3'-a_3)-b_3(a_2'-a_2))(b_4'(a_1'-a_1)-b_1'(a_4'-a_4)).
\end{eqnarray*}
The last six expressions above cancel altogether, and we are left with
\[
d_{11}= 4(b_1'b_2'b_3b_4 - b_1'b_3'b_2b_4 +b_1'b_4'b_2b_3+ b_2'b_3'b_1b_4- b_2'b_4'b_1b_3+b_3'b_4'b_1b_2).
\]
As $D_i=C_1^{n_i}C_2^{m_i}$ it follows by (\ref{explicitproduct}) that $b_i=x_1^{n_i}Q_{m_i-1}y_2+x_2'^{m_i}q_{n_i-1}y_1+c_i$ for some $c_i\in
K[X;Y]^{(3)}$. Also $b_i'b_j'b_kb_l=g(i,j,k,l) y_1y_2y_1'y_2'$ where
\begin{eqnarray*}
g(i,j,k,l) &=&
x_1^{n_l}x_1'^{n_j} q_{n_k-1}q_{n_i-1} x_2^{m_i}x_2'^{m_k} Q_{m_j-1}Q_{m_l-1}\\
&+& x_1^{n_k}x_1'^{n_i} q_{n_j-1}q_{n_l-1} x_2^{m_j}x_2'^{m_l} Q_{m_i-1}Q_{m_k-1}\\
&-& x_1^{n_k}x_1'^{n_j} q_{n_l-1}q_{n_i-1} x_2^{m_i}x_2'^{m_l} Q_{m_j-1}Q_{m_k-1}\\
&-& x_1^{n_l}x_1'^{n_i} q_{n_j-1}q_{n_k-1} x_2^{m_j}x_2'^{m_k} Q_{m_i-1}Q_{m_l-1}.
\end{eqnarray*}
Thus for $d_{11}=s_4(D_1,D_2,D_3,D_4)_{11}$ we have
\begin{eqnarray*} d_{11}&=&
(g(1,2,3,4)-g(1,3,2,4)+g(1,4,2,3))y_1y_2y_1'y_2'\\
&+& (g(2,3,1,4)-g(2,4,1,3)+g(3,4,1,2))y_1y_2y_1'y_2'.
\end{eqnarray*}
Expanding the sum of the $g(i,j,k,l)$ above we arrive at
\begin{eqnarray*}
& & Q_{m_2-1}Q_{m_3-1}q_{n_1-1}q_{n_4-1}(x_2^{m_4}x_2'^{m_1}-x^{m_1}x_2'^{m_4})(x_1^{n_3}x_1'^{n_2}-x_1^{n_2}x_1'^{n_3})\\
&+& Q_{m_1-1}Q_{m_4-1}q_{n_2-1}q_{n_3-1}(x_2^{m_3}x_2'^{m_2}-x_2^{m_2}x_2'^{m_3})(x_1^{n_4}x_1'^{n_1}-x_1^{n_1}x_1'^{n_4})\\
&+& Q_{m_2-1}Q_{m_4-1}q_{n_1-1}q_{n_3-1}(x_2^{m_1}x_2'^{m_3}-x_2^{m_3}x_2'^{m_1})(x_1^{n_4}x_1'^{n_2}-x_1^{n_2}x_1'^{n_4})\\
&+& Q_{m_1-1}Q_{m_3-1}q_{n_2-1}q_{n_4-1}(x_2^{m_2}x_2'^{m_4}-x_2^{m_4}x_2'^{m_2})(x_1^{n_3}x_1'^{n_1}-x_1^{n_1}x_1'^{n_3})\\
&+& Q_{m_1-1}Q_{m_2-1}q_{n_3-1}q_{n_4-1}(x_2^{m_4}x_2'^{m_3}-x_2^{m_3}x_2'^{m_4})(x_1^{n_2}x_1'^{n_1}-x_1^{n_1}x_1'^{n_2})\\
&+& Q_{m_3-1}Q_{m_4-1}q_{n_1-1}q_{n_2-1}(x_2^{m_1}x_2'^{m_2}-x_2^{m_2}x_2'^{m_1})(x_1^{n_3}x_1'^{n_4}-x_1^{n_4}x_1'^{n_3}).
\end{eqnarray*}
By the relations from Section 2: $(x_1^nx_1'^m - x_1^mx_1'^n) = (x_1-x_1')(q_nq_{m-1}-q_mq_{n-1})$ and
$(x_2^nx_2'^m-x_2^mx_2'^n)=(x_2-x_2')(Q_nQ_{m-1}-Q_mQ_{n-1})$ it follows $d_{11}=0$.
Now we prove that $s_4(D_1,D_2,D_3,D_4)_{21}=d_{21}=0$. The approach is similar to that of $d_{11}$. Computing the $(2,1)$-entry of $[D_1,D_2]\circ[D_3,D_4]$ we obtain
\begin{eqnarray*}
&& 2(b_2'(a_1'-a_1)-b_1'(a_2'-a_2))(b_3b_4'+b_3'b_4) \\
&+& 2(b_4'(a_3'-a_3)-b_3'(a_4'-a_4))(b_1b_2'+b_1'b_2)\\
&=& 2(a_1'-a_1) (b_2'b_3'b_4 + b_2'b_3b_4') - 2(a_2' - a_2) (b_1'b_3'b_4 + b_1'b_3b_4') \\
&+& 2(a_3'-a_3) (b_1b_2'b_4' + b_1'b_2b_4')- 2(a_4' - a_4)(b_1'b_2b_3' + b_1b_2'b_3').
\end{eqnarray*}
Permuting the indices we get that $(1/4)d_{21}$ equals
\begin{eqnarray*}
&& (a_1'-a_1) (b_2'b_3'b_4 - b_2'b_4'b_3 + b_3'b_4'b_2)
- (a_2'-a_2) (b_1'b_3'b_4 - b_1'b_4'b_3 + b_3'b_4'b_1)\\
&& +(a_3'-a_3) (b_1'b_2'b_4 - b_1'b_4'b_2 + b_2'b_4'b_1) - (a_4'-a_4)(b_1'b_2'b_3 - b_1'b_3'b_2 + b_2'b_3'b_1).
\end{eqnarray*}
Substitute $b_i=x_1^{n_i}Q_{m_i-1}y_2+x_2'^{m_i}q_{n_i-1}y_1+c_i$, and $a_i=x_1^{n_i}x_2^{m_i}+d_i+e_i$ as in the previous case. Here $c_i\in
K[X;Y]^{(3)}$, $d_i\in K[X;Y]^{(2)}$, and $e_i\in K[X;Y]^{(4)}$.
One has $b_i'b_j'b_k = f_0(i,j,k)y_1y_1'y_2'+g_0(i,j,k)y_2y_1'y_2'$ with
\begin{eqnarray*}
f_0(i,j,k)&=& x_2'^{m_k} q_{n_k-1} (x_2^{m_i} Q_{m_j-1}x_1'^{n_j} q_{n_i-1} - x_2^{m_j}Q_{m_i-1} x_1'^{n_i}q_{n_j-1})\\
g_0(i,j,k)&=& x_1^{n_k} Q_{m_k-1}(x_2^{m_i}Q_{m_j-1}x_1'^{n_j} q_{n_i-1} - x_2^{m_j}Q_{m_i-1}x_1'^{n_i}q_{n_j-1}).
\end{eqnarray*}
By applying the equality $(x_2^n x_2'^m - x_2^m x_2'^n)=(x_2'-x_2) (Q_nQ_{m-1} - Q_mQ_{n-1})$ and after manipulations we get that
$f_0(i,j,k)-f_0(i,k,j)+f_0(j,k,i)$ equals
\begin{eqnarray*}
&& q_{n_k-1} q_{n_i-1} Q_{m_j-1}x_1'^{n_j}(x_2'^{m_k} x_2^{m_i} -x_2'^{m_i} x_2^{m_k})\\
&+& q_{n_k-1}q_{n_j-1}Q_{m_i-1}x_1'^{n_i}(x_2'^{m_j} x_2^{m_k} -x_2'^{m_k}x_2^{m_j})\\
&+& q_{n_j-1} q_{n_i-1} Q_{m_k-1}x_1'^{n_k}(x_2'^{m_i}x_2^{m_j} -x_2'^{m_j} x_2^{m_i})\\
&=& Q_{m_i}Q_{m_j-1} Q_{m_k-1} q_{n_i-1}(q_{n_k-1} x_1'^{n_j} -q_{n_j-1}x_1'^{n_k}) (x_2'-x_2)\\
&+& Q_{m_i-1} Q_{m_j} Q_{m_k-1} q_{n_j-1} (q_{n_i-1}x_1'^{n_k} -q_{n_k-1} x_1'^{n_i})(x_2' - x_2)\\
&+& Q_{m_i-1} Q_{m_j-1} Q_{m_k} q_{n_k-1} (q_{n_j-1}x_1'^{n_i} -q_{n_i-1}x_1'^{n_j})(x_2'-x_2).
\end{eqnarray*}
Thus $d_{21}=4f(x)y_1y_1'y_2'+ 4g(x)y_2y_1'y_2'$. We shall prove $f(x)=g(x)=0$. But
\begin{eqnarray*} (1/4)f(x) &=&
(x_1'^{n_1}x_2'^{m_1}-x_1^{n_1}x_2^{m_1})(f_0(2,3,4)-f_0(2,4,3)+f_0(3,4,2))\\
&-& (x_1'^{n_2}x_2'^{m_2} - x_1^{n_2}x_2^{m_2}) (f_0(1,3,4)-f_0(1,4,3) +f_0(3,4,1))\\
&+& (x_1'^{n_3}x_2'^{m_3}-x_1^{n_3}x_2^{m_3}) (f_0(1,2,4) -f_0(1,4,2) +f_0(2,4,1))\\
&-& (x_1'^{n_4}x_2'^{m_4}-x_1^{n_4}x_2^{m_4}) (f_0(1,2,3) -f_0(1,3,2) +f_0(2,3,1)).
\end{eqnarray*}
But $(x_1'^nx_2'^m - x_1^n x_2^m) = x_1'^n(x_2' -x_2) Q_{m-1} +x_2^m (x_1'-x_1)q_{n-1}$; substituting the $f_0$ by their defining equalities in
$f_0(i,j,k)-f_0(i,k,j)+f_0(j,k,i)$ we get
$f(x)=(x_2'-x_2)^2f^{(1)}(x)+(x_2'-x_2)(x_1'-x_1)f^{(2)}(x)$.
Here
\begin{eqnarray*} f^{(1)}(x)&=& Q_{m_1-1} Q_{m_2} Q_{m_3-1} Q_{m_4-1} x_1'^{n_1}q_{n_2-1}(q_{n_4-1}x_1'^{n_3}-q_{n_3-1}x_1'^{n_4})\\
&+& Q_{m_1-1} Q_{m_2-1}Q_{m_3} Q_{m_4-1} x_1'^{n_1}q_{n_3-1} (q_{n_2-1} x_1'^{n_4}-q_{n_4-1}x_1'^{n_2})\\
&+& Q_{m_1-1} Q_{m_2-1} Q_{m_3-1} Q_{m_4} x_1'^{n_1} q_{n_4-1}(q_{n_3-1} x_1'^{n_2}-q_{n_2-1} x_1'^{n_3})\\
&-& Q_{m_1} Q_{m_2-1} Q_{m_3-1} Q_{m_4-1} x_1'^{n_2} q_{n_1-1}(q_{n_4-1} x_1'^{n_3}- q_{n_3-1} x_1'^{n_4})\\
&-& Q_{m_1-1} Q_{m_2-1} Q_{m_3} Q_{m_4-1} x_1'^{n_2} q_{n_3-1}(q_{n_1-1} x_1'^{n_4}- q_{n_4-1} x_1'^{n_1})\\
&-& Q_{m_1-1} Q_{m_2-1} Q_{m_3-1} Q_{m_4}x_1'^{n_2} q_{n_4-1}(q_{n_3-1} x_1'^{n_1}- q_{n_1-1}x_1'^{n_3})\\
&+& Q_{m_1} Q_{m_2-1} Q_{m_3-1} Q_{m_4-1} x_1'^{n_3} q_{n_1-1}(q_{n_4-1} x_1'^{n_2} - q_{n_2-1}x_1'^{n_4})\\
&+& Q_{m_1-1} Q_{m_2} Q_{m_3-1} Q_{m_4-1}x_1'^{n_3} q_{n_2-1}(q_{n_1-1} x_1'^{n_4}-q_{n_4-1} x_1'^{n_1})\\
&+& Q_{m_1-1} Q_{m_2-1} Q_{m_3-1} Q_{m_4}x_1'^{n_3} q_{n_4-1}(q_{n_2-1} x_1'^{n_1}-q_{n_1-1} x_1'^{n_2})\\
&-& Q_{m_1} Q_{m_2-1} Q_{m_3-1} Q_{m_4-1}x_1'^{n_4} q_{n_1-1}(q_{n_3-1} x_1'^{n_2}- q_{n_2-1}x_1'^{n_3})\\
&-& Q_{m_1-1} Q_{m_2} Q_{m_3-1} Q_{m_4-1}x_1'^{n_4} q_{n_2-1}(q_{n_1-1} x_1'^{n_3}- q_{n_3-1}x_1'^{n_1})\\
&-& Q_{m_1-1}Q_{m_2-1} Q_{m_3-1}Q_{m_4} x_1'^{n_4} q_{n_3-1}(q_{n_2-1} x_1'^{n_1} -q_{n_1-1}x_1'^{n_2}).
\end{eqnarray*}
After simple manipulations we obtain $f^{(1)}(x)=0$. As for $f^{(2)}(x)$ we have
\begin{eqnarray*}
f^{(2)}(x)&=& q_{n_1-1}x_2^{m_1} Q_{m_2} Q_{m_3-1} Q_{m_4-1} q_{n_2-1}(q_{n_4-1}x_1'^{n_3}- q_{n_3-1}x_1'^{n_4}) \\
&+& q_{n_1-1}x_2^{m_1} Q_{m_2-1} Q_{m_3} Q_{m_4-1} q_{n_3-1}(q_{n_2-1}x_1'^{n_4}- q_{n_4-1}x_1'^{n_2}) \\
&+&q_{n_1-1}x_2^{m_1}Q_{m_2-1} Q_{m_3-1} Q_{m_4}q_{n_4-1}(q_{n_3-1} x_1'^{n_2}- q_{n_2-1}x_1'^{n_3}) \\
&-&q_{n_2-1}x_2^{m_2} Q_{m_1}Q_{m_3-1} Q_{m_4-1}q_{n_1-1}(q_{n_4-1} x_1'^{n_3}-q_{n_3-1}x_1'^{n_4}) \\
&-&q_{n_2-1}x_2^{m_2}Q_{m_1-1} Q_{m_3}Q_{m_4-1} q_{n_3-1}(q_{n_1-1} x_1'^{n_4} -q_{n_4-1}x_1'^{n_1}) \\
&-&q_{n_2-1}x _2^{m_2}Q_{m_1-1}Q_{m_3-1} Q_{m_4}q_{n_4-1}(q_{n_3-1} x_1'^{n_1}-q_{n_1-1}x_1'^{n_3}) \\
&+&q_{n_3-1}x_2^{m_3}Q_{m_1} Q_{m_2-1}Q_{m_4-1} q_{n_1-1}(q_{n_4-1} x_1'^{n_2} - q_{n_2-1}x_1'^{n_4}) \\
&+&q_{n_3-1}x_2^{m_3}Q_{m_1-1} Q_{m_2}Q_{m_4-1}q_{n_2-1}(q_{n_1-1} x_1'^{n_4} -q_{n_4-1}x_1'^{n_1}) \\
&+&q_{n_3-1}x_2^{m_3}Q_{m_1-1} Q_{m_2-1}Q_{m_4} q_{n_4-1}(q_{n_2-1} x_1'^{n_1}- q_{n_1-1}x_1'^{n_2}) \\
&-&q_{n_4-1}x_2^{m_4}Q_{m_1}Q_{m_2-1} Q_{m_3-1} q_{n_1-1}(q_{n_3-1} x_1'^{n_2}- q_{n_2-1}x_1'^{n_3}) \\
&-&q_{n_4-1}x_2^{m_4}Q_{m_1-1} Q_{m_2}Q_{m_4-1}q_{n_2-1}(q_{n_1-1} x_1'^{n_3}-q_{n_3-1}x_1'^{n_1}) \\
&-&q_{n_4-1}x_2^{m_4}Q_{m_1-1}Q_{m_2-1} Q_{m_4}q_{n_3-1}(q_{n_2-1} x_1'^{n_1} -q_{n_1-1}x_1'^{n_2})
\end{eqnarray*}
which in turn equals
\begin{eqnarray*}
&& q_{n_1-1}q_{n_2-1}q_{n_3-1} Q_{m_4-1}x_1'^{n_4} x_2^{m_1}(Q_{m_2-1} Q_{m_3}-Q_{m_2}Q_{m_3-1})\\
&+&q_{n_1-1}q_{n_2-1}q_{n_3-1}Q_{m_4-1} x_1'^{n_4}x_2^{m_2}(Q_{m_1} Q_{m_3-1}-Q_{m_1-1}Q_{m_3})\\
&+&q_{n_1-1}q_{n_2-1}q_{n_3-1}Q_{m_4-1} x_1'^{n_4} x_2^{m_3}(Q_{m_1-1} Q_{m_2}-Q_{m_1}Q_{m_2-1})\\
&-&q_{n_1-1}q_{n_2-1}q_{n_4-1}Q_{m_3-1} x_1'^{n_3}x_2^{m_1}(Q_{m_2-1} Q_{m_4}-Q_{m_2}Q_{m_4-1})\\
&-&q_{n_1-1}q_{n_2-1}q_{n_4-1}Q_{m_3-1} x_1'^{n_3}x_2^{m_2}(Q_{m_1} Q_{m_4-1}-Q_{m_1-1}Q_{m_4})\\
&-&q_{n_1-1}q_{n_2-1}q_{n_4-1}Q_{m_3-1} x_1'^{n_3}x_2^{m_4}(Q_{m_1-1} Q_{m_2}-Q_{m_1}Q_{m_2-1})\\
&+&q_{n_1-1}q_{n_3-1}q_{n_4-1}Q_{m_2-1} x_1'^{n_2}x_2^{m_1}(Q_{m_3-1} Q_{m_4}-Q_{m_3}Q_{m_4-1})\\
&+&q_{n_1-1}q_{n_3-1}q_{n_4-1} Q_{m_2-1}x_1'^{n_2}x_2^{m_3}(Q_{m_1} Q_{m_4-1}-Q_{m_1-1}Q_{m_4})\\
&+&q_{n_1-1}q_{n_3-1}q_{n_4-1} Q_{m_2-1}x_1'^{n_2}x_2^{m_4}(Q_{m_1-1} Q_{m_3}-Q_{m_1}Q_{m_3-1})\\
&-&q_{n_2-1}q_{n_3-1}q_{n_4-1}Q_{m_1-1} x_1'^{n_1}x_2^{m_2}(Q_{m_3-1} Q_{m_4}-Q_{m_3}Q_{m_4-1})\\
&-&q_{n_2-1}q_{n_3-1}q_{n_4-1} Q_{m_1-1}x_1'^{n_1}x_2^{m_3}(Q_{m_2} Q_{m_4-1}-Q_{m_2-1}Q_{m_4})\\
&-&q_{n_2-1}q_{n_3-1}q_{n_4-1} Q_{m_1-1}x_1'^{n_1}x_2^{m_4}(Q_{m_2-1}Q_{m_3}-Q_{m_2}Q_{m_3-1}).
\end{eqnarray*}
Applying $(x_2^n x_2'^m- x_2^mx_2'^n) = (x_2- x_2')(Q_nQ_ {m-1} -Q_mQ_{n-1})$ one sees that the first three summands in the final expression cancel out.
Repeat the procedure for the remaining three groups of 3 summands in each thus getting $f^{(2)}(x)=0$.
In order to show that $g(x)=0$, we observe that
\begin{eqnarray*} (1/4)g(x) &=&
(x_1'^{n_1}x_2'^{m_1}-x_1^{n_1}x_2^{m_1}) (g_0(2,3,4)-g_0(2,4,3)+g_0(3,4,2))\\
&-& (x_1'^{n_2}x_2'^{m_2}-x_1^{n_2}x_2^{m_2}) (g_0(1,3,4)-g_0(1,4,3)+g_0(3,4,1))\\
&+& (x_1'^{n_3}x_2'^{m_3}-x_1^{n_3}x_2^{m_3}) (g_0(1,2,4)-g_0(1,4,2)+g_0(2,4,1))\\
&-& (x_1'^{n_4}x_2'^{m_4}-x_1^{n_4}x_2^{m_4}) (g_0(1,2,3)-g_0(1,3,2)+g_0(2,3,1)).
\end{eqnarray*}
Now consider the automorphism $\phi$ of $K[x_1,x_2,x_1',x_2']$ defined by $x_1\mapsto x_2'$, $x_2\mapsto x_1'$, $x_1'\mapsto x_2$ and $x_2'\mapsto x_1$.
Let us compute $\phi(f(x))$.
First observe that for each $i$, $\phi(x_1'^{n_i}x_2'^{m_i}-x_1^{n_i}x_2^{m_i})=-(x_1'^{m_i}x_2'^{n_i}-x_1^{m_i}x_2^{n_i})$ and for each $i$, $j$ and $k$,
\[
\phi(f_0(i,j,k))=-x_1^{m_k}Q_{n_k-1}(x_1'^{m_j}q_{m_i-1}x_2^{n_i}Q_{n_j-1}-x_1'^{m_i}q_{m_j-1}x_2^{n_j}Q_{n_i-1}).
\]
By the above observations, we can see that $-\phi(f(x))$ is equal to the expression obtained by $g(x)$ by permuting $m_i$ and $n_i$ for each $i$. Since
$f(x)=0$, for arbitrary integers $n_i$, $m_i$, $i\in \{1,2,3,4\}$, it follows that $g(x)=0$. Thus $s_4$ is an identity for $F$. \hfill$\diamondsuit$\medskip
\noindent \textbf{Remark 1} If one tries to prove the above Proposition by splitting $F$ replacing it with the algebra generated by
$\{x_ie_{11},x_i'e_{22},y_ie_{12},y_i'e_{21},i=1,2\}$, the resulting larger algebra does not satisfy $s_4$, since
\[s_4(y_1e_{12},y_1'e_{21},y_2e_{12},y_2'e_{21})=4y_1y_1'y_2y_2'(e_{11}+e_{22})\neq 0.\]
\noindent \textbf{Remark 2} Neither of the identities from Propositions~(\ref{1-2}), (\ref{s_4}) is a polynomial identity for $M_{11}(E)$. Clearly
$T(M_{11}(E))\subseteq T(F)$; it can be easily shown that all identities for $M_{11}(E)$ follow from $[[t_1,t_2][t_3,t_4],t_5]$ (though the latter is not
an identity for $M_{11}(E)$).
\section{The identities of $M_{11}(E)$}
Here we shall prove that the polynomials from Propositions~(\ref{1-2}), (\ref{s_4}) generate the T-ideal of $F$. To this end we make use of the results of Popov \cite{popov}. Namely we shall need not only the concrete form of the basis of $T(M_{11}(E))$ but also the structure of the corresponding relatively free algebra. In order to make our exposition more self-contained we recall here the results from \cite{popov}.
Let $\Gamma_n$ be the vector space of the proper multilinear polynomials in $t_1$, \dots, $t_n$ in the free associative algebra $K\langle T\rangle$. We work with unitary algebras over a field of characteristic 0 hence the T-ideal of any algebra $A$ is generated by the intersections $T(A)\cap \Gamma_n$. The vector space $\Gamma_n$ is a left module over the symmetric group $S_n$. The action of $S_n$ on $\Gamma_n$ is by permuting the variables. We refer the reader to \cite[Chapter 12]{drenskybook} for all necessary information concerning the representations of $S_n$ and their applications to PI theory.
Let $W$ be the variety of algebras defined by $M_{11}(E)$ and put $\Gamma_n(W) = \Gamma_n/(\Gamma_n\cap T(W))$. Then clearly $\Gamma_n(W)$ is an
$S_n$-module (with the induced action). The result from \cite{popov} we need is the decomposition of $\Gamma_n(W)$ into irreducible submodules. It is well
known that the irreducibles for $S_n$ are described in terms of partitions of $n$ and Young tableaux. We refer once again to \cite{drenskybook} for that
description. Recall that the polynomial representations of the general linear group $GL_m$ are also described in terms of partitions (of not more than $m$
parts) and Young diagrams, see \cite{drenskybook}. Sometimes it is convenient to use the ''symmetrized'' version of a generator of an irreducible
$S_n$-module; it generates an irreducible $GL_m$-module, and vice versa via linearisation. Recall that one can linearise and go back to the symmetrized version of a polynomial as char$K=0$.
The following polynomials were defined in \cite{popov}. Let $p\ge q\ge 2$ and $s\ge 0$. Set $\varphi_p^{(s)}=\varphi_p^{(s)}(t_1, \dots, t_p)$ and $\varphi_{p,q}^{(s)} = \varphi_{p,q}^{(s)}(t_1, \dots,t_p)$ as follows.
\begin{eqnarray*}
\varphi_p^{(s)}&=&\begin{cases} \sum_{\sigma\in S_p}(-1)^{\sigma}[t_{\sigma(1)}, t_{\sigma(2)}] \dots [t_{\sigma(p)}, x_1^{(s)}] & p \text{ odd,} \\
\sum_{\sigma\in S_p}(-1)^{\sigma}[t_{\sigma(1)},t_{\sigma(2)}]\dots[t_{\sigma(p-1)},t_{\sigma(p)}, t_1^{(s)}] & p \text{ even,} \\
\end{cases}\\
\varphi_{p,q}^{(s)}&=&
\begin{cases}
\sum_{\tau\in S_q}(-1)^{\tau}[t_{\tau(1)},t_{\tau(2)}]\dots[t_{\tau(q-1)},t_{\tau(q)}] \varphi_p^{(s)} & q\text{ even,} \\
\sum_{\sigma\in S_p}(-1)^{\sigma}[t_{\sigma(1)},t_{\sigma(2)}] \dots[t_{\sigma(p-1)},t_{\sigma(p)}] \varphi_q^{(s)} & p \text{ even, } q \text{ odd} \end{cases}
\end{eqnarray*}
and $\varphi_{p,q}^{(s)}=\sum (-1)^{\sigma\tau}[x_{\tau(1)},x_{\tau(2)}] \cdots[x_{\sigma(1)},x_{\sigma(2)}] \cdots
[x_{\sigma(p)}, x_1^{(s)}, x_{\tau(q)}]$ when both $p$ and $q$ are odd. Here the summation runs over $\sigma\in S_p$, $\tau\in S_q$, and $[a,b^{(s)}]$ stands for $[a,b,\ldots,b]$ with $s$ entries of $b$.
We observe that for $s=0$, some of these polynomials are not defined. For example, for $p$ even, $\varphi_p^{(0)}(x_1,\dots,x_p)$ is the standard
polynomial of degree $p$, while for $p$ odd, it is not defined, since the associated diagram is a single column, which is associated to the standard
polynomial of degree $p$ that is not a proper polynomial when $p$ is odd. In a similar way $\varphi_{p,q}^{(0)}=s_q(x_1,\dots,x_q)s_p(x_1,\dots,x_p)$, if
$p$ and $q$ are even and $\varphi_{p,q}^{(0)}=\sum (-1)^{\sigma\tau}[x_{\tau(1)},x_{\tau(2)}] \cdots[x_{\sigma(1)},x_{\sigma(2)}] \cdots
[x_{\sigma(p)},x_{\tau(q)}]$ if $p$ and $q$ are odd, and it is not defined for the remaining cases.
Set $M_p^{(s)}$ to be the $S_n$-submodule of $\Gamma_n(W)$ generated by $\varphi_p^{(s)}$, $n=p+s$, and $M_{p,q}^{(s)}$ the
$S_n$-submodule of $\Gamma_n(W)$ generated by $\varphi_{p,q}$, $n=p+q+s$.
Let $f(t_1,\ldots,t_n)\in \Gamma_n$ be a proper multilinear polynomial and suppose $d=D_\alpha$ is a Young tableau associated to a diagram $D$ of a partition of $n$. That is we fill the boxes of the diagram $D$, along the rows, with the numbers of the permutation $\alpha$ of $n$. Form the Young semi-idempotent $e(d)$ of $d$ and denote by $M(d,f)$ the $S_n$-module generated by $e(d) f$. It is well known that it is either 0 or irreducible.
The polynomials $\varphi_p^{(s)}$ and $\varphi_{p,q}^{(s)}$ are not multilinear but we consider their complete linearisations. The description of
$\Gamma_n(W)$ given in \cite{popov} is based on the following results.
\begin{enumerate}
\item
If $p\ge 2$, $s\ge 0$ then $\varphi_p^{(s)}$ and $\varphi_{p,q}^{(s)}$ are not polynomial identities for $M_{11}(E)$.
\item
Let $n=p+s$ and let $D=(s+1,1^{p-1})$ be a partition of $n$ with an associated Young tableau $d=D_\alpha$. If $f(t_1,\ldots,t_n)\in \Gamma_n(W)$ then $M(d,f)\subseteq M_p^{(s)}$
\item
Let $n=p+q+s$, $d=D_\alpha$ with $D=(s+2,2^{q-1},1^{p-q})$ and $f=f(x_1,\dots,x_n)\in \Gamma_n(W)$. Then $M(d,f)\subseteq M_{p,q}^{(s)}$.
\item
If the second row of a diagram $D$ contains at least 3 boxes then for every $f$ the module $M(d,f)$ is 0 in $\Gamma_n(W)$.
\item
The decomposition of $\Gamma_n(W)$ in irreducibles is
\[
\Gamma_n(W)=(\oplus_{p+s=n}M_{p}^{(s)}) \bigoplus (\oplus_{p+q+s=n} M_{p,q}^{(s)}).
\]
\end{enumerate}
Popov deduced, as a corollary to the above listed results, the main theorem of \cite{popov}, namely that the T-ideal of $M_{11}(E)$ is generated by the polynomials $[[t_1,t_2]^2, t_1]$ and $[t_1,t_2,[t_3,t_4],t_5]$.
\section{The identities of $F$}
\begin{theorem}
\label{mainthm}
Let $K$ be a field of characteristic 0. The polynomials
\begin{equation}
\label{basisF}
[[t_1,t_2][t_3,t_4],t_5], \quad [t_1,t_2] [t_3,t_4] [t_5,t_6], \quad s_4(t_1,t_2,t_3,t_4)
\end{equation}
form a basis of the polynomial identities for the algebra $F=K[C_1,C_2]$.
\end{theorem}
\textit{Proof}\textrm{. }
First we shall prove that the polynomials $\varphi_4^{(s)}$ lie in the T-ideal generated by the three polynomials from our theorem. Recall that
\[
\varphi_p^{(s)}=\begin{cases} \sum_{\sigma\in S_p}(-1)^{\sigma}[t_{\sigma(1)}, t_{\sigma(2)}] \dots [t_{\sigma(p)}, x_1^{(s)}] & p \text{ odd,} \\
\sum_{\sigma\in S_p}(-1)^{\sigma}[t_{\sigma(1)},t_{\sigma(2)}]\dots[t_{\sigma(p-1)},t_{\sigma(p)}, t_1^{(s)}] & p \text{ even.}
\end{cases}
\]
Also it is immediate that $\varphi_4^{(0)}(t_1,t_2,t_3,t_4) = 4s_4(t_1,t_2,t_3,t_4)$. Suppose $s\ge 1$. By the identity from Corollary~\ref{2times3} we
obtain that $(1/4) \varphi_p^{(s)}(t_1,t_2,t_3,t_4)$ equals
\begin{eqnarray*}
&&[t_1,t_2][t_3,t_4,t_1^{(s)}] + (-1)^{s}[t_3,t_4,t_1^{(s)}] [t_1,t_2] + [t_2,t_3] [t_1,t_4,t_1^{(s)}] \\
&+& (-1)^{s}[t_1,t_4,t_1^{(s)}][t_2,t_3] - [t_2,t_4] [t_1, t_3, t_1^{(s)}] - (-1)^{s}[t_1, t_3, t_1^{(s)}][t_2,t_4].
\end{eqnarray*}
Thus $(1/4) \varphi_p^{(s)}(t_1,t_1t_2,t_3,t_4)$ will be equal to
\begin{eqnarray*}
&& [t_1,t_3] t_2 [t_1,t_4,t_1^{(s)}] + [t_1,t_4] [t_1,t_3, t_1^{(s)}] t_2 - [t_1,t_4] t_2 [t_1,t_3,t_1^{(s)}] \\
&-& [t_1,t_3][t_1,t_4, t_1^{(s)}] t_2 - [t_1,t_4] [t_2, t_3, t_1^{(s+1)}] - [x_3,x_4] [x_1,x_2, x_1^{(s+1)}] \\ &+& [t_1,t_3] [t_2,t_4,t_1^{(s+1)}]
+ (1/4)t_1\varphi_4^{(s)}(t_1, t_2,t_3, t_4).
\end{eqnarray*}
Analogously for $(1/4) \varphi_p^{(s)}(t_1,t_2t_1,t_3,t_4)$ we obtain
\begin{eqnarray*}
&&t_2[t_1,t_3][t_1,t_4,t_1^{(s)}] + (-1)^s [t_1,t_4, t_1^{(s)}] t_2[t_1,t_3] - t_2[t_1,t_4][t_1, t_3, t_1^{(s)}]\\
&-&(-1)^s[t_1, t_3, t_1^{(s)}] t_2[t_1,t_4] - [t_1,t_2][t_3,t_4, t_1^{(s+1)}] - [t_2,t_3][t_1,t_4,t_1^{(s+1)}]\\
&+& [t_2,t_4] [t_1,t_3,t_1^{(s+1)}] + (1/4) \varphi_4^{(s)} (t_1, t_2,t_3, t_4) t_1.
\end{eqnarray*}
Therefore for $(1/4) ( \varphi_p^{(s)}(t_1,t_1t_2,t_3,t_4) + \varphi_p^{(s)}(t_1,t_2t_1,t_3,t_4))$ we have
\begin{eqnarray*}
& & (1/4)(t_1\circ \varphi_4^{(s)}(t_1,t_2, t_3,t_4)- \varphi_4^{(s+1)} + [t_1,t_3] t_2[t_1,t_4, t_1^{(s)}]\\
&-& (-1)^s[t_1,t_3,t_1^{(s)}]t_2[t_1,t_4] - [t_1,t_4]t_2 [t_1,t_3, t_1^{(s)}] +(-1)^s[t_1,t_4,t_1^{(s)}]t_2[t_1,t_3].
\end{eqnarray*}
An easy computation shows that the first two identities from the theorem, together with the fact that commutation by a fixed element is a derivation, imply the identity $[[t_1,t_2]t_3[t_4,t_5], t_6]=0$. Using the fact that the product of 3 commutators is 0 we obtain
$[t_1,t_2]t_3[t_4,t_5,t_6]+ [t_1,t_2,t_6]t_3[t_4,t_5]=0$. Repeating several times we arrive at
\begin{eqnarray*}
[t_1,t_3]t_2[t_1,t_4,t_1^{(s)}]- (-1)^s[t_1,t_3,t_1^{(s)}] t_2[t_1,t_4] &=&0,\\
{}[t_1, t_4] t_2[t_1,t_3,t_1^{(s)}]- (-1)^s[t_1,t_4,t_1^{(s)}] t_2[t_1,t_3]&=&0.
\end{eqnarray*}
In this way $\varphi_4^{(s+1)}(t_1,t_2,t_3,t_4)$ equals
\[\varphi_4^{(s)}(t_1,t_2,t_3,t_4)\circ t_1-\varphi_4^{(s)}(t_1,t_1t_2,t_3,t_4)-\varphi_4^{(s)}(t_1,t_2t_1,t_3,t_4),\]
and we may proceed by induction on $s$ since $\varphi_4^{(0)} = 4s_4$.
\medskip
As we work with unitary algebras and char$K=0$ we can consider only the multilinear proper identities. Denote by $I$ the T-ideal generated by the identities from the theorem and let $V$ be the variety of unitary algebras determined by $I$; we shall study the $S_n$-module $\Gamma_n(V) = \Gamma_n/\Gamma_n\cap I$. But the T-ideal of $M_{11}(E)$ is contained in $I$ hence $\Gamma_n(V)$ is a homomorphic image of $\Gamma_n(W)$, and we have to determine which of the irreducibles in $\Gamma_n(W)$ vanish modulo the T-ideal $I$.
But it is easy to see that the polynomials $\varphi_p^{(s)}$, $p\ge 5$, are products of at least three commutators, and as such they follow from the second identity of the theorem. The same holds for $\varphi_{p,q}^{(s)}$ whenever $p+q\ge 5$. Also we showed above that $\varphi_4^{(s)}$ lies in $I$ for every $s\ge 0$. Therefore
\[
\Gamma_n(V) = M_2^{(n-2)} \oplus M_3^{(n-3)} \oplus M_{2,2}^{(n-4)}.
\]
Hence in order to complete the proof it suffices to see that the generators of the irreducible modules from the above decomposition are all non-zero modulo $I$. But $\varphi_2^{(s)} (t_1,t_2) = 2[t_1,t_2, t_1^{(s)}]$ and $\varphi_2^{(s)}(C_1,C_2)\ne 0$ according to \cite[Lemma 9]{pktcm}.
Analogously $\varphi_3^{(s)} (t_1,t_2,t_3) = 2([t_1,t_2][t_3,t_1^{(s)}] - [t_1,t_3] [t_2,t_1^{(s)}])$ and we obtain
$\varphi_3^{(s)}(C_1,C_2,[C_1,C_2])=-4[C_1,C_2][C_2,C_1^{(s+1)}]\ne 0$ due to \cite[Lemma 9]{pktcm}.
Finally $\varphi_{2,2}^{(s)}=4[t_1,t_2][t_1,t_2,t_1^{(s)}]$ and $\varphi_{2,2}^{(s)}(C_1,C_2)=4[C_1,C_2][C_1,C_2,C_1^{(s)}]$ is non-zero for the same
reason as above. \hfill$\diamondsuit$\medskip
In \cite{gordienko}, A. Gordienko studied the identities satisfied by the algebra $A_1\subseteq UT_3(K)$ where $UT_3(K)$ are the upper triangular matrices
of order 3 over $K$. The algebra $A_1$ consists of all matrices whose $(1,1)$ and $(3,3)$ entries are equal. He deduced that $T(A_1)$ is generated by the
same three identities as in our Theorem~\ref{mainthm}. Moreover Gordienko described basis of the vector space $P_n(A_1)$ and $\Gamma_n(A_1)$ of the
multilinear and the proper multilinear elements of degree $n$ modulo the identities of $A_1$, respectively. Combining our result with Gordienko's theorem
we obtain the following corollary.
\begin{corollary}
The algebras $F = K[C_1,C_2]$ and $A_1$ are PI equivalent over a field of characteristic 0.
\end{corollary}
It will be interesting to know whether these two algebras remain PI equivalent if the field $K$ is infinite and of characteristic $p>2$.
\medskip
\section{A description of the subvarieties}
In this section we shall describe the subvarieties of the variety $V$ of unitary algebras defined by the identities of the algebra $F$. We shall work over a field of characteristic 0. Recall that $\Gamma_n(V) = M_2^{(n-2)} \oplus M_3^{(n-3)} \oplus M_{2,2}^{(n-4)}$. Therefore we have to find the consequences of degree $n+1$ in $\Gamma_{n+1}(V)$ of the polynomials $\varphi_2^{(n-2)}$, $\varphi_{2,2}^{(n-4)}$, $\varphi_3^{(n-3)}$.
\begin{proposition}
\label{conseq}
The consequences of degree $n+1$ are as follows.
(1) The polynomials $\varphi_2^{(n-1)}$, $\varphi_{2,2}^{(n-3)}$, and $\varphi_3^{(n-2)}$, follow from $\varphi_2^{(n-2)}$.
(2) The polynomials $\varphi_{2,2}^{(n-3)}$ and $\varphi_3^{(n-2)}$ follow from $\varphi_{2,2}^{(n-4)}$.
(3) The polynomials $\varphi_{2,2}^{(n-3)}$ and $\varphi_3^{(n-2)}$ follow from $\varphi_3^{(n-3)}$.
\end{proposition}
\textit{Proof}\textrm{. } Let $u_2^{(n-2)}(t_1,t_2,t_3)$ be the multihomogeneous component of the polynomial $\varphi_2^{(n-2)}(t_1+t_3,t_2)$ that is linear in $t_3$, and let
$u_{2,2}^{(n-4)}(t_1,t_2,t_3)$ be the component that is linear in $t_2$ and in $t_3$ of $\varphi_{2,2}^{(n-4)}(t_1,t_2+t_3)$. Clearly the identities
$u_2^{(n-2)}$ and $\varphi_2^{(n-2)}$ are equivalent, and also $u_{2,2}^{(n-4)}$ and $\varphi_{2,2}^{(n-4)}$ are equivalent, as the former are partial
linearisations of the latter. Then
\begin{eqnarray*}
u_2^{(n-2)} &=& -2([t_2,t_3,t_1^{(n-2)}]+(n-3)[t_2,t_1,t_3,t_1^{(n-3)}]+[t_2,t_1^{(n-2)},t_3]);\\
u_{2,2}^{(n-4)} &=& -4([t_1,t_2][t_3,t_1^{(n-3)}]+[t_1, t_3] [t_2, t_1^{(n-3)}]).
\end{eqnarray*}
In order to prove the proposition we verify which of the generators of the irreducible modules of $\Gamma_{n+1}(M_{1,1})$ are consequences of each
generator of the irreducible modules of $\Gamma_n(M_{1,1})$.
1. We compute directly that $\varphi_2^{(n-1)} = \varphi_2^{(n-2)}t_1 - t_1\varphi_2^{(n-2)}$. Similarly
\begin{eqnarray*}
\varphi_3^{(n-2)}(t_1,t_2,t_3) &= & (-1)^n (u_2^{(n-2)}(t_1, t_3, t_1t_2) - u_2^{(n-2)}(t_1, t_2,t_1t_3) \\
&+& t_1(u_2^{(n-2)} (t_1,t_2,t_3)- u_2^{(n-2)}(t_1, t_3, t_2))/(n-2)\\
&+& (n-1)(\varphi_2^{(n-2)}(t_1,t_2)t_3- \varphi_2^{(n-2)}(t_1, t_3) t_2)).
\end{eqnarray*}
Also $\varphi_{2,2}^{(n-3)} = u_2^{(n-2)}(t_1,t_1t_2, t_2) +2\varphi_2^{(n-2)}t_2-t_1u_2^{(n-2)}(t_1,t_2,t_2)$ when $n$ is odd, and similarly $\varphi_{2,2}^{(n-3)} = (u_2^{(n-1)}(t_1, t_2, t_2) - [\varphi_2^{(n-2)}(t_1,t_2), t_2])/(n-2) - u_2^{(n-2)}(t_1, t_2,[t_1,t_2])$ if $n$ is even.
2. One computes directly that
\begin{eqnarray*}
\varphi_{2,2}^{(n-3)} & = & t_1 u_{2,2}^{(n-4)} (t_1,t_2,t_2)- u_{2,2}^{(n-4)}(t_1,t_1t_2, t_2);\\
\varphi_3^{(n-3)} & = & (u_{2,2}^{(n-4)}(t_1,t_2,t_1t_3) - u_{2, 2}^{(n-4)} (t_1,t_1t_2,t_3))/2.
\end{eqnarray*}
Moreover $\varphi_2^{(n-1)}$ is not a consequence of $\varphi_{2,2}^{(n-4)}$. In order to see this observe that $\varphi_{2,2}^{(n-4)}(C_1,C_2)$ is strongly central in $F$. Hence all its consequences will be central but $\varphi_2^{(n-2)}(C_1,C_2)$ is not central.
3. Finally, for $\varphi_{2,2}^{n-4}$, one checks directly that
\begin{eqnarray*}
\varphi_3^{(n-2)}& = & t_1 \varphi_3^{(n-3)}+ \varphi_3^{(n-3)} t_1 -\varphi_3^{(n-3)}(t_1,t_1t_2,t_3)- \varphi_3^{(n-3)}(t_1, t_2t_1, t_3);\\
\varphi_{2,2}^{(n-3)} & = & \varphi_3^{(n-3)}(t_1,t_2,[t_1,t_2]).
\end{eqnarray*}
As in (2), $\varphi_2^{(n-1)}$ is not a consequence of $\varphi_3^{(n-3)}$.
\hfill$\diamondsuit$\medskip
\medskip
Let $I$ and $J$ be two T-ideals in $K\langle X\rangle$. Following \cite{Kemer2}, we say that $I$ and $J$ are \textsl{asymptotically equivalent} if for all
sufficiently large $n$ it holds $I\cap B_n=J\cap B_n$, where $B_n$ states for the proper polynomials of degree $n$. As we consider unitary algebras over a field of characteristic 0 we can
substitute $B_n$ by $\Gamma_n$.
\begin{corollary}
\label{asympt} If $U$ is a proper subvariety of $V$ then $U$ is asymptotically equivalent to either var$(K)$ or to var$(UT_2(K))$.
\end{corollary}
\textit{Proof}\textrm{. }
The variety $U$ satisfies, for some $n$, at least one of the identities $\varphi_2^{(n-2)}(x_1,x_2)$, $\varphi_3^{(n-3)}(x_1,x_2,x_3)$,
$\varphi_{2,2}^{(n-4)}(x_2,x-2)$.
If it satisfies some $\varphi_2^{(n-2)}$ then by Proposition~\ref{conseq} every commutator of sufficiently large degree will vanish on $U$. Therefore $U$ is asymptotically equivalent to var$(K)$. (Recall the latter is generated by the identity $[t_1,t_2]$.)
If on the other hand $U$ satisfies $\varphi_{2,2}^{(n-4)}$ or $\varphi_3^{(n-3)}$ then every proper polynomial of sufficiently large degree, that is a product of two commutators, will vanish on $U$. But the T-ideal of $UT_2(K)$ is generated by the product of two commutators, hence $V$ is asymptotically equivalent to var$(UT_2(K))$.
\hfill$\diamondsuit$\medskip
\section{Identities in two variables for $M_{11}(E)$}
In this section we apply some of the results obtained above in order to describe the identities in two variables for the algebra $M_{11}(E)$ over a field of characteristic 0. Recall that in this way we determine the identities in two variables for $E\otimes E$ as well. The identities in two variables for $M_2(K)$, char$K=0$ were described by Nikolaev in \cite{nikolaev}. We shall employ some of the results obtained in \cite{nikolaev}.
A word about the notation we shall use here. In order not to accumulate indices we shall use the letters $x$ and $y$ for the variables.
The main theorem in \cite{nikolaev} states that the identities in two variables for $M_2(K)$ all follow from the Hall polynomial $h(x,y) = [[x,y]^2,x]$. Recall that $T(M_2(K))$ is generated by $h$ and by $s_4$; one usually writes $h(x,y,z) = [[x,y]^2,z]$. This form of $h$ is equivalent to the one above \textsl{modulo} the standard polynomial $s_4$, see for example \cite{jcpk}; otherwise the two forms of the Hall polynomial are not equivalent.
Let $H$ be the variety of (unitary) algebras determined by the polynomial $h$. As in the previous section we shall work with the proper multilinear elements only. Define
\[
f_{kmn} = [x,y]^k [x,y,x^{(m)}, y^{(n)}]; \qquad d_{k,l}(x,y) = [x,y]^k [x,y,x^{(l)}].
\]
Nikolaev proved that the polynomials $f_{kmn}$, $k$, $m$, $n\ge 0$ span the vector space of all proper multilinear polynomials modulo the identity $h$.
Moreover he proved that $\Gamma_n(H)=\oplus M_{k,l}$ where the sum is over all $k$, $l\ge 0$ such that $2k+l+2=n$. The irreducible $S_n$-modules $M_{k,l}$
are generated by the complete linearisations of the polynomials $d_{k,l}$ given above.
But $h(x,y)=[[x,y]^2,x]$ vanishes on $M_{1,1}(E)$. Therefore $\Gamma_n(M_{11}(E))$ is a homomorphic image of $\Gamma_n(H)$. We denote by $J$ the T-ideal of the identities in two variables for $M_{11}(E)$, thus we will have a decomposition $\Gamma_n(M_{1,1})=\oplus M_{k,l}\pmod{J}$. Here the sum is taken over (some of the) $k$ and $l$. Thus we have to check which of the polynomials $d_{k,l}$ are identities for $M_{11}(E)$ and which are not.
\begin{lemma}
The polynomial $d_{k,l}$ is not an identity for $M_{1,1}(E)$ when $k < 2$.
\end{lemma}
\textit{Proof}\textrm{. } Suppose $k=0$, then $d_{0,l}=[x,y,x^{(l)}]$, and $d_{0,l}(C_1,C_2)\ne 0$. Therefore $d_{0,l}$ is not an identity for $M_{11}(E)$. If $k=1$ we have
$d_{1,l}=[x,y][x,y,x^{(l)}]$ and as above $d_{1,l}$ is not an identity for $M_{11}(E)$. \hfill$\diamondsuit$\medskip
\begin{lemma}
\label{dkl}
The polynomials $d_{k,l}$, $k\ge 2$, are identities for $M_{1,1}(E)$. If $k\ge 3$ then $d_{k,l}$ follows from $d_{2,l}$.
\end{lemma}
\textit{Proof}\textrm{. }
Suppose $k\ge 2$, then for every $l\ge 0$ the polynomial $d_{k,l}$ is a product of three commutators. Therefore $d_{k,l}(C_1,C_2)=0$ and consequently $d_{k,l}\in T(M_{11}(E))$. The second statement is immediate.
\hfill$\diamondsuit$\medskip
Thus we have the following corollary.
\begin{corollary}
\label{manypi} All identities in two variables for $M_{11}(E)$ are consequences of $[[x,y]^2,x]$ and the polynomials $d_{2,l} = [x,y]^2[x,y,x^{(l)}]$,
$l\ge 0$.
\end{corollary}
\begin{theorem}
\label{twovar}
All identities in two variables for $M_{11}(E)$ follow from the two polynomials $h=[[x,y]^2,x]$ and $d=[x,y]^3$.
\end{theorem}
\textit{Proof}\textrm{. }
Clearly $h$ and $d$ are identities for $M_{11}(E)$. In order to prove the theorem it suffices to show, according to Lemma \ref{dkl}, that all polynomials $d_{2,l} = [x,y]^2 [x,y,x^{(l)}]$, are consequences of $d$ and $h$. We shall induct on $l$. The base of the induction is $l=0$ when $d_{2,0}=d$. Write then
\begin{eqnarray*}
[x,y]^2[x,y,x^{(l)}] &=& [x,y]^2([x,y,x^{(l-1)}]x -x[x,y,x^{(l-1)}])\\ &=&[x,y]^2[x,y,x^{(l-1)}]x-[x,y]^2x[x,y,x^{(l-1)}].
\end{eqnarray*}
As $[x,y]^2$ commutes with $x$ (and with $y$) we have that $d_{2,l}$ follows from $d_{2,l-1}$ and we are done.
\hfill$\diamondsuit$\medskip
\begin{center}
\textbf{Acknowledgements}
\end{center}
\noindent
We thank the Referee (of a journal) whose valuable and precise comments improved significantly the exposition and made precise several statements. In particular, the inclusion of the complete version of the proof of Proposition 7, and the inclusion of Remark 1 in Section 3 were done following the Referee's suggestions.
|
hep-th/9810001
|
\section{Introduction}
\vspace{1cm}
Lagrangian quantization (LQ) remains one of the most attractive
approaches to quantize gauge theories. Its main advantage is a
direct construction of the quantum effective action avoiding the
long detour of the Hamiltonian approach through canonical
quantization with subsequent integration over momenta in the path
integral, and producing directly the vacuum
expectation value of the S-matrix in the presence of external
sources (see, for example, \cite{GT}). The main difficulty
faced by the implementation of the LQ program lies in the complicated
structure of the
gauge symmetries corresponding to the initial classical action of a theory,
as well as in the ambiguities existing in the choice of possible gauges. It
is well known that gauge theories with closed algebra of
linearly independent generators of gauge transformations -- when considered
in the class of certain admissible (although restricted) gauges -- permit
the quantum effective action $S_{eff}$ to be constructed directly by means
of the Faddeev-Popov (FP) rules \cite{1a}. The overwhelming majority of
physically interesting theories -- such as Yang-Mills theories, theory of
gravity, etc -- belong to the realm of the application of this
quantization method. According to the FP rules, the action $S_{eff}$
depends on an enlarged set of classical fields $\phi$, composed, apart
from the basic classical fields $A$, also by the sets of auxiliary fields
$B$ (Nakanishi-Lautrup fields) related to the gauge-fixing, as well as by
the sets of FP ghost and anti-ghost fields $C,\;\bar{C}$: altogether
they are given by $\phi=(A,B,C,\bar{C})$.
The action $S_{eff}$ constructed according to the
FP rules is invariant under the (global) nilpotent BRST
transformations \cite{2a,3a}. Assigning to each field $\phi$ the
corresponding BRST--source, or anti--field, $\phi^*$, one
observes that the BRST
symmetry results in the nonlinear Zinn-Justin equation \cite{4a}
determining the quantum action
$S$ which ensures complete description of $S_{eff}$.
Attempts to go beyond the scope of the above mentioned
restrictions on the structure
of gauge symmetries of the initial classical action -- as well as beyond the
restrictions on the above mentioned classes of gauge conditions -- have
resulted in a number of new schemes of LQ, based of the
so-called ${\em master \;equations}$ (analogous to the Zinn-Justin
equation).
Their solutions determine the quantum action $S$,
which satisfies a set of
natural boundary conditions related to the initial classical action. One
of the first schemes of this kind was the Batalin-Vilkovisky (BV) method
\cite{5a,6a}, which allows to consider gauge theories with
open gauge algebra. In the case of irreducible theories
\cite{5a}, corresponding to a linearly independent set of generators, the
configuration space $\phi$ of the BV method coincides with that of the FP
approach, whereas in the case of reducible theories \cite{6a} one has to
introduce additional (sometimes infinite) sets of auxiliary and ghost
fields. Afterwards the so-called $Sp(2)$ symmetric scheme has been
developed \cite{7a,8a,9a}, in which the ghost and anti--ghost
fields are treated on
equal footing (in contrast to the BV method, where the anti-ghost fields
appear only at the stage of gauge--fixing), and in which the
(in)finite towers
of auxiliary and ghost fields are classified in terms of irreducible
representations of the $Sp(2)$ group. In fact, such a quantization scheme
realizes a combined BRST -- anti-BRST symmetry principle (the latter
symmetry was first discovered in \cite{10a,11a} for Yang-Mills
theories in the framework of the FP method). Within the $Sp(2)$ scheme, the
quantum action $S$ depends on an extended set of field variables, which
includes, apart from the variables $\phi$
also three
sets of anti--fields, namely, $Sp(2)$--doublets $\phi^*_a,\;a=1,2$ (sources
of BRST and anti-BRST transforms) and $Sp(2)$--scalars $\bar{\phi}$
(sources of mixed transforms).
In a recent work \cite{12a} Batalin and Marnelius presented a
new possible generalization of the $Sp(2)$ method -- related to an additional
extension of the configuration space of the quantum action $S$ -- and aimed
at providing equal treatment for all anti--fields of the $Sp(2)$ approach.
In particular, it is suggested to consider the fields $\pi^a$ (which
parametrize gauges in the path integral of the $Sp(2)$ method) as
anticanonically conjugated to the anti--fields $\bar{\phi}$, with the
corresponding redefinition of the extended anti--brackets. Thus, along
with the $Sp(2)$--singlet $\bar \phi$, the theory also contains an
$Sp(2)$--doublet
$\pi^a$, which thus accounts for the fact that the corresponding scheme
is referred to as triplectic quantization (TQ).
An essential original point
of the TQ scheme consists in dividing the entire task of constructing the
quantum effective action $S_{eff}$ into the following two steps: first,
the construction of the quantum action $S$, and second the construction
of the corresponding gauge-fixing functional.
Either problem is solved by means of an appropriate master equation.
Despite considering these new ideas as very promising,
as to their concrete realization we propose a different,
modified scheme of the TQ, which -- especially from some
geometrical viewpoint --
changes the meaning of the latter. Namely, remaining in the same
configuration space of fields, and accepting the idea of a separate treatment
of the two above mentioned actions, we propose to change both systems of
master equations by using a new set of two $Sp(2)$--doublets of generating
operators: $V^a$ and $U^a$. Such a modification is inspired by our experience
in the superfield formulation of the $Sp(2)$ method \cite{13a}, in which the
above operators acquire the geometrical interpretation of the generators of
(super)transformations in a superspace spanned by fields and anti--fields.
In
this approach, the first master equation, determining the quantum action $S$,
is defined by means of the operators $V^a$, whereas the other master equation,
determining the gauge fixing functional $X$,
is defined by means of the operators
$U^a$. As in the original TQ scheme, we may expect that the generating
functional of Green's functions does not depend on the choice of gauge. It is
important to emphasize that within the modified TQ scheme the entire
information contained in the initial classical action of the theory is
conveyed to the quantum effective action via the corresponding boundary
conditions. At the same time, the classical action obeys the first modified
master equation in complete analogy with all previously known schemes of LQ.
The original TQ scheme gives no explicit relation to the initial classical
action. If one assumes that such a classical action occurs, as usual,
in the boundary condition to the solution of the master equation (with
vanishing auxiliary fields and quantum corrections), then this classical
action does not obey the master equation.
The purpose of this paper is to elaborate a complete description of TQ
scheme -- within the framework of the above mentioned
modifications -- re\-ve\-al\-ing, at the same time, the points which make
such a description similar to, or different from, the original TQ scheme.
\section{Main Definitions}
Let us denote by $\phi^A$, $\varepsilon(\phi^A)\equiv\varepsilon_A$,
the complete set of fields which span the configuration space
corresponding to a certain initial gauge theory. The explicit structure
of the fields $\phi^A$ is not essential for the purposes of the following
treatment (for details, see the original papers \cite{5a,6a}).
According to refs.~\cite{7a,12a} we further introduce the set of antifields
$\phi^*_{Aa}$, $\varepsilon(\phi^*_{Aa})=\varepsilon_A+1$; $\bar{\phi}_A$,
$\varepsilon(\bar{\phi}_A)=\varepsilon_A$, accompanied \cite{12a} by the set
of fields $\pi^{Aa}$, $\varepsilon(\pi^{Aa})=\varepsilon_A+1$. In the entire
space of variables $\phi^A$, $\phi^*_{Aa}$, $\pi^{Aa}$, $\bar{\phi}_A$
we define extended antibrackets, given, for any two functionals $F$, $G$,
by the rule
\begin{eqnarray}
(F,G)^a=\frac{\delta F}{\delta\phi^A}\frac{\delta G}{\delta\phi^{*}_{Aa}}
+\varepsilon^{ab}
\frac{\delta F}{\delta\pi^{Ab}}\frac{\delta G}{\delta\bar{\phi}_{A}}
-(F\leftrightarrow G)(-1)^{(\varepsilon(F)+1)(\varepsilon(G)+1)}\;,
\end{eqnarray}
where $\varepsilon^{ab}$ is the constant antisymmetric second-rank tensor,
$\varepsilon^{12}=1$. Eq.~(1) coincides with the definition of the
extended antibrackets given in the method of triplectic quantization
\cite{12a}, as well as with the definition of the extended antibrackets given
in the method of superfield quantization \cite{13a}. Speaking of the
algebraic properties of the extended antibrackets, we will only mention the
generalized Jacobi identities
\begin{eqnarray}
((F,G)^{\{a},H)^{b\}}(-1)^{(\varepsilon(F)+1)(\varepsilon(H)+1)}
+{\rm cycle}\,(FGH)\equiv 0.
\end{eqnarray}
Here and elsewhere the curly brackets stand for symmetrization over
$a$, $b$.
In solving the functional equations determining the effective action we
make use of the operators $\Delta^a$, $V^a$ and $U^a$
\begin{eqnarray}
\Delta^a&=&(-1)^{\varepsilon_A}\frac{\delta_l}{\delta\phi^A}
\frac{\delta}{\delta\phi^{*}_{Aa}}
+(-1)^{\varepsilon_A+1}\varepsilon^{ab}\frac{\delta_l}{\delta\pi^{Ab}}
\frac{\delta}{\delta\bar{\phi}_A},\\
V^a&=&\varepsilon^{ab}\phi^{*}_{Ab}\frac{\delta}{\delta\bar\phi_A}\;,\\
U^a&=&(-1)^{\varepsilon_A+1}\pi^{Aa}\frac{\delta_l}{\delta\phi^A}\;.
\end{eqnarray}
Notice that the operators $\Delta^a$ have already appeared both within the
scheme of triplectic quantization \cite{12a} and, virtually, within the
scheme of superfield quantization \cite{13a}. Even though the operators
$V^a$ in eq.~(4) differ from the corresponding operators of the triplectic
quantization \cite{12a}, they coincide, at the same time, with the operators
applied in the framework of the $Sp(2)$ method \cite{7a}. The use of the
operators $U^a$ in eq.~(5) (an analog of these operators has been introduced
in the method of superfield quantization) exhibits an essentially new
feature as compared to both the $Sp(2)$ method and the TQ \cite{12a}.
One easily establishes the following algebra of the operators (3)--(5):
\begin{eqnarray}
&&\Delta^{\{a}\Delta^{b\}}=0,\\
&&V^{\{a}V^{b\}}=0,\\
&&\Delta^{\{a}V^{b\}}+V^{\{a}\Delta^{b\}}=0,\\
&&U^{\{a}U^{b\}}=0,\\
&&\Delta^{\{a}U^{b\}}+U^{\{a}\Delta^{b\}}=0,\\
&&V^aU^b+U^bV^a=0,\\
&&\Delta^aV^b+V^b\Delta^a+\Delta^aU^b+U^b\Delta^a=0.
\end{eqnarray}
The action of the operators $\Delta^a$ (3) on the product of any two
functionals $F$, $G$,
\begin{eqnarray}
\Delta^a(F\cdot G)=(\Delta^aF)\cdot G+F\cdot(\Delta^a G)
(-1)^{\varepsilon(F)}
+(F,G)^a(-1)^{\varepsilon(F)},
\end{eqnarray}
may serve as an independent definition of the extended antibrackets (1).
The action of each of the operators $\Delta^a$, $V^a$ and $U^a$ (3)--(5)
on the extended antibrackets is given by the rule
($D^a=(\Delta^a,V^a,U^a)$)
\begin{eqnarray}
D^{\{a}(F,G)^{b\}}=(D^{\{a}F,G)^{b\}}-(F,D^{\{a}G)^{b\}}
(-1)^{\varepsilon(F)}.
\end{eqnarray}
Apart from $\Delta^a$, $V^a$, we also introduce the operators
\begin{eqnarray}
\bar{\Delta}^a&\equiv&\Delta^a+\frac{i}{\hbar}V^a,\\
\tilde{\Delta}^a&\equiv&\Delta^a-\frac{i}{\hbar}U^a.
\end{eqnarray}
From eqs. (6)--(12) it follows that the algebra of these operators has
the form
\begin{eqnarray}
&&\bar{\Delta}^{\{a}\bar{\Delta}^{b\}}=0,\\
&&\tilde{\Delta}^{\{a}\tilde{\Delta}^{b\}}=0,\\
&&\bar{\Delta}^{\{a}\tilde{\Delta}^{b\}}+
\tilde{\Delta}^{\{a}\bar{\Delta}^{b\}}=0.
\end{eqnarray}
\section{Generating Functional, Extended BRST \\Symmetry and Gauge
Independence}
Let us denote by $S=S(\phi,\phi^*,\pi,\bar{\phi})$ the quantum action,
corresponding to the initial classical theory with the action $S_0$, and
defined as a solution of the following master equations:
\begin{eqnarray}
\frac{1}{2}(S,S)^a+V^aS=i\hbar\Delta^aS,
\end{eqnarray}
with the standard boundary condition
\begin{eqnarray}
\left.S\right|_{\phi^*=\bar{\phi}=\hbar=0}=S_0.
\end{eqnarray}
Eq.~(20) can be represented in the equivalent form
\begin{eqnarray}
\bar{\Delta}^a\exp\left\{\frac{i}{\hbar}S\right\}=0.
\end{eqnarray}
Let us further define the vacuum functional as the following functional
integral:
\begin{eqnarray}
Z_X=\int d\phi\,d\phi^*d\pi\,d\bar{\phi}\,d\lambda\exp\left\{
\frac{i}{\hbar}\left(S+X+\phi^*_{Aa}\pi^{Aa}\right)\right\},
\end{eqnarray}
where $X=X(\phi,\phi^*,\pi,\bar{\phi},\lambda)$ is a bosonic functional
depending also on the new variables $\lambda^A$,
$\varepsilon(\lambda)=\varepsilon_A$, which serve as gauge-fixing
parameters. We require that the functional $X$ satisfies
the following master equation:
\begin{eqnarray}
\frac{1}{2}(X,X)^a-U^aX=i\hbar\Delta^aX,
\end{eqnarray}
or, equivalently,
\begin{eqnarray}
\tilde{\Delta}^a\exp\left\{\frac{i}{\hbar}X\right\}=0.
\end{eqnarray}
Notice that the generating equations determining the quantum action $S$ in
eq.~(20) (or (22)) and the gauge-fixing functional $X$ in eq.~(24)
(or (25)) differ---along with the vacuum functional $Z$ in
eq.~(23)---from the corresponding definitions applied in the method of
TQ \cite{12a}.
One can easily obtain the simplest solution of eq.~(24) (or eq.~(25))
determining the gauge-fixing functional $X$
\begin{eqnarray}
X&=&\left(\bar{\phi}_A-\frac{\delta F}{\delta\phi^A}\right)\lambda^A-
\frac{1}{2}\varepsilon_{ab}U^aU^bF=\nonumber\\
&=&\left(\bar{\phi}_A-\frac{\delta F}{\delta\phi^A}\right)\lambda^A
-\frac{1}{2}\varepsilon_{ab}\pi^{Aa}{\frac {\delta^2 F}{\delta\phi^A
\delta\phi^B}}\pi^{Bb},
\end{eqnarray}
where $F=F(\phi)$ is a bosonic functional depending only on the fields
$\phi^A$. As a straightforward exercise, one makes sure that the
functional $X$ in eq.~(26) does satisfy eq.~(24). If we further demand
that the quantum action $S$ does not depend on the fields $\pi^A$, then
the functional (26) becomes exactly the vacuum functional of the $Sp(2)$
quantization scheme \cite{7a,8a}.
Let us consider a number of properties inherent in the present scheme of
triplectic quantization, i.e. modified according to eq.~(20)--(25).
In the first place, the vacuum functional (23) is invariant under the
following transformations:
\begin{eqnarray}
\delta\Gamma=(\Gamma,-S+X)^a\mu_a+\mu_a(V^a+U^a)\Gamma,
\end{eqnarray}
where $\mu_a$ is an $Sp(2)$ doublet of constant anticommuting parameters,
and $\Gamma$ stands for any of the variables $\phi$, $\phi^*$, $\pi$,
$\bar{\phi}$. Eq.~(27) defines the
transformations of extended BRST symmetry, realized on the space of the
variables $\phi$, $\phi^*$, $\pi$, $\bar{\phi}$. In the particular case,
corresponding to the gauge-fixing boson chosen as in eq.~(26), we have
\begin{eqnarray}
\delta\phi^A&=&-\left(\frac{\delta S}{\delta\phi^*_{Aa}}
-\pi^{Aa}\right)\mu_a,\\
\delta\phi^*_{Aa}&=&\mu_a\left(\frac{\delta S}{\delta\phi^A}
+\frac{\delta^2F}{\delta\phi^A\delta\phi^B}\lambda^B
+\frac{1}{2}(-1)^{\varepsilon_A}\varepsilon_{bc}\pi^{Bb}
\frac{\delta^3F}{\delta\phi^B\delta\phi^A\delta\phi^C}\pi^{Cc}\right)\!,\\
\delta\pi^{Aa}&=&\varepsilon^{ab}\left(
\frac{\delta S}{\delta\bar{\phi}_A}
-\lambda^A\right)\mu_b,\\
\delta\bar{\phi}_A&=&\mu_a\varepsilon^{ab}\left(
\frac{\delta S}{\delta\pi^{Ab}}+\phi^*_{Ab}\right)
+\mu_a\frac{\delta^2F}{\delta\phi^A\delta\phi^B}\pi^{Ba}\,.
\end{eqnarray}
Consider now the question of gauge dependence in the case of the vacuum
functional $Z$, eq.~(23). Any admissible variation $\delta X$ should satisfy
the equations
\begin{eqnarray}
(X,\delta X)^a-U^a\delta X=i\hbar\Delta^a\delta X.
\end{eqnarray}
It is convenient to consider an $Sp(2)$ doublet of operators
$\hat{S}^a(X)$, defined by the rule
\begin{eqnarray}
(X,F)^a\equiv\hat{S}^a(X)\cdot F,
\end{eqnarray}
and possessing the properties
\begin{eqnarray}
\hat{S}^{\{a}(X)\hat{S}^{b\}}(X)=\hat{S}^{\{a}\left
(\frac{1}{2}(X,X)^{b\}}\right),
\end{eqnarray}
which follow from the generalized Jacobi identities (2). Eq.~(32) can be,
consequently, represented in the form
\begin{eqnarray}
\hat{Q}^a(X)\delta X=0,
\end{eqnarray}
where we have introduced an $Sp(2)$ doublet of operators $\hat{Q}^a$,
defined by the rule
\begin{eqnarray}
\hat{Q}^a(X)=\hat{S}^a(X)-i\hbar\tilde{\Delta}^a.
\end{eqnarray}
With allowance for eq.~(24) the operators $\hat{Q}^a$ form a set of
nilpotent anticommuting operators, i.e.
\begin{eqnarray}
\hat{Q}^{\{a}(X)\hat{Q}^{b\}}(X)=0.
\end{eqnarray}
By virtue of eq.~(37), any bosonic functional of the form
\begin{eqnarray}
\delta X=\frac{1}{2}\varepsilon_{ab}\hat{Q}^a(X)\hat{Q}^b(X)\delta Y,
\end{eqnarray}
with an arbitrary bosonic functional $\delta Y$, is a solution of eq.~(35).
Moreover, by analogy with the theorems proved in ref.~\cite{9a}, one
establishes the fact that any solution of eq.~(35)---vanishing when all the vari
in $\delta X$ are equal to zero---has the form (38), with a certain
bosonic functional $\delta Y$. In the particular case of the gauge
functional $X$ (26), its variation $\delta X$ can be easily represented
in the form of eq.~(38), i.e.
\begin{eqnarray}
\delta X=-\frac{\delta(\delta F)}{\delta\phi^A}\lambda^A
-\frac{1}{2}\varepsilon_{ab}\pi^{Aa}{\frac{\delta^2(\delta F)}
{\delta\phi^A\delta\phi^B}}\pi^{Bb}
=-\frac{1}{2}\varepsilon_{ab}\hat{Q}^a(X)\hat{Q}^b(X)\delta F
\end{eqnarray}
with $\delta Y=-\delta F$.
Let us denote by $Z_X\equiv Z$ the value of the vacuum functional (23)
corresponding to the gauge condition chosen as a functional $X$.
In the vacuum functional $Z_{X+\delta X}$ we first make the change of
variables (27), with $\mu_a=\mu_a(\Gamma,\lambda)$, and then,
accompanying it with a subsequent change of variables
\begin{eqnarray}
\delta\Gamma=(\Gamma,\delta Y_a)^a,\;\;\;\varepsilon(\delta Y_a)=1,
\end{eqnarray}
with $\delta Y_a=-i\hbar\mu_a(\Gamma,\lambda)$, we arrive at
\begin{eqnarray}
Z_{X+\delta X}=\int d\phi\,d\phi^*d\pi\,d\bar{\phi}\,d\lambda
\exp\left\{\frac{i}{\hbar}\bigg(S+X+\delta X+\delta X_1
+\phi^*_{Aa}\pi^{Aa}\bigg)\right\},
\end{eqnarray}
In eq.~(41) we have used the notation
\begin{eqnarray}
\delta X_1=2\bigg((X,\delta Y_a)^a-U^a\delta Y_a-i\hbar\Delta^a\delta
Y_a\bigg)=2\hat{Q}^a(X)\delta Y_a.
\end{eqnarray}
Let us choose the functional $\delta Y_a$ in the form
\begin{eqnarray}
\delta Y_a=\frac{1}{4}\varepsilon_{ab}\hat{Q}^b\overline{\delta Y},\;\;\;
\varepsilon(\overline{\delta Y})=0.
\end{eqnarray}
Then, representing $\delta X$ as in eq.~(38), and identifying
$\delta Y=-\overline{\delta Y}$, we find that
\begin{eqnarray}
Z_{X+\delta X}=Z_X,
\end{eqnarray}
i.e. the vacuum functional (and hence, by virtue of the equivalence
theorem \cite{14a}, also the $S$ matrix) does not depend on the choice
of gauge. Note that in the particular case (39) we have
$\overline{\delta Y}=\delta F$.
\section{Concluding Remarks}
The reader may profit by considering the original \cite{12a} version
of TQ as compared to the modified scheme,
proposed in the this paper. Thus, both versions are based on extended
BRST symmetry. Both versions apply the vacuum functional and the $S$
matrix not depending on the choice of gauge, while admitting of gauges
which reproduce the results of the $Sp(2)$ symmetric quantization. Both
versions implement the idea of separate treatment of the quantum action
and the gauge-fixing functional, based each on appropriate master
equations. The principal distinctions concern a different form
of these equations as well as a different form of the vacuum
functional. The modification of the generating equations \cite{12a} permits
incorporating the information contained in the initial classical action by
means of the corresponding boundary conditions. In contrast to the original
version \cite{12a}, the classical action provides a solution of the modified
master equation. Thus, one establishes a connection with the previous
schemes of LQ. In particular, one easily reveals the
fact of equivalence with the $Sp(2)$ quantization, by means of explicit
realization of the corresponding class of boundary condition. In the original
version of TQ, however, these questions still remain open. Another
distinction of the two TQ schemes is connected with the explicit structure of
the corresponding master equations. Thus, the original version \cite{12a} of
TQ defined the generating equations for the quantum action and the vacuum
functional, using the operators
\begin{eqnarray}
V_{\rm BM}^a=\frac{1}{2}\left(
\varepsilon^{ab}\phi^{*}_{Ab}\frac{\delta}{\delta\bar\phi_A}
+\pi^{Aa}(-1)^{\varepsilon_A+1}\frac{\delta_l}{\delta\phi^A}\right)
=\frac{1}{2}(U^a+V^a).
\end{eqnarray}
The use of the generating equations determining the quantum action with
the help of the operators $V_{\rm BM}^a$ leads to the following
characteristic feature of the triplectic quantization \cite{12a}: the
classical action of the initial theory, defined as a limit of the quantum
action at $\hbar\to 0$ and $\phi^*=\bar{\phi}=\pi=0$, does not satisfy the
generating equations of the method. In turn, the proofs of the
existence theorems for the generating equations in all known methods
of LQ are based on the fact that the initial
classical action is a solution of the corresponding master equations.
Moreover, form the viewpoint of the superfield quantization \cite{13a},
which applies operators $V^a$, $U^a$, whose component representation is
\begin{eqnarray}
V^a&=&\varepsilon^{ab}\phi^{*}_{Ab}\frac{\delta}{\delta\bar\phi_A}
-J_A\frac{\delta}{\delta\phi^{*}_{Aa}},\nonumber\\
\\
U^a&=&(-1)^{\varepsilon_A+1}\pi^{Aa}
\frac{\delta_l}{\delta\phi^A}
+ (-1)^{\varepsilon_A}\varepsilon^{ab}\lambda^A
\frac{\delta_l}{\delta\pi^{Ab}}\nonumber
\end{eqnarray}
(with $J_A$ being the sources to the fields $\phi^A$), the operators (45)
have no precise geometrical meaning, whereas the $V^a$ and $U^a$ in
eq.~(46) serve as generators of supertranslations---in superspace spanned
by superfields and superantifields---along additional (Grassmann)
coordinates. In turn, the operators $V^a$ (4) and $U^a$ (5), applied in
this paper, can be considered as limits (at $J_A=0$, $\lambda^A=0$) of the
operators (46), which possess a clear geometrical meaning. The present
modified scheme of triplectic quantization enjoys every attractive feature
of the quantization \cite{12a}: the theory possesses extended BRST
transformations; the vacuum functional and the $S$ matrix do not depend on
the choice of the gauge-fixing functional; there exists such a choice
of the gauge-fixing functional and solutions of the generating
equations that reproduces the results of the $Sp(2)$ method.
\section*{Acknowledgments}
The work of one of the authors (PML) has been supported by the Russian
Foundation for Basic Research (RFBR), project 96--02--16017, as well as
by grant INTAS 96-0308 and by grant RFBR-DFG 96--02--00180. PML
appreciates the kind hospitality extended to him by the Center of
Advanced Study (NTZ) of Leipzig University and by the Institute
of Physics of the University of S\~{a}o Paulo. DMG thanks
Brazilian Foundation CNPq and FAPESP for partial support.
|
cond-mat/9810028
|
\section{Introduction}
Reactive turbulent flow is important in a variety of natural
processes, ranging from the production of smog in the
atmosphere\cite{Bilger} to the feeding habits of certain oceanic
creatures\cite{Kiorboe}. The mixing effects of turbulence are put to
use in a variety of engineering applications, including combustion
reactors, fluid catalytic cracking units, and polymerization
reactors. The behavior of reactive turbulent flow is usually
analyzed, numerically or analytically, with the continuum
reaction/transport equations \cite{Bilger,Hill}. These mean-field
equations fail, however, in two dimensions at low densities of
reactants. This is understood in a general way,
since the upper critical dimension for bimolecular reactions,
even in the absence of turbulence, is two \cite{Peliti,Lee2,Deem1,Deem2}.
A renormalization group treatment of a model of
reactive turbulent flow has recently predicted a regime of
``superfast'' reactivity for the $A + A \to \emptyset$ reaction in
two dimensions \cite{Deem4}. The superfast reactivity occurs when a
certain degree of quenched potential disorder is added to the
turbulent system. In the superfast regime, the reactant concentration
is predicted to decay more quickly than in a well-mixed reaction:
\emph{i.e.}\
$c(t) \sim a/t^{1+x}$ at long times, with $x>0$. This rapid decay occurs
due to a subtle combination of the effects of turbulent transport,
trapping by disorder, and reaction. It shows up in the
analytical calculations as a renormalization of the effective
reaction rate. It was anticipated that this result may have
technological implications for certain thin-film reactors or fluidized
MEMS devices.
The renormalization group calculations were carried out with a
statistical model of turbulence, in the same spirit as in treatments
of the turbulent transport of passive scalars
\cite{Fisher,Kravtsov1,Majda2}. A review of this approach to the
dynamics of a passive scalar can be found in \cite{Bouchaud3}. The
statistics of the turbulent velocity field in the model
were chosen to reproduce
the correct transport properties at long times. There was freedom
to choose either a time-dependent or time-independent velocity field,
and the time-independent choice was made. This choice corresponds to
G.\ I.\ Taylor's hypothesis of frozen turbulence.
In this paper, we perform further studies to confirm the
existence of the superfast reaction regime.
In Section 2, we describe a
method for simulating reactive turbulent flow with random,
time-independent velocity fields. In Section 3, we present and
discuss our computational results. The existence of the superfast
regime is verified, and additional features arising from
higher-loop
contributions in the renormalization group are found. In Section 4,
we present a renormalization group treatment of reactive
turbulent flow with a more realistic,
time-dependent model of the
velocity field. When the appropriate type of
potential disorder is included, the superfast regime is found to
persist. We conclude in Section 5.
\section{Simulation Methodology}
We consider the reaction
\beq{1}
A+A
~{\mathrel{\mathop{\to}\limits^{\lambda}}}~ \emptyset
\end{equation}
occurring in a model of a two-dimensional, turbulent fluid. In the
absence of reaction, the $A$ particles are advected by the fluid
streamlines. This chaotic motion is superimposed upon the natural
Brownian motion of the particles. In addition, the particles
experience a force due to the spatially-varying, random potential. In
the presence of reaction, two $A$ particles react with the conventional
reaction rate constant $\lambda$ when in contact.
We consider this reaction to proceed
on a square $N \times N$ lattice, both for
computational convenience and to allow a direct comparison with the
field-theoretic results. How the configuration of reactants on the
lattice changes with time is specified by a master equation. The
master equation defines transition rates for all of the possible
changes to a given configuration of reactants on the lattice:
\bey{2}
\fl
\frac{\partial P(\{n_i\},t)}{\partial t} = \sum_{ij}
[\tau_{ji}^{-1} (n_j +1)P(...,n_i -1,n_j +1,...,t)
- \tau_{ij}^{-1} n_i P]
\nonumber \\
+ \frac{\lambda}{2 h^2} \sum_i \left[(n_i+2)(n_i +1)
P (...,n_i +2,...,t)
-n_i (n_i -1)P \right] \ ,
\end{eqnarray}
where $n_i$ is the number of $A$ particles on lattice site $i$,
and $h$ is the lattice spacing. The rate of
hopping from lattice site $i$ to lattice site $j$ is given by
$\tau_{ij}^{-1}$, which will be defined below. The summation over
$i$ is over all sites on the lattice, and the summation over $j$ is
over all nearest neighbors of site $i$. The particles are initially
placed at random on the lattice, with average density $n_0$. The
initial concentration at any given site is, therefore, a Poisson random
number:
\bey{3}
P(n_i) =
\frac{[n_0 h^2]^{n_i}}{n_i !}
{\rm e}^{-n_0 h^2}
\nonumber \\
\langle n_i /h^2 \rangle = n_0 \ .
\end{eqnarray}
The turbulence and potential disorder enter this master equation
through the hopping rates $\tau_{ij}^{-1}$. These rates are chosen
so that the master equation reduces to the conventional transport
equation when there is no reaction:
\beq{4}
\frac{\partial c}{\partial t} = D \nabla^2 c + \beta D \nabla
\cdot \left[ c \nabla u - c \nabla \times \phi\right] \ ,
\end{equation}
where $c(\bi{r}, t)$ is the concentration of the $A$ particles, $D$ is the
diffusion constant, $\beta = 1/(k_\mathrm{B} T)$ is the inverse
temperature, $u(\bi{r})$
is the random potential, and $\phi (\bi{r})$ is proportional to
the random stream function. A particularly simple form for the
transfer rate from position $i=\bi{ r}$ to the position $j=\bi{ r} +
\Delta \bi{ r}$ is
\beq{5}
\tau_{ij}^{-1} = \frac{D}{h^2} [ 1 - \beta \Delta \bi{ r} \cdot (\nabla u
- \nabla \times \phi)/2] \ .
\end{equation}
When equation (\ref{5}) is used in equation (\ref{2}), one finds via
a Taylor series expansion in $h$ that the conventional transport equation
(\ref{4}) is reproduced. This form of the transfer rates (\ref{5}) is the
one conventionally used
in theoretical calculations \cite{Deem1}.
In our simulations we choose, instead, to use the form
\beq{6}
\tau_{ij}^{-1} = \frac{D}{h^2} \exp\{\beta(u_i - u_j)/2 \}
\exp \{\beta (\phi_{j'} - \phi_i)/2 \} \ .
\end{equation}
Here $j$ and $j'$ are nearest neighbors of site $i$. The identity of site
$j'$ is derived from site $j$ by a counter-clockwise rotation of
$\pi/2$ about site $i$. This form leads to a transition rate in the
simulation that is
bounded and non-negative, in contrast to the simpler form of equation (\ref{5}).
A Taylor series expansion of equation (\ref{6}) leads directly to the
simpler equation (\ref{5}). Since both of these forms are equivalent in the
limit of a small lattice spacing, they are expected to lead
to identical scaling in the long-time regime.
Following the usual statistical approach to turbulence,
we take the stream function and potential to be Gaussian random variables
with correlation functions given by
\bey{7}
\hat \chi_{\phi \phi} ( \bi{ k}) = \frac{\sigma}{k^{2+y}}
\nonumber \\
\hat \chi_{u u} ( \bi{ k}) = \frac{\gamma}{k^{2+y}} \ ,
\end{eqnarray}
where the Fourier transform of a function in two dimensions
is given by
$\hat f(\bi{ k}) = \int \rmd^2 \bi{ r} f(\bi{ r}) \exp( \rmi
\bi{ k} \cdot \bi{ r})$.
The random stream function and potential are generated first in
Fourier space, where the values at different wave vectors are
independent Gaussian random variables, and then converted to real
space with a fast Fourier transform \cite{Deem5}. Periodic boundary
conditions are used on the lattice. The quality of the random number
generator is of importance when simulating systems with long-ranged
correlations \cite{Deem5}. We use a sum of three linear
congruential generators method \cite{Byte}.
Isotropic, fully-developed turbulence is modeled by $y = 8/3$ and
$\gamma = 0$. Developing turbulence is modeled by smaller, positive
values of $y$. Potential disorder of the form we consider here could
arise in a reaction between ionic species confined to a thin film of
fluid between spatially-addressable electrodes or media with
quenched, charged disorder. The ionic species would move in the
random potential generated by the electrodes or quenched disorder.
The electrodes or quenched disorder could be devised so as to
reproduce the correlation function $ \chi_{u u}(r)$.
The master equation (\ref{2}) can be exactly solved by a Markov
Poisson process \cite{Doering}. In such a process,
we consider the motion of discrete
$A$ particles on the lattice. For any given configuration
of reactants at time $t$,
there are $4 \sum_i n_i$ possible hopping events. Each occurs with a
rate given by equation (\ref{6}). In addition, there are $\sum_i n_i
(n_i-1)/2$ possible reaction events on the lattice. Each of these
occurs with the rate $\tau_\mathrm{rxn}^{-1} = \lambda / h^2$.
The Markov process is started by initially placing the reactants on
the lattice at random with average density $n_0$. Each step of the
Markov process consists of randomly picking one of the possible
reaction or diffusion events and incrementing time appropriately.
The probability of event $\alpha$ occurring, out of all the possible
diffusion and reaction moves, is
\beq{8}
P({\rm event~ }\alpha) = \frac{\tau_\alpha ^{-1}}{\sum_\gamma \tau_\gamma
^{-1}} \ .
\end{equation}
After performing the chosen event, time is incremented by
\beq{9}
\Delta t = \frac{-\log \zeta}{\sum_\gamma \tau_\gamma^{-1}} \ ,
\end{equation}
where $\zeta$ is a random number uniformly distributed between 0 and 1.
The Markov chain is continued until zero or one reactants remain on
the lattice, at which point no more reaction events will occur.
This Markov process, when averaged over initial conditions and
trials, exactly reproduces the predictions of the master equation.
Averages over the statistics of the turbulence and potential disorder
are taken by directly averaging over sufficiently many
instances of the random fields.
Since the
asymptotic scaling regime is reached only for long times, we
make two approximations to facilitate the computations. First, we
set the reaction rate to infinity. That is, when a particle moves to
a site with another particle already present, the reaction occurs
immediately. Physically, we do not expect this to modify the
long-time decay law, since at long times the reaction will be in
a transport-limited regime. Indeed, the renormalization group prediction
for the decay exponent is independent of the reaction rate
\cite{Deem4}. The predicted prefactor also has a well-defined value
in the infinite reaction rate limit. Second, we assume that each
particle on the lattice is equally likely to undergo a hopping event:
\beq{10}
P({\rm moving~particle} ~ \alpha) = \frac{1}{n} \ ,
\end{equation}
where $n=\sum_i n_i$ is the total number of particles on the lattice.
After the particle to move is chosen, the probabilities of each of the
four possible hopping events are derived from
equation (\ref{6}). In particular, another uniform random
number is generated for comparison against the four different hop
probabilities:
\beq{11}
P({\rm hop} ~ i) = \frac{\tau_i^{-1}}{\sum_j \tau_j^{-1}} \ .
\end{equation}
After the chosen hop is performed, time is incremented by
$\Delta t = N^2/(n \sum_{ij} \tau_{ij}^{-1})$. The uniform
choice of particles to move and the approximate time incrementation
are not expected to influence the long-time exponents. In
particular, these
approximations are exact in the long-time limit if the hopping
rates of equation (\ref{5}) are used.
\section{Simulation Results and Discussion}
The renormalization group calculations make a prediction for the
concentration decay exponent, the $\alpha$ in $c(t) \sim a
t^{-\alpha}$ \cite{Deem4}. The predictions depend on the properties
of the turbulence and potential disorder through the
parameters $y$, $\sigma$, and
$\gamma$. The predictions for several values of
$y$ are shown in figure \ref{fig1}.
The decay exponent initially rises with increasing potential disorder
and eventually decreases. The maximum reaction rate of $\alpha = 1 +
y/(6-y)$ occurs for $\sigma = 3 \gamma$. The decay rate depends on
$\sigma$ and $\gamma$ only through the combination $\gamma/\sigma$.
These predictions come from a one-loop renormalization group
calculation, and they are strictly valid only for small $y$. In
addition, the predictions
should be more accurate for small $\gamma$, because it
is known that higher-loop corrections lead to a modification of the
flow diagram for large $\gamma$.
We find that our Poisson process efficiently solves the master
equation for values of $y$ near unity. For smaller values of $y$,
the renormalization of the effective reaction rate occurs
slowly. This means that the predicted asymptotic scaling occurs
only at long times, times longer than we can reach in our
simulation. For larger values of $y$, significant lattice effects
occur, due to our choice of transition rates, equation (\ref{6}), and
correlation functions, equation (\ref{7}).
Shown in figure \ref{fig2} are the decay exponents observed in our
simulation for $y=3/4$. The decay exponents observed for $y=1$ and
$y=5/4$ are shown in figures \ref{fig3} and \ref{fig4},
respectively. Each of the data points in the figure is an average
over three different runs on three different instances of the
turbulence and potential disorder. The standard deviations estimated
from these three runs are shown as the error bars.
We made a few canonical choices for constants in our simulation. So
as to reach the asymptotic scaling regime, we used $4096 \times 4096$
square lattices. Simulations on $2048 \times 2048$ lattices gave
similar results.
Finite size effects should be most noticeable for small values of
$y$, $\sigma$, or $\gamma$, since in that regime the renormalization of
the effective reaction rate is most slow. Indeed,
finite size effects only appear to be present
in figure \ref{fig2} for small $\gamma/\sigma$, where
the slow renormalization of $\lambda(l)$ leads to an observed
decay exponent that is below the correct asymptotic value of unity.
While additional finite size effects could be present, the agreement
at small $\gamma/\sigma$ among
curves for different $\sigma$ but same $\gamma/\sigma$ and $y$, and the
agreement between the results for $2048 \times 2048$ and
$4096 \times 4096$ lattices, argues against this.
We used a lattice spacing of $h=1$, which we can
enforce by a spatial rescaling, and a diffusion constant of $D=1$,
which we can enforce by a temporal rescaling. The long-time decay
exponent is not affected by these rescalings. We collected the
time-dependent concentration data on the lattice by binning the
results into a
histogram with a temporal bin width of $\delta t = 1$. We continued each
simulation until the concentration was so low, $c \approx
(\mathrm{const}) /N^2$, that lattice effects were obvious.
We show in figure \ref{fig6} the simulation results for a typical run.
The average of the slopes determined in three such runs
gives one of the data points in figures \ref{2}-\ref{4}.
Since the renormalization group predictions depend only on
$\gamma/\sigma$, we fixed $\sigma$ and varied $\gamma$ for each
simulation at a fixed value of $y$. Under these conditions, the
effective strength of turbulence is expected to flow to its fixed
point value, $\sigma \to \sigma^* \approx 2 \pi y (2 \pi / h)^y$ \cite{Deem4}.
In each
of the figures \ref{fig2}-\ref{fig4} we show results for three different
``bare''
values of $\sigma$: $\sigma=\sigma^*$, $\sigma=\sigma^*/2$,
and $\sigma=3 \sigma^*/2$.
We use a $\ln c(t)$ \emph{versus} $\ln t$ plot to obtain the
decay exponent. This plot has many more data at large $\ln t$,
which leads to an over-weighting of the long-time
regime. To counteract this effect, we use data exponentially spaced
in time. We exclude both short-time data, which are not in the
asymptotic regime, and long-time data, which show finite-size
effects. Use of exponentially spaced data allows one to obtain a
reliable estimate of the error in the exponent determined for a given
realization of the disorder. We find that this statistical error is
smaller than the systematic deviation that occurs for
each different realization of the disordered streamlines
and potential. The error bars shown in the
figures encompass both the statistical and the systematic errors.
We see that there is a general agreement between the simulation
results and the renormalization group predictions. In particular,
the superfast reaction regime, $\alpha > 1$, is observed. This was a
dramatic analytical prediction, and confirmation of this regime is
significant. The exponent is observed to
reach roughly the maximum value
of $1 + y/(6-y)$ predicted by theory.
The agreement between the simulation results and the one-loop
renormalization group predictions is better for smaller $y$.
For small $y$, the location of the peak in reactivity is
trending towards the predicted value of $\gamma/\sigma = 1/3$. For
finite values of $y$, the location of the peak is shifted to the
left. Indeed, the whole curve is shifted to the left for finite $y$.
One consequence of this is that the observed decay exponent decreases below
unity for $\gamma/\sigma$ close to, but less, than one, in contrast to the
one-loop predictions.
The agreement between the simulation results and the one-loop
predictions is also better for small $\gamma$. For small
$\gamma/\sigma$, the simulation results for different values of
$\sigma$ fall on the same curve. This is the universal curve
predicted at one-loop, albeit compressed to the left.
For large $\gamma$, the simulation results
depend on both $\gamma$ and $\sigma$, not simply on the ratio
$\gamma/\sigma$. This dependence is presumably due to corrections that
would enter in a higher-loop calculation.
\section{Renormalization Group Treatment of Dynamic Turbulence}
The simulations, and the previous renormalization group calculations,
were performed for a model of turbulence with streamlines random in
space, but constant in time. For this model, both the analytical and
computational studies predict a regime of superfast reaction in the
presence of potential disorder. This regime was understood to occur
as a general consequence of the interplay among turbulent transport,
trapping by the random potential, and a non-linear dependence of the
reaction rate on local reactant concentration. This regime was not
believed to occur, for example, due to some subtle, artificial interaction
between the quenched potential and quenched streamlines.
Even so, it would be interesting to predict the existence of the
superfast reaction regime for a dynamic model of turbulence. As we
have seen, reliable simulations require large lattices and long
times, even for quenched turbulence. Moreover, simulations were
possible only for an intermediate range of $y$, due to the competing
considerations of minimizing lattice effects and accessing
the asymptotic regime. Renormalization group theory is,
however, a viable approach to studying
reactive turbulent flow in a dynamic model of turbulence.
We, again, use a statistical model of turbulence \cite{Majda2}.
We now assume that the stream function is random in both space
and time:
\beq{12}
\hat \chi_{\phi \phi} ( \bi{ k}, t_1-t_2) = \frac{\sigma}{k^{2+y_\sigma}}
\vert t_1 - t_2 \vert ^{-\rho} \Theta(t_1-t_2) \ ,
\end{equation}
where $\Theta(t)$ is the Heavyside step function. A family of models
for turbulence is generated by varying $\rho$ and $y_\sigma$.
Isotropic turbulence is modeled when $3 y_\sigma - 2 \rho = 8$, at
least for small $\rho$. We include a random potential, as before:
\beq{12a}
\hat \chi_{u u} ( \bi{ k}) = \frac{\gamma}{k^{2+y_\gamma}} \ .
\end{equation}
We will find that the most interesting regime occurs for $y_\gamma
< y_\sigma$, due to the weakening of the random stream function
through decorrelation over time.
As in previous studies \cite{Deem4}, we map the master equation
(\ref{2}) onto a field theory using the coherent state representation
\cite{Peliti,Lee1}. For this operation, we use the transition rates
given by equation (\ref{5}). The random and stream function are
incorporated with the replica trick \cite{Kravtsov1}, using $N$
replicas of the original problem. The concentration of reactants at
time $t$, averaged over the initial conditions, $c(\bi{ r},t)$, is
given by
\beq{13}
c(\bi{ r},t) = \lim_{N \to 0} \langle a(\bi{ r},t) \rangle \ ,
\end{equation}
where the average is taken with respect to $\exp(-S)$, with
\bey{14}
\lo{S =} \sum_{\alpha=1}^N
\int \rmd^2 \bi{ r} \int_0^{t_\mathrm{f}} \rmd t
\bar a_\alpha(\bi{ r},t) \left[
\partial_t - D \nabla^2 + \delta(t)
\right]
a_\alpha(\bi{ r},t)
\nonumber \\
+\frac{\lambda}{2} \sum_{\alpha=1}^N
\int \rmd^2 \bi{ r} \int_0^{t_\mathrm{f}} \rmd t \bigg[
2 \bar a_\alpha(\bi{ r},t)
a_\alpha^2(\bi{ r},t)
+ \bar a_\alpha^2(\bi{ r},t)
a_\alpha^2(\bi{ r},t)
\bigg]
\nonumber \\
-n_0 \sum_{\alpha=1}^N \int \rmd^2 \bi{ r} \bar a_\alpha(\bi{ r},0)
\nonumber \\
-\frac{\beta^2 D^2}{2}
\sum_{\alpha_1,\alpha_2=1}^N
\int \rmd t_1 d t_2 \int_{\bi{ k}_1 \bi{ k}_2 \bi{ k}_3 \bi{ k}_4}
(2 \pi)^2 \delta(\bi{ k}_1+\bi{ k}_2+\bi{ k}_3+\bi{ k}_4)
\nonumber \\
\times
\hat{\bar a}_{\alpha_1}(\bi{ k}_1, t_1)
\hat{ a}_{\alpha_1}(\bi{ k}_2, t_1)
\hat{\bar a}_{\alpha_2}(\bi{ k}_3, t_2)
\hat{ a}_{\alpha_2}(\bi{ k}_4, t_2)
\nonumber \\
\times \bigg[
\bi{ k}_1 \cdot (\bi{ k}_1+\bi{ k}_2)
\bi{ k}_3 \cdot (\bi{ k}_3+\bi{ k}_4)
\hat\chi_{uu}(\vert \bi{ k}_1+\bi{ k}_2\vert)
\nonumber \\
+
\bi{ k}_1 \times \bi{ k}_2~
\bi{ k}_3 \times \bi{ k}_4
\hat\chi_{\phi \phi}(\vert \bi{ k}_1+\bi{ k}_2\vert, t_2-t_1) \bigg] \ .
\end{eqnarray}
The notation
$\int_\bi{ k}$ stands for $\int \rmd^2 \bi{ k} / (2 \pi)^2$.
The upper time limit in the action is required only to satisfy
$t_\mathrm{f} \ge t$.
Using renormalization group theory, we can derive the flow equations
for this model. We find that the turbulence contributes directly to
the dynamical exponent, and therefore to the transport properties,
but not directly
to the effective reaction rate. The potential disorder contributes
directly to both the dynamical exponent and the effective reaction
rate. Finally, the dynamical exponent contributes indirectly to the effective
reaction rate as well.
We define $g_\sigma =
\beta^2 \sigma \Lambda ^{2\rho - y_\sigma}D^\rho \Gamma(1-\rho)/ (4 \pi)$
and
$g_\gamma = \beta^2 \gamma \Lambda^{- y_\gamma} / (4 \pi)$,
where $\Lambda = 2 \pi /h$ is the cutoff in Fourier space, and
$\Gamma(x)$ is the standard Gamma function. The precise definition
of $g_\sigma$ depends on the regularization of $\chi_{\phi \phi}$
at $t=0$.
At one loop, we find the flow equations to be
\bey{15}
\frac{d \ln n_0}{d l} = 2
\nonumber \\
\frac{d \ln \lambda}{d l} = -
\frac{\lambda}{4 \pi D} +3 g_\gamma - g_\sigma
\nonumber \\
\frac{d \ln g_\gamma}{d l} = y_\gamma - 2 g_\sigma
\nonumber \\
\frac{d \ln g_\sigma}{d l} = y_\sigma - \rho z - 2 g_\sigma \ .
\end{eqnarray}
The dynamical exponent is given by
\beq{16}
z = 2 + g_\gamma- g_\sigma \ .
\end{equation}
To achieve the superfast regime, we must set
$y_\gamma = y_\sigma - \rho z$.
Without this choice, the potential disorder will either dominate or be
irrelevant. It is natural that the appropriate
$y_\gamma$ should be weaker than
$y_\sigma$, since the effective turbulence strength is weakened by
decorrelation effects over time. With this choice, we find that the
ratio $g_\gamma(l)/g_\sigma(l)$ remains constant under the
renormalization group flows. The parameter $g_\sigma(l)$, however,
flows to the fixed point $g_\sigma^* = y_\gamma/2$. Moreover, we
find that when $g_\gamma/g_\sigma > 1/3$, the effective reaction rate
flows to a finite fixed point $\lambda^* = 4 \pi D(3 g_\gamma^* -
g_\sigma^*)$. When $g_\gamma/g_\sigma < 1/3$, the effective reaction rate
renormalizes to zero.
With the choice of
$y_\gamma = y_\sigma - \rho z$,
the flow equations are
the same as for the static model of turbulence \cite{Deem4}. The
matching is the same, as well. In terms of these new variables,
therefore, the predicted concentration dependence is the same.
\emph{In particular, the superfast reaction regime is predicted to
exist for this dynamic model of turbulence.} The explicit, long-time
concentration dependence for weak disorder is
\bey{18}
c(t) \sim \left[ \frac{1}{4 \pi D (g_\sigma^* - 3 g_\gamma^*)}
+ \frac{1}{\lambda_0} \right] \frac{1}{t}
\left( \frac{t}{t_0} \right)^{-
2 g_\gamma^*/(2 + g_\gamma^* - g_\sigma^*)} , ~~~(3 g_\gamma < g_\sigma)
\ ,
\end{eqnarray}
where $\lambda_0$ is the bare value of the reaction
rate. The matching time$, t_0$, is given roughly by
$t_0 \approx h^2/(2 D)$. For strong disorder, the concentration decays as
\bey{19}
c(t) \sim \frac{1}{4 \pi D(3 g_\gamma^* - g_\sigma^*) t}
\left(\frac{t}{t_0}\right)^{
(g_\gamma^* - g_\sigma^*)/(2 + g_\gamma^* - g_\sigma^*)} ,
~~~(3 g_\gamma > g_\sigma) \ .
\end{eqnarray}
At the location of the maximum reaction rate, there is a
logarithmic correction to power-law decay:
\beq{20}
c(t) \sim \frac{\ln (t/t_0)}{8 \pi (1 -y_\gamma/6) D t}
t^{-y_\gamma/(6-y_\gamma)}
, ~~~(3 g_\gamma = g_\sigma) \ .
\end{equation}
\section{Conclusion}
A regime of reaction rates faster than that for a well-mixed reaction
was observed in simulations of a static model of turbulence plus
potential disorder. Qualitative predictions of previous
renormalization group studies of this model were reproduced.
Interesting departures from the one-loop predictions, such as a
compression of the decay exponent curve to the left, were observed.
These features are expected to be reproduced in higher-loop
analytical calculations.
Renormalization group calculations for a more
realistic, dynamic model of turbulence show that the superfast regime
persists. Indeed, these calculations suggest that our general
understanding of the phenomenon of superfast reactivity is correct. The
random potential tends to attract reactants to localized regions in
space, and by doing so increases the reaction rate in a non-linear
manner in these regions. The reactants, therefore, annihilate at a
faster than average rate in these regions. The turbulent velocity
fields transport reactants to these regions rapidly, although the
transport is slowed by the ruggedness of the random potential
landscape. The net effect of the trapping and the turbulent transport is
superfast reactivity whenever the mean-square displacement is
super-linear.
The existence of a superfast reaction regime is of great interest
from a practical point of view. Often, it is assumed in reactor design
that the highest reaction rate is obtained for a well-mixed system
\cite{Jensen}. Our results show that under certain conditions,
the highest reaction rate is achieved, instead, for an inhomogeneous
system. This behavior has been observed experimentally.
Indeed,
an enhancement due to inhomogeneous reactant concentrations
of the effective reaction rate between ions
in a chaotically-mixed,
two-dimensional,
fluid with attractors has recently been observed
\cite{Tabeling}.
The superfast regime may be relevant to a variety of reactors,
including thin-film reactors, certain combustion reactors, and
certain types of micro-electro-mechanical (MEMS) devices.
\ack
This research was supported by the National Science Foundation
through grants CHE--9705165 and CTS--9702403.
\bigskip
|
0812.2977
|
\section{\label{sec:intro} Introduction}
The physics of strongly correlated lattice electrons is complex as well as interesting. The electronic properties of very many exciting materials (such as, the transition metal oxides, rare earths etc.) exhibit this physics in a variety of different ways. A minimal model for understanding these systems requires one to consider, at least, the local coulomb repulsion, in addition to the tight-binding hopping of electrons. In an effective one-band system, the local repulsion, $U$, is the energy cost for putting two electrons with opposite spins together on the same site (Wannier orbital). This model, famously called the Hubbard model, has been a subject of great interest. It poses one of the most difficult problems in quantum many body theory. Although the model has been long solved exactly in one dimension (1D) using Bethe ansatz~\cite{Lieb_Wu}, and also shown to be integrable~\cite{Shastry}, it still evades an exact analytical solution in higher dimensions. In recent times, the interest in the Hubbard model has been further reinforced by the high-T$_C$ superconductivity in cuprates, and also by the need to understand other phenomena in strongly correlated electrons~\cite{Imada}.
There has been a long-standing interest in understanding the metallic-ferromagnetism through Hubbard model. The limit of infinite local repulsion in one-band Hubbard model presents an interesting case study in this context. In this limit, the ground state on certain lattices (for example, square lattice) was shown to be saturated metallic-ferromagnetic for a single hole in an otherwise half-filled system (Nagaoka-Thouless theorem)~\cite{Nagaoka,Thouless}. Subsequent variational and numerical studies have shown that the Nagaoka ferromagnetism survives for finite hole densities away from half-filling, up to a (lattice dependent) critical doping~\cite{Nagaoka_SKA, Linden, Becca, DMFT}. The Lieb-Mattis theorem, however, rules out the existence of ferromagnetism in 1D~\cite{Lieb_Mattis_Theorem}. For the infinite-$U$ Hubbard model with nearest-neighbor hopping, it implies the lack of Nagaoka ferromagnetism, which is borne out by the analytic studies of this problem for one and two holes explicitly. These investigations either use the Bethe ansatz approach~\cite{Doucot_Wen}, or work with an effective spin Hamiltonian in the presence of a single hole~\cite{Haerter_Shastry}, starting with a Gutzwiller projected Hamiltonian.
Recently, we have developed a new approach to the infinite-$U$ Hubbard problem~\cite{BK_new_rep}. In our formulation, we canonically represent an electron in terms of a spinless fermion and the spin-1/2 (Pauli) operators. We then write the Hubbard model in this representation. Finally, we take the limit $U\rightarrow\infty$, and get the infinite-$U$ Hamiltonian, $H_\infty$. Although our prescription is applicable to any lattice, it clearly distinguishes the bipartite lattices from others. By exploiting the two sublattice structure of a bipartite lattice, we can write $H_\infty$ in a beautiful form (resembling the Anderson-Hasegawa Hamiltonian, but different from it and fully quantum mechanical) which reveals the phenomenon of metallic-ferromagnetism in the infinite-$U$ Hubbard model in a very transparent way. (Otherwise, we all know that the Nagaoka ferromagnetism is a non-obvious strong correlation effect.)
In the present study, we investigate 1D infinite-$U$ Hubbard problem within our approach, and exactly solve it by means of the suitably constructed unitary transformation. From this exact solution, we conclude that its ground state is `correlated metallic' and 'ideal paramagnetic' for arbitrary density of electrons. Ours is a `non-Bethe' method which also solves a class of very general models in 1D. The infinite-$U$ Hubbard model happens to be one among them. This paper is organized as follows. First we present the exact solution of $H_\infty$ in 1D. Then, we identify a general class of models which can be exactly solved by our method. We also briefly discuss the Anderson-Hasegawa problem and the minimal coupling lattice Hamiltonian in the light of ideas developed here. Finally, we conclude with a summary.
\section{\label{sec:1D_Hubbard} Infinite-$U$ Hubbard model in 1D}
According to a recently developed canonical (and invertible) representation, an electron can be described in terms of a spinless fermion and the Pauli operators~\cite{BK_new_rep}. On a bipartite lattice, we can represent the electrons on different sublattices in two different, but equivalent, ways. (In principle, we can generate infinitely many equivalent representations through unitary transformations, but our purpose is served by the following two forms.)
The electronic operators in this representation are: $ \hat{f}^\dag_{l,\uparrow}=\hat{\phi}^{ }_l\sigma^+_l$, $\hat{f}^\dag_{l,\downarrow}=(i\hat{\psi}_l-\hat{\phi}_l\sigma^z_l)/2$
on the odd-numbered sites ($l=1,3,5,\dots$), and $\hat{f}^\dag_{l,\uparrow}=i\hat{\psi}_l\sigma^+_l$, $\hat{f}^\dag_{l,\downarrow}=(\hat{\phi}_l-i\hat{\psi}_l\sigma^z_l)/2$ on the even-numberd sites, where $\hat{\phi}_l=(\hat{a}^\dag_l+\hat{a}_l)$ and $i\hat{\psi}=(\hat{a}^\dag_l-\hat{a}_l)$ are the Majorana fermions. Here, $\hat{a}^{ }_l$ and $\hat{a}^\dag_l$ are the spinless fermion operators, and $\vec{\sigma}_l$ the Pauli operators, on the $l^{th}$ site.
Moreover, the electronic number operator is written as: $\hat{N}_e = L+\sum_{l=1}^L(1-\hat{n}_l)\sigma^z_l$, and the `physical spin' of an electron on the $l^{th}$ site is given by: $\vec{{\bf S}}_l = \hat{n}_l\vec{\sigma}_l/2$, where $\hat{n}_l = \hat{a}^\dag_l\hat{a}^{ }_l$. For completeness, also note the local mapping: $\left|0\rangle\right.=\left|\stackrel{-}{_\circ}\right\rangle$, $\left|\uparrow\rangle\right.=\left|\stackrel{+}{_\bullet}\right\rangle$, $\left|\downarrow\rangle\right.=\left|\stackrel{-}{_\bullet}\right\rangle$, and $\left|\uparrow\downarrow\rangle\right.=\left|\stackrel{+}{_\circ}\right\rangle$ (on an odd-numbered site) and $=-\left|\stackrel{+}{_\circ}\right\rangle$ (on an even-numbered site). Here, $\left|0\rangle\right.$, $\left|\uparrow\rangle\right.$, $\left|\downarrow\rangle\right.$ and $\left|\uparrow\downarrow\rangle\right.$ are the local electronic states (in the usual notation). The states $|\circ\rangle$ and $|\bullet\rangle$ denote the empty and filled site, respectively, for the spinless fermion, and $\{|+\rangle,|-\rangle\}$ is the basis set of a single Pauli spin. Note that $|+\rangle$ and $|-\rangle$ represent the actual electronic spin on a site when it is occupied by a spinless fermion.
Now, consider the Hubbard model with nearest-neighbor hopping on an open chain. (The reason for working with an `open' chain will become clear shortly.) Since the hopping process is bipartite, we can use the above two forms of the representation for electrons to convert the Hubbard model into a corresponding `spin-fermion' model. In the limit of infinite-$U$, we get the following Hamiltonian.
\begin{equation}
H_\infty = -t\sum_{l=1}^{L-1}\hat{X}^{ }_{l,l+1}\left(\hat{a}^\dag_{l}\hat{a}^{ }_{l+1} + \hat{a}^\dag_{l+1}\hat{a}^{ }_l\right) \label{eq:H_infty}
\end{equation}
For the details of its derivation, please refer to Ref.~[11].
Here, $\hat{X}^{ }_{l,l+1}=(1+\vec{\sigma}_l\cdot\vec{\sigma}_{l+1})/2$ is the Dirac-Heisenberg exchange operator, and $L$ is the total number of lattice sites.
\subsection{\label{subsec:exact_sol} Exact analytic solution}
For the moment, we forget the physical origin and purpose of $H_\infty$, and just take it as a given spin-fermion model in one dimension. Our immediate goal is then to find its eigenvalues and eigenstates. There are two key features that we make use of in exactly solving this problem. First is the property, $\hat{X}^{2}_{l,l+1}=\mathbb{1}$, of the exchange operators. And, the second is the open boundary condition of the 1D lattice (similar to the $XY$ spin-1/2 chain~\cite{XYchain}).
We exploit the former to construct a unitary operator which, on an open chain, transforms Eq.~(\ref{eq:H_infty}) to a tight-binding model of the spinless fermions only.
We first define the following unitary operator on the first bond [that is, for the pair of sites (1,2)].
\begin{equation}
{\cal U}_{1,2}=\left(1-\hat{n}_2\right) + \hat{n}_2 \hat{X}_{2,1} \label{eq:U12}
\end{equation}
Here, $\hat{X}^{ }_{2,1}=\hat{X}^{ }_{1,2}$. Clearly, ${\cal U}^\dag_{1,2}={\cal U}^{ }_{1,2}$ and ${\cal U}^2_{1,2}=\mathbb{1}$. Thus, ${\cal U}_{1,2}$ is both Hermitian as well as unitary. Moreover, it has the following important property.
\begin{equation}
{\cal U}^\dag_{1,2}\left( \hat{a}^\dag_1 \hat{X}^{ }_{1,2} \hat{a}^{ }_2\right){\cal U}^{ }_{1,2} = \hat{a}^\dag_1\hat{a}^{ }_2
\end{equation}
In the above equation, ${\cal U}_{1,2}$ leaves $\hat{a}_1$ and $\hat{X}_{1,2}$ unaffected while transforming $\hat{a}_2\rightarrow \hat{X}_{2,1}\hat{a}_2$. Thus, ${\cal U}_{1,2}$ gets rid of the exchange operator $\hat{X}_{1,2}$, and what remains is the hopping of the fermions alone. As it happens, we will get rid of the exchange operators on each bond by carefully following this approach.
Before constructing a similar ${\cal U}_{2,3}$ for the next bond, it is important to consider the effect of ${\cal U}_{1,2}$ on other terms in $H_\infty$. Clearly, ${\cal U}_{1,2}$ leaves the operators on other bonds unaffected, except for the bond (2,3). The net effect of this unitary transformation on $H_\infty$ is the following.
\begin{widetext}
\begin{equation}
{\cal U}_{1,2}^\dag \, H_\infty \, {\cal U}_{1,2} = -t\left\{\left(\hat{a}^\dag_1\hat{a}^{ }_2 + \hat{a}^\dag_2\hat{a}^{ }_1 \right) + \left(\hat{a}^\dag_2 \, \hat{X}_{1,2}\hat{X}_{2,3} \, \hat{a}^{ }_3 + \hat{a}^\dag_3 \, \hat{X}_{3,2}\hat{X}_{2,1} \, \hat{a}^{ }_2\right) + \sum_{l=3}^{L-1}\left(\hat{a}^\dag_l \, \hat{X}^{ }_{l,l+1} \, \hat{a}^{ }_{l+1} + \hat{a}^\dag_{l+1} \, \hat{X}^{ }_{l+1,l} \, \hat{a}^{ }_l \right)\right\}
\label{eq:H_12}
\end{equation}
\end{widetext}
The ${\cal U}_{1,2}$ transfers $\hat{X}_{1,2}$ from the first bond to the second.
We define $\hat{\cal X}_{2,3}=\hat{X}_{1,2}\hat{X}_{2,3}$ and $\hat{\cal X}_{3,2}=\hat{X}_{3,2}\hat{X}_{2,1}$, which replace $\hat{X}_{2,3}$ on bond (2,3). Note that $\hat{\cal X}^\dag_{2,3}=\hat{\cal X}_{3,2}\neq\hat{\cal X}_{2,3}$ and $\hat{\cal X}^\dag_{2,3}\hat{\cal X}^{ }_{2,3}=\hat{\cal X}^{ }_{3,2}\hat{\cal X}^{ }_{2,3}=\mathbb{1}$. Thus, $\hat{\cal X}_{2,3}$ is unitary but not Hermitian (unlike $\hat{X}_{2,3}$).
Now we define the following unitary operator for getting rid of $\hat{\cal X}_{2,3}$ and $\hat{\cal X}_{3,2}$ from the terms inside the second parentheses in Eq.~(\ref{eq:H_12}).
\begin{equation}
{\cal U}_{2,3}=\left(1-\hat{n}_3\right) + \hat{n}_3\hat{\cal X}_{3,2} \label{eq:U23}
\end{equation}
Note that ${\cal U}_{2,3}$ is unitary but not Hermitian (unlike ${\cal U}_{1,2}$). We can show that $\hat{\cal X}_{2,3}$ is invariant under ${\cal U}_{2,3}$, while
\begin{equation}
{\cal U}_{2,3}^\dag \, \hat{a}^{ }_3 \, {\cal U}^{ }_{2,3} = \hat{a}^{ }_3 \, \hat{\cal X}^{ }_{3,2}.
\end{equation}
Therefore,
\begin{equation}
{\cal U}^\dag_{2,3}\left( \hat{a}^\dag_2 \, \hat{\cal X}^{ }_{2,3} \, \hat{a}^{ }_3\right){\cal U}^{ }_{2,3} = \hat{a}^\dag_2\hat{a}^{ }_3.
\end{equation}
Moreover, ${\cal U}^\dag_{2,3}\,(\hat{a}^\dag_{3} \hat{X}_{3,4} \hat{a}^{ }_4)\, {\cal U}^{ }_{2,3} = \hat{a}^\dag_{3}\hat{\cal X}_{3,4}\hat{a}^{ }_4$, where $\hat{\cal X}_{3,4}=\hat{\cal X}_{2,3}\hat{X}_{3,4}=\hat{X}_{1,2}\hat{X}_{2,3}\hat{X}_{3,4}$. It is clear by now that we can continue this process, and get rid of all the exchange operators in $H_\infty$. To achieve this, we define the following general unitary operators,
\begin{eqnarray}
{\cal U}_{l,l+1} &=& \left(1-\hat{n}_{l+1}\right) + \hat{n}_{l+1}\hat{\cal X}_{l+1,l} \\
{\cal U} &=& \prod_{l=1}^{L-1}{\cal U}_{l,l+1} \label{eq:full_U}
\end{eqnarray}
where $\hat{\cal X}_{l,l+1} = \hat{\cal X}_{l-1,l}\hat{X}_{l,l+1}=\prod_{m=1}^{l}\hat{X}_{m,m+1}$, for $l=2, L-1$. We can show that the full unitary operator ${\cal U}$, of Eq.~(\ref{eq:full_U}), transforms $H_\infty$ to a Hamiltonian of free spinless fermions. That is,
\begin{eqnarray}
{\cal U}^\dag \, H_\infty \, {\cal U} &=& -t\sum_{l=1}^{L-1}\left(\hat{a}^\dag_l\hat{a}^{ }_{l+1} + \hat{a}^\dag_{l+1}\hat{a}^{ }_l\right) \label{eq:H_freefermi} \\
&=&\sum_k\epsilon_k~\hat{a}^\dag_k\hat{a}^{ }_k \label{eq:H_diag}
\end{eqnarray}
where $\epsilon_k = -2t\cos{k}$, and $k$ is the momentum. The operators $\{\hat{a}_k\}$ are the fermions in the momentum space. Since Eq.~(\ref{eq:H_freefermi}) is derived on a chain with open boundary condition, the Fourier transformation between $\{\hat{a}_l\}$ and $\{\hat{a}_k\}$ is defined as: $\hat{a}_l= \sqrt{\frac{2}{L+1}}\sum_k \hat{a}_k\, \sin{kl}$, where $k=n\pi/(L+1)$ for $n=1,2,\cdots, L$. Hence, the exact solution of $H_\infty$ in 1D.
To this end, we also note that
\begin{equation}
{\cal U}^\dag\,\hat{N}\,{\cal U} = \hat{N} ~~\mbox{and}~~ {\cal U}^\dag\,\hat{M}_\sigma \,{\cal U} = \hat{M}_\sigma,
\label{eq:N_and_Msigma}
\end{equation}
where $\hat{N} = \sum_{l=1}^L\hat{n}_l$ is the number operator of the spinless fermions, and $\hat{M}_\sigma = \sum_{l=1}^L\sigma^z_l$ is the total $\sigma^z$ operator. Therefore, ${\cal U}$ diagonalizes a more general Hamiltonian: $H=H_\infty -\lambda\hat{N} -\eta\hat{M}_\sigma$, where $\lambda$ is the chemical potential of the spinless fermions and $\eta$ is the external `magnetic' field acting on the spins.
The above exercise presents a rigorous and transparent case of the complete decoupling of the Pauli and Fermi attributes of the electron.
As a result of this decoupling, the energy eigenvalues of $H_\infty$ become independent of spins, giving rise to an extensive entropy in every eigenstate. Just as the spinless fermions $\{\hat{a}_l\}$ transform under ${\cal U}$, the spins $\{\vec{\sigma}_l\}$ also transform to new spins. However, the total $\sigma^z$ is invariant under ${\cal U}$, as noted in Eq.~(\ref{eq:N_and_Msigma}). It is these transformed spins that are absent in the free spinless fermion Hamiltonian [Eq.~(\ref{eq:H_freefermi})].
\subsection{\label{subsec:no_Nagaoka} Absence of the Nagaoka ferromagnetism in 1D}
In order to discuss the ground state of the infinite-$U$ Hubbard model, first let us recall the complete connection between the problem worked out in the previous subsection and the physical Hubbard model. Technically, the infinite-$U$ Hubbard model in our representation is not just $H_\infty$, but $H_\infty + \infty\sum_l(\frac{1}{2}-\hat{n}_l)$. That is, the physical problem is described by $H_\infty$ with an infinite chemical potential for the spinless fermions (please look into Ref.~[11] for details). Next, we note that $N=N_e-2N_D$, where $N$ is the total number of spinless fermions (that is, the number of singly occupied sites in terms of the electrons), $N_e$ is the total number of electrons,
and $N_D$ is the total number of doubly occupied sites. Clearly, $N$ is a conserved quantity of $H_\infty$ [evident from Eq.~(\ref{eq:H_infty})]. So is $N_e$, and hence $N_D$. It is evident from the following explicit form of $H_\infty$ in terms of the electron operators.
\begin{equation}
H_\infty = -\frac{t}{2}\sum_{l=1}^{L-1}\sum_s^{\uparrow,\downarrow}\left(\hat{n}_{l,s}-\hat{n}_{l+1,s}\right)^2\left[\hat{f}^\dag_{l,\bar{s}}\hat{f}^{ }_{l+1,\bar{s}} + h.c \right]
\end{equation}
Since $N_e=N+2N_D$, we can label the sectors of states for a given $N_e$ in terms of the partitions: $(N,N_D)$. The physical validity of a partition, however, depends upon whether $N_e\le L$ (less than or equal to half-filling) or $N_e > L$ (more than half-filled case), subject to the natural constraint: $N+N_D \le L$. That is, an arbitrary partition of the integer $N_e$ into two other integers $N$ and $N_D$ does not necessarily denote a physical sector. For $0\le N_e\le L$, the constraint is guaranteed to be satisfied. Therefore, all partitions are valid physical sectors. For example, if $N_e=7$ and $\le L$, then the corresponding sectors of states are: $(7,0)$, $(5,1)$, $(3,2)$, and $(1,3)$. However, when $N_e>L$, the constraint will disqualify many partitions. For example, if $L=4$ and $N_e=7$, then the only physical sector is: $(1,3)$, as the other partitions such as $(3,2)$ don't respect the constraint. In general, the physical sectors for $N_e\le L$ are given by the set: $\{(N_e-2N_D,N_D),~ \forall ~ N_D=0, 1,\cdots ,[N_e/2]\}$, where $[N_e/2]=N_e/2$ for even values of $N_e$ and $=(N_e-1)/2$ for odd values. For $N_e>L$, we use the relations: $N[2L-N_e] = N[N_e]$, and $N_D[2L-N_e] = L-N_e+N_D[N_e]$, to find the physical sectors. These relations are a consequence of the particle-hole transformation on the electronic operators. Here, $N[N_e]$ and $N_D[N_e]$ denote the dependence of $N$ and $N_D$ on $N_e$.
The exact solution of $H_\infty$ gives a highly disordered ground state in terms of the spinless fermions and the Pauli spins. However, we need to carefully translate its meaning for the electrons. Interestingly, we are able to show that the ground state of the infinite-$U$ Hubbard model for a given $N_e$, is a fermi-sea which is $2^{N^*}$-fold degenerate [where $N^*$ is defined in Eq.~(\ref{eq:Nstar})]. Physically, it means that the ground state is metallic and ideally paramagnetic. In other words, it is not Nagaoka ferromagnetic. Nor it is a kinematic singlet (like a normal electronic fermi-sea).
Now, the {\em proof}.
Due to the fact that $U=\infty$ and it is the chemical potential of the spinless fermions, the ground state of the Hubbard problem, for a given $N_e$, lies in the sector $(N^*,(N_e-N^*)/2)$, where $N^*$ is the maximum allowed value of $N$ for the given $N_e$.
\begin{equation}
N^* = \left\{\begin{array}{lcl} N_e &,& 0\le N_e\le L\\ 2L-N_e &,& L\le N_e\le 2L\end{array}\right.
\label{eq:Nstar}
\end{equation}
Since $\hat{N}$ is invariant under ${\cal U}$ [Eq.~(\ref{eq:N_and_Msigma})], the infinite-$U$ ground state corresponds to the fermi-sea of $N^*$ spinless fermions with dispersion $\epsilon_k$ [Eq.~(\ref{eq:H_diag})]. We derive the following exact expression for the ground state energy.
\begin{equation}
E_g[N^*] = -2t\cos{\left(\frac{\pi}{2}\frac{N^*+1}{L+1}\right)}\frac{\sin{\left(\frac{\pi}{2}\frac{N^*}{L+1}\right)}}{\sin{\left(\frac{\pi}{2}\frac{1}{L+1} \right)}}
\end{equation}
Expectedly, $E_g=0$ for $N_e=0$ and $2L$ (trivial cases: empty and fully filled bands, respectively), and also for $N_e=L$ (the half-filled case for $U=\infty$). In the thermodynamic limit ($L\rightarrow\infty$), for a finite electron density, $n_e=N_e/L$, the ground state energy density, $e_g=E_g/L$, can be written as~\cite{fnote_eg}:
\begin{equation} e_g[n_e]=-\frac{2t}{\pi}| \sin(\pi n_e)|. \label{eq:eg_ne} \end{equation}
Now, we enumerate the spin degeneracy in the ground state, which will decide the magnetic nature of the ground state. Let us discuss $N_e\le L$ case first. In this case, the ground sector is $(N_e,0)$. That is, the number of electrons is completely exhausted by the number of spinless fermions. As $N_D=0$, the remaining $L-N_e$ sites must be empty. The state of an empty site is uniquely $\left|\stackrel{-}{_\circ}\right\rangle$. However, the state of a site, occupied by a single electron, is either $\left|\stackrel{+}{_\bullet}\right\rangle$ or $\left|\stackrel{-}{_\bullet}\right\rangle$, corresponding to the fact that it could be an $\uparrow$ or $\downarrow$ spin electron. Therefore, corresponding to any given distribution of $N_e$ spinless fermions on $L$ sites (any one of the $^LC_{N_e}$ combinations), there are exactly $2^{N_e}$ states (a total of $^LC_{N_e}\times2^{N_e}$ states in the ground sector). For example, on a chain with $L=7$ and $N_e=3$, the states in the ground sector are like: $\left|\stackrel{\pm}{_\bullet}\stackrel{\pm}{_\bullet}\stackrel{\pm}{_\bullet}\stackrel{-}{_\circ}\stackrel{-}{_\circ}\stackrel{-}{_\circ}\stackrel{-}{_\circ}\right\rangle$, where the filled sites could be in any one of the $^7C_3$ combinations. On each filled site, the Pauli spin could be $+$ or $-$ (without affecting the number of electrons). Hence, $2^3$ different $M_\sigma$ states.
Coming back to the general situation, these $2^{N_e}$ states can be grouped according to their $M_\sigma$ values ($^{N_e}C_{M_\sigma}$ states for a given $M_\sigma$). We know that $M_\sigma$ is a conserved quantity of $H_\infty$, and it is also invariant under ${\cal U}$ [Eq.~(\ref{eq:N_and_Msigma})]. Therefore, $2^{N_e}$ different $M_\sigma$ states in the ground sector will be degenerate, as the exact energy eigenvalues of $H_\infty$ are independent of $M_\sigma$. Hence, the ground state, in the sector $(N_e,0)$, is a $2^{N_e}$-fold degenerate fermi-sea of $N_e$ spinless fermions. Since there are only empty or singly occupied sites, the $2^{N_e}$-fold degeneracy is strictly due to the physical spin of electrons. Therefore, we conclude that the exact ground state of the infinite-$U$ Hubbard model is ideally paramagnetic and metallic (more correctly, {\em strange} or correlated metallic, as it is not a fermi-sea of the normal electrons).
Furthermore, we note that the arguments for $N_e>L$ are the same as that for $N_e\le L$. The (only) key difference between the two is that the for $N_e>L$, there are only singly or doubly occupied sites in the ground sector, while for $N_e<L$ the sites are either empty or singly occupied. For example, when $N_e=11$ and $L=7$, then a typical state will be of the form: $\left|\stackrel{\pm}{_\bullet}\stackrel{\pm}{_\bullet}\stackrel{\pm}{_\bullet}\stackrel{+}{_\circ}\stackrel{+}{_\circ}\stackrel{+}{_\circ}\stackrel{+}{_\circ}\right\rangle$. This state is the counterpart of a previously mentioned state for less than half-filling. The physics of the Hubbard model on a bipartite lattice for $N_e$ electrons is same as that for $2L-N_e$ electrons. Without going into the (repetitive) details of the analysis all over again, we conclude that there is no Nagaoka ferromagnetism in the one-dimensional inifinte-$U$ Hubbard model, and that the exact ground state is a $2^{N^*}$-fold degenerate fermi-sea of $N^*$ spinless fermions, where $N^*$ is defined by Eq.~(\ref{eq:Nstar}).
To this end, we would like to make four comments. First is that the state labeling and counting procedure presented above is applicable to all lattices. It is not specific to 1D (although the solution is). Second comment is that, in the cases different from the present 1D problem, it will be impossible to completely get rid of the exchange operators. Due to which, the different $M_\sigma$ states are not guaranteed to be degenerate. Therefore, we stand a clear chance of finding some sort of metallic magnetism (ferro or antiferro~\cite{Haerter_Shastry} or something else) on other lattices (and hopping geometries). Third is a minor comment about $U=-\infty$ problem. In this case, the ground sector corresponds to $N_D=N_e/2$ (for even $N_e$) or $(N_e-1)/2$ (for odd $N_e$). The ground state (say for even $N_e$) is a $^LC_{N_D}$-fold degenerate hard-core bosonic state with no kinetic energy gains (due to spinless fermion). The final comment is about the finite temperature calculations. Since $U$ is infinite, the ground sector (for a given $N_e$) is the only part of the Hilbert space which is accessible by finite temperatures. The thermodynamics of this problem can therefore be worked out in the canonical ensemble of the spinless fermions. The Pauli spins remain ideally paramagnetic down to absolute zero temperature. This sets the quantum coherence temperature for electrons to be zero (even though there is this Fermi temperature scale for the spinless fermions)~\cite{TVR_comment}.
\subsection{\label{subsec:spin_orbit} Spin-orbital model}
In a different incarnation, the Hamiltonian $H_\infty$ can be considered as a one-dimensional model of the coupled spin and orbital degrees of freedom. By applying the Jordan-Wigner transformation on the spinless fermions, we can derive the following Hamiltonian,
\begin{equation}
H_{SO} = -t\sum_{l=1}^{L-1}\hat{X}_{l,l+1}\left(\tau^+_l\tau^-_{l+1} + \tau^-_l\tau^+_{l+1}\right) - \frac{1}{2}\sum_l(\lambda\tau^z_l + \eta\sigma^z_l)
\label{eq:spin_orbit}
\end{equation}
where $\{\hat{a}_l\}$ have been changed to $\{\tau^-_l\}$, and $\tau^z_l = 2\hat{n}_l-1$. In a transition metal ion with two-fold orbital degeneracy, the orbital degrees of freedom can be described in terms of the Pauli operators. Let $\vec{\tau}$ denote this orbital degree in the present discussion. The model $H_{SO}$ is special case of the more general Kugel-Khomskii type models~\cite{Kugel_Khomskii}. Equation~(\ref{eq:spin_orbit}) thus presents an exactly solvable spin-orbital lattice model, in which the spins and the orbitals behave as decoupled. While the orbital part acts as an $XY$ chain, the spins become paramagnetic.
\section{\label{sec:general_models} General class of exactly solvable models}
While transforming $H_\infty$ to the tight-binding model of spinless fermions, it became clear that a very general class of models can be solved exactly by our method. A Hamiltonian in this class can be written as:
\begin{equation}
H = -t\sum_{l=1}^{L-1}\left(\hat{a}^\dag_{l} \hat{T}^{ }_{l,l+1}\hat{a}^{ }_{l+1} + \hat{a}^\dag_{l+1}\hat{T}^{ }_{l+1,l}\hat{a}^{ }_l\right) \label{eq:H_general}
\end{equation}
Here, $\hat{T}_{l,l+1}$ is some unitary operator on the bond $(l,l+1)$, and by defintion $\hat{T}_{l+1,l}=\hat{T}^\dag_{l,l+1}$. Moreover, $\hat{T}_{l,l+1}$ doesn't have to commute with $\hat{T}_{l-1,l}$ and $\hat{T}_{l+1,l+2}$, while strictly commuting with the $\hat{T}$ operators on other bonds, and also with the fermions, $\{\hat{a}_l\}$. With a few careful steps of algebra, this Hamiltonian can be transformed to a tight-binding model of the fermions [Eq.~(\ref{eq:H_freefermi})] with the help of ${\cal U} = \prod_{l=1}^{L-1}{\cal U}_{l,l+1}$, where ${\cal U}_{l,l+1}=(1-\hat{n}_{l+1}) + \hat{n}_{l+1}\hat{\mathcal{T}}_{l+1,l}$. Here, $\hat{\mathcal{T}}_{l,l+1} = \prod_{m=1}^{l}\hat{T}_{m,m+1}$ and $\hat{\mathcal{T}}_{l+1,l}=\hat{\mathcal{T}}^\dag_{l,l+1}$.
For a special case in which $\hat{T}_{l,l+1}$ are just the phase factors $e^{i\xi_{l,l+1}}$, we can get rid of these on a Caley tree of arbitrary coordination, $z$ (for a nearest-neighbor chain, $z=2$). In this case, it can be done not only for the fermions but also for the bosons.
\subsection{\label{subsec:spin-1} Spin-fermion model for higher spins}
As an academic exercise, we construct the spin-fermion models for higher spins, which belong to this general class of exactly solvable models. We achieve this by constructing the `correct' analog of the exchange operator for a pair of higher spins. By correct exchange we mean that $|m_1\rangle|m_2\rangle$ must become $|m_2\rangle |m_1\rangle$ under the exchange operator. For clarity, we work it out explicitly for the spin-1 case. Here, $S^z|m\rangle =m|m\rangle$, with $m=1,0,\bar{1}$ (-1 is denoted as $\bar{1}$; for spin-$S$, $m=\bar{S}$, $\bar{S}+1$, $\dots$, $S$). The usual spin-spin interaction, ${\bf S}_1\cdot{\bf S}_2$, does not really exchange $m_1$ and $m_2$. Hence, we construct an operator $\hat{X}_{1,2}$ such that $\hat{X}_{1,2}|m_1\rangle|m_2\rangle = |m_2\rangle|m_1\rangle$. For example, $|0\rangle|0\rangle$ remains unaffected under $\hat{X}_{1,2}$, while $|1\rangle|\bar{1}\rangle$ becomes $|\bar{1}\rangle|1\rangle$ and vice versa. Explicitly, in terms of the spin operators, this spin-1 exchange operator can be written as:
\begin{eqnarray}
\hat{X}_{1,2} &=& 1-\left(S_{1z}^2 + S_{2z}^2\right) + \frac{1}{2}{\bf S}_1\cdot{\bf S}_2+\frac{1}{2}\left({\bf S}_1\cdot{\bf S}_2\right)^2 + \nonumber\\
& &({\bf S}_1\cdot{\bf S}_2)(S_{1z}S_{2z}) + \frac{i}{2}({\bf S}_1\times{\bf S}_2)_z(S_{1z}-S_{2z}) \nonumber\\
& &-\frac{1}{2}[({\bf S}_1\times{\bf S}_2)_z]^2
\end{eqnarray}
where $({\bf S}_1\times{\bf S}_2)_z = \frac{i}{2}(S_{1+}S_{2-}-S_{1-}S_{2+})$.
It is clear that we can similarly construct the exchange operators for higher spins. Since $\hat{X}^2_{1,2} = \mathbb{1}$, just like in the spin-1/2 problem, the corresponding spin-fermion model [that is, Eq.~(\ref{eq:H_infty})] can be diagonalized in the same way.
\subsection{\label{subsec:AH_Peierls} Some physical corrolaries}
\subsubsection{Anderson-Hasegawa problem}
Our method of getting rid of the unitary factors has an interesting consequence for the Anderson-Hasegawa (AH) problem. The AH model, $H_{AH}=-t\sum_{\langle l,m\rangle} \sqrt{\frac{1+{\bf \Omega}_l\cdot{\bf \Omega}_m}{2}}(\hat{a}^\dag_l e^{i\Phi_{l,m}}\hat{a}^{ }_m + h.c.)$, describes the motion of locally spin-projected electrons on a lattice, with classical spins, $\{{\bf \Omega}_l\}$, in the background. Very often, the phases $\Phi_{l,m}$ arising due to the spin-projection along ${\bf \Omega}_l$ and ${\bf \Omega}_m$ are ignored without proper justification. There have been studies which rightly emphasize on taking into consideration the effects of these phases while computing physical properties~\cite{Pinaki}. The special case on a Caley tree discussed earlier, however, implies that indeed the original $H_{AF}$ can be transformed to $H_{AF}=-t\sum_{\langle l,m\rangle} \sqrt{\frac{1+{\bf \Omega}_l\cdot{\bf \Omega}_m}{2}}(\hat{a}^\dag_l \hat{a}^{ }_m + h.c.)$ on Caley trees. Thus, we give a reason for dropping the phases in $H_{AF}$, at least on a Caley tree. On an arbitrary lattice, however, one must keep these phases.
\subsubsection{Minimal coupling Hamiltonian in 1D}
We now briefly discuss the Peierls minimal coupling of the quantized electromagnetic radiation to the lattice fermions, in the light of these gauge removing tricks. Consider tight-binding electrons with nearest neighbor hopping on an open 1D lattice. The corresponding gauge-invariant Hamiltonian can be written as:
\begin{equation}
H=-t\sum_{l=1}^{L-1}\sum_{s}^{\uparrow,\downarrow}\left(\hat{f}^\dag_{l+1,s} \hat{f}^{ }_{l,s} e^{i\frac{e}{\hbar}\int_l^{l+1}A_x dx} + h.c.\right) + H_{F}
\label{eq:minimal}
\end{equation}
Here, $e=-|e|$ is the electronic charge, $H_F=\sum_{{\bf q},\lambda}\omega_{\bf q} (\hat{\alpha}^\dag_{{\bf q}\lambda}\hat{\alpha}^{ }_{{\bf q}\lambda} +\frac{1}{2})$ is the field Hamiltonian (where $\hat{\alpha}_{{\bf q}\lambda}$, $\hat{\alpha}^\dag_{{\bf q}\lambda}$ are the Bose operators for an electromagnetic field of wavevector ${\bf q}$ and the polarization $\lambda$), and $A_x$ is the $x$-component of the vector potential. We have chosen the $x$-axis to be along the chain ($y=z=0$ line). In general, the vector potential can be written as:
\begin{equation}
\vec{A}({\bf r}) = i\sum_{{\bf q},\lambda} \sqrt{\frac{\hbar}{2\omega_{\bf q}\epsilon_0V}}\left[\vec{u}_{{\bf q}\lambda}({\bf r})\hat{\alpha}^\dag_{{\bf q}\lambda} - \vec{u}^*_{{\bf q}\lambda}({\bf r})\hat{\alpha}_{{\bf q}\lambda}\right]
\end{equation}
where $\vec{u}_{{\bf q}\lambda}$ is a normal mode (vector) function (including the information on polarization). Clearly, the vector potential operators at different spatial points commute with each other. The corresponding electric field operator is given by: $\vec{E}({\bf r})=\frac{i}{\hbar}[\vec{A}({\bf r}),H_F]$.
Now again, we get rid of the field dependent factors from the hopping by applying the following unitary transformation:
\begin{equation}
{\cal U} = \exp\left\{i\frac{e}{\hbar}\sum_{l=2}^L\hat{n}_l \int_{1}^l A_x dx\right\}
\end{equation}
where $\hat{n}_l = \hat{n}_{l,\uparrow} + \hat{n}_{l,\downarrow}$. Under this ${\cal U}$, we get the following transformed Hamiltonian:
\begin{equation}
{\cal U}^\dag H\,{\cal U} = -t\sum_{l=1}^{L-1}\sum_s^{\uparrow,\downarrow}(\hat{f}_{l+1,s}^\dag\hat{f}^{ }_{l,s} + h.c.) + {\cal U}^\dag H_F {\cal U}
\end{equation}
While the hopping becomes simple, the field Hamiltonian transforms to:
\begin{eqnarray}
{\cal U}^\dag H_F {\cal U} &=& H_F - e\sum_{l=2}^L\hat{n}_l\int_1^l E_x dx + \nonumber \\
&& \frac{e^2}{2\epsilon_0}\sum_{l,l^\prime=2}^L V^{ }_{l,l^\prime} \hat{n}^{ }_l \hat{n}^{ }_{l^\prime}
\end{eqnarray}
Here, the second term is the potential energy in the presence of electromagnetic field. Suppose $E_x$ is the electric field of a free radiation propagating in the $z$ direction. Then, $e\sum_{l=2}^L\hat{n}_l\int_1^l E_x dx$ is same as $P_x E_x$, the dipole interaction. Here, $P_x=e\sum_{l=2}^L (l-1)\hat{n}_l$ is the electric polarization operator. Furthermore, $V_{l,l^\prime}=\frac{1}{V}\int_1^l dx\int_1^l dx^\prime \sum_{{\bf q}\lambda} \Re{[u^*_{x,{\bf q}\lambda}u_{x^\prime,{\bf q}\lambda}]}$ is the `Coulomb' repulsion between electrons, generated by the `exchange' of the photon (facilitated by ${\cal U}$). Thus, we have derived the gauge independent Coulomb and the dipole-field interactions, starting from the minimal coupling Hamiltonian on a lattice, without any approximations. Since the vector potential commutes at different points, in principle, we can do the same on a Caley tree as well.
\section{\label{sec:conclude} Conclusion}
To summarize, we have exactly solved the infinite-$U$ Hubbard model with nearest neighbor hopping on an open chain. We use a newly developed canonical representation for electrons, in which the Hubbard model becomes a spin-fermion model. This spin-fermion model is exactly solved by applying a non-local unitary transformation. Under this transformation, the Pauli spins completely decouple from the fermions, as a result of which, the ground state is correlated metallic and ideal paramagnetic for arbitrary density of electrons.
This method solves a class of very general models. Guided by this observation, we have also constructed spin-fermion models for higher spins, by suitably extending the notion of `exchange' operators for higher spins. It is explicitly worked out for spin-1. (The spin only models, using our definition of the exchange operator for higher spins, exhibit interesting properties. These calculations will be discussed elsewhere.) By using the ideas developed here, we have shown that the phase factors in the Anderson-Hasegawa model can be droped on Caley trees. We have also derived the `dipole' interaction and Coulomb repulsion starting with Peierls minimal coupling Hamiltonian.
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0812.4279
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\section{Introduction}
In finite games correlated equilibria are simpler than Nash equilibria in several senses -- mathematically at least, if not conceptually. The set of correlated equilibria is a convex polytope, described by finitely many explicit linear inequalities, while the set of (mixed) Nash equilibria need not be convex or connected and can contain components which look like essentially any real algebraic variety (set described by polynomial equations on real variables) \cite{d:une}. The existence of correlated equilibria can be proven by elementary means (linear programming or game theoretic duality \cite{hs:ece}), whereas the existence of Nash equilibria seems to require nonconstructive methods (e.g., fixed point theorems as in \cite{nash:ncg,glicksberg:cg}) or the analysis of complicated algorithms \cite{lh:epbg,fy:rt,gl:gncfyrt}. Computing a sample correlated equilibrium or a correlated equilibrium optimizing some quantity such as social welfare can be done efficiently \cite{gz:ncescc,p:ccempg}; strong evidence in complexity theory suggests that the corresponding problems for Nash equilibria are hard \cite{dgp:ccne,cd:nashcomp, gz:ncescc}.
There are several exceptional classes of games for which the above problems about Nash equilibria become easy. The most important here are the zero-sum games. Broadly speaking, Nash equilibria of these games have complexity similar to correlated equilibria of general games. In particular, the set of Nash equilibria is an easily described convex polytope, existence can be proven by duality, and a sample equilibrium can be computed efficiently.
The situation in games with infinite strategy sets is not nearly so clear. For the computational sections of this paper we restrict attention to the simplest such class of games, those with finitely many players, strategy sets equal to $[-1,1]$, and polynomial utility functions. We make this restriction for several reasons.
The first is conceptual and notational simplicity. Results similar to ours will hold when the strategy sets are general compact semialgebraic (described by finitely many polynomial equations and inequalities) subsets of $\mathbb{R}^n$. However, dealing with this additional level of generality requires machinery from computational real algebraic geometry and does little to illuminate our basic methods.
The second is generality. Much of the study of games with infinite strategy sets is fraught with assumptions of concavity or quasiconcavity which appear to be motivated not by natural game theoretic premises, but rather by the inadequacy of available tools for games without these properties. While polynomiality assumptions and concavity assumptions are both rigid in their own ways, polynomials have the benefit of being dense in the space of all continuous functions, and thus suitable for approximating a much wider class of games.
The third reason is convenience. The algebraic structure we gain by restricting attention to polynomials allows us to use recent advances in semidefinite programming and real algebraic geometry to construct efficient algorithms and gain conceptual insights.
Little is known about correlated equilibria of these polynomial games, but much is known about Nash equilibria. Most importantly, the set of mixed Nash equilibria is nonempty and admits a finite-dimensional description in terms of the moments of the players' mixed strategies \cite{sop:slrcg}.
This set of moments can be described explicitly in terms of polynomial equations and inequalities \cite{sop:slrcg}. The Nash equilibrium conditions are expressible via first order statements, so the set of all moments of Nash equilibria is a real algebraic variety and can be computed in theory, albeit not efficiently in general. In the two-player zero-sum case, the set of Nash equilibria can be described by a semidefinite program (an SDP is a generalization of a linear program which can be efficiently solved; see the appendix), hence we can compute a sample Nash equilibrium or one which optimizes some linear functional in polynomial time \cite{pp:polygames}. A summary of the results described so far is shown in Table \ref{tab:eqsummary}.
\begin{table}
\label{tab:eqsummary}
\begin{center}
\begin{tabular}{c|c|c|c|}
& Nash equilibria & Nash equilibria & correlated equilibria \\
& (non-zero sum) & (zero sum) & \\
\hline
Finite games & Semialgebraic set \cite{lm:nevpe} & LP & LP \cite{a:ceebr}\\
\hline
Polynomial games & Semialgebraic set \cite{sop:slrcg} & SDP \cite{pp:polygames} & ? \\
\hline
\end{tabular}
\end{center}
\caption{Comparison of the simplest known description of different classes of equilibrium sets in finite and polynomial games.}
\end{table}
\paragraph{Contributions}The impetus for this paper was to address the bottom right cell of Table \ref{tab:eqsummary}, the one with the question mark. The table seems to suggest that the set of correlated equilibria of a polynomial game should be describable by a semidefinite program. We will see that this is approximately true, but not exactly. The contribution of this paper is twofold.
\begin{itemize}
\item First, we present several new characterizations of correlated equilibria in games with continuous utility functions (polynomiality is not needed here). In particular we show that the standard definition of correlated equilibria in terms of measurable departure functions is equivalent to other definitions in which the utilities are integrated against all test functions in some class (Theorem \ref{thm:correqmainchar}). This characterization does not have any obvious game theoretic significance, but it is extremely useful analytically and it forms the base for our other contributions.
\item Second, we present several algorithms for approximating correlated equilibria within an arbitrary degree of accuracy. We present one inefficient linear programming based method as a benchmark, followed by two semidefinite programming based algorithms which perform much better in practice. The first SDP algorithm, called adaptive discretization, iteratively computes a sequence of approximate correlated equilibria supported on finite sets (Section \ref{sec:adaptivedisc}). We enlarge the support sets at each iteration using a heuristic which guarantees convergence in general and yields fast convergence in practice. The second SDP algorithm, called moment relaxation, does not discretize the strategy spaces but instead works in terms of joint moments. It produces a nested sequence of outer approximations to the set of joint moments of correlated equilibrium distributions, and these approximate equilibrium sets are described by semidefinite programs (Section \ref{subsec:moment}). These relaxations depend crucially on one of the correlated equilibrium characterizations we have developed.
\end{itemize}
\paragraph{Related literature} The questions we address and the techniques we use are inspired by existing literature in two main areas. First, our work is related to a number of papers in the game theory literature.
\begin{itemize}
\item Aumann defines correlated equilibria in his famous paper \cite{a:scrs}, focusing on finite games to establish the basic properties and important examples. He obtains existence as a consequence of Nash's theorem on the existence of Nash equilibria in finite games \cite{nash:ncg}.
\item Hart and Schmeidler show that existence of correlated equilibria in finite games can be proven directly by a duality argument \cite{hs:ece}. They then use a careful limiting argument to prove existence of correlated equilibria in continuous games with compact Hausdorff strategy spaces (Theorem $3$ of that paper). The germs of ideas in this limiting argument are developed further in Section \ref{sec:char} of the present paper to yield various characterizations of correlated equilibria. It is worth noting that in \cite{hs:ece} the authors also consider part \eqref{item:char} of Corollary \ref{cor:correqmultiplierchar} as a candidate definition of correlated equilibria. They discard it is as not obviously capturing the game theoretic idea of correlated equilibrium, but we prove that it is nonetheless an equivalent definition in games with continuous utilities.
\item Stoltz and Lugosi study learning algorithms which converge to correlated equilibria in continuous games \cite{sl:lcegcss}. These algorithms have a game theoretic interpretation as avoiding ``regret'' in a repeated game setting. Each player can carry out these procedures separately without knowledge of his opponents' utilities. These are conceptual advantages over our methods, which merely aim for efficient computation.
However, these advantages come at a cost. The learning procedures require each player to solve a fixed point equation at each iteration. In general finding fixed points is as hard as finding Nash equilibria \cite{dgp:ccne,cd:nashcomp}, so these procedures do not seem to lead directly to efficient methods for computing correlated equilibria of continuous games.
There exist classes of fixed point equations which can be solved efficiently (i.e., the steady-state distribution of a Markov chain, which is defined by a linear program). To our knowledge there has been no work on whether the equations of \cite{sl:lcegcss} fall into such a class.
Furthermore, each of these learning algorithms either makes concavity-type assumptions about the utility functions, which we seek to avoid for modeling flexibility, or discretizes the players' strategy spaces a priori. We will see in Section \ref{sec:staticdisc} that such discretization without regard to the structure of the game can result in slow convergence.
However, we will make use of some of the tools which Stoltz and Lugosi have created. In particular, they consider replacing the class of all measurable departure functions with a smaller class, such as simple or continuous departure functions, and study when this yields an equivalent equilibrium notion. One result of this type is stated as Lemma \ref{lem:simplecorreq} below and used to prove our characterization theorems.
\item Germano and Lugosi prove the existence of correlated equilibria with small support in finite games \cite{gl:essce}. To prove this they analyze the extreme points of the set of correlated equilibria. Such an analysis cannot carry over directly to polynomial games because the set of correlated equilibria of polynomial games may have extreme points with arbitrarily large finite support or with infinite support \cite{sop:sece}. Support bounds for correlated equilibria of polynomial games are proven in \cite{s:mastersthesis} using similar tools, but assuming a finitely supported Nash equilibrium is on hand as a starting point. Since Nash equilibria are generally assumed to be harder to compute than correlated equilibria, these results do not apply in the present setting where the goal is efficient computation.
\item Separately from the literature on correlated equilibria, Dresher, Karlin, and Shapley study the structure of Nash equilibria in zero-sum games with polynomial or separable (polynomial-like) utility functions. They show how to cast separable games as finite-dimensional ``convex games'' by replacing the infinite-dimensional mixed strategy spaces with finite-dimensional spaces of moments \cite{dks:pg} and prove existence of equilibria via fixed point arguments \cite{dk:scgfp}. There always exist finitely supported equilibria in separable games as can be shown using the finite-dimensonality of the moment spaces. The rich geometry of these spaces is studied in \cite{ks:gms}. Most of these results as well as ad hoc methods for computing equilibria in simple cases are summarized in Karlin's book \cite{karlin:tig}. The authors of the present paper study generalizations and extensions of these results to nonzero-sum separable games in \cite{sop:slrcg}.
\end{itemize}
Second, our work is related to results from the optimization and computer science literature.
\begin{itemize}
\item Aumann showed that the set of correlated equilibria of a finite game is defined by polynomially many (in the size of the payoff tables) linear inequalities \cite{a:ceebr}. However, it was not clear whether this meant they could be computed in polynomial time. This question was settled in the affirmative when Khachian proved that linear programs could be solved in polynomial time; for an overview of this and other more efficient algorithms, see \cite{bt:ilo}. Papadimitriou extends this result in \cite{p:ccempg}, showing that correlated equilibria can be computed efficiently in many classes of games for which the payoffs can be written succinctly, even if the explicit payoff tables would be exponential in size.
\item The breakthrough in optimization most directly related to the work in this paper is the development of semidefinite programming, a far-reaching generalization of linear programming which is still polynomial-time solvable (for an overview, see the appendix and \cite{vb:sdp}). More specifically, the development of sum of squares methods has allowed many optimization problems involving polynomials or moments of measures to be solved efficiently \cite{p:phd}. Parrilo applies these techniques to efficiently compute Nash equilibria of two-player zero-sum polynomial games in \cite{pp:polygames}.
\end{itemize}
The remainder of this paper is organized as follows. In Section \ref{sec:char} we define the classes of games we study and correlated equilibria thereof, then prove several characterization theorems. We present algorithms for approximating sample correlated equilibria and the set of correlated equilibria of polynomial games in Section \ref{sec:comp}. Finally, we close with conclusions and directions for future work.
\section{Characterizations of Correlated Equilibria}
\label{sec:char}
In this section we will define finite and continuous games along with correlated equilibria thereof. We will present several known characterizations of correlated equilibria in finite games and show how these naturally extend to continuous games.
Some notational conventions used throughout are that subscripts refer to players, while superscripts are frequently used for other indices (it will be clear from the context when they represent exponents). If $S_j$ are sets for $j=1,\ldots,n$ then $S = \Pi_{j=1}^n S_j$ and $S_{-i} = \Pi_{j\neq i} S_j$. The $n$-tuple $s$ and the $(n-1)$-tuple $s_{-i}$ are formed from the points $s_j$ similarly. The set of regular Borel probability measures $\pi$ over a compact Hausdorff space $S$ is denoted by $\Delta(S)$. For simplicity we will write $\pi(s)$ in place of $\pi(\{s\})$ for the measure of a singleton $\{s\}\subseteq S$. All polynomials will be assumed to have real coefficients.
\subsection{Finite Games}
\label{subsec:finite}
We start with the definition of a finite game.
\begin{definition}
A \textbf{finite game} consists of \textbf{players} $i = 1,\ldots, n$, each of whom has a finite \textbf{pure strategy set} $C_i$ and a \textbf{utility} or \textbf{payoff function} $u_i: C\rightarrow \mathbb{R}$, where $C = \Pi_{j=1}^n C_j$.
\end{definition}
Each player's objective is to maximize his (expected) utility. We now consider what it would mean for the players to maximize their utility if their strategy choices were correlated. Let $R$ be a random variable taking values in $C$ distributed according to some measure $\pi\in\Delta(C)$. A realization of $R$ is a \textbf{pure strategy profile} (a choice of pure strategy for each player) and the $i^{\text{th}}$ component of the realization $R_i$ will be called the recommendation to player $i$. Given such a recommendation, player $i$ can use conditional probability to form a posteriori beliefs about the recommendations given to the other players. A distribution $\pi$ is defined to be a correlated equilibrium if no player can ever expect to unilaterally gain by deviating from his recommendation, assuming the other players play according to their recommendations.
\begin{definition}
\label{def:finitegamecorreq}A \textbf{correlated equilibrium} of a finite game is a joint probability measure $\pi\in\Delta(C)$ such that if $R$ is a random variable distributed according to $\pi$ then
\begin{equation*}
\label{eq:conditionalcorreq}
\mathbb{E}\left[u_i(t_i,R_{-i}) - u_i(R)\vert R_i = s_i\right] \equiv \sum_{s_{-i}\in C_{-i}} \prob(R = s | R_i = s_i)\left[u_i(t_i,s_{-i})-u_i(s)\right] \leq 0
\end{equation*}
for all players $i$, all $s_i\in C_i$ such that $\prob(R_i = s_i) > 0$, and all $t_i\in C_i$.
\end{definition}
While this definition captures the idea we have described above, the following characterization is easier to apply and visualize.
\begin{proposition}
\label{prop:finitecorreqchar}
A joint probability measure $\pi\in\Delta(C)$ is a correlated equilibrium of a finite game if and only if
\begin{equation}
\label{eq:finitecorreqcond}
\sum_{s_{-i}\in C_{-i}} \pi(s) \left[u_i(t_i,s_{-i}) - u_i(s)\right] \leq 0
\end{equation}
for all players $i$ and all $s_i,t_i\in C_i$.
\end{proposition}
This proposition shows that the set of correlated equilibria is defined by a finite number of linear equations and inequalities (those in \eqref{eq:finitecorreqcond} along with $\pi(s) \geq 0$ for all $s\in C$ and $\sum_{s\in C} \pi(s) = 1$) and is therefore convex and even polyhedral. It can be shown via linear programming duality that this set is nonempty \cite{hs:ece}. This can be shown alternatively by appealing to the fact that Nash equilibria exist and are the same as correlated equilibria which are product distributions.
We can think of correlated equilibria as joint distributions corresponding to recommendations which will be given to the players as part of an extended game. The players are then free to play any function of their recommendation as their strategy in the game.
\begin{definition}
A function $\zeta_i: C_i\rightarrow C_i$ is called a \textbf{departure function}.
\end{definition}
If it is a Nash equilibrium of this extended game for each player to play his recommended strategy (i.e. if no player has an incentive to unilaterally deviate from using the identity departure function), then the distribution is a correlated equilibrium. This interpretation is due to Aumann \cite{a:ceebr} and is justified by the following alternative characterization of correlated equilibria.
\begin{proposition}
\label{prop:finitecorreqchar2}
A joint probability measure $\pi\in\Delta(C)$ is a correlated equilibrium of a finite game if and only if
\begin{equation}
\label{eq:finitecorreqcond2}
\sum_{s\in C} \pi(s)\left[u_i(\zeta_i(s_i),s_{-i})-u_i(s)\right] \leq 0
\end{equation}
for all players $i$ and all departure functions $\zeta_i$.
\end{proposition}
For examples and more discussion of the basics of correlated equilibria, including the ideas behind the equivalence of these characterizations, see \cite{a:scrs,a:ceebr}.
\subsection{Continuous Games}
Again we begin with the definition of this class of games.
\begin{definition}
A \textbf{continuous game} consists of an arbitrary (possibly infinite) set $I$ of players $i$, each of whom has a pure strategy set $C_i$ which is a compact Hausdorff space and a utility function $u_i: C\rightarrow \mathbb{R}$ which is continuous.
\end{definition}
Note that any finite set forms a compact Hausdorff space under the discrete topology and any function out of such a set is continuous, so the class of continuous games includes the finite games. Another class of continuous games are the polynomial games, which are our primary focus when we study computation of correlated equilibria in the sections which follow. The theorems and proofs below can safely be read with polynomial games in mind, ignoring such topological subtleties as regularity of measures. However the extra generality of arbitrary continuous games requires little additional work in the proofs of the characterization theorems, so we will not formally restrict our attention to polynomial games here.
\begin{definition}
A \textbf{polynomial game} is a continuous game with $n < \infty$ players in which the pure strategy spaces are $C_i = [-1,1]$ for all players and the utility functions are polynomials.
\end{definition}
Defining correlated equilibria in continuous games requires somewhat more care than in finite games. Because of the technical difficulties of dealing with conditional distributions on continuous spaces, it is preferable not to formulate our new definition by generalizing Definition \ref{def:finitegamecorreq} directly. An obvious thing to try would be to replace the sum in Proposition \ref{prop:finitecorreqchar} with an integral and to choose that as the definition of correlated equilibria in continuous games. That would be simple enough, but this leads to a notion which is very weak and uninformative. Since we would be integrating over ``slices'' our candidate definition would be met, for example, by any continuous probability distribution regardless of the game chosen. Thus we have to use a different approach.
The standard definition of correlated equilibria in continuous games (as used in \cite{hs:ece}) instead follows Proposition \ref{prop:finitecorreqchar2}. In this case we must add the additional assumption that the departure functions be Borel\footnote{The Borel $\sigma$-algebra is the $\sigma$-algebra generated by the topology on $C_i$, which was assumed given in the definition of a continuous game.} measurable to ensure that the integrals are defined. For finite games this assumption is vacuous so this definition is equivalent to Definition \ref{def:finitegamecorreq}.
\begin{definition}
\label{def:correq}
A \textbf{correlated equilibrium} of a continuous game is a joint probability measure $\pi\in\Delta(C)$ such that
\begin{equation*}
\int\left[u_i(\zeta_i(s_i),s_{-i}) - u_i(s)\right]\,d\pi(s) \leq 0
\end{equation*}
for all $i$ and all Borel measurable departure functions $\zeta_i$.
\end{definition}
Before stating and proving alternative characterization theorems, we will discuss some of the difficulties of working with correlated equilibria in continuous games and this definition in particular. The goal here is to motivate the need for alternative characterizations.
The problem of computing Nash equilibria of polynomial games can be formulated exactly as a finite-dimensional nonlinear program or as a system of polynomial equations and inequalities \cite{sop:slrcg}. The key feature of the problem which makes this possible is the fact that it has an explicit finite-dimensional formulation in terms of the moments of the players' mixed strategies.
To see this, suppose that player $1$ chooses his action $x\in [-1,1]$ according to a mixed strategy $\sigma$ (a probability distribution over $[-1,1]$). Each player's utility function is a multivariate polynomial which only contains terms whose degree in $x$ is at most some constant integer $d$. Then regardless of how everyone chooses their strategies, their expected utility will only depend on $\sigma$ through the moments $\int x\,d\sigma(x),\int x^2\,d\sigma(x),\ldots, \int x^d\,d\sigma(x)$. Therefore player $1$ can switch from $\sigma$ to any other mixed strategy with the same first $d$ moments without affecting game play, and we can think of the Nash equilibrium problem as one in which each player seeks to choose moments which correspond to an actual probability distribution and form a Nash equilibrium.
On the other hand there is no exact finite-dimensional characterization of the set of correlated equilibria in polynomial games; for a counterexample see \cite{sop:sece}. Given the characterization of Nash equilibria in terms of moments, a natural attempt would be to try to characterize correlated equilibria in terms of the joint moments, i.e. the values $\int s_1^{k_1}\cdots s_n^{k_n}\,d\pi$ for nonnegative integers $k_i$ and joint measures $\pi$. In fact we will be able to obtain such a characterization below, albeit in terms of infinitely many joint moments. The reason this attempt fails to yield a finite dimensional formulation is that the definition of a correlated equilibrium implicitly imposes constraints on the conditional distributions of the equilibrium measure. A finite set of moments does not contain enough information about these conditional distributions to check the required constraints exactly. Therefore we also consider approximate correlated equilibria.
\begin{definition}
\label{def:epscorreq}
An \textbf{$\epsilon$-correlated equilibrium} of a continuous game is a joint probability measure $\pi\in\Delta(C)$ such that
\begin{equation*}
\int\left[u_i(\zeta_i(s_i),s_{-i}) - u_i(s)\right]\,d\pi(s) \leq \epsilon
\end{equation*}
for all $i$ and all Borel measurable departure functions $\zeta_i$. This definition reduces to that of a correlated equilibrium when $\epsilon = 0$.
\end{definition}
That is to say, $\epsilon$-correlated equilibria are distributions of recommendations in which no player can improve his expected payoff by more than $\epsilon$ by deviating from his recommendation unilaterally. Compare this definition to the main characterization theorem for $\epsilon$-correlated equilibria below (Theorem \ref{thm:correqmainchar}). This theorem shows that $\epsilon$-correlated equilibria can equivalently be defined by integrating the utilities against any sufficiently rich class of test functions, instead of by using measurable departure functions. Intuitively, the advantage of this characterization is that the product $f_i(s_i)u_i(t_i,s_{-i})$ is a ``simpler'' mathematical object than the composition $u_i(\zeta_i(s_i),s_{-i})$, especially when $f_i$, $t_i$, and $\zeta_i$ are allowed to vary. While this characterization does not have an obvious game theoretic interpretation, it allows us to compute correlated equilibria both algorithmically (Section \ref{sec:comp}) and analytically \cite{sop:sece}.
There also exist a variety of characterizations in which the departure functions are restricted to lie in a particular class (e.g., Lemma \ref{lem:simplecorreq} below and similar results in \cite{sl:lcegcss}) and no test functions are used. These characterizations have the advantages of conceptual simplicity and ease of interpretation. However, any characterization involving departure functions suffers from the difficulty that compositions of the utilities and the departure functions must be computed and these will likely be complex even if the departure functions are restricted to a simple class. The difficulty is magnified by the fact that even these restricted classes of departure functions are large and often difficult to parametrize in a way which is amenable to computation. Therefore it seems that departure function characterizations of correlated equilibria cannot be applied directly to yield effective computational procedures.
\begin{theorem}
\label{thm:correqmainchar}
A probability measure $\pi\in\Delta(C)$ is an $\epsilon$-correlated equilibrium of a continuous game if and only if for all players $i$, positive integers $k$, strategies $t_i^1,\ldots,t_i^k\in C_i$, and functions $f_i^1,\ldots,f_i^k: C_i\rightarrow [0,1]$ in one of the classes
\begin{enumerate}
\item \label{item:char} Weighted measurable characteristic functions,
\item \label{item:simp} Measurable simple functions (i.e., functions with finite range),
\item \label{item:meas} Measurable functions,
\item \label{item:cont} Continuous functions,
\item \label{item:poly} Squares of polynomials (if $C_i\subset \mathbb{R}^{k_i}$ for some $k_i$ for all $i$).
\end{enumerate}
such that $\sum_{j=1}^k f_i^j(s_i)\leq 1$ for all $s_i\in C_i$, the inequality
\begin{equation}
\label{eq:correqmainchar}
\sum_{j=1}^k\int f_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi \leq \epsilon
\end{equation}
holds.
\end{theorem}
To prove this, we need several approximation lemmas.
\begin{lemma}[A special case of Lemma $20$ in \cite{sl:lcegcss}]
\label{lem:simplecorreq}
Simple departure functions (those with finite range) suffice to define $\epsilon$-correlated equilibria in continuous games. That is to say, a joint measure $\pi$ is an $\epsilon$-correlated equilibrium if and only if
\begin{equation*}
\int\left[u_i(\xi_i(s_i),s_{-i}) - u_i(s)\right]\,d\pi(s) \leq \epsilon
\end{equation*}
for all players $i$ and all Borel measurable simple departure functions $\xi_i$.
\end{lemma}
\begin{proof}
The forward direction is trivial. To prove the reverse, first fix $i$. Then choose any measurable departure function $\zeta_i$ and let $\delta>0$ be arbitrary. By the continuity of $u_i$ and compactness of the strategy spaces there exists a finite open cover $U^1,\ldots,U^k$ of $C_i$ such that $s_i,s'_i\in U^j$ implies $\lvert u_i(s_i,s_{-i}) - u_i(s'_i,s_{-i})\rvert < \delta$ for all $s_{-i}\in C_{-i}$ and $j=1,\ldots, k$. Fix any $s_i^j\in U^j$ for all $j$. Define a simple measurable departure function $\xi_i$ by $\xi_i(s_i) = s_i^j$ where $j = \min \{l: \zeta_i(s_i)\in U^l\}$. Then $\lvert u_i(\zeta_i(s_i),s_{-i})-u_i(\xi_i(s_i),s_{-i})\rvert<\delta$ for all $s\in C$, so
\begin{equation*}
\begin{split}
\int & \left[u_i(\zeta_i(s_i), s_{-i})-u_i(s)\right]\,d\pi(s)
\leq \int\left[u_i(\xi_i(s_i),s_{-i}) + \delta -u_i(s)\right]\,d\pi(s)
\leq \epsilon + \delta.
\end{split}
\end{equation*}
Letting $\delta$ go to zero completes the proof.
\end{proof}
\begin{lemma}
\label{lem:generallusin}
If $C$ is a compact Hausdorff space, $\mu$ is a finite regular Borel measure on $C$, $f^1,\ldots,f^k: C\rightarrow [0,1]$ are measurable functions such that $\sum_{j=1}^k f^j \leq 1$, and $\delta > 0$, then there exist continuous functions $g^1,\ldots,g^k: C\rightarrow [0,1]$ such that $\mu(\{x\in C: f^j(x)\neq g^j(x)\}) < \delta$ for all $j$ and $\sum_{j=1}^k g^j \leq 1$.
\end{lemma}
\begin{proof}
We can apply Lusin's theorem which states exactly this result in the case $k = 1$ \cite{r:rca}. If $k > 1$, then we can apply the $k=1$ case with $\frac{\delta}{k}$ in place of $\delta$ to each of the $f^j$. Call the resulting continuous functions $\tilde{g}^j$. Then $\mu(\{x\in C:f^j(x)\neq \tilde{g}^j(x)\text{ for some }j\}) < \delta$. But $\sum_{j=1}^k f^j \leq 1$, so $\mu(\{x\in C: \sum_{j=1}^k \tilde{g}^j(x) > 1\}) <\delta$. Let $h(x) = \max\{1,\sum_{j=1}^k \tilde{g}^j(x)\}$ so $h: C\rightarrow [1,\infty)$ is a continuous map. Define $g^j(x) = \frac{\tilde{g}^j(x)}{h(x)}$. Then the $g^j$ are continuous, sum to at most unity, and are equal to the $f^j$ wherever all of the $\tilde{g}^j$ equal the $f^j$, i.e. except on a set of measure at most $\delta$.
\end{proof}
\begin{lemma}
\label{lem:contpolyapprox}
If $C\subset\mathbb{R}^d$ is compact, $f^1,\ldots,f^k: C\rightarrow [0,1]$ are continuous functions such that $\sum_{j=1}^k f^j \leq 1$, and $\delta > 0$, then there exist polynomials $p^1,\ldots,p^k: C\rightarrow [0,1]$ which are squares such that $\lvert f^j(x) - p^j(x)\rvert \leq \delta$ for all $x\in C$ and $\sum_{j=1}^k p^j \leq 1$.
\end{lemma}
\begin{proof}
By the Stone-Weierstrass theorem, any continuous function on a compact subset of $\mathbb{R}^d$ can be approximated by a polynomial arbitrarily well with respect to the sup norm. Approximating the square root of a nonnegative function $f$ using this theorem and squaring the resulting polynomial shows that a nonnegative continuous function on a compact subset of $\mathbb{R}^d$ can be approximated arbitrarily well by a square of a polynomial with respect to the sup norm.
Let $\tilde{p}^j$ be a square of a polynomial which approximates $f^j$ within $\frac{\delta}{2k}$ in the sup norm. Since $f^j$ takes values in $[0,1]$, $\tilde{p}^j$ takes values in $\left[0,1+\frac{\delta}{2k}\right]$. Let $p^j = \frac{\tilde{p}^j}{1+\frac{\delta}{2}}$. Then for all $x\in C$ we have $p^j(x)\leq \tilde{p}^j(x)$ and
\begin{equation*}
\tilde{p}^j(x) - p^j(x) = \tilde{p}^j(x) - \frac{\tilde{p}^j(x)}{1+\frac{\delta}{2}} = \tilde{p}^j(x)\left(\frac{\frac{\delta}{2}}{1+\frac{\delta}{2}}\right) \leq \left(1+\frac{\delta}{2k}\right)\left(\frac{\frac{\delta}{2}}{1+\frac{\delta}{2}}\right) \leq \frac{\delta}{2},
\end{equation*}
so $p^j(x)$ is within $\frac{\delta}{2}$ of $\tilde{p}^j(x)$ for all $x\in C$. By the triangle inequality $p^j$ approximates $f^j$ within $\delta$ in the sup norm. Furthermore for all $x\in C$ we have
\begin{equation*}
\sum_{j=1}^k p^j(x) = \frac{1}{1+\frac{\delta}{2}}\sum_{j=1}^k \tilde{p}^j(x) \leq \frac{1}{1+\frac{\delta}{2}}\sum_{j=1}^k \left(f^j(x)+\frac{\delta}{2k}\right)\leq \frac{1}{1+\frac{\delta}{2}}\left(1+\frac{\delta}{2}\right) = 1.\qedhere
\end{equation*}
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:correqmainchar}]
First we prove that if $\pi$ is an $\epsilon$-correlated equilibrium then \eqref{eq:correqmainchar} holds in the case where the $f_i^j$ are simple. We can choose a partition $B_i^1,\ldots,B_i^l$ of $C_i$ into disjoint measurable sets such that $f_i^j = \sum_{m=1}^l c_{jm}\chi_{B_i^m}$ where $c_{jm}\in [0,1]$ and $\chi_{B_i^m}$ denotes the indicator function which is unity on $B_i^m$ and zero elsewhere. Define a departure function $\zeta_i:C_i\rightarrow C_i$ piecewise on the $B_i^m$ as follows. If
\begin{equation*}
\int_{B_i^m\times C_{-i}}\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi
\end{equation*}
is nonnegative for some $j$ define $\zeta_i(s_i) = t_i^j$ for all $s_i\in B_i^m$ where $j$ is chosen to maximize the above integral. If the integral is negative for all $j$ define $\zeta_i(s_i) = s_i$ for all $s_i\in B_i^m$. Then we have
\begin{equation*}
\sum_{j=1}^k c_{jm}\int_{B_i^m\times C_{-i}}\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi\leq \int_{B_i^m\times C_{-i}}\left[u_i(\zeta_i(s_i),s_{-i})-u_i(s)\right]\,d\pi
\end{equation*}
for all $m$. Summing over $m$ and using the definition of an $\epsilon$-correlated equilibrium yields \eqref{eq:correqmainchar} in the case where the $f_i^j$ are simple.
Conversely suppose that \eqref{eq:correqmainchar} holds for all measurable simple functions. Let $\zeta_i: C_i\rightarrow C_i$ be any simple departure function. Let $t_i^1,\ldots,t_i^k$ be the range of $\zeta_i$ and $B_i^j = \zeta_i^{-1}(\{t_i^j\})$. Defining $f_i^j = \chi_{B_i^j}$, \eqref{eq:correqmainchar} says exactly that $\pi$ satisfies the $\epsilon$-correlated equilibrium condition for the departure function $\zeta_i$. By Lemma \ref{lem:simplecorreq}, $\pi$ is an $\epsilon$-correlated equilibrium.
Any simple function can be written as a sum of weighted characteristic functions, so by making several of the $t_i^j$ the same, we see that \eqref{eq:correqmainchar} for weighted characteristic functions is the same as \eqref{eq:correqmainchar} for simple measurable functions. If the inequality \eqref{eq:correqmainchar} holds for all simple measurable functions, a standard limiting argument proves that it holds for all measurable $f_i^j$, hence for all continuous $f_i^j$.
Suppose conversely that \eqref{eq:correqmainchar} holds for all continuous $f_i^j$. Fix any measurable $f_i^j$ satisfying the assumptions of the theorem. Define a signed measure $\pi_i^j$ on $C_i$ by $\pi_i^j(B_i) = \int_{B_i\times C_{-i}}\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi$. Let $\mu_i = \sum_{j=1}^k \lvert \pi_i^j\rvert$ and fix any $\delta > 0$. Then by the Lemma \ref{lem:generallusin} there exist continuous functions $g_i^j: C_i\rightarrow [0,1]$ which sum to at most unity and equal the $f_i^j$ except on a set of $\mu_i$ measure at most $\delta$. Therefore
\begin{equation*}
\begin{split}
&\left\lvert \sum_{j=1}^k\int f_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi - \sum_{j=1}^k\int g_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi \right\rvert \\ & \leq \sum_{j=1}^k\int \lvert f_i^j(s_i) - g_i^j(s_i)\rvert \,d\pi_i^j\leq 2k\delta,
\end{split}
\end{equation*}
so
\begin{equation*}
\sum_{j=1}^k\int f_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi \leq \epsilon + 2k\delta.
\end{equation*}
But $\delta$ was arbitrary, so \eqref{eq:correqmainchar} holds for all measurable $f_i^j$.
Finally assume $C_i\subset\mathbb{R}^{k_i}$ for some $k_i$. If \eqref{eq:correqmainchar} holds for all continuous $f_i^j$, then it holds for all squares of polynomials. Suppose conversely that it holds for all squares of polynomials. Let $f_i^j$ be any continuous functions satisfying the assumptions of the theorem and $\delta >0$. Let $p_i^j$ be polynomials squares which approximate the $f_i^j$ within $\delta$ in the sup norm and satisfy the assumptions of the theorem, as provided by Lemma \ref{lem:contpolyapprox}. Then
\begin{equation*}
\begin{split}
&\left\lvert \sum_{j=1}^k\int f_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi - \sum_{j=1}^k\int p_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi \right\rvert \\ & \leq \sum_{j=1}^k\int \lvert f_i^j(s_i) - p_i^j(s_i)\rvert \,d\pi_i^j\leq \delta\sum_{j=1}^k \int d\pi_i^j,
\end{split}
\end{equation*}
so
\begin{equation*}
\sum_{j=1}^k\int f_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]\,d\pi \leq \epsilon + \delta\sum_{j=1}^k \int d\pi_i^j.
\end{equation*}
But $\delta$ was arbitrary and the integrals on the right are finite, so \eqref{eq:correqmainchar} holds for all continuous $f_i^j$.
\end{proof}
Several simplifications occur when specializing Theorem \ref{thm:correqmainchar} to the $\epsilon = 0$ case, yielding the following characterization. We will use the polynomial condition of this corollary in Section \ref{subsec:moment} to develop algorithms for computing (approximate) correlated equilibria. The characteristic function condition is used to compute extreme correlated equilibria of an example game in \cite{sop:sece}.
\begin{corollary}
\label{cor:correqmultiplierchar}
A joint measure $\pi$ is a correlated equilibrium of a continuous game if and only if
\begin{equation}
\label{eq:correqdef3}
\int f_i(s_i)\left[u_i(t_i,s_{-i}) - u_i(s)\right]\,d\pi(s) \leq 0
\end{equation}
for all $i$ and $t_i\in C_i$ as $f_i$ ranges over any of the following sets of functions from $C_i$ to $[0,\infty)$:
\begin{enumerate}
\item Characteristic functions of measurable sets,
\item Measurable simple functions,
\item Bounded measurable functions,
\item Continuous functions,
\item Squares of polynomials (if $C_i\subset \mathbb{R}^{k_i}$ for some $k_i$ for all $i$).
\end{enumerate}
\end{corollary}
\begin{proof}
When $\epsilon = 0$ the $k = 1$ case of equation \eqref{eq:correqmainchar} implies the $k > 1$ cases. Furthermore $\epsilon = 0$ makes \eqref{eq:correqmainchar} homogeneous, so it is unaffected by positive scaling of the $f_i^j$, which allows us to drop the assumption $f_i\leq 1$.
\end{proof}
Theorem \ref{thm:correqmainchar} also has important topological implications for the structure of $\epsilon$-correlated equilibria. Recall that the weak* topology on the set of probability distributions $\Delta(C)$ over a compact Hausdorff space is the weakest topology which makes $\pi \mapsto \int f \,d\pi$ a continuous functional whenever $f: C\rightarrow\mathbb{R}$ is a continuous function.
\begin{corollary}
\label{cor:epscorreqcompact}
The set of $\epsilon$-correlated equilibria of a continuous game is weak* compact.
\end{corollary}
\begin{proof}
By the continuous test function condition in Theorem \ref{thm:correqmainchar}, the set of $\epsilon$-correlated equilibria is defined by conditions of the form $\int f \,d\pi \leq \epsilon$ where $f$ ranges over continuous functions of the form $\sum_{j=1}^k f_i^j(s_i)\left[u_i(t_i^j,s_{-i})-u_i(s)\right]$. By definition this presents the set of $\epsilon$-correlated equilibria as the intersection of a family of weak* closed sets. Hence the set of $\epsilon$-correlated equilibria is a closed subset of $\Delta(C)$. But $\Delta(C)$ is compact by the Banach-Alaoglu theorem \cite{r:fa}, so the set of $\epsilon$-correlated equilibria is compact.
\end{proof}
\begin{corollary}
\label{cor:limitcorreq}
If $\pi^k$ is a sequence of $\epsilon^k$-correlated equilibria and $\epsilon^k\rightarrow 0$, then the sequence $\pi^k$ has a weak* limit point\footnote{It is important to note that here we use the term limit point to refer to a limit point of a sequence, which is slightly different from a limit point of the underlying set of values which appear in the sequence. The difference is essentially that the singleton set $\{\pi\}$ has no limit points (in the sense of, say, \cite{m:t}), but we would like to say that $\pi$ is a limit point of the constant sequence $\pi,\pi,\pi,\ldots$. Rigorously, we say that $\pi$ is a limit point of the sequence $\pi^1,\pi^2,\pi^3,\ldots$ if for any neighborhood $U$ of $\pi$, there are infinitely many indices $i$ such that $\pi^i\in U$. Equivalently (at least in a Hausdorff space) $\pi$ is a limit point of the sequence if and only if $\pi$ appears infinitely often or $\pi$ is a limit point of the underlying set $\{\pi^k\vert k\in\mathbb{N}\}$.} and any such limit point is a correlated equilibrium.
\end{corollary}
\begin{proof}
If there is some $\pi$ such that $\pi^k = \pi$ for infinitely many $k$, then $\pi$ is a limit point of the sequence. Also $\pi$ is an $\epsilon^k$-correlated equilibrium for arbitrarily small $\epsilon^k$, so it is a correlated equilibrium and we are done. Otherwise, the sequence $\pi^k$ contains infinitely many points. The space $\Delta(C)$ with the weak* topology is compact by the Banach-Alaoglu theorem \cite{r:fa}, hence any infinite set has a limit point. Let $\pi\in\Delta(C)$ be a limit point of the sequence $\pi^k$. For any $\epsilon > 0$ there exists $k_0$ such that for all $k\geq k_0$, $\pi^k$ is an $\epsilon$-correlated equilibrium. The set $\Delta(C)$ is Hausdorff \cite{r:fa}, so $\pi$ is also a limit point of the set $\{\pi^k\}_{k\geq k_0}$. Since the set of $\epsilon$-correlated equilibria is compact by Corollary \ref{cor:epscorreqcompact}, the limit point $\pi$ must be an $\epsilon$-correlated equilibrium for all $\epsilon > 0$, i.e.\ a correlated equilibrium.
\end{proof}
Finally, we consider $\epsilon$-correlated equilibria which are supported on some finite subset. In this case, we obtain another generalization of Proposition \ref{prop:finitecorreqchar} which we will use in the algorithms presented in Section \ref{sec:adaptivedisc}.
\begin{proposition}
\label{prop:sampledepscorreqchar}
A probability measure $\pi\in\Delta(\tilde{C})$, where $\tilde{C} = \Pi_{j\in I} \tilde{C}_j$ is a finite subset of $C$, is an $\epsilon$-correlated equilibrium of a continuous game if and only if there exist $\epsilon_{i,s_i}$ such that
\begin{equation*}
\sum_{s_{-i}\in\tilde{C}_{-i}} \pi(s)\left[u_i(t_i,s_{-i}) - u_i(s)\right] \leq \epsilon_{i,s_i}
\end{equation*}
for all players $i$, all $s_i\in\tilde{C}_i$, and all $t_i\in C_i$, and
\begin{equation*}
\sum_{s_i\in\tilde{C}_i} \epsilon_{i,s_i} \leq \epsilon
\end{equation*}
for all players $i$.
\end{proposition}
\begin{proof}
If we replace $t_i$ with $\zeta_i(s_i)$ in the first inequality then sum over all $s_i\in\tilde{C}_i$ and combine with the second inequality, we get that
\begin{equation}
\label{eq:sampledepscorreq}
\sum_{s\in\tilde{C}} \pi(s)\left[u_i(\zeta_i(s_i),s_{-i}) - u_i(s)\right] \leq \epsilon
\end{equation}
holds for all $i$ and any function $\zeta_i:\tilde{C}_i\rightarrow C_i$. This is exactly the definition of an $\epsilon$-correlated equilibrium in the case when $\pi$ is supported on the finite set $\tilde{C}$.
Conversely if $\pi$ satisfies \eqref{eq:sampledepscorreq} for all $\zeta_i:\tilde{C}_i\rightarrow C_i$ then let
\begin{equation*}
\epsilon_{i,s_i} = \max_{t_i\in C_i}\sum_{s_{-i}\in\tilde{C}_{-i}} \pi(s)\left[u_i(t_i,s_{-i}) - u_i(s)\right].
\end{equation*}
For each $s_i\in\tilde{C}_i$, let $\zeta_i(s_i)$ be any $t_i\in C_i$ which achieves this maximum; such a $t_i$ exists by compactness and continuity. Substituting this $\zeta_i$ into \eqref{eq:sampledepscorreq} shows that $\pi$ satisfies the assumptions of the theorem.
\end{proof}
\section{Computing Correlated Equilibria}
\label{sec:comp}
We focus in this section on developing algorithms that can compute approximate correlated equilibria with arbitrary accuracy. We consider three types of algorithms, which we will illustrate in turn using the example below.
\savecounter{correqex1}
\begin{example}
\label{ex:correqex1}
Consider the polynomial game with two players, $x$ and $y$, each choosing their strategies from the interval $C_x = C_y = [-1,1]$. Their utilities are given by
\begin{equation*}
\begin{split}
u_x(x,y) & = 0.596x^2 + 2.072xy - 0.394y^2 + 1.360x
-1.200y + 0.554 \text{ and}\\
u_y(x,y) & = -0.108x^2 + 1.918xy - 1.044y^2 - 1.232x
+ 0.842y - 1.886.
\end{split}
\end{equation*}
The coefficients have been selected at random. This example is convenient, because as Figure \ref{fig:g1moment} shows, the game has a unique correlated equilibrium (the players choose $x=y=1$ with probability one). For the purposes of visualization and comparison, we will project the computed equilibria and approximations thereof into expected utility space, i.e. we will plot pairs $\left(\int u_x\,d\pi,\int u_y\,d\pi\right)$.
\end{example}
\subsection{Static Discretization Methods}
\label{sec:staticdisc}
The static discretization methods we present here are slow in practice and should be taken as a benchmark against which to compare the methods of later sections. The techniques in this section are general enough to apply to arbitrary continuous games with finitely many players, so we will not restrict our attention to polynomial games here.
The basic idea of static discretization methods is to select some finite subset $\tilde{C}_i \subset C_i$ of strategies for each player and limit his strategy choice to that set. Restricting the utility functions to the product set $\tilde{C} = \Pi_{i=1}^n \tilde{C}_i$ produces a finite game, called a \textbf{sampled game} or \textbf{sampled version} of the original continuous game. The simplest computational approach is then to consider the set of correlated equilibria of this sampled game. This set is defined by the linear inequalities in Proposition \ref{prop:finitecorreqchar} along with the conditions that $\pi$ be a probability measure on $\tilde{C}$.
The complexity of this approach in practice depends on the number of points in the discretization.
The question is then: what kind of approximation does this technique yield? In general the correlated equilibria of the sampled game may not have any relation to the set of correlated equilibria of the original game. The sampled game could, for example, be constructed by selecting a single point from each strategy set, in which case the unique probability measure over $\tilde{C}$ is automatically a correlated equilibrium of the sampled game but is a correlated equilibrium of the original game if and only if the points chosen form a pure strategy Nash equilibrium. Nonetheless, it seems intuitively plausible that if a large number of points were chosen such that any point of $C_i$ were near a point of $\tilde{C}_i$ then the set of correlated equilibria of the finite game would be ``close to'' the set of correlated equilibria of the original game in some sense, despite the fact that each set might contain points not contained in the other.
To make this precise, we will show how to choose a discretization so that the correlated equilibria of the finite game are $\epsilon$-correlated equilibria of the original game.
\begin{proposition}
\label{prop:staticdisc} Consider a continuous game with finitely many players, strategy sets $C_i$, and payoffs $u_i$. For any $\epsilon > 0$, there exists a finite open cover $U_i^1,\ldots,U_i^{l_i}$ of $C_i$ such that if $\tilde{C}_i\subseteq C_i$ is a finite set chosen to contain at least one point from each $U_i^l$, then all correlated equilibria of the finite game with strategy spaces $\tilde{C}_i$ and utilities $u_i|_{\tilde{C}}$ will be $\epsilon$-correlated equilibria of the original game.
\end{proposition}
\begin{proof}
Note that the utilities are continuous functions on a compact set, so for any $\epsilon > 0$ we can choose
a finite open cover $U_i^1,\ldots,U_i^{l_i}$ such that if $s_i$ varies within one of the $U_i^l$ and $s_{-i}\in C_{-i}$ is held fixed, the value of $u_i$ changes by no more than $\epsilon$. Let $\tilde{C}$ satisfy the stated assumption and let $\pi$ be any correlated equilibrium of the corresponding finite game. Then by Proposition \ref{prop:finitecorreqchar},
\begin{equation*}
\sum_{s_{-i}\in\tilde{C}_{-i}} \pi(s)\left[u_i(t_i,s_{-i}) - u_i(s)\right] \leq 0
\end{equation*}
for all $i$ and all $s_i,t_i\in\tilde{C}_i$. Any $t_i\in C_i$ belongs to the same $U_i^l$ as some $\tilde{t}_i\in\tilde{C}_i$, so
\begin{equation*}
\sum_{s_{-i}\in\tilde{C}_{-i}} \pi(s)\left[u_i(t_i,s_{-i}) - u_i(s)\right] \leq \sum_{s_{-i}\in\tilde{C}_{-i}} \pi(s)\left[u_i\left(\tilde{t}_i,s_{-i}\right) - u_i(s) + \epsilon\right] \leq \epsilon\sum_{s_{-i}\in\tilde{C}_{-i}} \pi(s) = \epsilon.
\end{equation*}
Therefore the assumptions of Proposition \ref{prop:sampledepscorreqchar} are satisfied with $\epsilon_{i,s_i} = \epsilon\sum_{s_{-i}\in\tilde{C}_{-i}}\pi(s)$.
\end{proof}
Though our primary goal here is to compute correlated equilibria, not prove existence, it is worth noting that Proposition \ref{prop:staticdisc}, Corollary \ref{cor:limitcorreq}, and the existence of correlated equilibria in finite games \cite{hs:ece} combine to prove the existence of correlated equilibria in continuous games with finitely many players. Indeed, this is proven in \cite{hs:ece} (along with the extension to an arbitrary set of players) with an argument along similar lines. One can view much of the contents of the present paper up to this point as expanding on this argument from \cite{hs:ece}.
The proof of Proposition \ref{prop:staticdisc} shows that if the utilities are Lipschitz functions, such as polynomials, then the $U_i^l$ can in fact be chosen to be balls with radius proportional to $\epsilon$. If the strategy spaces are $C_i = [-1,1]$ as in a polynomial game, then $\tilde{C}_i$ can be chosen to be uniformly spaced within $[-1,1]$. In this case $\epsilon = O\left(\frac{1}{d}\right)$ where $d=\max_i\left|\tilde{C}_i\right|$.
\usesavedcounter{correqex1}
\begin{example}[continued]
Figure \ref{fig:g1static} is a sequence of static discretizations for this game for increasing values of $d$, where $d$ is the number of points in $\tilde{C}_x$ and $\tilde{C}_y$. These points are selected by dividing $[-1,1]$ into $d$ subintervals of equal length and letting $\tilde{C}_x = \tilde{C}_y$ be the set of midpoints of these subintervals. For this game it is possible to show that the rate of convergence is in fact $\Theta\left(\frac{1}{d}\right)$ so the worst case bound on convergence rate is achieved in this example.
\end{example}
\setcounter{theorem}{\value{tempthm}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{fig1.pdf}
\caption{Computing a sequence of $\epsilon$-correlated equilibria of the game in Example \ref{ex:correqex1} by static discretization. Each point represents the (unique) correlated equilibrium of the finite game where players are restricted to strategies chosen from a finite set of $d$ strategies equally spaced in $[-1,1]$. The axes represent the utilities received by players $x$ and $y$. It can be shown that the convergence in this example happens at a rate $\epsilon = \Theta\left(\frac{1}{d}\right)$. This is slow enough that it is not obvious visually that the sequence of points is converging to the exact correlated equilibrium payoff, though we can prove that it is (e.g., by combining Proposition \ref{prop:staticdisc} with Figure \ref{fig:g1moment}, which shows that the equilibrium payoff is unique).}
\label{fig:g1static}
\end{figure}
In fact we can improve this convergence rate to $\epsilon = O\left(\frac{1}{d^2}\right)$ if we include the endpoints $\pm 1$ in $\tilde{C}_i$ as well and assume that the utilities have bounded second derivatives. We omit the proof for brevity.
\subsection{Adaptive Discretization Methods}
\label{sec:adaptivedisc}
\subsubsection{A family of convergent adaptive discretization algorithms}
\label{sec:convalgs}
In this section we consider continuous games with finitely many players and provide two algorithms (the second is in fact a parametrized family of algorithms which generalizes the first) to compute a sequence of $\epsilon^k$-correlated equilibria such that $\lim_{k\rightarrow\infty}\epsilon^k = 0$. By Corollary \ref{cor:limitcorreq} any limit point of this sequence is a correlated equilibrium. We will show that for polynomial games these algorithms can be implemented efficiently using semidefinite programming.
Informally, these algorithms work as follows. Each iteration $k$ begins with a finite set $\tilde{C}_i^k\subseteq C_i$ of strategies which each player $i$ is allowed to play with positive probability in that iteration; the initial choice of this set at iteration $k=0$ is arbitrary. We then compute the ``best'' $\epsilon$-correlated equilibrium in which players are restricted to use only these strategies, i.e., the one which minimizes $\epsilon$ (subject to some extra technical conditions needed to ensure convergence).
Given the optimal objective value $\epsilon^k$ and optimal probability distribution $\pi^k$, there is some player $i$ who can improve his payoff by $\epsilon^k$ if he switches from his recommended strategies to certain other strategies. We interpret these other strategies as good choices for that player to use to help make $\epsilon^k$ smaller in later iterations $k$. Therefore we add these strategies to $\tilde{C}_i^k$ to get $\tilde{C}_i^{k+1}$ and repeat this process for iteration $k+1$.
\begin{algorithm}
\label{alg:specificadaptivedisc}
Fix a continuous game with finitely many players. Let $k = 0$ and for each player fix a finite subset $\tilde{C}_i^0\subseteq C_i$.
\begin{itemize}
\item Let $\pi^k$ be an $\epsilon^k$-correlated equilibrium of the game having minimal $\epsilon^k$ subject to two extra conditions. First, $\pi^k$ must be supported on $\tilde{C}^k$. Second, we require that $\pi^k$ be an exact correlated equilibrium of the finite game induced when deviations from the recommended strategies are restricted to the set $\tilde{C}^k$, i.e. when we replace the condition $t_i\in C_i$ in Proposition \ref{prop:sampledepscorreqchar} with $t_i\in\tilde{C}_i^k$.
That is to say, let $\epsilon^k$ be the optimal value of the following optimization problem, and $\pi^k$ be an optimal assignment to the decision variables.
\begin{equation*}
\hskip -0.3in
\begin{array}{rl}
\text{minimize} & \epsilon \\
\text{subject to} & \\
\displaystyle\sum_{s_{-i}\in\tilde{C}_{-i}^k} \pi(s)\left[u_i(t_i,s_{-i}) - u_i(s)\right] \leq 0 & \text{for all } i\text{ and } s_i,t_i\in\tilde{C}_i^k \\
\displaystyle\sum_{s_{-i}\in\tilde{C}_{-i}^k} \pi(s)\left[u_i(t_i,s_{-i}) - u_i(s)\right] \leq \epsilon_{i,s_i} & \text{for all } i\text{, }s_i\in\tilde{C}_i^k \text{and }t_i\in C_i \\
\displaystyle\sum_{s_i\in\tilde{C}_i^k} \epsilon_{i,s_i} \leq \epsilon & \text{for all } i \\
\pi(s) \geq 0 & \text{for all } s\in\tilde{C} \\
\displaystyle\sum_{s\in\tilde{C}^k} \pi(s) = 1 &
\end{array}
\end{equation*}
\item If $\epsilon^k = 0$, terminate.
\item For each player $i$ for whom $\sum_{s_i\in\tilde{C}_i^k} \epsilon_{i,s_i} = \epsilon$, form $\tilde{C}_i^{k+1}$ from $\tilde{C}_i^k$ by adding in, for each $s_i\in\tilde{C}_i^k$ such that $\epsilon_{i,s_i}>0$, at least one strategy $t_i$ which makes
\[
\sum_{s_{-i}\in\tilde{C}_{-i}^k} \pi(s)\left[u_i(t_i,s_{-i}) - u_i(s)\right] = \epsilon_{i,s_i}.
\]
\item For all other players $i$, let $\tilde{C}_i^{k+1} = \tilde{C}_i^k$.
\item Let $k = k+1$ and repeat.
\end{itemize}
\end{algorithm}
Note that all steps of this algorithm are well-defined. First, the optimization problem is feasible. To see this let $\pi^k$ be any exact correlated equilibrium of the finite game with strategy spaces $\tilde{C}_i^k$ and utilities $u_i$ restricted to $\tilde{C}^k$; such an equilibrium exists because all finite games have correlated equilibria \cite{hs:ece}. The $u_i$ are bounded on $C$ (being continuous functions on a compact set), so by making $\epsilon$ and the $\epsilon_{i,s_i}$ large, we see that $\pi^k$ is a feasible solution of the problem.
Second, the optimal objective value is achieved by some $\pi^k$ because the space of probability measures on $\tilde{C}^k$ is compact, the constraints are closed, and $\epsilon$ is bounded below by zero.
Third, the set of new strategies added in the third bullet is nonempty. Suppose for a contradiction that this set were empty for each $i$ such that $\sum_{s_i\in\tilde{C}_i^k} \epsilon_{i,s_i} = \epsilon$ and each $s_i\in\tilde{C}_i^k$ such that $\epsilon_{i,s_i}>0$. By continuity of $u_i$ and compactness of $C_i$, the left-hand side of the $\epsilon$-correlated equilibrium constraint achieves its maximum as a function of $t_i\in C_i$. If this maximum value were less than $\epsilon_{i,s_i}$, then the value of $\epsilon_{i,s_i}$ could be decreased. If this could be done for all $i$ such that $\sum_{s_i\in\tilde{C}_i^k} \epsilon_{i,s_i} = \epsilon$ then $\epsilon$ itself could be decreased, contradicting optimality of $\pi^k$.
Fourth, this set of new strategies added in the third bullet consists only of strategies which are not in $\tilde{C}_i^k$ because we have the constraint that the deviations in utility be nonpositive for $t_i\in\tilde{C}_i^k$.
To show that Algorithm \ref{alg:specificadaptivedisc} converges, we will view it as a member of the following family of algorithms with the parameters set to $\alpha = 0$ and $\beta = 1$. Varying these parameters corresponds to adding some slack in the exact correlated equilibrium constraints and allowing some degree of suboptimality in the choice of strategies added to $\tilde{C}_i^k$ to form $\tilde{C}_i^{k+1}$. Such changes make little conceptual difference, but could be helpful in practice by making the optimization problem strictly feasible and allowing it to be solved to within a known fraction of the optimal objective value rather than all the way to optimality. We will prove that all algorithms in this family converge, that is, with these algorithms $\epsilon^k$ converges to zero in the limit.
\begin{algorithm}
\label{alg:generaladaptivedisc}
Fix a continuous game with finitely many players and parameters $0\leq \alpha < \beta \leq 1$. Let $k = 0$ and for each player fix a finite subset $\tilde{C}_i^0\subseteq C_i$.
\begin{itemize}
\item Choose $\epsilon^k$ to be the smallest number for which there exists $\pi^k$ such that:
\begin{itemize}
\item $\pi^k$ is a probability distribution supported on $\tilde{C}^k$,
\item $\pi^k$ is an $\epsilon^k$-correlated equilibrium of the game,
\item $\pi^k$ is not an $\epsilon$-correlated equilibrium for any $\epsilon < \epsilon^k$,
\item $\pi^k$ is an $\alpha\epsilon^k$-correlated equilibrium of the game when strategy deviations are restricted to $\tilde{C}^k$ (i.e., when the condition $t_i\in C_i$ is changed to $t_i\in\tilde{C}_i^k$ in Proposition \ref{prop:sampledepscorreqchar}).
\end{itemize}
\item If $\epsilon^k = 0$, terminate.
\item For at least one value of $i$, form $\tilde{C}_i^{k+1}$ from $\tilde{C}_i^k$ by adding strategies $t_{i,s_i} \in C_i$ such that
\begin{equation*}
\sum_{s\in\tilde{C}^k}\pi^k(s)\left[u_i(t_{i,s_i},s_{-i}) - u_i(s)\right] \geq \beta\epsilon^k.
\end{equation*}
\item For all other values of $i$, let $\tilde{C}_i^{k+1} = \tilde{C}_i^k$.
\item Let $k = k+1$ and repeat.
\end{itemize}
\end{algorithm}
\begin{proposition}
The steps of Algorithm \ref{alg:generaladaptivedisc} are well-defined.
\end{proposition}
\begin{proof}
It is not immediately obvious that the first step of the algorithm is well-defined, i.e., that a minimal $\epsilon^k$ (or any $\epsilon^k$ for that matter) satisfying these conditions exists. To see this let $\pi^{k,1}$ be an exact correlated equilibrium of the finite game induced when strategy deviations are restricted to $\tilde{C}^k$, and let $\epsilon^{k,1}\geq 0$ be the smallest value such that $\pi^{k,1}$ is an $\epsilon^{k,1}$-correlated equilibrium. Then the pair $(\pi^{k,1},\epsilon^{k,1})$ satisfies the four conditions under the first bullet above. This shows that the set of $\epsilon^k$ values satisfying these conditions is nonempty.
Choose some sequence $(\pi^{k,l},\epsilon^{k,l})$, $l = 1,2,\ldots$, of pairs satisfying these conditions such that the limit $\epsilon^k = \lim_{l\rightarrow\infty}\epsilon^{k,l}$ is the infimum over all $\epsilon^k$ values of pairs satisfying these conditions. Passing to a subsequence if necessary we can assume without loss of generality that the $\pi^{k,l}$ converge to some $\pi^k$. It is clear from the proof of Corollary \ref{cor:limitcorreq} that $\pi^k$ is an $\epsilon^k$-correlated equilibrium supported on $\tilde{C}^k$ which is an $\alpha\epsilon^k$-correlated equilibrium when deviations are restricted to $\tilde{C}^k$.
From Proposition \ref{prop:sampledepscorreqchar} we see that for a fixed support $\tilde{C}^k$, the minimal value of $\epsilon$ for which a probability measure $\pi$ on $\tilde{C}^k$ is a correlated equilibrium of the game varies continuously with the probabilities $\pi(s)$ for $s\in\tilde{C}^k$.
Therefore $\pi^k$ is not an $\epsilon$-correlated equilibrium for any $\epsilon<\epsilon^k$. Note that this final step depends crucially on the fact that $\tilde{C}^k$ is finite and fixed while $l$ varies. Also note that this subtlety disappears if $\alpha = 0$ because in that case it wouldn't matter if the limiting distribution had a smaller $\epsilon$ value. It is clear that the remaining steps of the algorithm are well-defined.
\end{proof}
\begin{theorem}
\label{thm:adaptconv}Fix a continuous game with finitely many players. Algorithms \ref{alg:specificadaptivedisc} and \ref{alg:generaladaptivedisc} converge to the set of correlated equilibria, i.e., they converge in the sense that $\epsilon^k\rightarrow 0$.
\end{theorem}
\begin{proof}
Suppose not, so there exists $\epsilon > 0$ and infinitely many values of $k$ such that $\epsilon^k\geq \epsilon$. For each $i$ let $B_i^1,\ldots, B_i^{l_i}$ be a finite open cover of $C_i$ such that $u_i(s_i,s_{-i}) - u_i(t_i,s_{-i}) \leq \frac{1}{2}(\beta - \alpha)\epsilon$ when $s_i$ and $t_i$ belong to the same set $B_i^l$ and $s_{-i}\in C_{-i}$. Such a cover exists by the compactness of the $C_i$ and the continuity of the $u_i$. There are finitely many sets $B_i^l$ so there is some iteration $k$, which we can take to satisfy $\epsilon^k\geq\epsilon$, such that for all $i$ all of the sets $B_i^l$ which will ever contain an element of $\tilde{C}_i^k$ at some iteration $k$ already do.
Note that $\pi^k$ is an $\alpha\epsilon^k$-correlated equilibrium when strategy choices are restricted to $\tilde{C}_i^k$, and $\epsilon^k > 0$ so we have $\beta\epsilon^k>\alpha\epsilon^k$. By the minimality of $\epsilon^k$, the set $\tilde{C}_i^{k+1}\setminus\tilde{C}_i^k$ is nonempty for some player $i$ (that is to say, it is always possible to perform the third step of the algorithm). Choose such an $i$ and $t_{i,s_i}\in \tilde{C}_i^{k+1}$ which satisfy
\begin{equation*}
\sum_{s\in\tilde{C}^k} \pi^k(s)\left[u_i(t_{i,s_i},s_{-i}) - u_i(s)\right] \geq \beta\epsilon^k.
\end{equation*}
By assumption, for any choice of $r_{i,s_i}\in\tilde{C}_i^k$ we have
\begin{equation*}
\sum_{s\in\tilde{C}^k} \pi^k(s)\left[u_i(r_{i,s_i},s_{-i}) - u_i(s)\right] \leq \alpha\epsilon^k,
\end{equation*}
so
\begin{equation*}
\sum_{s\in\tilde{C}^k}\pi^k(s)\left[u_i(t_{i,s_i},s_{-i}) - u_i(r_{i,s_i},s_{-i})\right] \geq (\beta - \alpha)\epsilon^k.
\end{equation*}
By construction of $k$, we can choose $r_{i,s_i}\in\tilde{C}_i^k$ to lie in the same set $B_i^l$ as $t_{i,s_i}$ for each $s_i\in\tilde{C}_i^k$. Thus
\begin{equation*}
\begin{split}
(\beta - \alpha)\epsilon & \leq (\beta - \alpha)\epsilon^k \\ & \leq \left\lvert \sum_{s\in\tilde{C}^k}\pi^k(s)\left[u_i(t_{i,s_i},s_{-i}) - u_i(r_{i,s_i},s_{-i})\right] \right\rvert \\ & \leq \sum_{s\in\tilde{C}^k}\pi^k(s)\left\lvert u_i(t_{i,s_i},s_{-i}) - u_i(r_{i,s_i},s_{-i})\right\rvert \\ &\leq \sum_{s\in\tilde{C}^k} \pi^k(s)\frac{(\beta - \alpha)\epsilon}{2} = \frac{(\beta - \alpha)\epsilon}{2},
\end{split}
\end{equation*}
a contradiction.
\end{proof}
Now we will illustrate Algorithm \ref{alg:specificadaptivedisc} on two examples.
\usesavedcounter{correqex1}
\begin{example}[continued]
In Figure \ref{fig:g1adaptive} we illustrate Algorithm \ref{alg:specificadaptivedisc} initialized with $\tilde{C}_x^0 = \tilde{C}_y^0 = \{0\}$. In this case convergence is obtained in three iterations, significantly faster than the static discretization method. The resulting strategy sets were $\tilde{C}_x^2 = \tilde{C}_y^2 = \{0,1\}$.
\end{example}
\setcounter{theorem}{\value{tempthm}
\begin{figure}
\centering
\includegraphics[width=.7\textwidth]{fig2.pdf}
\caption{Convergence of Algorithm \ref{alg:specificadaptivedisc} (note the change in scale from Figure \ref{fig:g1static}). At each iteration, the expected utility pair is plotted along with the computed value of $\epsilon$ for which that iterate is an $\epsilon$-correlated equilibrium of the game. In this case convergence to $\epsilon = 0$ (to within numerical error) occurred in three iterations.}
\label{fig:g1adaptive}
\end{figure}
\begin{example}
For a more complex illustration, we consider a polynomial game with three players, choosing strategies $x, y,\text{ and }z\in [-1,1]$. The utilities were chosen to be polynomials with terms up to degree $4$ in all the variables and the coefficients were chosen independently according to a normal distribution with zero mean and unit variance (their actual values are omitted). Algorithm \ref{alg:specificadaptivedisc} proceeds as in Table \ref{tab:3playerex}, which shows the value of $\epsilon^k$ and the new strategies added on each iteration. The terminal probability distribution $\pi^6$ does not display any obvious structure; in particular it is not a Nash equilibrium (product distribution).
\begin{table}
\begin{center}
\begin{tabular}{c|c|c|c|c}
$k$ & $\epsilon^k$ & $\tilde{C}_x^k\setminus \tilde{C}_x^{k-1}$ & $\tilde{C}_y^k\setminus\tilde{C}_y^{k-1}$ & $\tilde{C}_z^k\setminus\tilde{C}_z^{k-1}$ \\
\hline
$0$ & $0.99$ & $\{0\} $& $\{0\}$ & $\{0\}$ \\
$1$ & $4.16$ & & & $\{0.89\}$ \\
$2$ & $5.76$ & $\{-1\}$ & & \\
$3$ & $0.57$ & & $\{1\}$ & \\
$4$ & $0.28$ & $\{0.53\}$ & & $\{0.50,0.63\}$ \\
$5$ & $0.16$ & & $\{0.49,0.70\}$ & \\
$6$ & $10^{-7}$& & $\{-1,0.60\}$ & $\{-0.60,0.47\}$ \\
\end{tabular}
\end{center}
\caption{Output of Algorithm \ref{alg:specificadaptivedisc} on a three player polynomial game with utilities of degree $4$ and randomly chosen coefficients.}
\label{tab:3playerex}
\end{table}
\end{example}
\subsubsection{Implementing these algorithms with semidefinite programs}
\label{sec:imp}
To implement these algorithms for polynomial games, we must be able to do two things. First, we need to solve optimization problems with finitely many decision variables, linear objective functions and two types of constraints: nonnegativity constraints on linear functionals of the decision variables, and nonnegativity constraints on univariate polynomials whose coefficients are linear functionals of the decision variables. That is to say, we must be able to handle constraints of the form $p(t)\geq 0$ for all $t\in [-1,1]$, where the coefficients of the polynomial $p$ are linear in the decision variables. Second, we need to extract values of $t$ for which such polynomial inequalities are tight at the optimum.
Both of these tasks can be done simultaneously by casting the problem as a \textbf{semidefinite program (SDP)}. For an overview of semidefinite programs and a summary of the necessary results (both of which are classical), see the appendix.
In the optimization problem in Algorithm \ref{alg:specificadaptivedisc} we have a finite number of univariate polynomials in $t_i$ whose coefficients are linear in the decision variables $\pi(s)$ and $\epsilon_{i,s_i}$. We wish to constrain these coefficients to allow only polynomials which are nonnegative for all $t_i\in [-1,1]$. By Propositions \ref{prop:posintsos} and \ref{prop:sossdp} in the appendix this is the same as asking that these coefficients equal certain linear functions of matrices (i.e., sums along antidiagonals) which are constrained to be symmetric and positive semidefinite. Therefore we can write this optimization problem as a semidefinite program.
As a special case of convex programs, semidefinite programs have a rich duality theory which is useful for theoretical and computational purposes. In particular, SDP solvers keep track of feasible primal and dual solutions in order to determine when optimality is reached. It can be shown that the dual data obtained by an SDP solver run on this optimization problem will encode the values of $t_i$ making the polynomial inequalities tight at the optimum \cite{p:phd}.
The process of generating an SDP from the optimization problem in the algorithms above, solving it, and extracting an optimal solution along with $t_i$ values from the dual can all be automated. We have done so using the SOSTOOLS MATLAB toolbox for the pre- and post-processing and SeDuMi for solving the semidefinite programs efficiently \cite{sostools, sedumi}.
\subsubsection{A nonconvergent limiting case}
\label{sec:nonconvalgs}
Note that in the algorithms above the convergence of the sequence $\epsilon^k$ is not necessarily monotone. If we were to let $\alpha = \beta$ (a case we did not allow above), the sequence would become monotone nonincreasing. If we were to furthermore fix $\alpha = \beta = 1$, then the condition that $\pi$ be an exact (or $\alpha\epsilon^k$-) correlated equilibrium when deviations are restricted to $\tilde{C}_i^k$ would become redundant and could be removed.
These changes would simplify the behavior of Algorithm \ref{alg:generaladaptivedisc} conceptually as well as reducing the size of the SDP solved at each iteration, so we would like to adopt them if possible. However, the resulting algorithm may not converge, in the sense that $\epsilon^k$ may remain bounded away from zero
\begin{table}
\begin{center}
\begin{tabular}{c|c|c|c|}
& $a$ & $b$ & $c$ \\
\hline
$a$ & $0$ & $1$ & $0$ \\
\hline
$b$ & $1$ & $5$ & $7$ \\
\hline
$c$ & $0$ & $7$ & $0$ \\
\hline
\end{tabular}
\caption{A finite symmetric game with identical utilities for which Algorithm \ref{alg:generaladaptivedisc} with $\alpha = \beta = 1$ does not converge when started with strategy sets $\tilde{C}_1^0 = \tilde{C}_2^0 = \{a\}$.}
\label{tab:adaptivediscbad}
\end{center}
\end{table}
\begin{example}
Consider the game shown in Table \ref{tab:adaptivediscbad}, which is symmetric and has identical utilities for both players. Let $\tilde{C}_1^0 = \tilde{C}_2^0 = \{a\}$ and apply Algorithm \ref{alg:specificadaptivedisc}, but remove the condition that $\pi^k$ be an exact correlated equilibrium when deviations are restricted to $\tilde{C}_i^k$. The only probability distribution supported on $\tilde{C}^0$ is $\delta_{(a,a)}$ which has an objective value of $\epsilon^0 = 1$. It is easy to see that $\tilde{C}_i^1$ is formed by simply adding each player's best response to $a$, so that $\tilde{C}_1^1 = \tilde{C}_2^1 = \{a,b\}$. We will argue that the unique solution to the optimization problem in iteration $k = 1$ is also $\delta_{(a,a)}$, hence $\tilde{C}_i^2 = \tilde{C}_i^1$ and the algorithm gets ``stuck'', so that $\epsilon^k = \epsilon^0 = 1$ for all $k$.
For a probability distribution $\pi$, let $\pi^{T}$ denote $\pi$ with the players interchanged. By symmetry and convexity, if $\pi$ is an optimal solution then so is $\frac{\pi+\pi^T}{2}$, which is a symmetric probability distribution with respect to the two players. Hence an optimal solution which is symmetric always exists. We will parametrize such distributions by $\pi = p\delta_{(a,a)} + q\delta_{(a,b)} + q\delta_{(b,a)} + r\delta_{(b,b)}$, where $p,q,r\geq 0$ and $p + 2q + r = 1$. Define a departure function $\zeta: C_1\rightarrow C_1$ by $\zeta(a) = b$, $\zeta(b) = \zeta(c) = c$. Then for $\pi$ to be an $\epsilon$-correlated equilibrium it must satisfy the following condition:
\begin{equation*}
\begin{split}
\epsilon & \geq \sum_{s_1\in \tilde{C}_1^1} \epsilon_{1,s_1} \geq \sum_{s \in \tilde{C}^1} \pi(s)\left[u_1(\zeta(s_1),s_2) - u_1(s_1,s_2)\right] \\
& = p + 4q - q + 2r = 1 + q + r.
\end{split}
\end{equation*}
We know we can achieve $\epsilon = 1$ with $p = 1$ (i.e. $\pi = \pi^0 = \delta_{(a,a)}$), and this inequality shows that if $p < 1$ then $\epsilon > 1$. Therefore the minimal $\epsilon$ value in iteration $k = 1$ is unity and is achieved by $\pi = \delta_{(a,a)}$. Furthermore we have shown that this is the unique symmetric probability distribution which achieves the minimal value of $\epsilon$. Hence any other (not necessarily symmetric) optimal solution $\hat{\pi}$ satisfies $\frac{\hat{\pi} + \hat{\pi}^T}{2} = \delta_{(a,a)}$. But $\delta_{(a,a)}$ is an extreme point of the convex set of probability distributions on $\tilde{C}^1$, so we must in fact have $\hat{\pi} = \delta_{(a,a)}$. Therefore $\pi^1 = \pi^0 = \delta_{(a,a)}$ is the unique optimal solution on iteration $k = 1$, so the procedure must get stuck as claimed. That is, $\tilde{C}_i^k = \{a,b\}$ and $\epsilon^k = 1$ for all $k\geq 1$.
\end{example}
The same behavior can occur in polynomial games, as can be shown by ``embedding'' the above finite game in a polynomial game. For example, we can take $C_x = C_y = [-1,1]$ and
\begin{equation*}
\begin{split}
u_x(x,y) = u_y(x,y) =\ & (1-x^2)(3y^2 + 6y + 5) \\ +\ & (1-y^2)(3x^2+6x+5).
\end{split}
\end{equation*}
Then if $\tilde{C}_x^0 = \tilde{C}_y^0 = \{-1\}$ the same analysis as above shows that $\tilde{C}_x^k = \tilde{C}_y^k = \{-1,0\}$ and $\epsilon^k = 2$ for all $k\geq 1$.
\begin{example}
If we run Algorithm \ref{alg:specificadaptivedisc} on this polynomial game, the iterations proceed as in Table \ref{tab:2playerex}. The correlated equilibrium obtained in iteration $2$ is
\begin{equation*}
\begin{split}
\pi^2 = &\ 0.4922\delta(x=0,y=1) + 0.4922\delta(x=1,y=0) \\
&+ 0.0156\delta(x=1,y=1),
\end{split}
\end{equation*}
i.e., a probability of $0.4922$ is assigned to each of the outcomes $(x,y) = (0,1)$ and $(x,y) = (1,0)$ and a probability of $0.0156$ is assigned to $(x,y) = (1,1)$.
\begin{table}
\begin{center}
\begin{tabular}{c|c|c|c}
$k$ & $\epsilon^k$ & $\tilde{C}_x^k\setminus \tilde{C}_x^{k-1}$ & $\tilde{C}_y^k\setminus\tilde{C}_y^{k-1}$ \\
\hline
$0$ & $2$ & $\{-1\}$ & $\{-1\}$ \\
$1$ & $4$ & $\{0\}$ & $\{0\}$ \\
$2$ & $0$ & $\{1\}$ & $\{1\}$ \\
\end{tabular}
\end{center}
\caption{Output of Algorithm \ref{alg:specificadaptivedisc} for a polynomial game on which Algorithm \ref{alg:generaladaptivedisc} with $\alpha = \beta = 1$ does not converge to a correlated equilibrium.}
\label{tab:2playerex}
\end{table}
\end{example}
\subsection{Moment Relaxation Methods}
\label{subsec:moment}
In this section we again consider only polynomial games. The \textbf{moment relaxation methods} for computing correlated equilibria have a different flavor from the discretization methods discussed above. Instead of using tractable finite approximations of the correlated equilibrium problem derived via discretizations, we begin with the alternative exact characterization given in condition \ref{item:poly} of Corollary \ref{cor:correqmultiplierchar}. In particular, a measure $\pi$ on $C$ is a correlated equilibrium if and only if
\begin{equation}
\label{eq:correqpolychar}
\int p^2(s_i)\left[u_i(t_i,s_{-i}) - u_i(s)\right]\,d\pi(s) \leq 0
\end{equation}
for all $i$, $t_i\in C_i$, and polynomials $p$. If we wish to check all these conditions for polynomials $p$ of degree less than or equal to $d$, we can form the matrices
\begin{equation*}
S_i^d = \left[\begin{array}{ccccc}1 & s_i & s_i^2 & \cdots & s_i^d \\s_i & s_i^2 & s_i^3 & \cdots & s_i^{d+1} \\s_i^2 & s_i^3 & s_i^4 & \cdots & s_i^{d+2} \\\vdots & \vdots & \vdots & \ddots & \vdots \\s_i^d & s_i^{d+1} & s_i^{d+2} & \cdots & s_i^{2d}\end{array}\right].
\end{equation*}
Let $c$ be a column vector of length $d+1$ whose entries are the coefficients of $p$, so $p^2(s_i) = c' S_i^d c$. If we define
\begin{equation*}
M_i^d(t_i) = \int S_i^d\left[u_i(t_i,s_{-i}) - u_i(s)\right] \,d\pi(s),
\end{equation*}
then \eqref{eq:correqpolychar} is satisfied for all $p$ of degree at most $d$ if and only if $c' M_i^d(t_i) c\leq 0$ for all $c\in\mathbb{R}^{d+1}$ and $t_i\in C_i$, i.e. if and only if $M_i^d(t_i)$ is negative semidefinite for all $t_i\in C_i$.
The matrix $M_i^d(t_i)$ has entries which are polynomials in $t_i$ with coefficients which are linear in the joint moments of $\pi$. By Proposition \ref{prop:psdinterval} in the appendix, $M_i^d(t_i)$ is negative semidefinite for all $t_i\in [-1,1]$ for a given $d$ and a fixed $\pi$ if and only if there exists a certificate of a certain form proving this condition holds. We can write a semidefinite program (again, see Proposition \ref{prop:psdinterval} in the appendix) in which the decision variables represent such a certificate, so we can check this condition by solving the semidefinite program. As $d$ increases we obtain a sequence of semidefinite relaxations of the correlated equilibrium problem and these converge to the exact condition for a correlated equilibrium. That is to say, for a measure to be a correlated equilibrium it is necessary and sufficient that its moments be feasible for all of these semidefinite programs.
We can also let the measure $\pi$ vary by replacing the moments of $\pi$ with variables and constraining these variables to satisfy some necessary conditions for the moments of a joint measure on $C$ (see appendix). These conditions can be expressed in terms of semidefinite constraints and there is a sequence of these conditions which converges to a description of the exact set of moments of a joint measure $\pi$. Thus we obtain a nested sequence of semidefinite relaxations of the set of moments of measures which are correlated equilibria, and this sequence converges to the set of correlated equilibria
\usesavedcounter{correqex1}
\begin{example}[continued]
Figure \ref{fig:g1moment} shows moment relaxations of orders $d=0,1,\text{ and }2$. Since moment relaxations are outer approximations to the set of correlated equilibria (having been defined by necessary conditions which correlated equilibria must satisfy) and the $2^{\text{nd}}$ order moment relaxation corresponds to a unique point in expected utility space, all correlated equilibria of the example game have exactly this expected utility. In fact, the set of points in this relaxation is a singleton (even before being projected into utility space), so this proves that this example game has a unique correlated equilibrium.
\end{example}
\setcounter{theorem}{\value{tempthm}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{fig3.pdf}
\caption{Semidefinite relaxations approximating the set of correlated equilibrium payoffs. The second order relaxation is a singleton, so this game has a unique correlated equilibrium payoff (and in fact a unique correlated equilibrium).}
\label{fig:g1moment}
\end{figure}
\section{Future Work}
These results leave several open questions. For any continuous game, the set of correlated equilibria is nonempty, and this can be proven constructively as in \cite{hs:ece}. Under the same assumptions we can prove the existence of a Nash equilibrium, but the proof is nonconstructive, or at least does not seem to give an efficient algorithm for constructing an equilibrium \cite{sop:slrcg}. In the case of polynomial games, existence of a Nash equilibrium immediately gives existence of a finitely supported Nash equilibrium by Carath\'{e}odory's theorem, which is constructive \cite{sop:slrcg}. Therefore there exists a finitely supported correlated equilibrium of any polynomial game. Is there a constructive way to prove this fact directly, without going through Nash equilibria? Such a proof could potentially lead to a provably efficient algorithm for computing a sample correlated equilibrium of a polynomial game.
While the adaptive discretization and moment relaxation algorithms converge in general and work well in practice, we do not know of any results regarding rate of convergence. If we regard the probability distributions produced by these algorithms at the $k^{\text{th}}$ iteration as $\epsilon^k$-correlated equilibria, how fast does $\epsilon^k$ converge to zero?
Finally, we note that we have merely shown that the adaptive discretization algorithm converges to the set of correlated equilibria, not to a particular correlated equilibrium (of course it will do so along some subsequence by compactness). Could the algorithm be modified to converge to a single correlated equilbrium? Or even better, could one assure convergence to a correlated equilibrium with some desirable properties, such as one which maximizes the social welfare or (in the polynomial case) is finitely supported? This seems plausible given that the algorithm is itself optimization-based, but these problems remain open.
\section*{Acknowledgements}
The authors would like to thank Professor Muhamet Yildiz for a productive discussion which led to an early formulation of the characterization theorems in Section \ref{sec:char} as well as the moment relaxation methods presented in Section \ref{subsec:moment}. Figures were produced using the SeDuMi package for MATLAB \cite{sedumi}.
|
0812.2894
|
\section{Introduction}
The discovery of superconductivity in the Fe pnictides has opened an area of
research that is attracting considerable attention.\cite{Fe-SC, chen1, chen2, wen, chen3, 55, ren1, ren2}
In the early stages of these investigations, the layered structure of the Fe pnictides
superconductors,\cite{Fe-SC, chen1, chen2, wen, chen3, 55, ren1, ren2}
the existence of a magnetic spin striped state revealed by neutron scattering in the undoped limit,\cite{neutrons1,neutrons2}
and their large superconducting critical temperatures~\cite{Fe-SC, chen1, chen2, wen, chen3, 55, ren1, ren2}
motivated discussions on a possible close relation between these new
Fe-based materials and the high-temperature cuprate superconductors.
However, it was clear from the initial investigations that there were substantial differences as well: for example,
the resistivity vs. temperature curves of the parent compounds~\cite{Fe-SC, chen1, chen2, wen, chen3, 55, ren1, ren2} do not show the characteristic
Mott gapped behavior of, e.g., LaCuO$_4$. In fact LaOFeAs behaves
as a bad metal or semiconductor,\cite{Fe-SC, chen1, chen2, wen, chen3, 55, ren1, ren2} but not as an insulator. Moreover, the magnetic moment in the spin striped
state of LaOFeAs is
much smaller than expected.\cite{neutrons1,neutrons2}
Although further neutron scattering research has shown that the magnetic order parameters
are larger in other Fe pnictides,\cite{neutrons4}
their values are still below those anticipated from
band structure calculations~\cite{singh,first, xu, cao, fang2}
or from the large Hubbard $U$ limit of model Hamiltonians~\cite{daghofer} (unless couplings
are in a spin frustrated regime~\cite{si}). In summary, the parent compounds of the
Fe superconductors behave in a manner different
from the parent compounds of the Cu-oxide superconductors
because the zero temperature resistivity is finite and the magnetic order weak. However, the pnictides
are also different
from BCS materials, where the normal state is a non-magnetic metal with low resistivity.
Then, the Fe superconductors appear to be
in an {\it intermediate} regime of couplings, somewhere in between, e.g., MgB$_2$ and the Cu oxide
superconductors.\cite{Jaro,basov} The ``antiferromagnetic metallic''
nature of LaOFeAs is clearly different from an
antiferromagnetic insulator or a non-magnetic metal.
Further confirming the need to focus on the intermediate coupling regime for the Fe pnictides parent compounds,
a pseudogap has been observed~\cite{Y.Ishida,T.Sato,liu2,L.Zhao} in their density of states (DOS), which is
different from the featureless DOS of
a good metal or the gapped DOS of an insulator.
Much of the current theoretical
effort~\cite{daghofer,si,kuroki,mazin,FCZhang,han,korshunov,baskaran,yao,
xu2,plee,yildirim,scalapino,hu,zhou,lorenzana,sk,parish,choi,wang,yang,shi,2orbitals,calderon}
has focused thus far on two well defined limits. On one hand,
band structure calculations have reported the existence of Fermi pockets,\cite{singh,first, xu, cao, fang2}
which were confirmed by
photoemission
experiments.\cite{hashimoto,arpes,arpes2,C.Martin,T.Chen,parker,arpes3}
On the other hand,
intuition on the physics of the model Hamiltonians is often gained
by investigating the large $U$ regime.
Although results by some of us for a two-orbital model
using numerical techniques
have already shown that a small magnetic order can be accommodated
at intermediate couplings,\cite{daghofer}
this range of couplings
is typically the most difficult to handle using computer simulations
and, moreover, more bands are expected to be of relevance for a better quantitative
description of the Fe pnictides.
In this publication, multi-band Hubbard models are investigated
using mean-field and numerical techniques. Our most important result is the discovery of an
intermediate Hubbard $U$ coupling regime where the ground state is an antiferromagnetic metal.
More specifically, for $U$ larger than a critical value $U_{\rm c1}$
the spin striped order develops with continuity from zero.
In spite of a gap at particular momenta, the overall state remains metallic (there is a nonzero
weight at the chemical potential in the DOS) due
to the phenomenon of band overlaps. Further increasing the coupling to $U_{\rm c2}$, a fully gapped
insulator is stabilized. Thus, the intermediate regime $U_{\rm c1}$$<$$U$$<$$U_{\rm c2}$ is
simultaneously (i) magnetic with a small order parameter and (ii) metallic. Moreover,
the DOS reveals the existence of a pseudogap in this regime. All these properties are compatible with our
current knowledge of the Fe pnictides parent compounds.
While these results are interesting, they were obtained using mean-field approximations. For the
realistic case of four or five orbitals, it is difficult to obtain reliable
numerical results
to confirm the mean-field predictions.
However, for the case of two orbitals, calculations can be carried out using
simultaneously the mean-field technique, that also reveals
an intermediate coupling regime similar to that
of the four and five orbitals models,
together with the Exact Diagonalization (ED)\cite{RMP} and Variational Cluster Approximation (VCA)\cite{Aic03,Pot03} methods used
before.\cite{daghofer} Below, it is reported that the results using these
computational methods are compatible with
those of the mean-field approximation, providing
confidence that the mean-field method
may have captured the essence of the problem. However, it is certainly desirable that
future investigations confirm our main results.
For completeness, here some related previous literature is briefly mentioned.
The band overlap mechanism for an insulator to metal transition has been extensively studied before using band structure
calculations in a variety of contexts,
such as TlCl and TlBr,\cite{samara} solid hydrogen,\cite{hydrogen}
and bromine under high pressure.\cite{bromine}
Closer to the present results, the existence
of an intermediate Hubbard $U$ regime with an antiferromagnetic metallic
state was previously discussed by Duffy and one of the authors (A.M.)~\cite{duffy}
in the context of the high temperature Cu-oxide superconductors and using the one-band Hubbard
model, after introducing hopping terms $t'$ and $t''$ between next-nearest-neighbor sites. Density functional methods
were also used before to discuss AF-metallic states and band-overlap insulator-metal transitions.\cite{sander,callaway}
Within dynamical mean-field theory, an AF-metallic state has also been discussed.\cite{marcelo}
Experimentally, itinerant antiferromagnetic states were found in the pyrite NiS$_{2-x}$Se$_x$,\cite{miyasaka,niklowitz}
in heavily doped manganites,\cite{moritomo} in ruthenates,\cite{cao97}
in organic conductors,\cite{organic,organic2} and in several other materials.
An incommensurate spin density wave was also reported in
metallic V$_{2-y}$O$_3$.\cite{bao}
The results discussed in this manuscript establish an interesting connection
between the Fe pnictides and the materials mentioned in this paragraph.
The organization of the paper is as follows. In Section II,
results for a four-orbital model are presented. This
includes a discussion of the model, the mean-field technique, and the
results, with emphasis on the intermediate coupling
state. The photoemission predictions for this state are discussed. In
Section III, we show results for the two-orbital model using
ED\cite{RMP} and VCA methods\cite{Aic03,Pot03} in addition
to mean-field approximations.
Section IV contains our main conclusions.
\section{Results for a four-orbital model}
In this section, we will describe a possible minimal four-orbital model for
the Fe-based superconductors. This model presents a Fermi surface similar to that
obtained with band structure calculations.
\subsection{The four-orbital model}
Previous studies have suggested that the Fe-As planes are the most
important substructures of the full crystal that must be analyzed in order
to reproduce the physical properties of the Fe pnictides
close to the Fermi surface. We consequently focus on these
Fe-As planes in the present study.
The effective Fe-Fe hopping Hamiltonian, using As as a bridge,
can be obtained within the
framework of the Slater-Koster (SK) formalism.\cite{slater} By this
procedure, here we will construct a minimal model defined on a Fe square lattice,
consisting of the four Fe $d$ orbitals $xz$, $yz$, $xy$, and
$x^2-y^2$. It is assumed that the $d_{3z^2-r^2}$ orbital lies at a
substantially lower energy and is thus
always filled with two electrons.\cite{singh,first, xu, cao, fang2} While the SK procedure
is not as quantitatively accurate as a full band-structure calculation, it can
still provide the proper model Hamiltonian, because it correctly takes
into account the geometry of the system and illustrates which orbitals are
connected to one another at different lattice sites. Thus, our procedure here will be to use the SK method
to construct the formal model, and then obtain the actual numerical values of the
hopping parameters via comparison with band-structure calculations.
The Hamiltonian $H=H_0 + H_{\rm int}$ includes two parts: the hopping term $H_0$ and the
interaction term $H_{\rm int}$. The hopping term in real space reads
\begin{eqnarray}\label{E.H0r}
H_0 &=& \sum_{\bf i,j}\sum_{\mu,\nu}\sum_\sigma \left( T^{\mu,\nu}_{\bf i,j}
d^\dagger_{{\bf i},\mu,\sigma} d_{{\bf j},\nu,\sigma} + h.c. \right),
\end{eqnarray}
where $d^\dagger_{{\bf i},\mu,\sigma}$ creates an electron
at site ${\bf i}$
with spin $\sigma$ on the $\mu$-th orbital ($\mu=1,2,3,4$ stands for the
$xz$, $yz$, $xy$, and $x^2$-$y^2$ orbitals, respectively). Here, hoppings
at nearest-neighbors (NN) and also at next-nearest-neighbors (NNN) along the plaquette
diagonals were considered. The hopping
tensor $T^{\mu,\nu}_{\bf i,j}$ has a complicated real-space structure
that will not be reproduced here. $H_0$ has a
simpler form when transformed to momentum space:
\begin{eqnarray}\label{E.H0k}
H_0 &=& \sum_{\mathbf{k}} \sum_{\mu,\nu}\sum_\sigma T^{\mu,\nu}(\mathbf{k})
d^\dagger_{\mathbf{k},\mu,\sigma} d_{\mathbf{k},\nu,\sigma},
\end{eqnarray}
with
\begin{eqnarray}
T^{11} &=& -2t_2\cos k'_x -2t_1\cos k'_y -4t_3 \cos k'_x \cos k'_y, \label{eq:t11}\\
T^{22} &=& -2t_1\cos k'_x -2t_2\cos k'_y -4t_3 \cos k'_x \cos k'_y, \label{eq:t22}\\
T^{12} &=& -4t_4\sin k'_x \sin k'_y, \label{eq:t12}\\
T^{33} &=& -2t_5(\cos(k'_x+\pi)+\cos(k'_y+\pi)) \nonumber\\
& & -4t_6\cos(k'_x+\pi)\cos(k'_y+\pi) +\Delta_{xy}, \\
T^{13} &=& -4it_7\sin k'_x + 8it_8\sin k'_x \cos k'_y, \\
T^{23} &=& -4it_7\sin k'_y + 8it_8\sin k'_y \cos k'_x, \\
T^{44} &=& -2t_{17}(\cos(k'_x+\pi)+\cos(k'_y+\pi)) \nonumber\\
& & -4t_9\cos(k'_x+\pi)\cos(k'_y+\pi) +\Delta_{x^2-y^2}, \\
T^{14} &=& -4it_{10}\sin k'_y, \\
T^{24} &=& ~~4it_{10}\sin k'_x, \\
T^{34} &=& ~~0.
\end{eqnarray}
Equation~(\ref{E.H0k}) represents a
matrix in the basis $\{d^{\dagger}_{{\bf k},\mu,\sigma}\}$ where
${\bf k=k'}$ if $\mu=1$ or 2 and ${\bf k=k'+Q}$ if $\mu=3$ or 4 with
$-\pi<k_x, k_y\leq\pi$, ${\bf k'}$ is defined
in the reduced
Brillouin zone corresponding to the two-Fe unit cell, and
$\mathbf{Q}=(\pi,\pi)$.
In other words, the above expressions
couple states with momentum
$\mathbf{k'}$ for orbitals $xz$ and $yz$ to states with
momentum $\mathbf{k'+Q}$ for orbitals $xy$ and $x^2-y^2$.
The momentum $\mathbf{Q}$ appears
after considering the staggered location
of the As atoms above and below the plane defined by the Fe atoms.
As already mentioned,
the mathematical form of this model arises directly from
the Slater-Koster considerations.
This problem is equivalent to an eight-orbital model with
a Hamiltonian expanded in the basis $\{d^{\dagger}_{{\bf k'},\mu,\sigma},
d^{\dagger}_{{\bf k'+Q},\mu,\sigma}\}$.
As a result, the eight-orbital
band structure and Fermi surface (Fig.~\ref{F.Band-bis})
are obtained by ``folding'' the results in
the equivalent four-orbital problem (Fig.~\ref{F.BandU0.0}).
The actual values of the hopping parameters could in principle be
obtained from the overlap integrals in the SK
formalism.\cite{slater,Moreoetal08}
However, to properly reproduce the Fermi surface obtained in the
Local Density Approximation (LDA)~\cite{singh,first, xu, cao, fang2} it is better to fit
the values of those hoppings. The
parameters used, as well as the on-site energies $\Delta_\mu$ for the
$xy$ and $x^2-y^2$ orbitals, are
listed in Table~\ref{T.hoppara}. The on-site energy term is
given by $\sum_{{\bf i},\mu} \Delta_{\mu} n^{\mu}_{\bf i}$ (standard notation)
and it is part of the tight-binding Hamiltonian.
\begin{table}[h]
\caption{Fitted hopping parameters and on-site energies for the
four-orbital model used in this section (in eV units).}\label{T.hoppara} \centering
\vskip 0.3cm
\begin{tabular}{||c|c||c|c||}
\hline
$\Delta_{xy}$ & -0.600 & $\Delta_{x^2-y^2}$ & -2.000 \\
\hline
$t_1$ & ~0.500 & $t_2$ & ~0.150 \\
\hline
$t_3$ & -0.175 & $t_4$ & -0.200 \\
\hline
$t_5$ & ~0.800 & $t_6$ & -0.450 \\
\hline
$t_7$ & ~0.460 & $t_8$ & ~0.005 \\
\hline
$t_9$ & -0.800 & $t_{10}$ & -0.400 \\
\hline
$t_{17}$ & ~0.900 & & \\
\hline
\end{tabular}
\end{table}
\begin{figure}[h]
\begin{center}
\vskip -0.5cm
\centerline{\includegraphics[width=8.0cm,clip,angle=0]{BandDispU0e.eps}}
\vskip -0.7cm
\centerline{\includegraphics[width=6.5cm,clip,angle=0]{FermiSurfU0.eps}}
\vskip -0.7cm
\caption{(Color online) (a) Band structure corresponding to the four-orbital
tight-binding Hamiltonian Eq.~(\ref{E.H0k}), using the LDA fitted values of the
hopping parameters provided in Table~\ref{T.hoppara}. The Fermi surface
is formed by two hole-like bands ($\alpha_1$, $\alpha_2$) and
one electron-like band ($\beta_1$). The chemical potential is at
$0$. (b) The topology of the corresponding Fermi surface. }
\label{F.BandU0.0}
\vskip -1.0cm
\end{center}
\end{figure}
In Figs.~\ref{F.BandU0.0} and
\ref{F.Band-bis}, we show the band structure and the
corresponding Fermi surface for the four-orbital tight-binding
Hamiltonian in Eq.~(\ref{E.H0k}), using Table~\ref{T.hoppara}. Since we assume that the
$3z^2-r^2$ orbital is always doubly occupied, the chemical potential
in the undoped case is determined by locating $n=4$ electrons per site in the
four bands considered here. As shown in Fig.~\ref{F.BandU0.0}(b),
two hole pockets centered
at $(0,0)$ (arising from the $\alpha_1$ and $\alpha_2$ bands) and
four pieces of
two electron pockets centered at $(0,\pi)$ and $(\pi,0)$ (from the
$\beta_1$ band) are obtained. The shape of the Fermi surface
qualitatively reproduces the band-structure
LDA calculations~\cite{singh,first, xu, cao, fang2}
after a $45^o$ rotation about the center of the First Brillouin Zone
(FBZ), as presented in Fig.~\ref{F.Band-bis}(b), due to the rotation
from the Fe-Fe axis to the Fe-As axis.
\begin{figure}[h]
\begin{center}
\vskip -0.5cm
\centerline{\includegraphics[width=8.0cm,clip,angle=0]{Fig1c.eps}}
\vskip -1.0cm
\centerline{\includegraphics[width=6.5cm,clip,angle=0]{Fig1d.eps}}
\vskip -0.7cm
\caption{(Color online)
(a) Band structure of the eight-orbital problem in the reduced BZ
obtained by ``folding'' the results presented in Fig.~\ref{F.BandU0.0}(a).
(b) Fermi surface
of the eight-orbital problem obtained by folding the FS obtained in
Fig.~\ref{F.BandU0.0}(b). The FS in
the first (second) BZ are indicated by continuous (dashed) lines.
}
\label{F.Band-bis}
\vskip -1.0cm
\end{center}
\end{figure}
To study the relation between the orbital hybridization and the Fermi surface topology,
the projected weight of each orbital at both the hole and
electron pockets were calculated. These weights are defined via the eigenvectors
of $H_0$: $W_{\mu,\lambda} (\mathbf{k}) =
\frac{1}{2} \sum_\sigma |U_{\mathbf{k},\mu,\sigma;\lambda}|^2$,
where $\lambda$ denotes the band index ($\alpha_1, \alpha_2,
\beta_1, \beta_2$), and $\mu$ refers to the four $d$ orbitals. The matrix
$U_{\mathbf{k},\mu,\sigma;\lambda}$ diagonalizes the system (see Eq.~(\ref{diago}) below).
An example of the angle-resolved weights in momentum space are shown in
Fig.~\ref{F.Project}. The two hole pockets centered at
$(0,0)$ mostly arise from the $xz$ and $yz$ orbitals, compatible with LDA~\cite{singh,first, xu, cao, fang2} and with
much simpler descriptions
based only on two orbitals.\cite{scalapino,daghofer} The electron
pocket centered at $(\pi,0)$ ($(0,\pi)$) arises mainly from the
hybridization of the $xz$ ($yz$) and $xy$ orbitals (not shown). These results are also
qualitatively consistent with those from the first-principles
calculations.\cite{fang2} However, there are some quantitative discrepancies that lead us
to believe that probably longer-range than NNN
plaquette-diagonal hoppings are needed to fully reproduce the
LDA results including orbital weights. Nevertheless, the discussion below on the
metallic magnetic phase at intermediate couplings is robust,
and we believe it will survive when more complex
multi-orbital models are used in the future.
Note that the eigenenergies (band dispersion) along
the $(0,0)\rightarrow(\pi,0)$ and
$(0,0)\rightarrow(0,\pi)$ directions are symmetric about $(0,0)$, but the
eigenvectors ($W_{\mu,\lambda}$) show a large anisotropy. For
instance, at the Fermi level the $\alpha_1$ band is almost $xz$-like
along the $(0,0)\rightarrow(\pi,0)$ direction but almost $yz$-like
along the $(0,0)\rightarrow(0,\pi)$ direction. Below, it will be discussed how
this anisotropy affects the mean-field results for the
interacting system.
\begin{figure}[h]
\vskip -0.3cm
\centerline{\includegraphics[width=9cm,clip,angle=0]{pwa.eps}}
\vskip -0.5cm
\caption{(Color online) The projected orbital weight
$W_{\mu,\lambda}$ of states at the Fermi surface. Shown, as example, are results for the
outer hole pocket
centered at $(0,0)$.
The
definition of $\Theta$ is given in the inset.
}
\vskip -0.3cm
\label{F.Project}
\end{figure}
Let us now consider the interaction term,\cite{daghofer} which reads
\begin{eqnarray}\label{E.Hint}
H_{\rm int} &=& U\sum_{{\bf i},\mu}n_{{\bf i},\mu,\uparrow}n_{{\bf i},\mu,\downarrow}
+(U'-{J\over{2}})\sum_{{\bf i},\mu\neq\nu} n_{{\bf i},\mu}n_{{\bf i},\nu}
\nonumber \\
& & -2J\sum_{{\bf i},\mu\neq\nu}\mathbf{S}_{{\bf i},\mu}\cdot\mathbf{S}_{{\bf i},\nu},
\end{eqnarray}
where $\mathbf{S}_{{\bf i},\mu}$ ($n_{{\bf i},\mu}$) is the spin (charge
density) of orbital $\mu$ at site ${\bf i}$, and
$n_{{\bf i},\mu}=n_{{\bf i},\mu,\uparrow}+n_{{\bf i},\mu,\downarrow}$. The first term
is a Hubbard repulsion for the electrons in the same orbital. The
second term describes an on-site inter-orbital repulsion, where the
standard relation $U'=U-J/2$ caused by rotational invariance is used.\cite{RMP01} The last term in
Eq.~(\ref{E.Hint}) is a Hund term with a ferromagnetic coupling $J$.
A complete description would also require a
pair-hopping interaction similar to the last term of
Eq.~(\ref{eq:Hint2}), where the interaction term for the
two-orbital model is shown. But
ED was used to test its impact in the case of two orbitals,
and it was not found to be important. Consequently, it was neglected in the
mean field treatment.
\subsection{The mean-field approach}
To study the ground state properties of the system, we
apply a mean-field approximation to the model Hamiltonian described
by Eqs.~(\ref{E.H0r}) to (\ref{E.Hint}).
We follow here the simple standard assumption of considering only the
mean-field values for the diagonal operators:\cite{nomura}
\begin{eqnarray}\label{E.MFA}
\langle d^\dagger_{{\bf i},\mu,\sigma} d_{{\bf j},\nu,\sigma'}\rangle =
\left(n_\mu+\frac{\sigma}{2}\cos(\mathbf{q}\cdot\mathbf{r}_{\bf i})m_\mu\right)
\delta_{\bf ij}\delta_{\mu\nu}\delta_{\sigma\sigma'},
\end{eqnarray}
where $\mathbf{q}$ is the ordering vector of the possible magnetic
order. $n_\mu$ and $m_\mu$ are mean-field parameters describing the
charge density and magnetization of the orbital $\mu$, and the rest of the
notation is standard. Applying
Eq.~(\ref{E.MFA}) to $H_{\rm int}$, the mean-field Hamiltonian in
momentum space can be written as
\begin{eqnarray}\label{E.HMF}
H_{\rm MF} = H_0 + C + \sum_{\mathbf{k},\mu,\sigma}
\epsilon_\mu d^\dagger_{\mathbf{k},\mu,\sigma}
d_{\mathbf{k},\mu,\sigma}\nonumber\\
+ \sum_{\mathbf{k},\mu,\sigma} \eta_{\mu,\sigma}
(d^\dagger_{\mathbf{k},\mu,\sigma} d_{\mathbf{k+q},\mu,\sigma} +
d^\dagger_{\mathbf{k+q},\mu,\sigma} d_{\mathbf{k},\mu,\sigma}),
\end{eqnarray}
where $\mathbf{k}$ runs over the extended FBZ, $H_0$ is
the hopping term in Eq.~(\ref{E.H0k}),
\begin{eqnarray}
C=&-&NU\sum_{\mu}\left(n^2_\mu-\frac{1}{4}m^2_\mu\right)
- N(2U'-J)\sum_{\mu\neq\nu}n_\mu n_\nu \nonumber \\
&+& \frac{NJ}{4} \sum_{\mu\neq\nu} m_\mu m_\nu \nonumber
\end{eqnarray}
is a constant, $N$ the lattice size, and we used the definitions
\begin{eqnarray}
\epsilon_\mu = Un_\mu + (2U'-J)\sum_{\nu\neq\mu}
n_\nu, \\
\eta_{\mu,\sigma} =
-\frac{\sigma}{2}\left(Um_\mu+J\sum_{\nu\neq\mu}m_\nu\right).
\end{eqnarray}
The above mean-field Hamiltonian can be numerically solved for a fixed
set of mean-field parameters using standard
library subroutines.
The parameters $n_\mu$ and $m_\mu$ are obtained in a
self-consistent manner by minimizing the energy. In practice an
initial guess for $n_\mu$ and $m_\mu$ serves as a set of
input parameters for a given value of the
couplings $U$ and $J$.
The mean-field Hamiltonian is then diagonalized
and $n_\mu$ and $m_\mu$ are reevaluated using Eq.~(\ref{E.MFA}).
This procedure is iterated until both $n_\mu$ and $m_\mu$ have converged.
During the iterative procedure
$\sum_\mu n_\mu$=$4$ was enforced at each step, such that
the total charge density is a constant. Note
that in the mean-field approximation an electron with momentum
$\mathbf{k}$ is coupled to an electron with momentum $\mathbf{k+q}$,
where $\mathbf{q}$ is the vector associated with the magnetic ordering.
Then, the Hamiltonian in Eq.~(\ref{E.HMF}) is solved
only in the magnetic reduced Brillouin
Zone with only half of the size of the unfolded FBZ.
The numerical solution of the mean-field Hamiltonian immediately allows for the
evaluation of the band structure, the density of states (DOS), and the
magnetization ($M=\sum_\mu m_\mu$) at the ordering wavevector
$\mathbf{q}$. Moreover, we can also calculate the photoemission
spectral function. Assuming that the mean-field Hamiltonian
Eq.~(\ref{E.HMF}) is diagonalized by the unitary transformation
\begin{eqnarray}\label{diago}
d_{\mathbf{k},\mu,\sigma} = \sum_\lambda
U_{\mathbf{k},\mu,\sigma;\lambda} \gamma_\lambda, \\
H_{\rm MF} = \sum_{\lambda}
\rho_\lambda\gamma^\dagger_\lambda\gamma_\lambda,
\end{eqnarray}
then the spectral function is given by
\begin{eqnarray}\label{E.Akw}
A(\mathbf{k},\omega) = \sum_\lambda\sum_{\mu,\nu,\sigma}
U_{\mathbf{k},\mu,\sigma;\lambda}
U^\dagger_{\lambda;\mathbf{k},\nu,\sigma}
\delta(\omega-\rho_\lambda).
\end{eqnarray}
In practice $\delta(\omega-\rho_\lambda)$ was here substituted by
$\frac{1}{\pi}\frac{\varepsilon}{\varepsilon^2+(\omega-\rho_\lambda)^2}$
with a broadening $\varepsilon=0.05$~eV.
\subsection{Mean-field results}
In this subsection, the mean-field results for the
four-orbital model previously described are presented. Experimentally,
a (bad) metallic phase with spin-stripe magnetic order at
wavevectors $(0,\pi)$ was observed in the undoped compound
LaOFeAs.\cite{neutrons1,neutrons2} It is then important to investigate
in the mean-field approximation the properties of such a
spin striped ordered state. Here
we show the numerical results for the mean-field approximation defined
on a $100\times100$ square lattice.
\subsubsection{Magnetic order}\label{sec:megn_order4}
To study in more detail the spin striped ordered state, the
ordering wavevector $\mathbf{q}$ is assumed to be $(0,\pi)$ in the mean-field
approximation. In Fig.~\ref{F.MagMF}(a), the evolution of the
magnetization vs. $U$, at $J$=$U/4$, is shown.
At a critical value $U_{\rm c1}=1.90$~eV, the $(0,\pi)$
order~\cite{noteorder} starts to grow continuously from zero. This ``stripe'' magnetization increases
slowly until it reaches $U_{\rm c2}=3.75$~eV, where it changes
discontinuously, showing the characteristic of a
first-order transition (see discussion for the origin of this transition later in this section). Note that for $U\leqslant U_{\rm c2}$, the
stripe magnetization for the $(0,\pi)$ state is smaller than $1.0$ (with a normalization such
that the maximum possible value is 4.0),
indicating that there is an intermediate $U$ regime that can accommodate
the rather weak striped magnetic order found in the neutron scattering
experiments.\cite{neutrons1,neutrons2}
In Fig.~\ref{F.MagMF}(b), magnetization curves for
the same state at various values of $J$ are presented. Two
transitions for all $J/U$ ratios studied are observed,
similarly as for $J=U/4$ in
(a).
\begin{figure}[h]
\begin{center}
\vskip -0.6cm
\centerline{\includegraphics[width=9cm,clip,angle=0]{MagnU0p.eps}}
\vskip -0.7cm
\centerline{\includegraphics[width=9cm,clip,angle=0]{MvsJ.eps}}
\vskip -0.5cm
\caption{(Color online) (a) The mean-field evolution with $U$ of the magnetization of
the state ordered at wavevector $(0,\pi)$, with $J=U/4$. Shown are the two critical
values: one where the magnetization becomes nonzero and a second one where a discontinuous behavior is
concomitant with a metal to insulator transition, as described in the text. (b) The same magnetization
curve as in (a), but for several values of $J/U$. The magnetization shown in (a) and (b) is normalized such that
the maximum value is 4, corresponding to four spin polarized electrons, one per orbital, at each site.}
\vskip -0.7cm
\label{F.MagMF}
\end{center}
\end{figure}
\subsubsection{Band structure and Fermi surfaces}
Let us analyze in more detail the spin striped ordered state. Since
varying $J/U$ does not significantly alter the results,
the value $J=U/4$ is adopted in
the rest of the analysis. In Figs.~\ref{F.Band},~\ref{F.shadow}, and \ref{F.FS}
the band structures and Fermi surfaces of this state
at several values of $U$ are extracted from the calculated
mean-field photoemission spectral function data. Both the band structure and
the Fermi surface at the critical point $U_{\rm c1}=1.90$~eV are, of course, identical
to those at $U=0$ in Fig.~\ref{F.BandU0.0}. This
non-interacting electronic description of the system changes gradually upon the establishment of
the striped magnetic order. As discussed in more detail below, gaps open
at particular momenta, while other bands crossing the Fermi surface do not
open a gap. Thus, this is a metallic regime with magnetic order.
Finally, at the magnetization discontinuity a full
gap develops.
\begin{figure}[h]
\begin{center}
\centerline{\includegraphics[width=7.5cm,clip,angle=0]{akwU1.90_s.eps}}
\centerline{\includegraphics[width=7.5cm,clip,angle=0]{akwU2.50_s.eps}}
\centerline{\includegraphics[width=7.5cm,clip,angle=0]{akwU3.20_s.eps}}
\centerline{\includegraphics[width=7.5cm,clip,angle=0]{akwU4.00_s.eps}}
\caption{(Color online) Mean-field photoemission band structure of
the spin-striped $(0,\pi)$ state in the
energy window [-2~eV,~2~eV] at $U=1.90$~eV, $U=2.50$~eV, $U=3.20$~eV,
and $U=4.00$~eV, from top to bottom, with $J$=$U/4$. The first $U$ is at the
first transition, and still has the shape of the noninteracting limit.
The second two values of $U$ are in the intermediate coupling
regime, and the presence of states at energy $0$ (location of the chemical
potential) indicate a metallic state. The last coupling, $U=4.00$~eV,
is above the second critical point, and thus already
in the gapped regime. In the two intermediate couplings, 2.50 and 3.20 eV, weak
bands not present in the non-interacting limit (top panel) are revealed. These are the bands
caused by the striped magnetic order, which should be observable in photoemission experiments.}
\label{F.Band}
\vskip -0.9cm
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\centerline{\includegraphics[width=9cm,clip,angle=0]{shadowU2.50_s.eps}}
\caption{(Color online) Mean-field photoemission band structure of
the spin-striped $(0,\pi)$ state in the
energy window [-0.8~eV,~0.8~eV] at $U=2.50$~eV.
Bands caused by magnetic order
that are not present in the non-magnetic case $U=0.0$ are here shown
in more detail than in Fig.~\ref{F.Band}.
}
\label{F.shadow}
\vskip -0.9cm
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\centerline{\includegraphics[width=6cm,clip,angle=0]{akwFSU1.90s.eps}}
\centerline{\includegraphics[width=6cm,clip,angle=0]{akwFSU2.50s.eps}}
\centerline{\includegraphics[width=6cm,clip,angle=0]{akwFSU3.20s.eps}}
\caption{(Color online) Mean-field photoemission Fermi surfaces in
the spin-striped $(0,\pi)$ state at $U=1.90$~eV, $U=2.50$~eV, and
$U=3.20$~eV, from top to bottom. The results are obtained via
$A({\bf k},\omega)$ using a window of 20~meV centered at the Fermi
energy. Shown are results obtained from an equal weight average of
data using $A({\bf k},\omega)$ and $A({\bf p},\omega)$, where ${\bf
k}$=$(k_x,k_y)$ and ${\bf p}$=$(k_x,-k_y)$. By this procedure the
results are properly symmetrized under rotations in the non-magnetic
phase. The anisotropy of the results in the spin-striped
intermediate $U$ region do appear because the spin order breaks
rotational invariance.}\label{F.FS}
\end{center}
\end{figure}
\subsubsection{Bands of magnetic origin in the spin striped state}
Due to the off-diagonal term in Eq.~(\ref{E.HMF}), an electron with
momentum $\mathbf{k}$ is coupled to another with momentum
$\mathbf{k+q}$ if their orbital characters are the {\it same}. This generally
leads to avoided level crossings and opens a gap proportional to the
magnetization. If the hole and electron Fermi surfaces can be
connected by the vector $\mathbf{q}$, then a magnetic state with spin
ordering at
$\mathbf{q}$ is stabilized over the non-magnetic state due to the
Fermi surface nesting effect. The magnetic
state gains energy via the opening of a gap near the Fermi level. This
magnetically ordered state significantly changes the band structure
of the non-interacting case, opening gaps and giving
rise to the emergence of new bands of magnetic origin, sometimes called the
``shadow bands'', as shown in Fig.~\ref{F.Band}(b) and (c), and with more detail
for a special case in Fig.~\ref{F.shadow}. Similar issues were discussed in the
context of high temperature superconductors,
where the existence of bands caused by the staggered magnetic order were
extensively studied before.\cite{haas}
In fact, experiments for undoped cuprates revealed a photoemission
spectral function in excellent agreement with theoretical expectations,\cite{wells} i.e.
containing the predicted bands generated by the magnetic order.
Thus, it is to be expected that the spin striped
order of the Fe pnictides should also produce magnetically-induced
bands in the undoped
limit, and even in the doped case if the magnetic correlation length
remains large enough.
\subsubsection{The first order transition at $U_{\rm c2}$}
Regarding the second transition at $U_{\rm c2}$, the
qualitative reason for its presence lies in the incompatibility of the intermediate
coupling metallic state with the large $U$ limit. The mechanism of nesting that
causes the special features of the intermediate $U$ state previously discussed,
including the survival of portions of the Fermi surface,
will lead to an energy that eventually cannot compete with a fully gapped
state at large $U$ at the electronic density considered here, thus
a transition must eventually occur.
But why is the second transition discontinuous? In Fig.~\ref{F.Band}
the coupling between the hole pockets and the electron pocket
at $(0,\pi)$ leads to the distortion of the Fermi surface in the
spin-striped state. Such a coupling between states with a specific
orbital symmetry and momentum also
accounts for the metallic nature of the spin-striped state: the
electron pocket at $(\pi,0)$ is almost undistorted for $U<U_{\rm c2}$.
However, when $U$ is approaching
the second critical value $U_{\rm c2}$, the peak at $(\pi,\pi)$ with occupied states
in the $\alpha_1$ band becomes
energetically closer to
the Fermi level (its energy is increasing with $U$). If the Fermi surface nesting effect could be
neglected at $U_{\rm c2}$, then a smooth behavior would be observed since
the charges could transfer continuously from
the peak at $(\pi,\pi)$, after crossing the Fermi level,
to the peak at $(0,0)$. However, note that
the valley of $\beta_1$ band at $(\pi,0)$
and the peak of the $\alpha_1$ band at $(\pi,\pi)$ have both a
partial $xy$ symmetry. Moreover, they are connected by the
vector $\mathbf{q}=(0,\pi)$. Thus, it will be
expected that a gap
close to the Fermi level will open to minimize the energy. At
$U_{\rm c2}$, the system
gains maximal energy by opening a finite gap at both $(\pi,0)$
and $(\pi,\pi)$, and
lowering the energy of
the $\alpha$ bands at $(0,0)$
such that they become
fully occupied. This leads to discontinuous changes in the
population of the individual orbitals, producing discontinuous
changes in the orbital magnetizations and, concomitantly,
a finite gap.
Since in the real undoped Fe-pnictide materials the full-gap regime
is not realistic, then we can proceed with
the rest of the analysis below without further consideration of
this discontinuity in the magnetization.
\subsubsection{Anisotropic Fermi surface \\
in the undoped parent compound}
The appearance of spin-striped magnetism with magnetically-induced bands
leads to an {\it anisotropic}
distortion of the Fermi surface in the magnetically ordered state. For instance,
consider the $\mathbf{q}$=$(0,\pi)$ mean-field state. To predict the
pattern of gaps in this state, the information previously discussed in, e.g.,
Fig.~\ref{F.Project}
is important. Through the projected weights, it was observed that the $\beta_1$ electron
pocket centered at $(0,\pi)$ has mainly $yz$+$xy$ symmetry,
while the $(\pi,0)$ electron pocket is mainly $xz$+$xy$.
A dominance of the $yz$ symmetry is found in the $\alpha_1$ hole pocket along
the $(0,0)\rightarrow(0,\pm \pi)$ directions, and in the $\alpha_2$ hole pocket
along the $(0,0)\rightarrow(\pm \pi,0)$ directions.
Then, for the magnetic state with $\mathbf{q}$=$(0,\pi)$, a gap should open in
the $(0,0)\rightarrow(0,\pm \pi)$ directions for the $\alpha_1$ (inner hole pocket) band and
for the electron pocket at $(0,\pi)$, but the $\alpha_2$ (outer hole pocket) band remains gapless.
In addition, a gap opens in the $(0,0)\rightarrow(\pm \pi,0)$ directions for the
$\alpha_2$ band, but both the inner hole pocket and the $(\pi,0)$
electron pocket remains gapless.
These results are indeed observed numerically as shown in Figs.~\ref{F.Band}(b,c)
Due to these anisotropic gaps, the Fermi surfaces of the $\alpha$ bands
change their topology from two pockets Fig.~\ref{F.FS}(a) to four arcs that
are very close to one another, as
shown in Fig.~\ref{F.FS}(b). In between the arcs, there is actually a nonzero but
weak intensity at the location of the original hole pockets.
For LaOFeAs, since a weak striped order
has been observed, the Fermi surface is expected to have a similar shape
as that shown in Fig.~\ref{F.FS}(b). However, topology of the Fermi
surface in this state is sensitive to the value of the coupling $U$:
further increasing this coupling
the four arcs merge into a single pocket Fig.~\ref{F.FS}(c).
Then,
ARPES experiments can provide valuable information about the exact
shape of the Fermi surface and, thus, the interaction strength.
Besides the anisotropic gaps near the
Fermi level, gaps far from the Fermi level also exists (see, for instance,
Fig.~\ref{F.Band}(b)). This cannot occur in a
one-band model because the opening of these gaps would not provide any
energy gain. However, such gaps are possible in more complex multi-orbital
models: since the off-diagonal term in Eq.~(\ref{E.HMF}) depends on the
magnetization that is contributed by each orbital, once the magnetic state
is stabilized by the opening of a gap near the Fermi level, the
off-diagonal term becomes non-zero even for the bands far from the Fermi
level.
A rather surprising result is that the intensities of the spectral
function of the two $\alpha$ pockets display an anisotropy even
in the non-magnetic phase (not shown).
This is puzzling since from Fig.~\ref{F.Project}, it can be observed that $\sum_{\mu}
W_{\mu,\lambda} (\mathbf{k})= 1$ for any $\mathbf{k}$. Naively, this
would lead to an isotropic $A(\mathbf{k},\omega)$. However, from
Eq.~(\ref{E.Akw}), $A(\mathbf{k},\omega)= \sum_{\lambda}
\delta(\omega-\rho_\lambda) W'_{\lambda}(\mathbf{k})$, where
\begin{eqnarray}\label{E.Wp}
W'_{\lambda}(\mathbf{k}) =
\sum_{\sigma}|\sum_{\mu}U_{\mathbf{k},\mu,\sigma;\lambda}|^2.
\end{eqnarray}
Here, note that an anisotropy could arise from the interference between
different orbitals. To observe this more explicitly, $W'_\lambda$ for
the two $\alpha$ hole pockets at $U=0$ is shown
in Fig.~\ref{F.Wp}. These functions are not
constant but oscillate between $0$ and $2$. From Fig.~\ref{F.Wp}, it is
observed that for the $\alpha_1$ band $W'$ reaches its maximum at, e.g.,
$-\pi/4$, corresponding to the $(0,0)\rightarrow(-\pi,\pi)$
direction, while for the $\alpha_2$ band, the maximum of $W'$ is at, e.g.,
$\pi/4$, i.e., in the $(0,0)\rightarrow(\pi,\pi)$ direction.
Such an anisotropy in $A(\mathbf{k},\omega)$ is not a
consequence of the mean-field approximation, but is related to the
multi-band nature of the model itself.
Interestingly, such an anisotropy in the topology of the Fermi surface
in the $A(\mathbf{k},\omega)$ data may account for the anisotropic
features of the hole pocket in the recent ARPES experiment on
LaOFeP,\cite{LaOFeP_ARPES} where a long-ranged striped magnetic order is
absent.\cite{LaOFeP_SDW}
However,
in Fig.~\ref{F.FS} this problem is avoided by a symmetrization procedure (see
caption of Fig.~\ref{F.FS}) that restores rotational invariance to the non-magnetic
state. Thus, the anisotropy of the important intermediate $U$ regime fully originates
in the lack of rotational invariance of the spin striped state with a wavevector
$(0,\pi)$ or $(\pi,0)$.
\begin{figure}
\begin{center}
\vskip -0.3cm
\centerline{\includegraphics[width=9cm,clip,angle=0]{pwp.eps}}
\vskip -0.5cm
\caption{(Color online) The angular resolved weight $W'$ at $U$=0
obtained using
Eq.~(\ref{E.Wp}) for the two $\alpha$ hole-like pockets. The
definition of $\Theta$ is the same as in
Fig.~\ref{F.Project}.}
\vskip -0.7cm
\label{F.Wp}
\end{center}
\end{figure}
\subsubsection{Metallic magnetically ordered phase at intermediate couplings and
existence of a pseudogap}
As already remarked,
Fig.~\ref{F.Band} indicates that for moderate $U$ the
magnetically ordered system is still gapless, i.e. it is in a metallic
phase with a finite Fermi surface due to the phenomenon
of ``band overlapping''
described in the Introduction. This intermediate state is also revealed via the
evolution of the DOS in Fig.~\ref{F.DOSMF}. As displayed in
Fig.~\ref{F.DOSMF}(a), a pseudogap in the DOS near the chemical
potential exists in the regime $U_{\rm c1}<U<U_{\rm c2}$. If the availability
of states at the Fermi level is assumed to be directly related with transport
properties, a DOS pseudogap suggests bad-metallic
characteristics in the intermediate $U$ regime. The pseudogap turns into a hard gap
at $U>U_{\rm c2}$, where the system becomes an insulator.
In Fig.~\ref{F.DOSMF}(b) the value of the DOS at the
chemical potential vs. $U$ is plotted.
Two transitions can be easily identified. For
$U<U_{\rm c1}$, $N(\mu)$ is a constant. It decreases continuously
for $U>U_{\rm c1}$, indicating a second-order transition from the
paramagnetic metallic phase to the metallic phase with striped
magnetic order. At $U>U_{\rm c2}$, it drops abruptly to a small value,
corresponding to the formation of a full gap in the insulating
phase (the finite DOS at large $U$ is simply caused by the
artificial broadening of delta functions to plot results).
\begin{figure}[h]
\begin{center}
\vskip -0.5cm
\centerline{\includegraphics[width=9.0cm,clip,angle=0]{dosU.eps}}
\vskip -0.7cm
\centerline{\includegraphics[width=9.0cm,clip,angle=0]{dosmuU.eps}}
\vskip -0.5cm
\caption{(Color online) (a) Density of states at various values of
$U$ showing the development of a pseudogap at the chemical potential
in the spin-striped $(0,\pi)$ state. (b) The evolution of $N(\omega)$, the DOS at the
chemical potential, indicates three well separated regions.
The non-zero values of $N(\omega=\mu)$ in the
insulating phase at $U>U_{\rm c2}$ arises from the finite broadening of
the raw DOS numerical data.}
\vskip -0.5cm
\label{F.DOSMF}
\end{center}
\end{figure}
\subsubsection{Stability of the striped phase}
Consider now the stability of the striped
spin-order state by analyzing two possible magnetically ordered
states, one with $\mathbf{q}$=$(0,\pi)$ and another one with
$(\pi,\pi)$. The $U$
dependence of the magnetization and the energy difference
between these two states are presented in Fig.~\ref{F.MagMF2}. The
$(\pi,\pi)$ order appears at $U\approx2.5$~eV, higher than the $U_{\rm c1}$
for the $(0,\pi)$ state. However, with increasing $U$, the $(\pi,\pi)$
staggered magnetization increases much
faster than for the striped state. As shown in the inset
of Fig.~\ref{F.MagMF2}, the $(\pi,\pi)$ state becomes more stable than the
$(0,\pi)$ state for $U\geqslant2.65$~eV. We have analyzed several other sets
of hopping parameters and ratios $J/U$ and the results are qualitatively the same:
the spin stripes are stable, but in a narrow region of couplings.
\begin{figure}[h]
\begin{center}
\vskip -0.3cm
\centerline{\includegraphics[width=9cm,clip,angle=0]{MagnU.eps}}
\vskip -0.4cm
\caption{(Color online) Main panel: the evolution of the magnetization
with $U$ for $\mathbf{q}$=$(0,\pi)$ and $(\pi,\pi)$. Inset: Energy
difference between the two states of the main panel.}
\vskip -0.5cm
\label{F.MagMF2}
\end{center}
\end{figure}
Hence, in our model the stripe-order ground state becomes unstable to
another antiferromagnetic state with $\mathbf{q}$=$(\pi,\pi)$ at
$U\geqslant2.65$~eV through a first-order transition. This is not too
surprising considering the values of the hopping parameters: the NN
and NNN hoppings are similar in magnitude. Thus, there is a competition
between different spin tendencies, as it occurs at intermediate couplings
in Heisenberg spin systems with
NN and NNN terms.
However, the neutron scattering experiments
show that the ground state of several undoped Fe-pnictides presents {\it only} the
striped spin order. To understand this predominance of one state
over the other, note that it has been argued that the emergence of the striped state is
closely related to a structural phase
transition~\cite{neutrons1,neutrons2} from space group $P4/nmm$ to
$P112/n$. The lattice distortion at the
transition breaks the four-fold rotational symmetry and lifts the
degeneracy of the $xz$ and $yz$ orbitals. Thus, according to recent
calculations that incorporate the lattice distortion, the system prefers the
$(0,\pi)$ state over the $(\pi,\pi)$ state.\cite{yildirim}
Since the effects of lattice distortions are not
included in our model, then the $(\pi,\pi)$ spin-ordered state
apparently strongly competes with the spin stripes, but this is misleading
and caused by the absence of lattice energetic considerations.
This discussion leads us to believe that simply analyzing the
spin-stripe state found in the mean-field approximation should be sufficient
to understand some of the electronic properties of the real system. However,
it is remarkable that even in the competing $(\pi,\pi)$
ordered state there are also two magnetically-ordered phases: a metallic phase for
moderate $U$ values, and an insulating phase with a finite gap at
larger $U$ values (see Fig.~\ref{F.BandG}).
This qualitatively agrees with the previous analysis
for the striped ordered phase. This suggests that the existence of
a metallic magnetically ordered phase at moderate $U$ is
an intrinsic property of the multi-orbital Hubbard model, treated in the mean-field
approximation, that is robust
varying the interactions, as discussed in more detail in the next subsection.
\begin{figure}[h]
\begin{center}
\centerline{\includegraphics[width=8.5cm,clip,angle=0]{akwGU2.75_s.eps}}
\centerline{\includegraphics[width=8.5cm,clip,angle=0]{akwGU4.50_s.eps}}
\caption{(Color online) Mean-field band structures of the spin-ordered
$(\pi,\pi)$ state in the
energy window [-2~eV,~2~eV]. The upper panel is obtained
at $U=2.75$~eV where band overlap indicates a metallic state, even if the
order parameter is nonzero. The lower panel is at
$U=4.50$~eV where the gap is fully developed.}
\vskip -0.9cm
\label{F.BandG}
\end{center}
\end{figure}
\subsubsection{Mean-field results for other models \\
with several active orbitals}
We have applied the mean-field technique to several other
multi-orbital models such as a five-orbital model,\cite{kuroki} another
four-orbital model,\cite{korshunov} and an effective
three-orbital model including the $xz$, $yz$, and $xy$ orbitals.\cite{plee} For all
these multi-band models a Coulombic interaction term similar to
Eq.~(\ref{E.Hint}) has been used. A robust conclusion of our mean-field analysis is that a
transition from a paramagnetic metal to a metallic striped
spin-order state is found in all the models considered here.
Let us discuss the results for the five-orbital model.\cite{kuroki}
This model does not give precisely the correct Fermi surface
topology in the undoped case, since it contains a pocket at $(\pi,\pi)$.
This problem can be fixed by slightly modifying the electronic concentration to, e.g., $n=6.2$.
However, in our study we will consider $n=6.0$ for consistency
with the rest of the analysis.
Within the mean-field approximation, the magnetization at
$n=6.0$ is shown in Fig.~\ref{F.Kurokin6.2}. In this case there is a broad region
of metallicity, and the second critical Hubbard coupling (also indicated)
does not involve a discontinuity.
The Fermi
surface in the inset shows pockets at $(0,0)$, $(\pi,0)$, $(0,\pi)$, and $(\pi,\pi)$.
We conclude that a variety of multi-orbital models show similar
features as the four-orbital model analyzed before, particularly
regarding an intermediate metallic magnetic phase.
\begin{figure}[h]
\begin{center}
\vskip -0.7cm
\centerline{\includegraphics[width=9cm,clip,angle=0]{Kurokin6.2n.eps}}
\vskip -0.5cm
\caption{(Color online) Main panel: the evolution of the magnetization
for the state with $\mathbf{q}$=$(0,\pi)$ in a five-orbital
model,\cite{kuroki} at electron filling $n=6.0$. Inset: the Fermi surface at $U=0$.}
\vskip -0.8cm
\label{F.Kurokin6.2}
\end{center}
\end{figure}
\section{Results for the two-orbital model}\label{sec:twoorbs}
The complexity of four- and five-orbital Hamiltonians leads to very
large Hilbert spaces even for small clusters and, as a consequence,
it is not possible to compare the mean field results against
Exact Diagonalization (ED)\cite{RMP} or Variational Cluster
Approach (VCA)\cite{Aic03,Pot03} results. For this reason
the two-orbital model, which was recently studied
with ED and VCA methods,\cite{daghofer} is here revisited to assess the validity
of the mean-field approximation. Our conclusion is that all
three techniques lead to similar results for the two-orbital model,
lending support to our claim for the
existence of an intermediate-coupling metallic
and magnetic state.
The two-orbital model has been widely analyzed in recent literature
and its derivation and
other properties will not be repeated here. It is given by
\begin{equation}
H_{2o}=H_{\rm K}+H_{\rm int},
\label{1}
\end{equation}
where the kinetic energy $H_\textrm{K}$
in real space is:\cite{daghofer,scalapino}
\begin{equation}\begin{split}
H_{\rm K}&=-t_1\sum_{{\bf i},\sigma}(d^{\dagger}_{{\bf i},x,\sigma}
d_{{\bf i}+\hat y,x,\sigma}+d^{\dagger}_{{\bf i},y,\sigma}
d_{{\bf i}+\hat x,y,\sigma}+h.c.)\\
&\quad -t_2\sum_{{\bf i},\sigma}(d^{\dagger}_{{\bf i},x,\sigma}
d_{{\bf i}+\hat x,x,\sigma}+d^{\dagger}_{{\bf i},y,\sigma}
d_{{\bf i}+\hat y,y,\sigma}+h.c.)\\
&\quad
-t_3\sum_{{\bf i},\hat\mu,\hat\nu,\sigma}(d^{\dagger}_{{\bf i},x,\sigma}
d_{{\bf i}+\hat\mu+\hat\nu,x,\sigma}+d^{\dagger}_{{\bf i},y,\sigma}
d_{{\bf i}+\hat\mu+\hat\nu,y,\sigma}+h.c.)\\
&\quad+t_4\sum_{{\bf i},\sigma}(d^{\dagger}_{{\bf i},x,\sigma}
d_{{\bf i}+\hat x+\hat y,y,\sigma}+d^{\dagger}_{{\bf i},y,\sigma}
d_{{\bf i}+\hat x+\hat y,x,\sigma}+h.c.)\\
&\quad-t_4\sum_{{\bf i},\sigma}(d^{\dagger}_{{\bf i},x,\sigma}
d_{{\bf i}+\hat x-\hat y,y,\sigma}+d^{\dagger}_{{\bf i},y,\sigma}
d_{{\bf i}+\hat x-\hat y,x,\sigma}+h.c.)\\
&\quad-\mu\sum_{\bf i}(n_{{\bf i},x}+n_{{\bf i},y}).
\end{split}\end{equation}
The form of $H_{\rm K}$ in momentum space was provided in
Refs.~\onlinecite{daghofer} and \onlinecite{scalapino} and is given by
Eqs.~(\ref{eq:t11})-(\ref{eq:t12}) of the four-orbital model.
The Coulomb interaction terms are
\begin{equation}\label{eq:Hint2}\begin{split}
H_{\rm int}&=
U\sum_{{\bf i},\alpha}n_{{\bf i},\alpha,\uparrow}n_{{\bf i},
\alpha,\downarrow}
+(U'-J/2)\sum_{{\bf i}}n_{{\bf i},x}n_{{\bf i},y}\\
&\quad-2J\sum_{\bf i}{\bf S}_{{\bf i},x}\cdot{\bf S}_{{\bf i},y}\\
&\quad+J\sum_{{\bf i}}(d^{\dagger}_{{\bf i},x,\uparrow}
d^{\dagger}_{{\bf i},x,\downarrow}d_{{\bf i},y,\downarrow}
d_{{\bf i},y,\uparrow+h.c.)},
\end{split}\end{equation}
where the notation is the same as for the case of the four-orbital model
but with $\alpha=x,y$ here denoting the orbitals $xz$ and $yz$.
The index $\hat\mu$ is a unit vector linking NN sites and takes the values
$\hat x$ or $\hat y$.
$\mu$ is the chemical potential. As for the case of four orbitals, the relation
$U'$=$U-2J$ originating in
rotational invariance\cite{RMP01} was used. In the
last term in Eq.~(\ref{eq:Hint2}), the same rotational invariance also establishes that
the pair hopping coupling $J'$ must be equal to $J$.
As before,
the hoppings are determined from the orbital integral overlaps within the
SK formalism or from LDA band
dispersion fittings.\cite{daghofer,scalapino}
Most of the properties of this two-orbital model are
formally similar to those of the four-orbital model and,
as a consequence, several details of the analysis of the previous section
do not need to be repeated here.
\subsection{Mean field approximation}
Following the same procedure used for the four-orbital model, here
we consider only the mean field values that are
diagonal with respect to the Fe site, orbital, and spin labels such that
\begin{equation}
\langle d^{\dagger}_{{\bf l},\alpha,\sigma}d_{{\bf l'},\alpha',\sigma'}\rangle=(n_{\alpha}
+{{\sigma}\over{2}}\cos({\bf q.r}_{\bf l})m_{\alpha})\delta_{\bf ll'}\delta_{\alpha\alpha'}
\delta_{\sigma\sigma'}.
\label{4}
\end{equation}
After introducing Eq.~(\ref{4}) to decouple the four-fermion interactions
in Eq.~(\ref{eq:Hint2}), and then transforming into momentum space, we obtain:
\begin{equation}\begin{split}\label{mf2}
H^{\rm MF}_{\rm int}&=
-UN\sum_{\alpha}(n^2_{\alpha}-{1\over{4}}m^2_{\alpha})-4(U'-{J\over{2}})Nn_xn_y\\
&\quad +\frac{JNm_xm_y}{2}+\sum_{{\bf k},\sigma}\left(Un_x+2(U'-{J\over{2}})n_y\right)n_{{\bf k}x\sigma}\\
&\quad+\sum_{{\bf k},\sigma}\left(Un_y+2(U'-{J\over{2}})n_x\right)n_{{\bf k}y\sigma}\\
&\quad-{1\over{4}}\sum_{{\bf k}\sigma}(Um_x+Jm_y)(d^{\dagger}_{{\bf k}x\sigma}d_{{\bf k+q}x\sigma}
+d^{\dagger}_{{\bf k+q}x\sigma}d_{{\bf k}x\sigma})\\
&\quad-{1\over{4}}\sum_{{\bf k}\sigma}(Um_y+Jm_x)(d^{\dagger}_{{\bf k}y\sigma}d_{{\bf k+q}y\sigma}
+d^{\dagger}_{{\bf k+q}y\sigma}d_{{\bf k}y\sigma}).
\end{split}\end{equation}
The four mean-field parameters $n_x$,
$n_y$, $m_x$, and $m_y$ are determined in the usual way by minimizing the
energy and by requesting that the system be half-filled. The half-filling
condition determines that $n_x=n_y=0.5$ while the values of $m_x$ and $m_y$
are a function of $U$ and $J$. We have observed that varying $U$, and
for fixed $J$, $m_{\alpha}$ becomes non-zero at a critical value $U=U_{\rm c1}$ where
a gap opens separating
the ``valence'' and ``conduction'' bands at particular momenta but, overall,
there is still an overlap in energy of some bands and the
chemical potential crosses both of them. Thus, the system has developed magnetic order
but it is still a metal. However, the bands no longer overlap when $U>U_{\rm c2}$ and, thus,
a metal insulator transition occurs. These results are similar to those obtained
using more orbitals
in the model.
\begin{figure}
\centerline{\includegraphics[width=10cm,clip,angle=0]{./Kien1.eps}}
\vskip -0.5cm
\caption{(Color online) Two-orbital model mean-field calculated spin-stripe magnetization $m$ vs.
$U$. Panel (a) is for the LDA fitted hoppings,\cite{scalapino} while panel (b) is for the
SK hoppings.\cite{daghofer} The solid red line is for
$J$=$0$. With a blue dashed line the results at $J$=$U/4$ are indicated. The dotted black line
denotes results for $J$=$U/8$. The large dot in each curve indicates the value of $m$ at
$U_{\rm c2}$
where the metal to insulator transition occurs (see text).
Note the absence of a first-order transition at $U_{\rm c2}$
as in the four-orbital model.
}
\label{kien1}
\end{figure}
In Fig.~\ref{kien1}, the stripe magnetization $m$=$m_x+m_y$ vs. $U$ is shown
for $J$=$0$, $U/4$, and $U/8$. Panel (a) contains the results for the LDA
fitted
hoppings\cite{scalapino} while panel (b) is for the SK hoppings\cite{daghofer} with
$pd\sigma=-0.2$. The critical coupling $U_{\rm c1}$ is the value of $U$ where
$m$ becomes different from zero. There is a second critical coupling
$U_{\rm c2}$ that is obtained by monitoring the density of states, and it separates a metallic from an
insulating regime. For completeness, this second coupling is indicated with a full circle for each case.
In panel (b), the shape of the curves is the same for the three values of $J$ investigated, and the actual
value of $U_{\rm c1}$ mildly depends on $J$.
The mean-field density of states (DOS) $N(\omega)$
is presented in Fig.~\ref{kien2} for some values of $U$ and $J$=$U/4$, and the
two sets of hoppings considered here.
The solid curve in both panels
shows results for $U<U_{\rm c1}$. In this case the system is metallic.
The dashed curve displays the DOS
for a value of $U$ in between the two critical points. Although the DOS varies continuously as $U$ increases,
it is clear that this regime is qualitatively different: A deep
pseudogap has developed at the chemical potential.
The system is still metallic in this regime, although
likely with ``bad metal'' characteristics.
Finally, the dotted curve shows the DOS for $U>U_{\rm c2}$. In this
case, there is a gap at the
chemical potential and the system has become
insulating. Interestingly, the transition at $U_{\rm c2}$ is not
first order for the two-orbital model, in contrast to the case with
four orbitals.
\begin{figure}
\centerline{\includegraphics[width=10cm,clip,angle=0]{Kien2.eps}}
\vskip -0.5cm
\caption{(Color online) Two-orbital model mean-field calculated density of states
(a) for the LDA fitted hoppings;\cite{scalapino} (b) for the
SK hoppings.\cite{daghofer} Panel (a): the solid red line is for
$U$=$2.5$, blue dashed line for $U=5.0$, and dotted
black line for $U=8.0$. Panel (b): red is for $U$=0.5,
blue is for $U$=0.8, and black is for $U$=2.0.
$J$=$U/4$ is used in both panels.}
\label{kien2}
\end{figure}
\subsection{Mean field results for the bands and Fermi surface}
\begin{figure}
\centerline{\includegraphics[width=9cm,clip,angle=0]{./Kien3_s.eps}}
\caption{(Color online) Two-orbital model mean-field
band structure along high symmetry directions in the extended FBZ.
The panels on the left column show results for the LDA fitted
hoppings,\cite{scalapino} while results on the right are for the SK
hoppings.\cite{daghofer} (a) $U$=$2.5$, (b) $U$=$5.0$, (c)
$U$=$8.0$, (d) $U$=$0.5$, (e) $U$=$0.8$, (f) $U$=$2.0$. In all
cases, $J$=$U/4$ and the magnetic order wavevector is $(\pi,0)$. }
\vskip -0.1cm \label{kien3}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=5.8cm,clip,angle=0]{Ds1_s.eps}
\includegraphics[width=5.8cm,clip,angle=0]{Ds3_s.eps}
\includegraphics[width=6cm,clip,angle=0]{YDs1_s.eps}
\includegraphics[width=6cm,clip,angle=0]{YDs3_s.eps}
\caption{(Color online) Mean-field photoemission Fermi surface
for the LDA fitted hoppings\cite{scalapino}
in the unfolded FBZ for (a) $U=1$, and (b) $U=3$, and in the folded FBZ
for (c) $U=1$, and (d) $U=3$. The ratio $J=U/4$ was used, and the
magnetic order wavevector is $(\pi,0)$. These results
were obtained via $A({\bf k},\omega)$ using the same
symmetrization procedure and energy window centered at the Fermi energy as in Fig.~\ref{F.FS}.
}
\vskip -0.5cm
\label{kien4}
\end{center}
\end{figure}
The mean-field band structure obtained by solving the mean-field
self-consistent equations is shown along high-symmetry directions in the Brillouin zone
in Fig.~\ref{kien3}. The panels on the left (right) column correspond to LDA fitted
(SK) hoppings.
The top row shows the band dispersion for $U<U_{\rm c1}$. The corresponding Fermi surfaced (FS) in the
extended and reduced Brillouin
zones (BZ) are shown in Figs.~\ref{kien4}~(a) and (c), and they agree with previous discussions
for the two-orbital model.\cite{daghofer}
The second row of panels in
Fig.~\ref{kien3} shows the band dispersion in the interesting regime
$U_{\rm c1}<U<U_{\rm c2}$. It can be observed that gaps have opened
along, e.g., the direction $(0,0)$-$(0,\pi)$ but not along
other high-symmetry directions. The partial gaps remove portions
of the original FS and
produce the $A({\bf k},\omega)$ disconnected features (arcs) at $\Gamma$
shown in panel (b) of Fig.~\ref{kien4}.
Note that in Fig.~\ref{F.FS}~(b), the results for the four-orbital
model also contained arcs, but they appeared in a more symmetric manner, namely with four arcs surrounding
the $\Gamma$ point. Two of those arcs were located in the inner hole pocket and two in the outer hole
pocket, while in Fig.~\ref{kien4}~(b) there are only two arcs around the $\Gamma$ point. However, as
often remarked in the context of the two-orbital model, to compare with either experiments or with results
of models with more orbitals, it is crucial to fold the results since by this mechanism the hole pocket
at $M$ becomes the outer hole pocket at $\Gamma$. Following this procedure,
in Fig.~\ref{kien4}~(d) the folded results
are shown and now there are {\it four arcs} around the $\Gamma$ point in good agreement with the
results of the four-orbital model. We conclude that in the interesting intermediate coupling regime,
both models give similar results upon folding of the extended Brillouin zone.
The last row of panels in Fig.~\ref{kien3} shows the band dispersion for $U>U_{\rm c2}$. Now the upper and lower bands no longer
overlap. The gap is complete and there is no FS.
The system has become an insulator, as it can be seen in the DOS shown in
Fig.~\ref{kien2}.
The intensity of the features determining the Fermi surface
should be calculated using the spectral function
$A({\bf k},\omega)$. Figure~\ref{kien3} only shows the eigenvalues of the mean-field
study, without incorporating the photoemission intensity of each state.
It is only when the strength of the coupling $U$ becomes very large that the spectral weights for all the bands will be
equal. Otherwise, some of the bands will produce strong FS while others will produce only
weak magnetically-induced ``shadow'' features that are hard to observe, as already shown
in Fig.~\ref{kien4}. To better visualize the bands induced
by magnetic order, in Fig.~\ref{kien5} the mean-field
spectral function $A({\bf k},\omega)$ is presented along
high-symmetry directions in the BZ for the two-orbital model with the SK parameters.\cite{daghofer} For $U<U_{\rm c1}$, panel (a), the spectral weight
resembles the non-interacting band structure, i.e. there is negligible
spectral weight in the magnetically-induced bands. For $U_{\rm c1}<U<U_{\rm c2}$,
panel (b),
the bands become distorted and bands of magnetic origin develop particularly at the
locations in which a gap opens. There are other
bands still crossing the Fermi energy, thus
the system is metallic.
Finally for $U>U_{\rm c2}$,
panel (c), the gap is
complete and the magnetic bands are well developed, i.e., four peaks can
be observed in $A({\bf k},\omega)$ for almost all values of ${\bf k}$.
Figure~\ref{kien7}
shows the results for the LDA fitted hoppings,\cite{scalapino}
at $J$=$U/4$: here a similar qualitative discussion applies.
Other values of $J$ such as $J$=$0$ and $J=U/8$
(not shown) were also considered, and the results are qualitatively the same.
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=8cm,clip,angle=0]{./Kien5_s.eps}
\caption{(Color online) Two-orbital model mean-field spectral function along high symmetry
directions
in the extended FBZ, using SK hoppings.\cite{daghofer}
(a) $U$=$0.5$, (b) $U$=$0.8$, (c) $U$=$2.0$. The Hund coupling is fixed to $J$=$U/4$ and
the magnetic order wavevector is $(\pi,0)$.
}
\vskip -0.5cm
\label{kien5}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\includegraphic
[width=8cm,angle=0,bbllx=14pt,bblly=14pt,bburx=241pt,bbury=383pt]{./Kien7_s.eps}
\caption{(Color online) Two-orbital model mean-field spectral function along high symmetry
directions
in the extended FBZ for the LDA fitted hoppings.\cite{scalapino}
The couplings used are (a) $U$=$2.5$, (b) $U$=$5$, (c) $U$=$8$. In all cases
$J$=$U/4$, and the magnetic order wavevector is $(\pi,0)$.
}
\vskip -0.5cm
\label{kien7}
\end{center}
\end{figure}
\subsection{Exact Diagonalization results}
To analyze the qualitative reliability of the mean-field results, we have performed
ED calculations on finite clusters. In our previous effort,\cite{daghofer}
it was discussed that due to the rapid growth of the
Hilbert space with the number of sites $N$, the largest cluster where
the two-orbital model can be exactly diagonalized has only $N$=$8$ sites. It is a tilted
$\sqrt{8}$$\times$$\sqrt{8}$ cluster, and when periodic
boundary conditions are implemented the available values
of the momenta are ${\bf k}=(0,0)$, $(\pm\pi/2,\pm\pi/2)$, $(0,\pi)$,
$(\pi,0)$, and $(\pi,\pi)$. This limited set of momenta
is not well suited to analyze band
dispersions in the BZ and the use of ``twisted'' boundary conditions (see below)
for the 8-site cluster would require too high a computational effort.
However, via the study of spin correlations
it has been observed that the spin-striped magnetic order
characteristic of this model is also apparent in the even smaller $2\times 2$ cluster. For
this very small system, the limited number of available momenta can be enlarged by
implementing ``twisted'' boundary conditions (TBC), namely requesting
that $d(N_i+1)$=$e^{i\phi}d(1)$ where $N_i$ is the number of sites along the
$i$=$x$ or $y$ direction in the square cluster, and $\phi$ is an arbitrary phase.
With these TBC
the values of momenta allowed are now $k_i$=${2\pi n_i+\phi\over{N_i}}$ with $n_i$
ranging from 0 to $N_i-1$. Thus, we can calculate the spectral functions for
a variety of values of ${\bf k}$ using this TBC approach applied to the $2 \times 2$
cluster. While the very small size is still a serious limitation, note that
there are simply no other procedures
available to contrast the mean-field results against
exact results at intermediate couplings. Our goal using this limited size
cluster is merely to analyze if mean-field conclusions stand against exact results.
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=8cm,height=8cm,clip,angle=0]{dougju4f1_s.eps}
\vskip -0.3cm
\caption{(Color online) Two-orbital model
spectral function along the $(0,0)$ to $(\pi,\pi)$ direction
in the extended FBZ for the LDA fitted
hoppings.\cite{scalapino}
(a) is for $U$=0.0, (b) for $U$=2.5, (c) for $U$=5.0, and (d) for $U$=8.0.
The Hund coupling is $J$=$U/4$. The method is ED and
the lattice is 2$\times$2 with TBC.
The arrows indicate the magnetic bands discussed in the text.}
\vskip -0.5cm
\label{doug_akju4}
\end{center}
\end{figure}
\begin{figure}[thbp]
\begin{center}
\vskip 0.5cm
\includegraphics[width=8cm,height=8cm,clip,angle=0]{dougju4f2_s.eps}
\vskip -0.3cm
\caption{(Color online) Two-orbital model
spectral function along the $(0,0)$ to $(\pi,0)$ and $(\pi,0)$ to $(0,\pi)$
directions
in the extended FBZ for the LDA fitted
hoppings.\cite{scalapino}
(a,b) is for $U$=0, while (c,d) is for $U$=2.5.
The Hund coupling is $J$=$U/4$. The method is ED and
the lattice is 2$\times$2 with TBC. This figure shows that the results barely change between the two
values of $U$, suggesting the survival of a metallic state at nonzero $U$.}
\vskip -0.9cm
\label{doug_akju4_other}
\end{center}
\end{figure}
In Fig.~\ref{doug_akju4}, the spectral function
$A({\bf k},\omega)$ is presented along the main diagonal
of the extended BZ for different values of $U$ and with $J$=$U/4$.
These data have to be compared with the mean-field prediction shown in
Fig.~\ref{kien7}.
We present the results for $U$=$0$ for comparison and to demonstrate that the
correct dispersion is obtained in spite of the fact that the cluster is so small.
The finite values
of $U$ have been chosen to be in the magnetic-metallic region ($U$=$2.5$ and $5.0$) and
in the insulating region ($U$=$8$) according to the mean-field results. The main point
of this figure is to report the development of bands induced by magnetic order with increasing $U$. A
representative momentum for these magnetically-induced bands is highlighted
with an arrow in the figure.
With increasing $U$, the magnetic bands smoothly develop (concomitant with
the development of spin stripe order at short distances, as shown later) and at least
along the main-diagonal direction in the extended FBZ that should occur simultaneously
with the opening of a gap.
Thus, our first conclusion is that the extra weak features in the one-particle spectral
function predicted by mean-field due to
the magnetic order do appear in the ED results. At large $U$ there is
no doubt that a substantial gap is observed, as in the mean-field approach.
We also performed calculations with other values of $J$ such as
$J=0$ and $J=U/8$ (not shown), and the conclusions are qualitatively the same
as for $J$=$U/4$. Qualitatively similar conclusions were reached using
the SK hoppings.\cite{daghofer}
Let us analyze now other directions in momentum space. In Fig.~\ref{doug_akju4_other},
the $(0,0)$ to $(\pi,0)$ and $(\pi,0)$ to $(0,\pi)$ directions are investigated
at $U$=0 and 2.5. The results indicate negligible changes along these
directions by turning on $U$: the system appears to remain metallic.
However, as shown below, the NN spin-stripe order
in this small cluster is already robust at $U$=2.5.
Thus, these results are
compatible with the concept of a state simultaneously metallic and magnetically
ordered.
Moreover, by monitoring the
opening of the complete gap we found that the metal-insulator transition occurs
at a value of $U_{\rm c2}$ in good agreement with the mean-field predictions.
The existence of a
$U_{\rm c1}$ is a more complicated issue but it can be inferred from
the development of the magnetic bands in
$A({\bf k},\omega)$ which also occurs in a range of $U$ consistent with
mean-field.
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=8cm,height=8cm,clip,angle=0]{./stfac_s8.eps}
\vskip -0.3cm
\caption{(Color online) The magnetic structure factor at ${\bf k}$=$(0,\pi)$
vs. $U$ calculated by ED of an 8-site cluster
for the values of $J$ indicated. (a) is for the SK hoppings;\cite{daghofer}
and (b) for the LDA fitted
hoppings.\cite{scalapino}
The insets show the results in a more extended range of $U$.}
\vskip -0.3cm
\label{struc_s8}
\end{center}
\end{figure}
In Fig.~\ref{struc_s8} we present the magnetic structure factor $S$ for
${\bf k}$=$(0,\pi)$ which is the value of the momentum for which it has a
maximum (degenerate with $(\pi,0)$ for this small cluster);
panel (a) shows results for the SK hoppings\cite{daghofer}
while panel (b) is for the LDA fitted hoppings.\cite{scalapino}
To reduce finite-size effects results for the
$N=8$ cluster are presented, although the results in the $2\times 2$ cluster are qualitatively
similar. For large $U$, the monotonic increase of $S$ with $U$ agrees with
the mean-field results. At small and intermediate $U$, the spin stripe correlations
are robust and for these small clusters this is equivalent to long-range order.
But the apparent lack of a gap at $U$=2.5 along particular directions in momentum
space (discussed before) lead us to believe that the ED results are compatible
with a metallic and magnetic phase at intermediate $U$.
\subsection{VCA results}
In this final subsection, numerical results for the spectral functions
and the density of states obtained with the Variational Cluster Approximation (VCA)
technique\cite{Aic03,Pot03} are presented.
This method embeds the ED solution of a small $2\times 2$
cluster into a very large system of a size comparable to the
$100\times 100$ momentum points used in the mean field, and thus
interpolates between the results obtained independently by the ED and
mean-field approaches.
The VCA results discussed here are for the SK parameters.\cite{daghofer}
Figure~\ref{dos} shows the VCA density of states. The
behavior with increasing $U$ is remarkably similar to that obtained in
our mean-field calculations presented in Fig.~\ref{kien2}~(b). Metallic, pseudogap,
and insulating regimes can be clearly observed.
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=8cm,clip,angle=0]{./dos_pdp-02_U05_08_2.eps}
\vskip -0.3cm
\caption{(Color online) VCA calculated density of states for different values of $U$ in the
metallic (good metal and pseudogap) and insulating regimes with the SK hopping parameters using
$pd\pi/pd\sigma$=$-0.2$ and $J$=$U/4$.
}
\vskip -0.3cm
\label{dos}
\end{center}
\end{figure}
The results for $A({\bf k},\omega)$ calculated with VCA are shown in
Fig.~\ref{vca}. Once again, a remarkable quantitative agreement with
the mean-field results of Fig.~\ref{kien5} is found.
\begin{figure}[thbp]
\begin{center}
\includegraphics[width=6.5cm,clip,angle=0]{maria.group.eps}
\vskip -0.3cm
\caption{(Color online) VCA calculated spectral functions along high symmetry directions in
the BZ for the SK hopping parameters with
$pd\pi/pd\sigma$=$-0.2$, $J$=$U/4$, and wavevector $(\pi,0)$.
(a) $U$=0.5;
(b) $U$=0.8; (c) $U$=2.0.
}
\vskip -0.9cm
\label{vca}
\end{center}
\end{figure}
\section{Conclusions}
In this investigation, the mean-field technique was applied to multi-orbital
Hubbard models for the Fe-pnictides. Varying $U$, three regions were observed. At
small coupling, the results are as in the non-interacting limit. In the other extreme of
very large $U$, the ground state has a robust gap and the magnetic spin-stripe
order parameter is large. The main result of our effort is the presence
of an intermediate $U$ coupling regime where magnetic spin-stripe order is shown to be
compatible with a metallic ground state due to band overlaps. This state has
similar characteristics as the parent compounds of the Fe-pnictide
superconductors.
Although further theoretical work is still needed to
firmly establish the existence of this interesting intermediate coupling regime,
for the case of two orbitals our conclusions were tested using
the ED and VCA methods, and the results are compatible with
mean-field.
The analysis of the intermediate $U$ regime allowed us to predict the results for
angle-resolved photoemission experiments. Interesting anisotropies manifested
as arcs at the Fermi surface. New bands of magnetic origin were also discussed.
The two- and four-orbital models lead to similar results
in this context.
Future work will address the optical properties
of the intermediate coupling regime, and the superconducting state
that may arise from its doping.
\section{Acknowledgments}
This work was mainly supported by the NSF grant DMR-0706020 and the
Division of Materials Science and Engineering, U.S. DOE, under contract
with UT-Battelle, LLC.
Computation for part of the work described in this paper was supported by the
University of Southern California Center for High Performance Computing
and Communications. S.H. and K.T. acknowledge financial support by the National
Science Foundation under grant DMR-0804914 and the Department of Energy
under grant DE-FG02-05ER46240.
|
0812.3320
|
\section{Introduction}
The computation of equilibrium statistical properties of
molecular systems is of great importance in materials science, computational
physics, chemistry, and biology~\cite{frenkelsmit,mcquarrie}.
These equilibrium statistical
properties are given by phase space integrals of the
form
\begin{equation}
\label{ps_aver}
\langle A \rangle = \int A(q,p) \, d\mu(q,p),
\end{equation}
where $q=(q_1,\ldots,q_N) \in {\mathbb R}^{N}$ and $p=(p_1,\ldots,p_N)
\in {\mathbb R}^{N}$ denote a set of positions and
momenta and $A(q,p)$ is an observable, a function defined over
the phase space and related to the macroscopic quantity under
study.
The computation of
integrals such as~(\ref{ps_aver}) is often a challenging problem, especially
when the number of degrees of freedom is large.
For molecular systems at fixed temperature $\theta,$
the measure $d\mu$ is the Gibbs measure for the canonical ensemble
\cite{frenkelsmit,mcquarrie}
\begin{equation}
\label{measure_can}
d\mu(q,p) =
\left[ \frac{\exp\left({-\beta H(q,p)}\right)}
{\displaystyle{ \int \exp\left({-\beta H(q,p)}\right) \ dq \, dp }} \right]
\, dq \, dp,
\end{equation}
where $H(q,p)$ is the Hamiltonian of the system and
$\beta$ is related
to the temperature $\theta$ by $\beta = 1/(k_B \theta)$ with $k_B$ denoting
the Boltzmann constant.
We will consider Hamiltonians of the general form
\begin{equation}
\label{H_qp}
H(q,p) = \frac {p^TM^{-1}(q)p}2+ V(q),
\end{equation}
where $M(q)\in {\mathbb R}^{N\times N}$ for $q\in {\mathbb R}^{N}$
is the generalized mass matrix and $V(q)$ is the potential energy.
We assume that the generalized mass matrix $M(q)\in {\mathbb R}^{N\times N}$
is symmetric and positive definite, so its inverse
$M^{-1}(q)\in {\mathbb R}^{N\times N}$
exists for all $q\in {\mathbb R}^{N}$ and is also symmetric and positive
definite.
Many methods have been proposed and utilized to approximate the
phase space integral~\eref{ps_aver}, including methods based on
stochastic or
deterministic dynamics for $(q,p).$
If the dynamics is ergodic with respect to the measure $d\mu$ given by
(\ref{measure_can}), then
the phase-space average
(\ref{ps_aver}) is equal to the time average
\begin{equation}
\label{ergo0}
\int A(q,p) \, d\mu(q,p)
=
\lim_{T \to +\infty} \frac{1}{T} \int_0^T A \left( q(t),p(t) \right) dt
\end{equation}
over a trajectory $(q(t),p(t))_{t \geq 0}$. Thus, the time average can be
approximated by
\begin{equation*}
\lim_{T \to +\infty} \frac{1}{T} \int_0^T A \left( q(t),p(t) \right)
dt \approx \lim_{{\mathcal N} \to +\infty} \frac{1}{\mathcal N}
\sum_{\ell=1}^{\mathcal N}
A(q_\ell,p_\ell),
\end{equation*}
where $(q_\ell,p_\ell)_{\ell \geq 1}$ is a numerical solution of the chosen
dynamics.
In this paper, we investigate the deterministic dynamics
known as Nos\'e-Hoover dynamics~\cite{Hoover}, which is
still widely used although variants have been developed with the goal
to improve its efficiency and overcome its
deficiencies~\cite{Martyna92,Tuckerman00,NPoincare99,RMT05}.
This dynamics has been first proposed in the form of a
Hamiltonian dynamics on an extended phase space~\cite{nose84}, the
Hamiltonian being chosen such that the marginal distribution of its
microcanonical density is the canonical Gibbs density for the physical
variables. The Nos\'e-Hoover dynamics is then
constructed by rescaling time and momentum to obtain a non-Hamiltonian
dynamics with physical time and momentum~\cite{Hoover}.
Stochastic
dynamics (such as the Langevin equation, or the recently proposed
Hoover-Langevin method~\cite{LeNoTh09}) can also be considered. See
\cite{comparisonNVT} for a review
of sampling methods of the canonical ensemble, along with a
theoretical and numerical comparison of their performances for molecular
dynamics.
The equality \eref{ergo0} relies on an ergodicity condition.
This condition has been rigorously proven neither for the
Nos\'e-Hoover dynamics, nor for any other deterministic method
commonly used in practice. In fact, there is numerical evidence that
shows that the Nos\'e-Hoover method is not ergodic for some systems
\cite{Hoover,Martyna92,Tuckerman00}, including the one-dimensional
harmonic oscillator. In \cite{LLM}, we have rigorously analyzed the
dynamics in this special case, and indeed proven the non-ergodicity, for
some regime of parameters.
In this article, we study more general systems. After
briefly recalling the Nos\'e-Hoover equations (see Section
\ref{background}), we first
consider a class of multidimensional systems (see Section
\ref{multiD}). Taking the limit of an infinite thermostat ``mass'' in the
Nos\'e-Hoover equations, we formally obtain an averaged dynamics,
for which we prove the existence of many invariants.
These theoretical results are illustrated by numerical simulations of a
specific system (see Section \ref{multiD_num}). We numerically observe
that, for finite thermostat mass, these invariants are of course
not exactly preserved, but still remain close to
their initial value. This prevents the Nos\'e-Hoover system from
thermalizing. In Section \ref{1Dgeneral},
we next turn to the one-dimensional case, for which we obtain stronger
results. We first prove non-ergodicity of the Nos\'e-Hoover
dynamics, when the mass of the thermostat is large enough (see
Section \ref{sec:1Dtheory}). Our
method extends the one we used to study the harmonic oscillator
case \cite{LLM}. Section \ref{1pendulum} describes an example of such a
one degree of freedom
problem. Again, numerical simulations illustrate the obtained theoretical
results.
\section{Nos\'e-Hoover dynamics}
\label{background}
The Nos\'e-Hoover dynamics involves the physical variables
$q$ and $p$ and
one additional scalar variable, $\xi$, which
represents the momentum of a thermal bath exchanging
energy with the system. The differential equations are:
\begin{equation}
\label{nh-dyn}
\begin{array}{rcl}
\dot{q} &=& \displaystyle{
\frac{\partial H}{\partial p}=M^{-1}(q)p
},
\\
\dot{p} &=& \displaystyle{
-\frac{\partial H}{\partial q}- \frac{\xi}{Q}\, p=-\nabla V(q) -\frac
{p^T\nabla M^{-1}(q)p}2- \frac{\xi}{Q}\, p
},
\\
\dot{\xi} &=& \displaystyle{
p^TM^{-1}(q)p - \frac{N}{\beta}
},
\end{array}
\end{equation}
where $\dot\_$ denotes the time-derivative. The parameter $Q$
represents the mass of the thermostat; it is a free parameter that the
user has to choose.
We recall that invariant measures $\rho(z) \, dz$ for a general
dynamical system
\begin{equation*}
\dot{z}=f(z)
\end{equation*}
are determined by the equilibrium equation
\begin{equation*}
\mbox{div}(\rho(z) f(z))=0.
\end{equation*}
It can be verified by direct computation that the dynamics
\eref{nh-dyn} preserves the measure
\begin{equation}
\label{invmeasure}
d\mu_{\rm NH} = \exp \left[ -\beta \left( H(q,p) + \frac{\xi^2}{2Q}
\right) \right] \ dq \ dp \ d\xi
\end{equation}
by using the fact that the kinetic energy
$\displaystyle \frac{p^TM^{-1}(q)p}2$ is quadratic in $p.$
If the dynamics (\ref{nh-dyn}) is ergodic with respect to $d\mu_{\rm
NH}$, then, by integrating out $\xi$, we have that
the dynamics $(q(t),p(t))$ is ergodic with respect to the
Gibbs measure. In this case, the time-average of a function $A(q,p)$
along a typical
Nos\'e-Hoover trajectory provides an estimate for
the space-average of $A$ with respect to Gibbs measure.
Unfortunately, the system is generally not ergodic. In \cite{LLM} we
proved non-ergodicity in the case of the one-dimensional harmonic
oscillator. Our aim here is to study more general
systems.
\section{Systems with first integrals}
\label{multiD}
In this section, we show how the presence of additional integrals for a
Hamiltonian system can impede ergodization of the Nos\'e-Hoover
dynamics.
\subsection{Homogeneous integrals}
\label{sec:homogeneous_integrals}
Consider a Hamiltonian system
\begin{equation}
\label{complete}
\dot{q} =\frac{\partial H}{\partial p},
\quad
\dot{p} = -\frac{\partial H}{\partial q},
\end{equation}
for energy (\ref{H_qp}) which admits a first integral other than $H$
itself. This means that there is a smooth function $F(q,p)$ whose
Poisson bracket with $H$ vanishes, i.e.,
\begin{equation}
\label{Poissonbracket}
\{H,F\} = H_q^T F_p - H_p^T F_q=0.
\end{equation}
If $F$ is a homogeneous function of the momentum variables, then it gives
rise to a first integral of the Nos\'e-Hoover system.
\begin{theo}\label{thhomogeneous}
If $F(q,p)$ is a first integral of (\ref{H_qp}) which is homogeneous of
degree $k$ with respect to the momentum variables, $p$, then
\begin{equation}
\label{homogeneous}
G(q,p,\xi) = \frac{\xi^2}{2Q} + H(q,p) - \frac{N}{\beta k}\ln |F(q,p)|
\end{equation}
is a first integral of the corresponding Nos\'e-Hoover system (\ref{nh-dyn}).
\end{theo}
The proof is a simple computation using (\ref{Poissonbracket}),
(\ref{nh-dyn}) and the fact that $F_p^T\,p = k F$.
\smallskip
Of course, the existence of such an integral immediately gives
non-ergodicity of the Nos\'e-Hoover system with respect to
\eref{invmeasure}. For a simple example of a system admitting such a
homogeneous integral, see Section~\ref{multiD_num}.
\subsection{Completely integrable systems and action-angle variables}
\label{sec:action_angle_multiD}
We now assume that the Hamiltonian dynamical system (\ref{complete})
is completely integrable, i.e., the system admits $N$ independent first
integrals which commute in the sense that the Poisson brackets of any
two of them vanish \cite{Arn}. The rest of this section is devoted to
showing that these integrals, even if they are not homogeneous, have a
deleterious effect on the ergodization of the corresponding
Nos\'e-Hoover system.
The non-degenerate level sets of the integrals are $N$-dimensional
manifolds and if they are compact then their connected components are
diffeomorphic to the $N$-dimensional torus $\mathbb{T}^{N}$. Moreover, such
a torus has a neighborhood $U\subset \mathbb{R}^{N}\times \mathbb{R}^{N}$ in which
one can introduce symplectic action-angle variables. More precisely,
there exist angle variables $\theta\in\mathbb{T}^{N},$
action variables $a\in D \subset \mathbb{R}^{N}$, and a symplectic diffeomorphism
$\psi:U\to \mathbb{T}^{N} \times D$
which transforms (\ref{complete}) to the form
\begin{equation}
\label{eq:newton_action}
\dot{\theta} = \omega(a),
\quad
\dot{a} = 0.
\end{equation}
Here $D$ is an open subset of $\mathbb{R}^{N}$.
Equivalently, the action-angle Hamiltonian
$\tilde H(\theta,\,a)=H(\psi^{-1}(\theta,\,a))$ is independent of
$\theta$ and
$\displaystyle \frac{\partial \tilde H(\theta,\,a)}{\partial a}=\omega(a).$
In what follows, it will be convenient to define the angle mapping
$\psi_1(q,\,p)\in \mathbb{R}^{N}$ and the action mapping
$\psi_2(q,\,p)\in\mathbb{R}^{N}$ by
$\psi(q,\,p)=(\psi_1(q,\,p),\,\psi_2(q,\,p))\in\mathbb{R}^{2N}.$
We will also use the abbreviated notation
$\displaystyle \partial_1 \psi_i=\frac{\partial \psi_i}{\partial q}
\in\mathbb{R}^{N\times N}$ and
$\displaystyle \partial_2 \psi_i=\frac{\partial \psi_i}{\partial p}
\in\mathbb{R}^{N\times N}.$ We then denote the Jacobian of $\psi$ by
\begin{equation*}
\mbox{D} \psi= \left(
\begin{array}{rcl}
\partial_1\psi_1 & \partial_2\psi_1
\\
\partial_1\psi_2 & \partial_2\psi_2
\end{array} \right).
\end{equation*}
We denote the inverse mapping of $\psi(q,p)$
by $\phi(\theta,a).$ The matrix $(\mbox{D} \psi)=(\mbox{D} \phi)^{-1}$
has a simple form since $\phi$ is symplectic:
\begin{equation}\label{sym5}
(\mbox{D} \psi)
=J^{-1} (\mbox{D} \phi)^T J = \left( \begin{array}{cc} (\partial_2
\phi_2)^T & -(\partial_2 \phi_1)^T \\ - (\partial_1 \phi_2)^T &
(\partial_1 \phi_1)^T \end{array} \right),
\end{equation}
where
\begin{equation*}
J= \left(
\begin{array}{rcl}
0 & I_N
\\
-I_N & 0
\end{array} \right).
\end{equation*}
The diffeomorphism $\psi$ transforms \eref{complete} to
\eref{eq:newton_action} by the chain rule
\begin{equation}
\label{first2}
\left( \begin{array}{c}
\dot{\theta} \\ \dot{a} \end{array} \right)=
\mbox{D} \psi \ \left( \begin{array}{c} \dot{q} \\ \dot{p} \end{array} \right).
\end{equation}
Since $\psi$ is symplectic, the dynamics \eref{eq:newton_action} is
obtained from the Hamiltonian $\tilde H(\theta,\,a)$. Hence,
\begin{equation}
\label{first3}
\left( \begin{array}{c}
\omega(a) \\ 0 \end{array} \right)=
\left( \begin{array}{c} \frac{\partial \tilde H}{\partial a}
\\ -\frac{\partial \tilde H}{\partial \theta} \end{array} \right)
= \mbox{D} \psi \ \left( \begin{array}{c} \frac{\partial H}{\partial p}
\\ -\frac{\partial H}{\partial q} \end{array} \right).
\end{equation}
\subsection{Recasting the Nos\'e-Hoover dynamics}
We now multiply the Nos\'e-Hoover equations~\eref{nh-dyn} for $\dot{q}$
and $\dot{p}$ by $\mbox{D} \psi$
to obtain from \eref{first2} and \eref{first3} that
\begin{equation*}
\left( \begin{array}{c} \dot{\theta} \\ \dot{a} \end{array} \right) =
\left( \begin{array}{c} \omega(a) \\ 0 \end{array} \right)
- \frac{\xi}{Q} \ \mbox{D} \psi \ \left( \begin{array}{c} 0 \\ p \end{array} \right).
\end{equation*}
As we are interested in the regime $Q \gg 1$, we rescale by
\begin{equation*}
\varepsilon = 1/\sqrt{Q}, \quad \alpha = \xi/\sqrt{Q}.
\end{equation*}
Using the symplectic property \eref{sym5} of $\psi,$ we
see that the Nos\'e-Hoover equation (\ref{nh-dyn}) can be then
given in the scaled angle-action variables by
\begin{equation}
\label{eq:NH_action_angle}
\left( \begin{array}{c} \dot{\theta} \\ \dot{a} \\ \dot{\alpha} \end{array}
\right) =
\left( \begin{array}{c} \omega(a) \\ 0 \\ 0 \end{array} \right)
+ \varepsilon \left( \begin{array}{c} \alpha (\partial_2 \phi_1)^T \phi_2 \\
-\alpha (\partial_1 \phi_1)^T \phi_2 \\
\phi_2^TM^{-1}(\phi_1)\phi_2 - N \beta^{-1} \end{array} \right).
\end{equation}
\subsection{Averaging the fast variables}
We next apply the averaging method to obtain an approximate system which
does not involve the fast variables. Rigorous results about averaging
for Hamiltonian systems with several degrees of freedom are fraught with
technical difficulties (see for example~\cite{AKN}). These arise from
the fact that $\omega(a)$, the frequency vector of the fast angles,
experiences resonances of the form $k\cdot \omega(a)=0$,
$k\in\mathbb{Z}^N$, for certain values of the action vector $a$. In fact,
these resonant actions are generally dense in the action domain $D$. In
spite of this, the averaged differential equations often provide a
useful first approximation to the behavior of the slow variables when
$\varepsilon$ is small.
In our problem, the averaged system for the slow variables is given by
\begin{equation}
\label{averaged}
\begin{array}{rcl}
\dot{a} &=& - \alpha \ S(a),
\\
\dot{\alpha} &=& k(a),
\end{array}
\end{equation}
where
\begin{equation}
\label{eq:def_S_k}
\begin{array}{rcl}
S(a) &=&\displaystyle{
\langle (\partial_1 \phi_1)^T \ \phi_2 \rangle(a)
=
\int_{\mathbb{T}^{N}}(\partial_1 \phi_1)^T (\theta,\,a)
\phi_2(\theta,\,a)\,d\theta
},
\\
k(a) &=& \displaystyle{
\langle \phi_2^TM^{-1}(\phi_1)\phi_2 \rangle(a) -
\frac{N}{\beta}
}
\\
&=& \displaystyle{
\int_{\mathbb{T}^{N}} \phi_2^T(\theta,\,a) M^{-1}
(\phi_1(\theta,\,a))\phi_2(\theta,\,a)
\,d\theta- \frac{N}{\beta}
}.
\end{array}
\end{equation}
We next show that, with an additional assumption on the action-angle
mapping $\phi$, we have
\begin{equation}\label{aa}
S(a) = a.
\end{equation}
Recall that a map $(q,p) = \phi(\theta,a)$ is symplectic if it preserves
the canonical differential two-forms, i.e.,
$\phi^*(\sum dp_i\wedge dq_i) = \sum da_i\wedge d\theta_i$
where $\phi^*$ indicates the pull-back, meaning that we write
$p_i, q_i, dp_i, dq_i$ in terms of the $\theta, a$ variables. It follows
from this that the difference of the corresponding canonical one-forms
$\phi^*(p^T\,dq) - a^T\,d\theta$ is a closed one-form on $ \mathbb{T}^{N}
\times D$. We will call $\phi$ {\em exact symplectic} if this closed
one-form is exact, i.e., if
\begin{equation}\label{exsym}
\phi^*(p^T\,dq) = a^T\,d\theta +dF(\theta, a)
\end{equation}
where $F(\theta,a)$ is a real-valued function on $\mathbb{T}^{N}\times D$. This
stronger condition holds for the action-angle coordinates associated to
many well-known integrable systems.
As an example, consider the one-dimensional harmonic oscillator $H(q,p)=
\frac12(p^2+q^2)$ which is a completely integrable system with $N=1$
degrees of freedom. In any annulus of the form $0<h_1\le (p^2+q^2)/2 \le
h_2$, we can introduce action-angle variables $(\theta,a)$ such that
\begin{equation*}
q = \sqrt{2a}\cos\theta, \qquad p = -\sqrt{2a}\sin\theta.
\end{equation*}
For $\phi(\theta,a) = (q,p)$, we have
\begin{equation*}
\phi^*(p\,dq)= -\sin\theta \cos\theta da +2a \sin^2\theta d\theta
\end{equation*}
and so
\begin{equation*}
\phi^*(pdq) - ad\theta = -\sin\theta \cos\theta da +a(2\sin^2\theta-1)
d\theta = dF
\end{equation*}
where
\begin{equation*}
F= -a\sin\theta\cos\theta.
\end{equation*}
It turns out that the action-angle variables constructed according to
the usual method of Arnold \cite{Arn} always have this exactness property.
To see this, recall that in Arnold's method, the tori given by fixing
the $N$-independent integrals of motion are parametrized by
angle variables $\theta= (\theta_1,\ldots,\theta_N)$ derived from the
$N$ commuting Hamiltonian flows defined by the integrals. Then the
action variables are given by
\begin{equation}\label{actionvars}
a_i = \int_{\gamma_i} p^T\,dq
\end{equation}
where $\gamma_i$ is the curve in the torus defined by holding $\theta_j
= \mbox{const}$ for $j\ne
i$ and letting $\theta_i$ run over $[0,1]$. The integral depends
on which torus is
considered, i.e., it is a function of the $N$ first integrals.
The usual proof shows that the map
$(q,p)=\phi(\theta,a)$ is defined and symplectic on some domain of the form
$\mathbb{T}^{N}\times D$. It follows that
\begin{equation*}
\nu = \phi^*(p^T\,dq) - a\,d\theta
\end{equation*}
is a closed differential one-form on $\mathbb{T}^{N}\times D$. Showing that
$\phi$ is exact
symplectic amounts to showing that $\nu$ is exact. For this, it suffices
to check that
its integral around any closed curve vanishes. In fact, since $\nu$ is
closed, it suffices to check the curves of the form
$C_i = \Gamma_i \times \{a_0\}, a_0\in D$
where $\Gamma_i = \{\theta:\theta_j= \mbox{const}, j\ne i\}$.
For such a curve, we have
\begin{equation*}
\int_{C_i} \nu = \int_{C_i} \phi^*(p\,dq) - a\,d\theta = \int_{\gamma_i} p\,dq -
\int_0^1 a_{0i}\,d\theta_i = a_{0i}-a_{0i} = 0.
\end{equation*}
Here we used the fact that under $\phi$ the curve $C_i$ maps to the curve $\gamma_i$
used in (\ref{actionvars}).
To prove \eref{aa} under the exact symplectic assumption \eref{exsym}, we first note that
\begin{equation*}
\phi^*(p^T\,dq) = \phi_2^T d\phi_1 =
\phi_2^T(\partial_1\phi_1\,d\theta + \partial_2\phi_1\,da).
\end{equation*}
Hence, \eref{exsym} reads
\begin{equation*}
\phi_2^T(\partial_1\phi_1\,d\theta + \partial_2\phi_1\,da) =
a^T\,d\theta +dF(\theta, a).
\end{equation*}
For any $j=1,\ldots,N$, we can integrate both sides with respect to
$\theta_j$, along the circular
loop $C_j$ as in the last paragraph. Using the
periodicity of $F$ in $\theta_j$, we obtain
\begin{equation}\label{explicit}
\sum_{i=1}^N\int_0^{1}\left(\frac {\partial \phi_{1i}}{\partial \theta_j}\right)
(\theta,\,a) \phi_{2i}(\theta,\,a)\,d\theta_j=a_j.
\end{equation}
We can then further integrate \eref{explicit} over the angles
$\theta_k$ for $k\ne j$ to obtain that
\begin{equation*}
S(a)_j=
\sum_{i=1}^N\int_{\mathbb{T}^{N}}\left(\frac {\partial \phi_{1i}}{\partial \theta_j}\right)
(\theta,\,a) \phi_{2i}(\theta,\,a)\,d\theta=a_j
\end{equation*}
for $j=1,\dots,N.$
\subsection{First integrals of the averaged Nos\'e-Hoover equations}
\label{sec:1st_int}
A direct calculation shows that a set of $N$ independent first integrals
for the averaged Nos\'e-Hoover equations
\begin{equation}\label{goodform}
\begin{array}{rcl}
\dot a &=& -\alpha a,\\
\dot \alpha &=& k(a),
\end{array}
\end{equation}
are given by
\begin{equation}\label{integ}
\begin{array}{rcl}
G_i(a,\,\alpha) &=& \displaystyle{ \frac{a_i}{a_N},\qquad i=1,\dots,N-1, }
\\
G_N(a,\,\alpha)&=& \displaystyle{
\frac{\alpha^2}2+\int^{a_N}\frac{k\left(s\frac{a_1}{a_N},
\dots,s\frac{a_{N-1}}{a_N},s\right)}s\,ds}.
\end{array}
\end{equation}
To prove that $G_N(a,\,\alpha)$ is a first integral, it is helpful to use
the fact that $G_i(a,\,\alpha)=a_i/a_N$ for $i=1,\dots,N-1$ is a first integral.
We summarize the result of this section in the following theorem.
\begin{theo} \label{thaveraged}
The averaged equations for the Nos\'e-Hoover dynamics for a
completely integrable Hamiltonian system
has $N$ independent first integrals.
\end{theo}
To the extent that the averaging method applies, we expect that $G_i(a(t),\,\alpha(t))$
evolves slowly for small $\varepsilon$ and so the sampling of the Gibbs measure is slow
even if the dynamics is ergodic. We will verify this numerically in an example in the
next section.
It turns out that $G_i(a(t),\,\alpha(t))$ remains quite close to its
initial value for fairly large values of $\varepsilon$ as well.
\section{A central force problem}
\label{multiD_num}
We consider here a two degrees of freedom system to illustrate the
theoretical results obtained in the previous section. We work with the
Hamiltonian (\ref{H_qp}) with $N=2$, the identity mass matrix $M(q) =
I_2$, and a potential $V(|q|)$ which depends only on the distance to the origin.
The Hamiltonian system (\ref{complete}) admits two first integrals, the
energy $H$ and the angular momentum
\begin{equation*}
L = q_1 p_2 - q_2 p_1,
\end{equation*}
which satisfy $\{H,L\} = 0$, and whose gradients are linearly
independent, except for values of $H$ and $L$ satisfying a condition of
the form $f(H,L) = 0$ for some function $f$. Hence, this system is
completely integrable.
Assume
that $V(r) \rightarrow +\infty$ as $r \rightarrow +\infty$. Then level
sets of $H$ are compact, hence the level sets
$\{ (q,p) \in \mathbb{R}^4; \ H(q,p) = h, L(q,p) = \ell \}$ are also compact,
hence there
exists action-angle variables for this system.
To describe the action variables, first introduce polar coordinates
$(r,\phi)$ in $\mathbb{R}^2$. The angular momentum is
\begin{equation*}
L= r^2 \dot\phi.
\end{equation*}
Fixing a value for $L$, we have a reduced Hamiltonian system for the
radial variables $(r,p_r)$, where $p_r=\dot r$, with Hamiltonian
\begin{equation*}
H_L(r,p_r) = \frac12 p_r^2 + \frac12 \frac{L^2}{r^2} + V(r).
\end{equation*}
This reduced system has one degree of freedom and can be understood by
the usual phase-plane method. Since $V(r)
\rightarrow +\infty$ as $r \rightarrow +\infty$, the level curves
\begin{equation*}
C_{(h,L)} = \{(r,p_r):H_L(r,p_r) = h\}, \qquad L\ne 0,
\end{equation*}
generically consist of one or more simple closed curves. For the
unreduced system, where we remember the angle $\phi$, each such curve
becomes an invariant torus $T_{(h,L)}$. It can be shown that the action
variables assigned to such a torus by Arnold's procedure are as follows:
$a_1(h,L)$ is the area in the $(r,p_r)$ plane enclosed by the simple
closed curve of the reduced system,
\begin{equation*}
a_1(h,L) = \int_{C_{(h,L)}} p_r \, dr,
\end{equation*}
and $a_2(h,L) = L$, the angular momentum. Note that $a_1(h,L)$ is easily
computable by standard numerical integration schemes.
Since $L$ is homogeneous of degree $k=1$ in the momentum variables,
Theorem~\ref{thhomogeneous} gives a first integral for the Nos\'e-Hoover
system:
\begin{equation}
\label{eq:inv1}
G(q,p,\xi) = \frac{\xi^2}{2Q} + H(q,p) -
\frac{2}{\beta} \ln \left| L(q,p) \right|.
\end{equation}
In addition, Theorem~\ref{thaveraged} provides additional integrals for
the averaged Nos\'e-Hoover equations, in particular the ratio of the
action variables
\begin{equation}
\label{eq:inv2}
G_1(q,p) = \frac{a_1(q,p)}{a_2(q,p)} = \frac{a_1(H(q,p),L(q,p))}{L(q,p)}.
\end{equation}
To the extent that the averaging method applies for this two-degrees of
freedom problem, this ratio should evolve only very slowly when $Q \gg
1$. In the sequel, we present some
numerical simulations showing first that, when $Q \gg 1$, this is indeed
the case, and second that, for $Q =1$, such a behaviour persists
to some extent.
\medskip
Consider the example with potential
\begin{equation*}
V(r) = r^2+r^4,
\end{equation*}
with an initial condition $(q_0,p_0,\xi_0)$ such that
$L(q_0,p_0) \neq 0$. We compute the trajectory of the Nos\'e-Hoover
dynamics \eref{nh-dyn} with the algorithm proposed in \cite{molphys96}.
On Figure~\ref{fig:Q100}, we plot
$G(q(t),p(t),\xi(t))$ and $G_1(q(t),p(t))$, where $G$ and $G_1$ are
defined by \eref{eq:inv1} and \eref{eq:inv2}, for $Q=100$. We indeed
observe that $G$ is preserved, whereas $G_1$ evolves slowly.
\begin{figure}[htbp]
\centerline{\input{figure1.tex}}
\caption{Plot of $G(q(t),p(t),\xi(t))$ and $G_1(q(t),p(t))$
(renormalized by their initial value) along the trajectory of
(\ref{nh-dyn}), for $Q=100$ ($\beta = 1$, initial
condition $q=(0;0.5)$, $p=(-1.5; 1.5)$, $\xi=0$).}
\label{fig:Q100}
\end{figure}
We now consider the value $Q=1$ and plot the same quantities as above on
Figure~\ref{fig:Q1}. Again, $G$ is preserved, whereas $G_1$ evolves in a
band which is still quite narrow, even for this small value of $Q$.
\begin{figure}[htbp]
\centerline{\input{figure2.tex}}
\caption{Plot of $G(q(t),p(t),\xi(t))$ and $G_1(q(t),p(t))$
(renormalized by their initial value) along the trajectory of
(\ref{nh-dyn}), for $Q=1$ ($\beta = 1$, initial
condition $q=(0;0.5)$, $p=(-1.5; 1.5)$, $\xi=0$).}
\label{fig:Q1}
\end{figure}
\medskip
Let us now derive another
quantity, which does not behave as well as $G_1$
for large $Q$, but happens to behave in a better way for small
$Q$\footnote{We have a clear
understanding of why $G_1$ behaves better than this quantity when $Q
\gg 1$. However, the situation for $Q=1$ is less clear.}.
From the Nos\'e-Hoover dynamics \eref{nh-dyn}, we compute that
\begin{eqnarray}
\label{Ldyn}
\dot{L} &= - \frac{\xi}{Q} L = - \varepsilon \alpha L,
\\
\label{Hdyn}
\dot{H} &= \displaystyle{ - \frac{\xi}{Q} p^T p = - \varepsilon \alpha \, p^T p
= - \varepsilon \alpha \, \phi_2^T(\theta,a) \, \phi_2(\theta,a) }.
\end{eqnarray}
Since $\theta$ are fast variables whereas $a$, $\alpha$ and $H$ are slow
ones, we again formally use the averaging method on \eref{Hdyn} and consider the dynamics
\begin{equation}
\label{eq:H_approx}
\dot{H} = - \varepsilon \alpha \, k_0(a)
\end{equation}
with
\begin{equation*}
k_0(a)= \int_{\mathbb{T}^2} \phi_2^T(\theta,a) \, \phi_2(\theta,a) \,
d\theta
\end{equation*}
(note that \eref{Ldyn} does not depend on the fast variables $\theta$).
Now recall that the action variables $a$ are functions of $H$ and
$L$. The equation \eref{eq:H_approx} hence reads
\begin{equation*}
\dot{H} = - \varepsilon \alpha \, k_0(H,L).
\end{equation*}
The averaged system is thus
\begin{equation}
\label{averaged_2d}
\begin{array}{rcl}
\dot{L} &=& - \alpha L,
\\
\dot{H} &=& - \alpha \, k_0(H,L),
\\
\dot{\alpha} &=& k_0(H,L) - 2 \beta^{-1}.
\end{array}
\end{equation}
Note that
\begin{equation}
\label{eq:def_E}
E(H,L,\alpha) = \frac{\alpha^2}2 + H -
\frac{2}{\beta} \ln \left| L \right|
\end{equation}
is a first integral of the above system. It is just the analogue of
\eref{eq:inv1} being a first integral for the Nos\'e-Hoover system (see
Theorem~\ref{thhomogeneous}).
On Figure~\ref{fig:k0_2d}, we plot the function $H \mapsto k_0(H,L)$,
for several values of $L$. We observe that $k_0(H,L)$
is almost a constant with respect to $L$, and can hence be
approximated\footnote{In practice, we have considered several energy values
$H_i$, and for each $H_i$, we have considered several configurations
$(q_{i,j},p_{i,j})$ with energy $H_i$ and angular momentum
$L_{i,j}$. We next have computed $k_0(H_i,L_{i,j})$ by averaging
$p(t)^T p(t)$ along a constant energy trajectory. Averaging these
$k_0(H_i,L_{i,j})$, we obtain $k_0^{\rm app}(H_i)$, which next leads
to $k_0^{\rm app}(H)$ for any $H$ by piecewise linear interpolation.}
by a function $k_0^{\rm app}(H)$.
\begin{figure}[htbp]
\centerline{\input{figure3.tex}}
\caption{Plot of $H \mapsto k_0(L,H)$. For each value of $H$, we have
considered several values of $L$.}
\label{fig:k0_2d}
\end{figure}
We hence approximate \eref{averaged_2d} by
\begin{equation}
\label{averaged_2d_bis}
\begin{array}{rcl}
\dot{L} &=& - \alpha L,
\\
\dot{H} &=& - \alpha \, k_0^{\rm app}(H),
\\
\dot{\alpha} &=& k_0^{\rm app}(H) - 2 \beta^{-1}.
\end{array}
\end{equation}
Now, it is natural to introduce the variable $\tau$ defined by
\begin{equation*}
\tau(H) = \exp \left( \int^{H} \frac{ds}{k_0^{\rm app}(s)} \right)
\end{equation*}
and its reciprocal $H(\tau)$, such that \eref{averaged_2d_bis} reads
\begin{equation}
\label{averaged_2d_ter}
\begin{array}{rcl}
\dot{L} &=& - \alpha L,
\\
\dot{\tau} &=& - \alpha \, \tau,
\\
\dot{\alpha} &=& k_0^{\rm app}(H(\tau)) - 2 \beta^{-1}.
\end{array}
\end{equation}
This system is in the form \eref{goodform}. Its two first integrals
are
\begin{equation*}
E_1(L,\tau) = \frac{\tau}{L}
\end{equation*}
and
\begin{eqnarray*}
E_2(\tau,\alpha) &=&
\frac{\alpha^2}2 + \int^\tau \frac{k_0^{\rm app}(H(s)) - 2
\beta^{-1}}{s} \, ds
\\
&=&
\frac{\alpha^2}2 + \int^\tau \frac{k_0^{\rm app}(H(s))}{s} \, ds -
\frac{2}{\beta} \ln \tau
\\
&=&
\frac{\alpha^2}2 + H(\tau) - \frac{2}{\beta} \ln \tau.
\end{eqnarray*}
The first invariant $E$ given by \eref{eq:def_E}
is not independent from $E_1$ and $E_2$: $E_2 = E - 2 \beta^{-1} \ln
|E_1|$.
We now consider the same trajectories of the Nos\'e-Hoover dynamics
that we considered on Figures~\ref{fig:Q100} and \ref{fig:Q1}, and we
plot
\begin{equation*}
E_1(q,p) = E_1 \left( L(q,p),\tau(H(q,p))\right).
\end{equation*}
We see on Figure~\ref{fig:G1_particular} that, for $Q=100$, this
quantity is almost preserved, and that, even for $Q=1$, it remains close
to its initial value.
\begin{figure}[htbp]
\centerline{\input{figure4.tex}}
\caption{$E_1(q(t),p(t))$ (renormalized by its initial
value) along the trajectory of
(\ref{nh-dyn}), for $Q=1$ and $Q=100$ ($\beta = 1$, initial
condition $q=(0;0.5)$, $p=(-1.5; 1.5)$, $\xi=0$).}
\label{fig:G1_particular}
\end{figure}
We finally consider an initial condition such that $L(q_0,p_0) =
0$. Along the trajectory of \eref{nh-dyn}, we have
$L(q(t),p(t))=0$ by \eref{Ldyn}, hence $E_1$ is not defined.
On Figure~\ref{fig:G2_particular}, we plot
\begin{equation*}
E_2(q,p,\xi) = E_2 \left( \tau(H(q,p)), \frac{\xi}{\sqrt{Q}} \right)
\end{equation*}
along two trajectories, obtained with the same initial condition and the
choices $Q=100$ and $Q=1$. We again observe that $E_2$ is almost
constant for $Q=100$, and that it remains close to its
initial value for $Q=1$.
\begin{figure}[htbp]
\centerline{\input{figure5.tex}}
\caption{$E_2(q(t),p(t),\xi(t))$ (renormalized by its initial
value) along the trajectory of
(\ref{nh-dyn}), for $Q=1$ and $Q=100$ ($\beta = 1$, initial
condition $q=(-0.5;0.5)$, $p=(-1; 1)$, $\xi=0$).}
\label{fig:G2_particular}
\end{figure}
On Figure~\ref{fig:H_L_null}, we plot the energy $H(q(t),p(t))$
along the same trajectory (for $Q=1$). We see that values $h \leq 1$ are
not sampled. However, there exist $(q,p) \in \mathbb{R}^4$
such that $L(q,p) = 0$ and $H(q,p)$ is as close to 0 as wanted. Hence,
the trajectory only samples a strict subset of the level set
$\{ (q,p); \ L(q,p) = 0 \}$.
\begin{figure}[htbp]
\centerline{\input{figure6.tex}}
\caption{$H(q(t),p(t))$ along the trajectory of
(\ref{nh-dyn}), for $Q=1$ ($\beta = 1$, initial
condition $q=(-0.5;0.5)$, $p=(-1; 1)$, $\xi=0$).}
\label{fig:H_L_null}
\end{figure}
\section{Systems with one degree of freedom}
\label{1Dgeneral}
Consider a Hamiltonian system of the form (\ref{H_qp}) with $N=1$ and
$M(q) = 1$. All such systems are
completely integrable since $H$ itself provides the required integral of motion. Suppose
there is
an interval of energies $I = [h_1, h_2]$ such that the level curves
$M(h) = \{(q,p):H(q,p)= h\}, h\in I,$ are all simple closed curves (one-dimensional tori)
which are non-degenerate in the sense that
the gradient of $H$ does not vanish. Then the
plane region $U = \{(q,p): h_1\le H(p,q) \le h_2 \}$ is diffeomorphic to an annulus,
and we can introduce action-angle variables $(a,\theta)$ in $U$, and an
exact symplectic map $\phi(\theta,a) = (q,p)$, as in Section
\ref{sec:action_angle_multiD}.
From Theorem~\ref{thaveraged}, we have $N=1$ first integrals for the averaged
Nos\'e-Hoover equations. Let us now rewrite
this first integral more explicitly. In view of \eref{integ}, we have
\begin{equation}
\label{eq:G}
G(a,\alpha) = \frac{\alpha^2}{2} + W(a)
\end{equation}
with
\begin{equation}
\label{eq:W1D}
W(a) = \int^a \frac{k(s)}{s} \, ds,
\end{equation}
where, in view of \eref{eq:def_S_k}, $k$ is given by
\begin{equation}
\label{eq:k1D}
k(a) = \int_{\mathbb{T}}
\phi^2_2(\theta,\,a) \, d\theta - \frac{1}{\beta}.
\end{equation}
As in the multidimensional case, this integral prevents rapid
ergodization, at least for small $\varepsilon$. But in the one-degree of
freedom case we go further and identify conditions on $H$ which
rigorously imply non-ergodicity. The method is essentially the one
used in \cite{LLM} where we treated the harmonic oscillator. Namely,
the integral $G$ leads to invariant tori of the averaged system which,
under certain assumptions, persist for small values of $\varepsilon$.
\subsection{Proof of non-ergodicity}
\label{sec:1Dtheory}
We will apply a KAM theorem to the Nos\'e-Hoover equations, in the formulation
\eref{eq:NH_action_angle}. Let us introduce the Poincar\'e return map,
$P_\varepsilon$, of the system \eref{eq:NH_action_angle} to the Poincar\'e
section defined by $\theta = 0 \mbox{ mod } 1$. It is convenient to rescale time
by $\omega(a)$, so that the return time when $\varepsilon = 0$ is 1. This just
alters the parametrization
of the solutions so that the return time to the Poincar\'e section is~1.
Since there is only one degree of freedom, the averaging method can be rigorously
justified.
Indeed we can eliminate the fast angle
$\theta$ of (\ref{eq:NH_action_angle}) by a change of variables.
We construct functions $g(\hat{a},\theta,\hat{\alpha})$
and $h(\hat{a},\theta,\hat{\alpha})$ and corresponding new variables
$(\hat{a},\hat{\alpha})$ defined by
\begin{equation}
\label{eq:change0}
\begin{array}{rcl}
a &=& \hat{a} + \varepsilon g(\hat{a},\theta,\hat{\alpha}),
\\
\alpha &=& \hat{\alpha} + \varepsilon h(\hat{a},\theta,\hat{\alpha}),
\end{array}
\end{equation}
so that in
the new variables $(\hat{a},\hat{\alpha})$, the dynamics
(\ref{eq:NH_action_angle}) is given (after replacing
$(\hat{a},\hat{\alpha})$ by $({a},{\alpha})$) by
\begin{equation}
\label{eq:NH_action_angle20}
\begin{array}{rcl}
\dot{\theta}&=&\omega(a)+O(\varepsilon),\\
\dot{a}&=&-\varepsilon \alpha S(a)+O(\varepsilon^2),\\
\dot{\alpha}&=&\varepsilon k(a)+O(\varepsilon^2),
\end{array}
\end{equation}
where $S(a)$ and $k(a)$ are the averages (\ref{eq:def_S_k}).
In view of \eref{eq:change0} and \eref{eq:NH_action_angle20},
the Poincar\'e map $P_\varepsilon(\hat{a},\hat{\alpha})$ is an $O(\varepsilon^2)$
perturbation of the time $\varepsilon$ advance map of the
averaged system \eref{averaged}, for which $G$ defined
by \eref{eq:G} is a first integral.
So we now make some assumptions about
the level curves of $G$.
Recall that we are working in a region $U$ of the
$(q,p)$-plane defined by an interval of energies $I$ which corresponds to an interval of
actions
$J = [a_1,a_2] $. We assume that $W(a)$ has at least one local minimizer
$a_0$ in $J$. We have
\begin{equation*}
0 = W'(a_0) = \frac{k(a_0)}{a_0},
\end{equation*}
hence $k(a_0) = 0$ and the point $P=(a_0,0)$ is an equilibrium point for
(\ref{averaged}). The parts of the level curves of $G$ which are near
$P$ are simple closed curves around $P$ in the $(a,\alpha)$-plane.
\begin{rem}
\label{rem:eq2solve}
If $a_0$ is a local minimizer of $W$, then $k(a_0) = 0$, hence
$\displaystyle \int_{\mathbb{T}} \phi^2_2(\theta,\,a_0) \, d\theta = \beta^{-1}$.
\end{rem}
Let $G_0 = G(a_0,0) = W(a_0)$ be the value of the integral $G$ at the
equilibrium point $P=(a_0,0)$. Choose constants $\tilde{G_1}$ and
$\tilde{G_2}$ such that $G_0 < \tilde{G_1} < \tilde{G_2} <
\min(G(a_1,0),G(a_2,0)) = \min(W(a_1),W(a_2))$
and let $\tilde K = [\tilde G_1,\tilde G_2]$ (see
Figure~\ref{fig:tilde_K}). Then the level curves $\{
(a,\alpha); \ G(a,\alpha) = c \}$, where $c \in \tilde{K}$, have connected
components which are simple closed curves near $P$. The union of these
components for $c \in
\tilde{K}$ forms a region $\tilde D$ near $P$ which is diffeomorphic to
an annulus.
\begin{figure}[htbp]
\centerline{\input{figure7.pstex_t}}
\caption{Schematic representation of the interval $\tilde{K} = [\tilde
G_1,\tilde G_2]$.}
\label{fig:tilde_K}
\end{figure}
The variable $G$ defines a natural action variable in $\tilde D$.
We construct the corresponding angle variable $\phi$ by following the
same method as in \cite{LLM}. Let $T_1(g)$ denote
the period of the periodic solutions of (\ref{averaged}) which
corresponds to the level curve $G = g \in \tilde K$. The averaged
differential equation (\ref{averaged}) becomes
\begin{equation}
\label{averaged3}
\begin{array}{rcl}
\dot{\phi} &=& 1/T_1(G),
\\
\dot{G} &=& 0.
\end{array}
\end{equation}
In these coordinates, the time $\varepsilon$ advance map takes the form
$(\phi,G) \mapsto (\phi_1,G_1)$ where
\begin{equation}
\label{timeeps}
\begin{array}{rcl}
\phi_1 &=& \phi + \varepsilon/T_1(G),
\\
G_1 &=& G.
\end{array}
\end{equation}
Call this map $Q_\varepsilon(\phi,G)$. Then $P_\varepsilon(\phi,G) = Q_\varepsilon(\phi,G) +
O(\varepsilon^2)$.
\begin{theo}
\label{theo:non_ergo}
Suppose the period function $T_1(G)$ is not identically
constant on the interval $\tilde K$. Then, for $\varepsilon$ sufficiently
small, the Poincar\'e map $P_\varepsilon$ has invariant circles in the region
$\tilde D$ and so the Nos\'e-Hoover system is not ergodic: trajectories
$(q(t),p(t))$ that solve (\ref{nh-dyn}) are not ergodic with respect to
the Gibbs measure \eref{measure_can}.
\end{theo}
\begin{proof}
As in \cite{LLM}, we apply Moser's twist theorem to the Poincar\'e
map $P_\varepsilon$. The details are similar to those in \cite{LLM}, so we
only sketch the argument here.
The fact that the Nos\'e-Hoover differential equation preserves the
invariant measure \eref{invmeasure} implies (as in \cite{LLM}) that
$P_\varepsilon$ preserves an
invariant measure in the $(a,\alpha)$ plane. It follows that the maps
$P_\varepsilon$ have the curve intersection property. The hypothesis on
$T_1(G)$ guarantees that, making $\tilde K$ and hence $\tilde D$ smaller
if necessary, we may assume that either $T_1'(G)>0$ or $T_1'(G)<0$
throughout $\tilde D$. This means that there are many invariant circles
in $\tilde D$ for which the rotation number under $Q_\varepsilon$ is Diophantine.
Moreover, the required twist condition holds. Moser's theorem guarantees
that such invariant circles perturb to nearby invariant circles for
$\varepsilon$ sufficiently small.
We now show that the existence of these invariant circles implies
non-ergodicity with respect to the Gibbs measure. First, note that
on a level curve ${\mathcal M} = \left\{ (a,\alpha); \ G(a,\alpha) = c
\right\}$, where $c \in \tilde{K}$, we have that $W(a)$ is bounded from
above, since $W(a) \leq G(a,\alpha) = c$. In view of the choice of
$\tilde{K}$ (see Figure~\ref{fig:tilde_K}), this implies that $a$ is
lower and upper bounded. Since $a'(h)$ is positive and bounded away from
0 for $h \in I = [h_1,h_2]$, this hence shows that $H(q,p)$ is lower and upper bounded
(that is,
$| H(q,p)|$ is bounded) on the invariant circle ${\mathcal M}$ of
$Q_\varepsilon$.
As a consequence, $|
H(q,p)|$ is bounded on the nearby invariant circles of $P_\varepsilon$. Hence,
the trajectory of (\ref{nh-dyn}) does not sample values of $H(q,p)$
larger than some threshold. This is a contradiction with $(q(t),p(t))$
that solves (\ref{nh-dyn}) being ergodic with respect to the Gibbs
measure \eref{measure_can}.
\end{proof}
\medskip
Because of the complicated series of coordinate changes leading from the
original Hamiltonian system to the averaged system, it is not easy to
state simple conditions on the original potential function $V(q)$ which
guarantee that the period function $T_1(G)$ is not constant. An
equilibrium point surrounded by periodic orbits of constant period is
called isochronous and various criteria for isochronicity have been
given. Our problem can be reduced to a Hamiltonian case for which a
simple criterion can be stated.
To carry out the reduction, replace $a$ in (\ref{averaged}) (that is,
\eref{goodform}) by $\sigma = \ln (a/a_0)$, where $a_0$ is a local
minimizer of $W$ (see Figure~\ref{fig:tilde_K}). The
differential equation (\ref{goodform}) becomes
\begin{equation}
\label{averaged4}
\dot \sigma = -\alpha,
\quad
\dot \alpha = U'(\sigma),
\end{equation}
where $U(\sigma) = W(a_0 \exp \sigma)$. Except for a reversal of time, this
is a classical Hamiltonian system with Hamiltonian
$G(\sigma,\alpha) = \alpha^2/2 + U(\sigma)$. It has an
equilibrium point at the origin $(\sigma,\alpha) = (0,0)$.
Now \cite{LL} discusses the problem of recovering the potential of such
a system from its period function (see also \cite{CMV}). Let $G_0 =
U(0)$ be the energy level of the equilibrium point at the origin. For
$G>G_0$ let $L(G)$ be the width of the potential well at energy $G$,
i.e., $L(G)= \sigma_2(G)-\sigma_1(G)$ where $\sigma_i(G)$ are the two
roots of $U(\sigma) = G$ near $\sigma=0$. Then $T_1(G)$ is constant
if and only if $\displaystyle L(G) = \frac{T_1}{\pi}\sqrt{2(G-G_0)}$.
This is just the formula for the width of the quadratic potential well
associated to a harmonic oscillator of period $T_1$. Clearly this is highly
exceptional and is easy to rule out, at least numerically.
Another way to show that $T_1(G)$ is non-constant is to observe that the
constancy of the period implies that the family of periodic orbits
surrounding the equilibrium point must fill the entire plane \cite{CMV}. If
this were not so, then there would be another equilibrium point on the
boundary of the maximal family which would force
$T_1(G)\rightarrow\infty$. For example, for certain values of
$\beta$, the pendulum equations (see Section \ref{1pendulum}) lead
to an averaged system with more than one equilibrium, and this
immediately implies that $T_1(G)$ is non-constant.
\subsection{The simple pendulum problem}
\label{1pendulum}
We consider here the numerical example of a simple pendulum whose
potential energy is given by
\begin{equation*}
V(q) = - \cos q.
\end{equation*}
We reduce $q$ modulo $2 \pi$.
By construction, the energy satisfies $h \geq -1$. The phase portrait is
shown on Figure~\ref{fig:pendulum}.
The above assumptions are satisfied for energies
in the interval $I=[h_1,h_2]$, with $-1<h_1<h_2<1$, or $1<h_1<h_2$.
\begin{figure}[htbp]
\centerline{\input{figure8.tex}}
\caption{Phase portrait of the simple pendulum.}
\label{fig:pendulum}
\end{figure}
First, we numerically compute $a(h)$ defined by
\eref{actionvars}. Note that, in this one-dimensional setting,
\begin{equation*}
a(h) = \int_{M(h)} p \ dq ,
\end{equation*}
where the line integral is taken in the direction of the Hamiltonian
flow, and $M(h) = \{(q,p) \in \mathbb{R}^2: \ H(q,p)= h\}$. We also compute
\begin{equation*}
k_0(a) = \int_{\mathbb{T}} \phi^2_2(\theta,\,a) \, d\theta,
\end{equation*}
which is independent of $\beta$ and satisfies $k(a) = k_0(a) - \beta^{-1}$,
with $k$ defined by \eref{eq:k1D}. In practice, $k_0$ is computed using
the fact that, for any energy level $h$,
\begin{equation*}
k_0(a(h)) = \lim_{T \to +\infty} \frac{1}{T} \int_0^T p^2(t) \, dt,
\end{equation*}
where $(q(t),p(t))$ solve the Newton equations of motion for the
pendulum at the constant energy $h$.
Results are shown on Figure~\ref{fig:a_h}.
\begin{figure}[htbp]
\centerline{
\input{figure9a.tex}
\input{figure9b.tex}
}
\caption{Numerically computed values of $a(h)$ and $k_0(a(h))$ (see text).}
\label{fig:a_h}
\end{figure}
We have seen that the action values $a$ such that $k_0(a) = \beta^{-1}$,
that is $k(a) = 0$,
play an important role (see Remark \ref{rem:eq2solve}).
In view of Figure~\ref{fig:a_h}, we see that, when $\beta^{-1} < \beta_c^{-1}$ for
some threshold $\beta_c$, then the equation $k_0(a) = \beta^{-1}$ has
three solutions. When $\beta^{-1} > \beta_c^{-1}$, then the equation
$k_0(a) = \beta^{-1}$ has a unique solution. In what follows, we detail
the numerical results obtained with the choice $\beta = 1$, which
corresponds to the first case. Similar results have been
obtained for
choices of $\beta$ corresponding to the other case. Hence, the
conclusions that we draw here are by no means restricted to the case
$\beta = 1$.
The function $W(a)$ defined by \eref{eq:W1D} is shown on
Figure~\ref{fig:w_h} for the choice $\beta = 1$. This function has two
local minimizers, $a_1 \approx 7.6$ and $a_2 \approx 16.17$. Following
the above theoretical analysis, we work close to one of them. We have
chosen to work close to $a_1$.
\begin{figure}[htbp]
\centerline{\input{figure10.tex}}
\caption{Numerically computed values of $W(a)$ for $\beta = 1$.}
\label{fig:w_h}
\end{figure}
On Figure~\ref{fig:level}, we plot the trajectory of the averaged dynamics
(\ref{averaged}) for different initial conditions. These trajectories have been
computed with the Symplectic Euler algorithm used on the Hamiltonian
formulation (\ref{averaged4}). As expected, the trajectory
is a simple closed curve around the equilibrium point $(a_1,0)$,
that corresponds to a level curve of $G$. These curves are also
invariant curves of the map $Q_\varepsilon$ defined in the previous section
(see map \eref{timeeps}).
\begin{figure}[htbp]
\centerline{\input{figure11.tex}}
\caption{Trajectories of (\ref{averaged}) for several
different values of $a(0)$ ($\beta = 1$).}
\label{fig:level}
\end{figure}
We now study how these curves persist upon perturbation.
We recall that the Poincar\'e return map $P_\varepsilon$ of the Nos\'e-Hoover
dynamics (\ref{eq:NH_action_angle}) on the section $\theta = 0 \mbox{ mod }
1$ is a perturbation of $Q_\varepsilon$.
Results for the Poincar\'e return map of the dynamics
(\ref{eq:NH_action_angle}) are shown on Figure~\ref{fig:level_full} for $Q
= 10^5$ (that is, $\varepsilon = \sqrt{10} \times 10^{-3}$), and on
Figure~\ref{fig:level_full_very_low} for $Q=1$ (that is, $\varepsilon = 1$),
for the same initial energies as for Figure~\ref{fig:level}.
These Poincar\'e return maps have been computed using the fact that the
section $\theta = 0 \mbox{ mod } 1$ corresponds to the section $q = 0 \mbox{ mod } 2
\pi$.
We see a good agreement between Figures~\ref{fig:level}
and~\ref{fig:level_full}.
The presence of invariant circles on Figures~\ref{fig:level_full} and
\ref{fig:level_full_very_low} shows that the system
(\ref{eq:NH_action_angle}) seems to have invariant curves, for $Q=10^5$
and $Q=1$.
\begin{figure}[htbp]
\centerline{\input{figure12.tex}}
\caption{Poincar\'e return map of (\ref{eq:NH_action_angle}) on the plane
$\theta = 0 \mbox{ mod } 1$ for several initial conditions ($Q = 10^5$, $\beta
= 1$).}
\label{fig:level_full}
\end{figure}
\begin{figure}[htbp]
\centerline{\input{figure13.tex}}
\caption{Poincar\'e return map of (\ref{eq:NH_action_angle}) on the plane
$\theta = 0 \mbox{ mod } 1$ for several initial conditions ($Q = 1$, $\beta
= 1$).}
\label{fig:level_full_very_low}
\end{figure}
Note that Theorem \ref{theo:non_ergo}, which states the non-ergodicity
of the Nos\'e-Hoover equations, relies on the important assumption
that the period $T_1(G)$ of the averaged equations is not constant. This
holds true for the pendulum case, in view of the discussion at the end of Section
\ref{sec:1Dtheory}. This is
also confirmed by numerical computations of $T_1(G)$ (see
Figure~\ref{fig:t1_g}).
\begin{figure}[htbp]
\centerline{\input{figure14.tex}}
\caption{Period $T_1(G)$ of the averaged equation \eref{averaged}
($\beta = 1$).}
\label{fig:t1_g}
\end{figure}
Let us now look at another criterion for ergodicity, namely what
energy values are sampled. We see on
Figure~\ref{fig:level_full_very_low} that small values of $a$
are not sampled: we have $a \geq 6$ for the three initial conditions
that we considered. In view of Figure~\ref{fig:a_h}, this
corresponds to small values of $H$ not being sampled. On
Figure~\ref{fig:energy_particular}, we plot the physical energy
$H(q(t),p(t))$ along the trajectory of
(\ref{eq:NH_action_angle}), for the value $Q=1$, and the initial
condition $q=0$, $p=1.5$, $\xi=0$, that corresponds to the initial value
$a(0) = 7.72$ that we studied on Figures~\ref{fig:level},
\ref{fig:level_full} and \ref{fig:level_full_very_low} (results are the same
for other initial conditions). We see that $H \geq
-0.4$. If the dynamics (\ref{eq:NH_action_angle}) was sampling the
canonical measure, then all values of $H$
would be attained. In particular, the smallest values $H \approx -1$ would be
the most frequent ones. Indeed, from the Gibbs measure
\eref{measure_can}, we compute the probability distribution function of
the energy, which reads $\rho(h) = z^{-1} \exp(-\beta h) \, a'(h)$, where
$z$ is a normalization constant. For the pendulum case, $a'(h)$ is
close to a constant (see Figure~\ref{fig:a_h}), hence the smallest
values of $h$ are the most frequent ones.
Hence, it seems that (\ref{eq:NH_action_angle}) is
not ergodic with respect to the canonical measure, even for the
value $Q=1$.
\begin{figure}[htbp]
\centerline{\input{figure15.tex}}
\caption{Energy $H(q(t),p(t))$ along the trajectory of
(\ref{nh-dyn}), for $Q=1$ and $\beta = 1$, and the initial
condition $q=0$, $p=1.5$, $\xi=0$ (that is, $a(0) = 7.72$).}
\label{fig:energy_particular}
\end{figure}
\ack
Part of this work was completed while the first author
was visiting the Institute for Mathematics and its Applications
(Minneapolis), whose hospitality is gratefully acknowledged.
The work of Fr\'ed\'eric Legoll was supported in part by the Agence Nationale de la
Recherche (INGEMOL non-thematic program) and by the Action Concert\'ee Incitative
``Nouvelles Interfaces des Math\'ematiques'' SIMUMOL (Minist\`ere de
la Recherche et des Nouvelles Technologies, France).
The work of Mitchell Luskin was supported in part by NSF Grants
DMS-0757355 and DMS-0811039, the Institute for Mathematics
and its Applications,
and by the University of Minnesota Supercomputing Institute.
This work is also based on
work supported by the Department of Energy under Award Number
DE-FG02-05ER25706.
Richard Moeckel was partially supported by NSF Grant
DMS-0500443.
\section*{References}
\bibliographystyle{plain}
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\section{Recollection of relevant results}
Using a construction from symplectic toric geometry (see \S 1 of
\cite{raz04:app_guil_abr_non_abel_group_act}) we proved the
following
\begin{theorem}[\cite{raz04:scal_cur_mul_free_act}]
Suppose $T^n$ acts on $\mathbb{C}^n$ (with its fixed standard symplectic
structure) via the standard linear action so that the moment
polytope $\Delta_{\mathbb{C}^n}$ of the moment map corresponding to this
action is the positive orthant $\bbR^n_{\geq 0}$ in $\mathbb{R}^n\cong(\mathbb{R}^n)^*$
with standard symplectic (action) coordinates
$(x_1,\dots,x_n)=\frac{1}{2}(|z_1|^2,\dots,|z_n|^2)$, where
$(z_1,\dots,z_n)$ are the standard complex coordinates on $\mathbb{C}^n
\setminus \{0\}$. Now consider the ${\rm SU}(n)$-invariant K\"ahler
metric $h_f$ on $\mathbb{C}^n \setminus \{0\}$ determined by the standard
complex structure on $\mathbb{C}^n$ and the K\"ahler potential $f(s)$
where $s=\sum_{i=1}^n|z_i|^2$. Then there exists a K\"ahler isometry
between the K\"ahler manifold $(\mathbb{C}^n \setminus \{0\},h_f)$ and the
K\"ahler manifold $(\mathbb{C}^n \setminus \{0\},h_g)$ where $h_g$ is the
${\rm SU}(n)$-invariant K\"ahler metric on $\mathbb{C}^n \setminus \{0\}$
determined by the standard symplectic structure on $\mathbb{C}^n$ and the
symplectic potential
\begin{equation}\label{spp}
g(x)=\frac{1}{2}\left(\sum_{i=1}^n x_i \ln x_i + F(t)\right)
\end{equation}
which is a smooth function defined on the interior
$\Delta^\circ_{\mathbb{C}^n}=\mathbb{R}^n_{>0}$ of $\Delta_{\mathbb{C}^n}$, where
\[
t=\sum_{i=1}^nx_i=2sf'(s)
\]
and
\[
F(t)= t \ln \left(s(t)t^{-1}\right) - 2f(s(t))
\]
which we call the $t$-potential of \eqref{spp}. Furthermore, the
scalar curvature of $h_g$ (and hence of $h_f$) is given (in the
coordinates $x$) by
\begin{equation}\label{ss}
S(g)=t^{1-n}\left(t^{n+1}F''(t)(1+tF''(t))^{-1}\right)''.
\end{equation}
Conversely, any function of the form $g(x)$ on
$\Delta^\circ_{\mathbb{C}^n}$ with $F(t)$ satisfying $ F''(t)>-t^{-1}$
determines such a K\"ahler metric.
\end{theorem}
Note that ${\rm SU}(n)$-invariant K\"ahler metrics on $\mathbb{C}^n \setminus
\{0\}$ are in fact ${\rm U}(n)$ (hence $T^n$)-invariant which is why the
above result comes about using ideas from toric geometry. Motivated
by the discussion in \S 3 of \cite{abr03:kah_geom_tor_man_sym_coor}
and the example in \S 6 of \cite{abr98:kah_geom_tor_var_ext_met} we
shall now use this theorem to (re)construct a family of extremal
K\"ahler metrics on certain $\mathbb{C}\mathbb{P}^1$ bundles over
$\mathbb{C}\mathbb{P}^{n-1}$ originally identified by Calabi
\cite{cal82:ext_kah_met}.
\section{Calabi's extremal K\"ahler metrics on
$\pmb{\widehat{\mathbb{C}\mathbb{P}}^n}$}
According to Abreu
\begin{theorem}[\cite{abr98:kah_geom_tor_var_ext_met}]
A toric K\"ahler metric determined by a symplectic potential $g(x)$
is extremal if and only if
\begin{equation}\label{excon}
\frac{\partial S}{\partial x_i}={\rm constant}
\end{equation}
for $i=1,\dots,n$ i.e. its scalar curvature
\begin{equation}\label{scalt}
S(g) = - \frac{1}{2}\sum^{n}_{i,j=1} \frac{\partial^2
G^{ij}}{\partial x_i \partial x_j}
\end{equation}
is an affine function of $x$. Here $G^{ij}$ is the $(i,j)$th entry
of inverted hessian matrix $G^{-1}$ of $g$.
\end{theorem}
In \S 3, pp.278-88 of \cite{cal82:ext_kah_met} Calabi constructed
extremal K\"ahler metrics of non-constant scalar curvature on
certain $\mathbb{C}\mathbb{P}^1$ bundles over $\mathbb{C}\mathbb{P}^{n-1}$. Since the total
space of these bundles is $\mathbb{P}(\mathscr{O}(-1)\oplus \mathbb{C})$ i.e.
the projectivization of the direct sum of the tautologous line
bundle over $\mathbb{C}\mathbb{P}^{n-1}$ with the trivial line bundle, one may
equally regard these manifolds as the blow-up of $\mathbb{C}\mathbb{P}^n$ at a
point, $\widehat{\mathbb{C}\mathbb{P}}^n$. Calabi constructed these metrics
using a scalar curvature formula ((3.9) in \S 3 of
\cite{cal82:ext_kah_met}) for K\"ahler metrics on $\mathbb{C}^n\setminus
\{0\}$ (considered as an open set in these bundles) invariant under
${\rm U}(n)$ (the maximal compact subgroup of the group of complex
automorphisms of these bundles) and then imposing the appropriate
boundary conditions. This scalar curvature formula was
\begin{equation}\label{cal}
S(f)= (n-1)v'({\widetilde{s}}) \left(f'({\widetilde{s}})\right)^{-1} +v''({\widetilde{s}})
\left(f''({\widetilde{s}})\right)^{-1}
\end{equation}
where $ v({\widetilde{s}})= n{\widetilde{s}}-(n-1)\ln f'({\widetilde{s}}) - \ln f''({\widetilde{s}})$ and $
{\widetilde{s}}=\ln s$. Using the symplectic (action) coordinate
$t=\sum_{i=1}^nx_i$ \eqref{cal} becomes of the extremely simple form
$\eqref{ss}$. We will now (re)construct Calabi's metrics using
\eqref{ss} and working on the Delzant moment polytope
$\Delta_{\widehat{\mathbb{C}\mathbb{P}}^n}$ of the symplectic toric manifold
$\widehat{\mathbb{C}\mathbb{P}}^n$. Our motivation for doing this came from the
discussion in \S 3 of \cite{abr03:kah_geom_tor_man_sym_coor}.
\subsection{The construction}
$\Delta_{\widehat{\mathbb{C}\mathbb{P}}^n}$ consists of $n+2$ facets and these
are determined by the affine functions
\begin{equation}\label{li}
l_i(x)=\left\{\begin{array}{cl}x_i & i=1,\dots,n
\\ t-a & i=n+1 \\ b-t & i=n+2 \end{array}\right.
\end{equation}
where $0<a<b$. The constants $a,b$ determine the cohomology class of
the extremal K\"ahler metrics we are about to construct. The
extremal condition \eqref{excon} in the ${\rm U}(n)$-invariant scenario
becomes $dS/dt={\rm constant}$ i.e. $S=At+B$ for constants $A,B$.
Thus we have \eqref{ss}$=At +B$. Solving this for $F''$ gives
\begin{equation}\label{text}
F''(t)=\frac{pt^{n-1}}{pt^n-\alpha} - \frac{1}{t}
\end{equation}
where $p=n(n+1)(n+2)$ and
\begin{equation}\label{alpha}
\alpha(t)=nAt^{n+2}+(n+2)Bt^{n+1}+p(Ct+D)
\end{equation}
for constants $C,D$. \eqref{text} is a general formula for the
$t$-potential of any extremal metric on $\mathbb{C}^n\setminus\{0\}$. We
denote by $F _{\widehat{\mathbb{C}\mathbb{P}}^n}$ the $t$-potential of the
extremal K\"ahler metrics we are after. By \eqref{spp} the
symplectic potential of these metrics is
$2g_{\widehat{\mathbb{C}\mathbb{P}}^n}=\sum_{i=1}^n x_i \ln x_i +
F_{\widehat{\mathbb{C}\mathbb{P}}^n}(t)$ which is
\begin{equation}\label{symp}
g_{\widehat{\mathbb{C}\mathbb{P}}^n}=\frac{1}{2}\left[\sum_{i=1}^{n+2} l_i \ln
l_i + h_{\widehat{\mathbb{C}\mathbb{P}}^n}(t)\right]
\end{equation}
in the form (2.9) of Abreu's Theorem 2.8 in
\cite{abr03:kah_geom_tor_man_sym_coor} with the $l_i$ given by
\eqref{li}. Therefore $h_{\widehat{\mathbb{C}\mathbb{P}}^n}(t) =
F_{\widehat{\mathbb{C}\mathbb{P}}^n}(t) - \sum_{i=n+1}^{n+2} l_i \ln l_i.$
Differentiating this twice gives
\begin{equation}\label{symp2}
h''_{\widehat{\mathbb{C}\mathbb{P}}^n}(t) = F''_{\widehat{\mathbb{C}\mathbb{P}}^n}(t) -
\frac{b-a}{(t-a)(b-t)}=\frac{pt^{n+1}-2apt^n+abpt^{n-1} -c \alpha}
{(pt^n-\alpha)(t-a)(t-b)} -\frac{1}{t}
\end{equation}
where $c=b-a$. Thus $a<t<b$ and the boundary conditions are that
as $ t \downarrow a$ and $t \uparrow b$ then
\begin{equation}\label{approx}
pt^{n+1}-2apt^n+abpt^{n-1} -c \alpha \approx
(pt^n-\alpha)(t-a)(b-t).
\end{equation}
Consider the former condition. Let $t=a+\epsilon$ for small
$\epsilon >0$ and ignore terms of $O(\epsilon^2)$ and above. Then
\eqref{alpha} gives $\alpha=\alpha_a + O(\epsilon^2)$ where
$\alpha_a=X_a+\epsilon Y_a$ with
\begin{equation}\label{x1}
X_a=na^{n+2}A+(n+2)a^{n+1}B+paC+pD
\end{equation}
and
\begin{equation}\label{y1}
Y_a=n(n+2)Aa^{n+1}+ (n+1)(n+2)Ba^n +pC.
\end{equation}
Now $(t-a)(b-t) = -c\epsilon +O(\epsilon^2)$ and $pt^n-\alpha_a =
pa^n+pna^{n-1}\epsilon -\alpha_a+O(\epsilon^2)$. Therefore $
(pt^n-\alpha)(t-a)(b-t) = -cpa^n\epsilon +c X_a\epsilon +
O(\epsilon^2)$. An analogous calculation shows that $
pt^{n+1}-2apt^n+abpt^{n-1} = cpa^n + cpa^{n-1}(n-1)\epsilon +
O(\epsilon^2)$. Hence \eqref{approx} gives $ cpa^n +
cpa^{n-1}(n-1)\epsilon -cX_a-cY_a\epsilon = -cpa^n\epsilon +c
X_a\epsilon$ which rearranges into $ pa^n + (pa^{n-1}(n-1) +
pa^n)\epsilon = X_a + (X_a+Y_a)\epsilon$. Comparing coefficients
shows that
\begin{equation}\label{x12}
X_a= pa^n
\end{equation}
and $ X_a+Y_a = pa^{n-1}(n-1) + pa^n $ i.e.
\begin{equation}\label{y12}
Y_a= (n-1)pa^{n-1}.
\end{equation}
The calculations for the second condition $t \uparrow b$ follow in
similar way i.e. we consider $t=b-\epsilon$ for small $\epsilon>0$.
By \eqref{alpha} we have that $\alpha=\alpha_b+O(\epsilon^2)$ with
$\alpha_b=X_b-\epsilon Y_b$ where $X_b$ and $Y_b$ are \eqref{x1} and
\eqref{y1}, respectively, with $a$ replaced by $b$. Also
$(t-a)(b-t)=c\epsilon +O(\epsilon^2)$ so that $
(pt^n-\alpha)(t-a)(b-t) = cpb^n\epsilon - cX_b\epsilon$.
Furthermore, $ pt^{n+1}-2apt^n+abpt^{n-1} =
pcb^n-(n+1)pcb^{n-1}\epsilon + O(\epsilon^2)$. By \eqref{approx} we
get $pcb^n-(n+1)pcb^{n-1} - cX_b+cY_b\epsilon = cpb^n\epsilon -
cX_b\epsilon $ which rearranges to give $ pcb^n-((n+1)pcb^{n-1} +
cpb^n)\epsilon = cX_b - c(X_b +Y_b)\epsilon$. Comparing coefficients
gives
\begin{equation}\label{x22}
X_b= pb^n
\end{equation}
and $ X_b+Y_b = ((n+1)pb^{n-1} + pb^n) $ i.e.
\begin{equation}\label{y22}
Y_b=(n+1)pb^{n-1}.
\end{equation}
Thus \eqref{x12}-\eqref{y22} give us four linear equations in the
unknowns $A,B,C,D$. Solving these simultaneously reveals
\begin{equation}\label{soln}
\begin{array}{c}
A=\frac{(n+1)(n+2)((ab)^{n-1}(na^2(n+1) +nb^2(n-1)
-2ab(n^2-1))-2a^{2n})}
{(ab)^n(2n(n+2)ab-(a^2+b^2)(n+1)^2)+a^{2(n+1)} + b^{2(n+1)}}, \\
\\
B=\frac{n(n+1)((ab)^{n-1}(a^2(nb(n+2)-a(n+1)^2) +
b^2(b(1-n^2)+a(n^2-4)))+3a^{2n+1}+b^{2n+1})}
{(ab)^n(2n(n+2)ab-(a^2+b^2)(n+1)^2)+a^{2(n+1)} + b^{2(n+1)}}, \\
\\
C=\frac{(ab)^{n-1}( (n+1)(a^{n+3}-ab^{n+2} -3ba^{n+2}) +
((n-1)b^{n+3} +2(n+2)b^2a^{n+1}))}
{(ab)^n(2n(n+2)ab-(a^2+b^2)(n+1)^2)+a^{2(n+1)} +
b^{2(n+1)}}, \\
\\
D=\frac{(ab)^n( b^{n+1}(n-b(n-2))-2a^nb^2(n+1)-na^{n+1}(a-3b))
}{(ab)^n(2n(n+2)ab-(a^2+b^2)(n+1)^2)+a^{2(n+1)} + b^{2(n+1)}}
\end{array}
\end{equation}
(cf. (3.13) in \cite{cal82:ext_kah_met}). Substituting \eqref{soln}
into \eqref{text} and \eqref{symp} gives the (second derivative of
the) $t$-potential $F_{\widehat{\mathbb{C}\mathbb{P}}^n}$ and the function
$h_{\widehat{\mathbb{C}\mathbb{P}}^n}$ of the extremal K\"ahler metrics,
respectively. We conclude that
\begin{coro}
There exists a family of ${\rm U}(n)$-invariant, extremal K\"ahler
metrics on $\widehat{\mathbb{C}\mathbb{P}}^n$ and these are determined by the
symplectic potential \eqref{symp} on $\Delta_{\widehat{\mathbb{C}\mathbb{P}}^n}$
where $h_{\widehat{\mathbb{C}\mathbb{P}}^n}(t) $ is a smooth function on
$[a,b]\in (0,\infty)$ given by \eqref{symp2} (as determined by
\eqref{soln}), and the real numbers $a,b$ parameterize the
cohomology class of these extremal K\"ahler metrics. Furthermore,
the scalar curvature of these extremal K\"ahler metrics is
$S(g_{\widehat{\mathbb{C}\mathbb{P}}^n})=At+B$ with $A,B$ as given by
\eqref{soln}.
\end{coro}
\begin{example}
{\rm In \S 6 of \cite{abr98:kah_geom_tor_var_ext_met} Abreu
derived a formula for the symplectic potential of Calabi's
extremal metric on $\widehat{\mathbb{C}\mathbb{P}}^2$ by Legendre transforming
the K\"ahler potential (as derived by Calabi) for this extremal
K\"ahler metric into symplectic coordinates. By his conventions
$0<a<1$, $b=1$ and $n=2$ i.e. $a$ is the amount by which
$\widehat{\mathbb{C}\mathbb{P}}^2$ is blown-up. Setting these values in
\eqref{soln} gives
\[
\begin{array}{c}
A = \frac{-24a}{a^3+3a^2-3a-1}, \;B =
\frac{6(3a^2-1)}{a^3+3a^2-3a-1}, \;C =
\frac{(3a^2-1)a}{a^3+3a^2-3a-1}, \;D =\frac{-2a^3}{a^3+3a^2-3a-1}.
\end{array}
\]
Substituting these into \eqref{symp2} gives
\[
h''_{\widehat{\mathbb{C}\mathbb{P}}^2}(t)=\frac{2a(1-a)}{2at^2+t-a^2t+2at+2a^2}
-\frac{1}{t}
\]
which is exactly Abreu's equation (24) in
\cite{abr98:kah_geom_tor_var_ext_met}.}
\end{example}
|
2105.06220
|
\section{Introduction}
\label{introduction}
Document layout analysis is a crucial step in automatic document understanding and enables many important applications, such as document retrieval~\cite{DBLP:journals/csur/BinMakhashenM20}, digitization~\cite{DBLP:conf/icpr/CorbelliBGC16} and editing.
Its goal is to identify the regions of interest in unstructured document and recognize the role of each region.
This task is challenging due to the diversity and complexity of document layouts.
Many deep learning models have been proposed on this task in both computer vision (CV) and natural language processing (NLP) communities.
Most of them consider either only visual features~\cite{DBLP:conf/icdar/HeCPKG17,DBLP:conf/icdar/ChenSLHI15,DBLP:conf/das/WickP18,DBLP:conf/icfhr/GatosLS14,DBLP:conf/icip/VoL16,DBLP:conf/icfhr/ZagorisPG14,DBLP:conf/cvpr/LiWTZBMMSF20,DBLP:conf/icdar/LiYXLOL19} or only semantic features~\cite{DBLP:conf/icdar/Conway93,DBLP:journals/pami/KrishnamoorthyNSV93,DBLP:conf/iccv/ShilmanLV05}.
However, information from both modalities could help recognize the document layout better. Some regions (\eg {Figure}, {Table}) can be easily identified by visual features, while semantic features are important for separating visually similar regions (\eg {Abstract} and {Paragraph}).
Therefore, some recent efforts try to combine both modalities~\cite{DBLP:conf/wacv/AggarwalSGK20,DBLP:conf/coling/LiXCHWLZ20,DBLP:conf/cvpr/YangYAKKG17,DBLP:journals/corr/abs-2002-06144}. Here we summarize them into two categories.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{images/architectore_comp.pdf}
\caption{Comparison of multimodal document layout analysis frameworks.
VSR supports both NLP-based and CV-based frameworks.
$\copyright$ and {\color{red}\textcircled{A}}
denote \textit{concatenation} and \textit{adaptive aggregation} fusion strategies.
Different colored regions in the prediction results indicate different semantic labels ({\color{red}{Paragraph}, \color{blue}{Figure}, \color{green}{Figure Caption}, \color{cyan}{Table Caption}}).}
\label{figure:architectore_comp}
\end{figure}
{NLP-based methods} (Fig~\ref{figure:architectore_comp} (a)) {model layout analysis as a sequence labeling task and }apply a bottom-up strategy.
They first serialize texts into 1D token sequence\footnote[1]{In the rest of this paper, we assume text is available. There are tools available to extract text from PDF documents (\eg PDFMiner~\cite{shinyama2015pdfminer}) and document images (\eg OCR engine~\cite{DBLP:conf/icdar/Smith07}).}.
Then, using both semantic and visual features (such as coordinates and image embedding) of each token, {they} determine token labels sequentially {through a sequence labeling model}.
{However, NLP-based methods show insufficient capabilities in layout modeling. For example in Fig.~\ref{figure:architectore_comp}(a), all texts in a paragraph should have consistent semantic labels ({{\color{red}Paragraph}}), but some of them are recognized as {{\color{green}Figure Caption}}, which are the labels of adjacent texts.}
{CV-based methods} (Fig~\ref{figure:architectore_comp} (b)) {model layout analysis as object detection or segmentation task, and} apply a top-down strategy.
{They first extract visual features by convolutional neural network and introduce semantic features through text embedding maps (at sentence-level~\cite{DBLP:conf/cvpr/YangYAKKG17} \textit{or} character-level~\cite{DBLP:journals/corr/abs-2002-06144}), which are directly concatenated as the representation of document. Then, detection or segmentation models (\eg Mask RCNN~\cite{DBLP:conf/iccv/HeGDG17}) are used to generate layout component candidates (coordinates and semantic labels).}
While capturing spatial information {better} compared to {NLP-based methods}, {CV-based methods} still have 3 limitations:
(1) \textit{limited semantics.}
Semantic information are embedded in {text} at different granularities, including characters (or words) and sentences, which could help identify different document elements. {For example, character-level features are better for recognizing components which need less context (\eg Author) while sentence-level features are better for contextual components (\eg Table caption).
Exploiting semantics at one granularity could not achieve optimal performances.}
(2) {\textit{simple and heuristic modality fusion strategy}. Features from different modalities contribute differently to component recognition. Visual features contribute more to recognizing visually rich components (such as {Figure} and {Table}), while semantic features are better at distinguishing text-based components ({Abstract} and {Paragraph}).
Simple and heuristic modality fusion by concatenation can not fully make use of complementary information between two modalities.}
(3) \textit{lack of relation modeling between components}. Strong relations exist in documents. For example, ``{Figure}'' and ``{Figure Caption}'' often appear together, {and ``{Paragraph}''s have aligned bounding box coordinates.} Such relations could be utilized to {boost layout analysis performances.}
In this paper, we propose a unified framework VSR for document layout analysis, combining \textit{\textbf{V}ision}, \textit{\textbf{S}emantics} and \textit{\textbf{R}elation modeling}, as shown in Fig~\ref{figure:architectore_comp} (c).
{This framework can be applied to both NLP-based and CV-based methods.}
{First, documents are fed into VSR in the form of images (vision) and text embedding maps (semantics at both character-level and sentence-level).}
{Then, modality-specific visual and semantic features are extracted through a two-stream network, which are {effectively} combined later in a multi-scale adaptive aggregation module.
Finally, a GNN(Graph Neural Network)-based relation module is incorporated to model relations between component candidates, and generate final results.
Specifically, for NLP-based methods, text tokens serve as the component candidates and relation module predicts their semantic labels.
While for CV-based methods, component candidates are proposed by detection or segmentation model (\eg Faster RCNN/ Mask RCNN) and relation module generates their refined coordinates and semantic labels.}
Our work makes four key contributions:
\begin{itemize}
\item We propose a unified framework VSR for document layout analysis, combining \textit{vision}, \textit{semantics} and \textit{relations} in documents.
\item To exploit vision and semantics effectively, we propose a \textit{two-stream} network to extract modality-specific visual and semantic features, and fuse them \textit{adaptively} through an adaptive aggregation module. Besides, we also explore document semantics at different granularities.
\item A GNN-based relation module is incorporated to model relations between document components, and it supports relation modeling in both NLP-based and CV-based methods.
\item We perform extensive evaluations of VSR, and on three public benchmarks, VSR shows significant improvements compared to previous models.
\end{itemize}
\section{Related Works}
\subsubsection{Document Layout Analysis.}
{
In this paper, we try to review layout analysis works from the perspective of {modality} used, namely, \textit{unimodal layout analysis} and \textit{multimodal layout analysis}.
\textit{Unimodal layout analysis} exploits either only visual features~\cite{DBLP:conf/icdar/LiYXLOL19,DBLP:conf/cvpr/LiWTZBMMSF20} (document image) or only semantic features (document texts) to understand document structures.
Using visual features, several works~\cite{DBLP:conf/icdar/ChenSLHI15,DBLP:conf/das/WickP18} have been proposed to apply CNN to segment various objects, \eg text blocks~\cite{DBLP:conf/icfhr/GatosLS14}, text lines~\cite{DBLP:conf/icip/VoL16,DBLP:conf/icdar/LeeHOU19}, words~\cite{DBLP:conf/icfhr/ZagorisPG14}, figures or tables~\cite{DBLP:conf/icdar/HeCPKG17,DBLP:conf/jcdl/SiegelLPA18}.
At the same time, there are also methods~\cite{DBLP:conf/icdar/Conway93,DBLP:journals/pami/KrishnamoorthyNSV93,DBLP:conf/iccv/ShilmanLV05} which try to address the layout analysis problem using semantic features.
However, all the above methods are strictly restricted to visual \textit{or} semantic features, and thus are not able to exploit complementary information from other modalities.
\textit{Multimodal layout analysis} tries to combine information from both visual and semantic modalities.
Related methods can be further divided into two categories, NLP-based and CV-based methods.
NLP-based methods work on low-level elements (\eg tokens) and model layout analysis as a sequence labeling task.
MMPAN~\cite{DBLP:conf/wacv/AggarwalSGK20} is presented to recognize form structures.
DocBank~\cite{DBLP:conf/coling/LiXCHWLZ20} is proposed as a large scale dataset of multimodal layout analysis and several NLP baselines have been released.
However, the above methods show insufficient capabilities in layout modeling.
CV-based methods introduce document semantics through text embedding maps, and model layout analysis as object detection or segmentation task.
MFCN~\cite{DBLP:conf/cvpr/YangYAKKG17} introduces sentence granularity semantics and inserts the text embedding maps at the decision-level (end of network), while \textit{dhSegment$^T$}\footnote[2]{\textit{dhSegment$^T$} means \textit{dhSegment} with inputs of image and text embedding maps.}~\cite{DBLP:journals/corr/abs-2002-06144} introduces character granularity semantics and inserts text embedding maps at the input-level.
Though showing great success, the above methods also bear the following limitations: limited semantics used, simple modality fusion strategy and lack of relation modeling between components.
To remedy the above limitations, we propose a unified framework VSR to exploit vision, semantics and relations in documents.
}
\subsubsection{Two-stream networks.}
{Two-stream networks are widely used to combine features in different modalities or representations~\cite{DBLP:journals/pami/BaltrusaitisAM19} effectively.}
In action recognition, two-stream networks are used to capture the complementary \textit{spatial} and \textit{temporal} information~\cite{DBLP:conf/cvpr/FeichtenhoferPZ16}.
In RGB-D saliency detection, the complete representations are fused from the deep features of the \textit{RGB} stream and \textit{depth} stream~\cite{DBLP:journals/tcyb/HanCLYL18}.
Also, two-stream networks are used to fuse different features of same input sample in sound event classification and image recognition~\cite{DBLP:conf/iccv/LinRM15}.
Motivated by their successes, we apply two-stream networks to capture complementary \textit{vision} and \textit{semantics} information in documents.
\subsubsection{Relation modeling.}
Relation modeling is a broad topic and has been studing for decades.
In natural language processing, dependencies between sequential texts are captured through \textit{RNN}~\cite{DBLP:journals/neco/HochreiterS97} or \textit{Transformer}~\cite{DBLP:conf//VaswaniSPUJGKP17} architectures.
In computer vision,
{non-local networks~\cite{DBLP:conf/cvpr/0004GGH18} and relation networks~\cite{DBLP:conf/cvpr/HuGZDW18} are presented to model long-range dependencies between pixels and objects.}
Besides, in document image processing, {relations between text and layout~\cite{DBLP:conf/kdd/XuL0HW020} or relations between document entities~\cite{DBLP:conf/icpr/YuLQG020,DBLP:conf/naacl/LiuGZZ19,DBLP:conf/mm/ZhangXCP0QNW20} are explored.}
As to multimodal layout analysis, NLP-based methods model it as a sequence labeling task and use \textit{RNN} to capture component relations, while CV-based methods model it as object detection task but lack relation modeling between layout components.
In this paper, we propose a GNN-based relation module, supporting relation modeling in both NLP-based or CV-based methods.
\section{Methodology}
\label{methodology}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{images/system_architecture.pdf}
\caption{The architecture overview of VSR. Best viewed in color.}
\label{figure:system_architecture}
\end{figure}
\subsection{Architecture Overview}
\label{overall_architecture}
{Our proposed framework has three parts:}
two-stream ConvNets, a multi-scale adaptive aggregation module and a relation module{ (as shown in Fig~\ref{figure:system_architecture})}.
{First, a two-stream convolutional network extracts modality-specific visual and semantic features, where
{visual stream and semantic stream take images and text embedding maps as input, respectively (Sec \ref{two_stream_convnets}).}
Next, instead of simply concatenating the visual and semantic features, we aggregate them via a multi-scale adaptive aggregation module (Sec \ref{sec_maa}).
{Then, a set of component candidates are generated.}
Finally, a relation module is incorporated to model relations {among those candidates} and generate final results (Sec \ref{sec_rm}).
{Notice that multimodal layout analysis can be modeled as sequence labeling (NLP-based methods) or object detection tasks (CV-based methods).
Our framework supports both modeling types. The only difference is what the component candidates are and how to generate them.
Component candidates are low-level elements (\eg text tokens) in NLP-based methods and can be generated by parsing PDFs, while candidates
are high-level elements (regions) generated by detection or segmentation model (\eg Mask RCNN) in CV-based methods. In the rest of this paper, we will illustrate how VSR is applied to CV-based methods,
and show it can be easily adapted to NLP-based methods in experiments on DocBank benchmark (Sec~\ref{main_result}).
}}
\subsection{Two-stream ConvNets}
\label{two_stream_convnets}
{{CNN is known to be good at learning deep features. However, previous multimodal layout analysis works~\cite{DBLP:conf/cvpr/YangYAKKG17,DBLP:journals/corr/abs-2002-06144} only apply it to extract visual features. Text embedding maps are directly used as semantic features.}
This single-stream network design could not make full use of document semantics.
Motivated by great success of two-stream network in various multimodal applications \cite{DBLP:conf/cvpr/FeichtenhoferPZ16,DBLP:conf/iccv/LinRM15}, we apply it to extract deep visual and semantic features.
\subsubsection{Visual stream ConvNet.}
This stream directly takes document images as input and extracts multi-scale deep features using CNN backbones like ResNet \cite{DBLP:conf/cvpr/HeZRS16}.
Specifically, for an input image $x \in \mathbb{R}^{H \times W \times 3}$, multi-scale features maps (denoted by $\left\{V_2, V_3, V_4, V_5\right\}$) are extracted,
where each $V_i \in \mathbb{R}^{\frac{H}{2^i} \times \frac{W}{2^i} \times C_i^V}$.
$H$ and $W$ are the height and width of input image $x$, $C_i^V$ is the channel dimension of feature map $V_i$, and $V_0=x$.
\subsubsection{Semantic stream ConvNet.}
Similar to~\cite{DBLP:conf/cvpr/YangYAKKG17,DBLP:journals/corr/abs-2002-06144}, we introduce document semantics through text embedding maps $S_0 \in \mathbb{R}^{H \times W \times C_0^S}$, which are the input of semantic stream ConvNet.
$S_0$ have same spatial sizes with document image $x$ ($V_0$) and $C_0^S$ denotes the initial channel dimension.
This type of representation not only encodes text {content}, but also preserves the 2D layout of a document.
{Previously, only semantics at one granularity is used (character-level~\cite{DBLP:journals/corr/abs-2002-06144} \textit{or} sentence-level\footnote[3]{Sentence is a group of words or phrases, which usually ends with a period, question mark or exclamation point. For simplicity, we approximate it with text lines.}~\cite{DBLP:conf/cvpr/YangYAKKG17}).
{However, semantics at different granularities contribute to identification of different components.}
Thus, $S_0$ consists of both character and sentence level semantics.
Next, we show how we build text embedding maps $S_0$.
}
The characters and sentences of a document page are denoted as $\mathbb{D}_c=\left\{\left( c_k, b_k^c\right) | k=0,\cdots, n\right\}$ and $\mathbb{D}_s=\left\{\left( s_k, b_k^s\right) | k=0,\cdots, m\right\}$, where $n$ and $m$ are the total number of characters and sentences.
$c_k$ and $b_k^c=\left(x_0, y_0, x_1, y_1\right)$ are the $k$-th character and its associated box, where $(x_0, y_0)$ and $(x_1, y_1)$ are top-left and bottom-right pixel coordinates.
{Similarly, $s_k$ and $b_k^s$ are the $k$-th sentence and its box location}.
Next, character embedding maps $CharGrid \in \mathbb{R}^{H \times W \times C_0^S}$ and sentence embedding maps $SentGrid \in \mathbb{R}^{H \times W \times C_0^S}$ can be constructed as follows.
\begin{equation}
CharGrid_{ij} = \left\{
\begin{array}{ll}
E^c(c_k) & \mbox{if}\ (i,j) \in b_k^c \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
\begin{equation}
SentGrid_{ij} = \left\{
\begin{array}{ll}
E^s(s_k) & \mbox{if}\ (i,j) \in b_k^s \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
All pixels in each $b_k^c$ ($b_k^s$) share the same character (sentence) embedding vector.
$E^c$ and $E^s$ are the mapping functions of $c_k\rightarrow\mathbb{R}^{C^S_0}$ and $s_k\rightarrow\mathbb{R}^{C^S_0}$.
In our implementation, $E^c$ is a typical word embedding layer and {we adopt pretrained language model BERT~\cite{DBLP:conf/naacl/DevlinCLT19} as $E^s$.}
Finally, the {text} embedding maps $S_0$ can be {constructed by applying LayerNorm normalization to the summation of $CharGrid$ and $SentGrid$, as shown in Eq.(\ref{eq-s0})}.
\begin{equation}
S_0 = LayerNorm\left(CharGrid + SentGrid\right)
\label{eq-s0}
\end{equation}
Similar to the visual stream, semantic stream ConvNet then takes text embedding maps $S_0$ as input and extracts multi-scale features $\left\{S_2, S_3, S_4, S_5\right\}$, which have the same spatial sizes and channel dimension with $\left\{V_2, V_3, V_4, V_5\right\}$.
\subsection{Multi-scale Adaptive Aggregation}
\label{sec_maa}
{Features from different modalities are important for identifying different objects.
Modality fusion strategy should adaptively aggregate visual and semantic features.
{Thus,} we design a multi-scale adaptive aggregation module that learns an attention map to combine visual features $\left\{V_2, V_3, V_4, V_5\right\}$ and semantic features $\left\{S_2, S_3, S_4, S_5\right\}$ adaptively.
At scale $i$, this module first concatenates $V_i$ and $S_i$, and then feed it into a convolutional layer to learn an attention map $AM_i$.
Finally, aggregated multi-modal features $FM_i$ is obtained.
All operations in this module are formulated by:
\begin{equation}
AM_i = h\left(g\left(\left[V_i, S_i\right]\right)\right)
\end{equation}
\begin{equation}
FM_i = AM_i \odot V_i + \left(1-AM_i\right) \odot S_i
\end{equation}
where $\left[\cdot\right]$ denotes the concatenation operation, $g\left(\cdot\right)$ is a convolutional layer with kernel size $1\times1\times\left(C_i^V+C_i^S\right)\times C_i^S$ and $h\left(\cdot\right)$ is a non-linear activation function.
$\odot$ denotes the element-wise multiplication.
Through this module, a set of fused multi-modal features $FM=\left\{FM_2, FM_3, FM_4, FM_5\right\}$ are generated, which serve as the multimodal multi-scale features of a document.
Then, FPN~\cite{DBLP:conf/cvpr/LinDGHHB17} (feature pyramid network) is applied on $FM$ and provides enhanced representations.
\subsection{Relation Module}
\label{sec_rm}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{images/relation_module_illutration.pdf}
\caption{Illustration of relation module. It captures relations between component candidates, and thus improves detection results (remove false {\color{blue}Figure} prediction, correct {\color{cyan}Table Caption} label and adjust {\color{red}Paragraph} coordinates).
The colors of semantic labels are: \color{blue}{Figure}, \color{red}{Paragraph}, \color{green}{Figure Caption}, \color{yellow}{Table}, \color{cyan}{Table Caption}. \color{black}{Best viewed in color.}}
\label{figure:rm_illustration}
\end{figure}
{
Given aggregated features $FM=\left\{FM_2, FM_3, FM_4, FM_5\right\}$, a standard object detection or segmentation model {(\eg Mask RCNN~\cite{DBLP:conf/nips/RenHGS15})} can be used to generate component candidates in a document.}
Previous works directly take those predictions as final results.
However, strong relations exist between layout components. {For example, bounding boxes of {Paragraphs} in the same column should be aligned; {Table} and {Table Caption} often appear together; there is no overlap between components.
{We find that such relations can be utilized to further refine predictions, as shown in Fig~\ref{figure:rm_illustration}, \ie adjusting regression coordinates for aligned bounding boxes, correcting wrong prediction labels based on co-occurrence of components and removing false predictions based on non-overlapping property.
Next, we show how we use GNN (graph neural network) to model component relations and how to use it to refine prediction results.}
We represent a document as a graph
$G=\left(O, E\right)$, {where $O=\left\{o_1,\cdots,o_N\right\}$ is the node set and $E$ is the edge set.
Each node {$o_j$} represents a component candidate generated by the object detection model previously, and each edge represents the relation between two component candidates.
Since remote regions in a document may also bear close dependencies (\eg a paragraph spans two columns), all regions constitute a neighbor relationship.
Thus, the document graph is a fully-connected graph and $E\subseteq O\times O$.
The key idea {of our relation module} is to update the hidden representations of each node by attending over its neighbors ($z_1, z_2, \cdots, z_8$$\rightarrow$$z_1'$, as shown in Fig~\ref{figure:rm_illustration}). With updated node features, we could predict its {refined} label and position coordinates.
Initially, each node, denoted by $o_j=\left(b_j, f_j\right)$, includes two pieces of information: position coordinates $b_j$ and deep features $f_j=RoIAlign(FM, b_j)$.
In order to incorporate both of them into node representation, we construct new node feature $z_j$ as follows,
\begin{equation}
\label{q_z_j}
z_j = LayerNorm(f_j + e^{pos}_j\left(b_j\right))
\end{equation}
where $e^{pos}_j\left(b_j\right)$ is the position embedding vectors of $j$-th node.
Then, instead of explicitly specifying the relations between nodes, inspired by \cite{DBLP:conf/iclr/VelickovicCCRLB18}, we apply \textit{self-attention} mechanism to automatically learn the relations, which has already shown great success in NLP and document processing~\cite{DBLP:conf/kdd/XuL0HW020,DBLP:conf/icpr/YuLQG020,DBLP:conf/naacl/LiuGZZ19,DBLP:conf/mm/ZhangXCP0QNW20}.
Specifically,} we adopt the popular scaled dot-product attention~\cite{DBLP:conf//VaswaniSPUJGKP17} to obtain sufficient expressive power.
{Scaled dot-product attention consists of queries $Q$ and keys $K$ of dimension $d_k$, and values $V$ of dimension $d_v$.}
The output $\widehat{O}$ is obtained by weighted sum over all values in $V$, where the attention weights are obtained using $Q$ and $K$, as shown in Eq.(\ref{self_att}). Please refer to \cite{DBLP:conf//VaswaniSPUJGKP17} for details.
\begin{equation}
\label{self_att}
\widehat{O}=Attention(Q,K,V)=softmax(\frac{QK^\mathsf{T}}{\sqrt{d_{k}}})V
\end{equation}
In our context, node feature set $Z=\left\{z_1,\cdots,z_N\right\}$ serves as $K$, $Q$ and $V$ and updated node feature set $Z'=\left\{z_1',\cdots,z_N'\right\}$ is the output $\widehat{O}$.
We apply multi-head attention to further improve representation capacity of node features.
Finally, given updated node features $Z'$, refined detection results of $j$-th node ($j$-th layout component candidate) $\widetilde{o_j}=\left( \widetilde{p_j^c}, \widetilde{b_j}\right)$ is computed as,
\begin{equation}
\widetilde{p_j^c} = Softmax(Linear_{cls}\left(z'_j\right))
\end{equation}
\begin{equation}
\widetilde{b_j} = Linear_{reg}\left(z'_j\right)
\end{equation}
where $\widetilde{p_j^c}$ is the probability of belonging to $c$-th class, $\widetilde{b_j}$ is its refined regression coordinates. $Linear_{cls}$ and $Linear_{reg}$ are projection layers.
{Relation module can be easily applied to NLP-based methods.
In this case, node feature $z_j$ in Eq.(\ref{q_z_j}) is the representation of $j$-th low-level elements (\eg tokens).
Then, GNN models pairwise relations between tokens and predicts their semantic labels ($\widetilde{p_j^c}$).}
\subsection{Optimization}
\label{sec_o}
Since multimodal layout analysis can be modeled as sequence labeling or object detection tasks, their optimization losses are different.
\textbf{Layout analysis as sequence labeling.}
The loss function is formulated as,
\begin{equation}
\label{losses_ce}
\mathcal{L}=-\frac{1}{T}\sum_{j=1}^{T}log\ \widetilde{p_j}\left(y_j\right)
\end{equation}
where, $T$ is the number of low-level elements and $y_j$ is the groundtruth semantic label of $j$-th element.
\textbf{Layout analysis as object detection.}
The loss function is generated from two parts,
\begin{equation}
\label{losses}
\mathcal{L}=\mathcal{L}_{DET} + \lambda\mathcal{L}_{RM}
\end{equation}
where $\mathcal{L}_{DET}$ and $\mathcal{L}_{RM}$ are the losses used in candidate generation process and relation module.
Both $\mathcal{L}_{DET}$ and $\mathcal{L}_{RM}$ consist of a cross entropy loss (classification) and a smooth L$_1$ loss (coordinate regression), as defined in~\cite{DBLP:conf/nips/RenHGS15}.
Hyper-parameters $\lambda$ controls the trade-off between two losses.
\section{Experiments}
\subsection{Datasets}
\label{dataset}
All three benchmarks provide document images and their original PDFs. {Therefore, text could be directly obtained {by parsing PDFs,}} allowing the explorations of multi-modal techniques.
To compare with existing solutions on each benchmark, we {use the same evaluation metrics as used by each benchmark}.
\textbf{Article Regions}~\cite{DBLP:conf/emnlp/SotoY19} consists of 822 document samples and 9 region classes {are annotated} ({Title, Authors, Abstract, Body, Figure, Figure Caption, Table, Table Caption and References}). The {annotation is in object detection format and the }evaluation metric is mean average precision (mAP).
\textbf{PubLayNet}~\cite{DBLP:conf/icdar/ZhongTJ19} is a large-scale document dataset recently released by IBM. It consists of 360K document samples and 5 {region classes} are annotated ({Text, Title, List, Figure, and Table}).
{The annotation is also in object detection format.}
{They use the same evaluation metric as used in the COCO competition, \ie the mean average precision (AP) @ intersection over union (IOU) [0.50:0.95].}
\textbf{DocBank}~\cite{DBLP:conf/coling/LiXCHWLZ20} is proposed by Microsoft. It contains 500K document samples with 12 {region classes} ({Abstract, Author, Caption, Equation, Figure, Footer, List, Paragraph, Reference, Section, Table} and {Title}). {It provides token-level annotations, and use F1 score as {official} evaluation metric.}
{Also, it provides object detection annotations, supporting object detection method.}
\subsection{Implementation Details}
\label{impl_detail}
{
Document image is directly used as input for visual stream. For semantic stream, we extract embedding maps (\textit{SentGrid} and \textit{CharGrid}) from text as input, where \textit{SentGrid} is generated by pretrained BERT model~\cite{DBLP:conf/naacl/DevlinCLT19}
and \textit{CharGrid} is obtained from a word embedding layer. They all have the same channel dimension size ($C^S_0=64$).
ResNeXt-101~\cite{DBLP:conf/cvpr/XieGDTH17} is used as backbone to extract both visual and semantic features (unless otherwise specified), which are later fused by a multi-scale adaptive aggregation and feature pyramid network.
For CV-based multimodal layout analysis methods, fused features are fed into RPN, followed by RCNN, to generate component candidates.
In RPN, 7 anchor ratios (0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0) are adopted to handle document elements that vary in sizes and scales.}
In relation module, dimension of each candidate is set to $1024$ and $2$ layers of multi-head attention with $16$ heads are used to model relations.
{We set $\lambda$ in Eq.(\ref{losses}) to be 1 in all our experiments.}
For NLP-based multimodal layout analysis methods, low-level elements parsed from PDFs (\eg tokens) serve as component candidates, and relation module predicts their semantic labels.
Our model is implemented under the PyTorch framework.
It is trained by the SGD optimizer with batchsize=$2$, momentum=$0.9$ and weight-decay=$10^{-4}$.
The initial learning rate is set to $10^{-3}$, which is divided by 10 every $10$ epochs on Article Regions dataset and $3$ epochs on the other two benchmarks.
The training of model on Article Regions lasts for $30$ epochs while on the other two benchmarks lasts for $6$.
All the experiments are carried out on Tesla-V100 GPUs.
Source code will be released in the near future.
\begin{table}[t]
\centering
\caption{Performance comparisons on Article Regions dataset}
\label{table:AR_dataset}
\begin{threeparttable}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|c|ccccccccc|c|}
\hline
Method & Title & Author & Abstract & Body & Figure & \begin{tabular}[c]{@{}c@{}}Figure\\ Caption\end{tabular} & Table & \begin{tabular}[c]{@{}c@{}}Table\\ Caption\end{tabular} & Reference & mAP \\ \hline
Faster RCNN~\cite{DBLP:conf/emnlp/SotoY19} & - & 1.22& - &87.49 & - & - & - & - &-& 46.38 \\
Faster RCNN \textit{w/ context}~\cite{DBLP:conf/emnlp/SotoY19} & - & 10.34 & - & 93.58 & - & - & - & 30.8 & - & 70.3 \\ \hline
Faster RCNN \textit{reimplement} & 100.0 & 51.1 & 94.8 &98.9 &94.2 &91.8 &97.3 &67.1 & 90.8 & 87.3 \\
\begin{tabular}[c]{@{}c@{}}Faster RCNN \textit{w/ context}\\ \textit{reimplement}\end{tabular}~\cite{DBLP:conf/emnlp/SotoY19} & 100.0 & 60.5 & 90.8 &98.5 &\textbf{96.2} &91.5 &\textbf{97.5} &64.2 & 91.2 & 87.8 \\ \hline
VSR & \textbf{100.0} & \textbf{94} &\textbf{95} &\textbf{99.1} &95.3 &\textbf{94.5} &96.1 &\textbf{84.6} &\textbf{92.3} &\textbf{94.5} \\ \hline
\end{tabular}}
\begin{tablenotes}
\scriptsize
\item {Note: missing entries are because those results are not reported in their original papers.}
\end{tablenotes}
\end{threeparttable}
\end{table}
\begin{table}[t]
\centering
\caption{Performance comparisons on PubLayNet dataset.}
\label{table:publaynet}
\resizebox{0.6\textwidth}{!}{
\begin{tabular}{|c|c|ccccc|c|}
\hline
Method & Dataset & Text & Title & List & Table & Figure & AP \\ \hline
Faster RCNN~\cite{DBLP:conf/icdar/ZhongTJ19} & \multirow{3}{*}{val} & 91 & 82.6 & 88.3 & 95.4 & 93.7 & 90.2 \\
Mask RCNN~\cite{DBLP:conf/icdar/ZhongTJ19} & & 91.6 & 84 & 88.6 & 96 & 94.9 & 91 \\
VSR & & \textbf{96.7} & \textbf{93.1} & \textbf{94.7} & \textbf{97.4} & \textbf{96.4} & \textbf{95.7} \\ \hline
Faster RCNN~\cite{DBLP:conf/icdar/ZhongTJ19} & \multirow{7}{*}{test} & 91.3 & 81.2 & 88.5 & 94.3 & 94.5 & 90 \\
Mask RCNN~\cite{DBLP:conf/icdar/ZhongTJ19} & & 91.7 & 82.8 & 88.7 & 94.7 & 95.5 & 90.7 \\
DocInsightAI & & 94.51& 88.31& 94.84& 95.77&97.52 & 94.19 \\
SCUT & & 94.3& 89.72& 94.25 &96.62 &97.68 &94.51 \\
SRK &&94.65&89.98&\textbf{95.14}&\textbf{97.16}&\textbf{97.95}&94.98 \\
SiliconMinds &&96.2&89.75&94.6&96.98&97.6&95.03 \\
VSR & &\textbf{96.69} & \textbf{92.27} &94.55 & 97.03 & 97.90 & \textbf{95.69} \\ \hline
\end{tabular}}
\end{table}
\subsection{Results}
\label{main_result}
\subsubsection{Article Regions.}
{We compare the performance of VSR on this dataset with two models: Faster RCNN and Faster RCNN \textit{with context}~\cite{DBLP:conf/emnlp/SotoY19}.
Faster RCNN \textit{with context} adds {limited} context (page numbers, region-of-interest position and size) as input in addition to document images.
In Table~\ref{table:AR_dataset}, we first show mAP as reported in their original papers~\cite{DBLP:conf/emnlp/SotoY19}. For fair comparison, we reimplement those two models using the same backbone (ResNet-101) and neck configuration as used in VSR.
We also report their performance after reimplementation.
We can see that our reimplemented models have much higher mAP than their original models.
We believe this is mainly because we use multiple anchor ratios {in RPN}, thus achieve better detection results on document elements with various sizes.
VSR makes full use of vision, semantics and relations between components, showing highest mAP on most classes.
On Figure and Table categories, VSR achieves comparable results and the slight performance drop will be further discussed in Sec~\ref{e_r_m}.}
\begin{table}[t]
\centering
\caption{Performance comparisons on DocBank dataset in F1 Score.}
\label{table:docbank_F1}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|c|cccccccccccc|c|}
\hline
Method & Abstract & Author & Caption & Equation & Figure & Footer & List & Paragraph & Reference & Section & Table & Title & \begin{tabular}[c]{@{}c@{}}Macro\\ Average\end{tabular} \\ \hline
BERT$_{base}$ & 92.94 & 84.84 & 86.29 & 81.52 & 100.0 & 78.05 & 71.33 & 96.19 & 93.10 & 90.81 & 82.96 & 94.42 & 87.70 \\
RoBERTa$_{base}$ & 92.88 & 86.18 & 89.44 & 82.48 & 100.0 & 80.14 & 73.53 & 96.46 & 93.41 & 93.37 & 83.89 & 95.11 & 88.91 \\
LayoutLM$_{base}$ & 98.16 & 85.95 & 95.97 & 89.47 & 100.0 & 89.57 & 89.48 & 97.88 & 93.38 & 95.98 & 86.33 & 95.79 & 93.16 \\
BERT$_{large}$ & 92.86 & 85.77 & 86.50 & 81.77 & 100.0 & 78.14 & 69.60 & 96.19 & 92.84 & 90.65 & 83.20 & 94.30 & 87.65 \\
RoBERTa$_{large}$ & 94.79 & 87.24 & 90.81 & 83.70 & 100.0 & 83.92 & 74.51 & 96.65 & 93.34 & 94.07 & 84.94 & 94.61 & 89.88 \\
LayoutLM$_{large}$ & 97.84 & 87.83 & 95.56 & 89.74 & \textbf{100.0} & 91.46 & 90.04 & 97.90 & 93.32 & 95.96 & 86.79 & 95.52 & 93.50 \\ \hline
X101 & 97.17 & 82.27 & 94.35 & 89.38 & 88.12 & 90.29 & 90.51 & 96.82 & 87.98 & 94.12 & 83.53 & 91.58 & 90.51 \\
X101+LayoutLM$_{base}$ & 98.15 & 89.07 & \textbf{96.69} & 94.30 & 99.90 & 92.92 & 93.00 & 98.43 & 94.37 & 96.64 & 88.18 & 95.75 & 94.78 \\
X101+LayoutLM$_{large}$ & 98.02 & 89.64 & 96.66 & 94.40 & 99.94 & 93.52 & 92.93 & 98.44 & 94.30 & 96.70 & 88.75 & 95.31 & 94.88 \\ \hline
VSR & \textbf{98.29} & \textbf{91.19} & 96.32 & \textbf{95.84} & 99.96 & \textbf{95.11} & \textbf{94.66} & \textbf{98.66} & \textbf{95.05} & \textbf{97.11} & \textbf{89.24} & \textbf{95.63} & \textbf{95.59} \\ \hline
\end{tabular}}
\end{table}
\begin{table}[t]
\centering
\caption{Performance comparisons on DocBank dataset in mAP.}
\label{table:docbank_map}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|c|cccccccccccc|c|}
\hline
Models & Abstract & Author & Caption & Equation & Figure & Footer & List & Paragraph & Reference & Section & Table & Title & mAP \\ \hline
\begin{tabular}[c]{@{}c@{}}Faster\\ RCNN\end{tabular} &96.2&88.9&93.9&\textbf{78.1}&85.4&\textbf{93.4}&86.1&67.8&89.9&76.7&77.2&\textbf{95.3}& 86.3 \\ \hline
VSR &\textbf{96.3}&\textbf{89.2}&\textbf{94.6}&77.3&\textbf{97.8}&93.2&\textbf{86.2}&\textbf{69.0}&\textbf{90.3}&\textbf{79.2}&\textbf{77.5}&94.9& \textbf{87.6} \\ \hline
\end{tabular} }
\end{table}
\subsubsection{PubLayNet.}
{In Table~\ref{table:publaynet}, we compare the performance of VSR on this dataset with two {pure image-based} methods, Faster RCNN~\cite{DBLP:conf/nips/RenHGS15} and Mask RCNN~\cite{DBLP:conf/iccv/HeGDG17}.}
While those two models present promising results (AP\textgreater90\%) on validation dataset, VSR improves the performance on all classes and increases the final AP by 4.7\%.
{VSR shows large performance improvements on text-related classes ({Text}, {Title} and {List}) since it also utilizes document semantics in addition to document image.}
On test dataset (also known as \textit{leaderboard of ICDAR2021 layout analysis recognition competition\footnote[4]{https://icdar2021.org/competitions/competition-on-scientific-literature-parsing/}}), VSR surpasses all participating teams and ranks first, with $4.99\%$ increase on AP compared with Mask RCNN baseline.
\subsubsection{DocBank.}
This dataset offers both token and detection annotations. {Therefore, we could treat layout analysis task either as sequence labeling task
or as object detection task, then compare VSR with existing solutions in both cases. }
\textbf{\textit{Layout analysis as sequence labeling}}.
Using token-level annotations, we compare VSR with BERT~\cite{DBLP:conf/naacl/DevlinCLT19}, RoBERTa~\cite{DBLP:journals/corr/abs-1907-11692}, LayoutLM~\cite{DBLP:conf/kdd/XuL0HW020}, Faster RCNN with ResNeXt-101~\cite{DBLP:conf/cvpr/XieGDTH17} and ensemble models (ResNeXt-101+LayoutLM) in Table~\ref{table:docbank_F1}.
{Even though highest F1 score of {Caption} and {Figure} are achieved by ensemble model (ResNeXt-101+LayoutLM) and LayoutLM respectively, VSR achieves comparable results with small gaps ($\leq$ 0.37\%).} {More importantly, VSR gets the highest scores on all other classes. }
This indicates that VSR is significantly better than BERT, RoBERTa and LayoutLM architectures on document layout analysis task.
\textbf{\textit{Layout analysis as object detection}}.
{Since both VSR and Faster RCNN with ResNeXt-101 can provide object detection results, we further compare them in object detection format using mAP as evaluation metric.
Results in Table~\ref{table:docbank_map} show that VSR outperforms Faster RCNN on most classes, {except {Equation}, {Footer} and {Title}.}
{Overall}, VSR shows $1.3\%$ gains in final mAP.}
\subsection{Ablation Studies}
\label{ablation_exp}
{
VSR introduces multi-granularity semantics, two-stream network with adaptive aggregation, and relation module. Now we explore how each of them contributes to VSR's performance improvement on Article Regions dataset.}
\subsubsection{Effects of multi-granularity semantic features.}
{To understand whether multi-granularity semantic features indeed improve VSR's performance, we compare 4 versions of VSR (\textit{vision-only}, \textit{vision+character}, \textit{vision+sentence},
\textit{vision+\\character+sentence}) in Table~\ref{table:xxgrid}.
Here \textit{character} and \textit{sentence} refer to semantic features at two different granularities.}
We can see that, introducing document semantics at each granularity alone can boost analysis performance while {combining both of them} leads to highest mAP.
This is consistent with {how humans comprehend documents. Humans can better recognize regions which require little context from characters/words (\eg {Author}) and those which need context from sentences (\eg {Table caption}).}
\begin{table}[t]
\centering
\caption{Effects of semantic features at different granularities.}
\label{table:xxgrid}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|ccc|ccccccccc|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{Vision}} & \multicolumn{2}{c|}{Semantics} & \multirow{2}{*}{Title} & \multirow{2}{*}{Author} & \multirow{2}{*}{Abstract} & \multirow{2}{*}{Body} & \multirow{2}{*}{Figure} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Figure\\ Caption\end{tabular}} & \multirow{2}{*}{Table} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Table\\ Caption\end{tabular}} & \multirow{2}{*}{Reference} & \multirow{2}{*}{mAP} \\ \cline{2-3}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{Char} & Sentence & & & & & & & & & & \\ \hline
$\surd$ & & & 100.0 & 51.1 & 94.8 &98.9 &94.2 &91.8 &97.3 &67.1 & 90.8 & 87.3 \\
$\surd$ & $\surd$& & 100.0&71.4 &\textbf{96.5} &98.9 &95.6 &\textbf{93.6} &96.9 &68.6 &89.9 & 90.2 \\
$\surd$ & &$\surd$ &100.0 &60.2 &95.5 &\textbf{99.0} &\textbf{97.8} &93.2 &98.9 &\textbf{73.0} &91.2& 89.8 \\
$\surd$ &$\surd$ &$\surd$ &\textbf{100.0} &\textbf{84.3} &96.1 &98.7 &95.7 &92.5 &\textbf{99.4} &71.4 &\textbf{92.4} & \textbf{92.3} \\ \hline
\end{tabular}}
\end{table}
\begin{table}[t]
\centering
\caption{Effects of two-stream network with adaptive aggregation.}
\label{table:two_stream_vs_single_stream}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|c|c|ccccccccc|c|c|}
\hline
\multicolumn{2}{|c|}{Method} & Title & Author & Abstract & Body & Figure & \begin{tabular}[c]{@{}c@{}}Figure\\ Caption\end{tabular} & Table & \begin{tabular}[c]{@{}c@{}}Table\\ Caption\end{tabular} & Reference & mAP & FPS \\ \hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Single-stream\\ at input level\end{tabular}} & R101 & 94.7&58.7&82.7&98.1&97.9&\textbf{96.3}&91.8&63.7&91.5&86.2 & 19.07 \\ \cline{2-2}
& R152 & 100.0 & 50.5& 85.3& 97.9& \textbf{98.0}& 94.4& 93.3& 62.6& 90.5& 85.8& 18.15 \\ \hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Single-stream\\ at decision level\end{tabular}} & R101 & 99.5&67.6&95.1&{98.8}&95.0&93.2&96.6&70.7&91.3&89.8 & \textbf{19.79} \\ \cline{2-2}
& R152 & 100.0& 80.2& 91.0& \textbf{99.4}& 96.0& 92.4& 98.3&\textbf{73.8}& 91.7& 91.4& 16.43 \\ \hline
{VSR} & R101 & \textbf{100.0} &\textbf{84.3} &\textbf{96.1} &98.7 &95.7 &92.5 &\textbf{99.4} &{71.4} &\textbf{92.4} & \textbf{92.3} & 13.94 \\ \hline
\end{tabular}}
\end{table}
\subsubsection{Effects of two-stream network with adaptive aggregation.}
We propose a two-stream network with adaptive aggregation module to combine vision and semantics of document.
To verify its effectiveness, we compare our VSR with its multimodal single-stream counterparts in Table~\ref{table:two_stream_vs_single_stream}.
{Instead of using extra stream to extract semantic features, single-stream networks directly use text embedding maps and concatenate them with visual features at input-level~\cite{DBLP:journals/corr/abs-2002-06144} or decision-level~\cite{DBLP:conf/cvpr/YangYAKKG17}.}
\cite{DBLP:journals/corr/abs-2002-06144} performs concatenation fusion in the input level and shows worse performances, while
\cite{DBLP:conf/cvpr/YangYAKKG17} fuses multimodal features in the decision level and achieves impressive performances ($89.8$ mAP).
{VSR first extracts visual and semantic features separately using two-stream network, and then fuses them adaptively.
This leads to highest mAP ($92.3$).
At the same time, VSR can run {at real-time (13.94 frames per second).}
We also experiment on larger backbone (ResNet-152) and reach consistent conclusions as shown in Table~\ref{table:two_stream_vs_single_stream}.
\iffalse
\begin{table}[t]
\centering
\caption{Effects of two-stream network with adaptive aggregation.}
\label{table:two_stream_vs_single_stream}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|c|c|ccccccccc|c|c|}
\hline
\multicolumn{2}{|c|}{Method} & Title & Author & Abstract & Body & Figure & \begin{tabular}[c]{@{}c@{}}Figure\\ Caption\end{tabular} & Table & \begin{tabular}[c]{@{}c@{}}Table\\ Caption\end{tabular} & Reference & mAP & FPS \\ \cline{1-2} \hline
\begin{tabular}[c]{@{}c@{}}Single-stream\\ at input level \cite{DBLP:journals/corr/abs-2002-06144}\end{tabular} & {\normalsize \textcircled{c}} & 94.7&58.7&82.7&98.1&97.9&\textbf{96.3}&91.8&63.7&91.5&86.2 & 19.07 \\ \cline{1-2}
\begin{tabular}[c]{@{}c@{}}Single-stream\\ at decision level \cite{DBLP:conf/cvpr/YangYAKKG17}\end{tabular} & {\normalsize \textcircled{c}} & 99.5&67.6&95.1&\textbf{98.8}&95.0&93.2&96.6&70.7&91.3&89.8 & \textbf{19.79} \\ \cline{1-2} \hline
\multirow{3}{*}{VSR} & {\normalsize \textcircled{c}} & 100.0 &70.6&90.8&97.8&96.3&94.3&96.3&60.8&90.0&88.5& 14.61 \\ \cline{2-2}
& {\normalsize \textcircled{s}} &100.0 &53.8&93.3&98.4&\textbf{98.5}&93.9&97.2&60.3&92.0&87.5& 14.65 \\ \cline{2-2}
& {\normalsize \textcircled{a}} & \textbf{100.0} &\textbf{84.3} &\textbf{96.1} &98.7 &95.7 &92.5 &\textbf{99.4} &\textbf{71.4} &\textbf{92.4} & \textbf{92.3} & 13.94 \\ \hline
\end{tabular}}
\end{table}
\fi
\begin{table}[t]
\centering
\caption{Effects of relation module.}
\label{table:erm}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|c|c|ccccccccc|c|}
\hline
\multicolumn{2}{|c|}{Method} & Title & Author & Abstract & Body & Figure & Figure caption & Table & Table caption & Reference & mAP \\ \hline
\multicolumn{1}{|l|}{\multirow{2}{*}{Faster RCNN}} & w/o RM & 1 & 51.1 & 94.8 & 98.9 & \textbf{94.2} & 91.8 & 97.3 & 67.1 & 90.8 & 87.3 \\ \cline{2-2}
\multicolumn{1}{|l|}{} & w/ RM & \textbf{1} & \textbf{88.4} & \textbf{99.1} & \textbf{99.1} & 85.4 & \textbf{92.6} & \textbf{98.0} & \textbf{79.2} & \textbf{91.6} & \textbf{92.6} \\ \hline
\multirow{2}{*}{VSR} & w/o RM & 1 & 84.3 & \textbf{96.1} & 98.7 & \textbf{95.7} & 92.5 & \textbf{99.4} & 71.4 & \textbf{92.4} & 92.3 \\ \cline{2-2}
& w/ RM & \textbf{1} & \textbf{94} & 95 & \textbf{99.1} & 95.3 & \textbf{94.5} & 96.1 & \textbf{84.6} & 92.3 & \textbf{94.5} \\ \hline
\end{tabular}
}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{images/result_vis.pdf}
\caption{Qualitative comparison between \textit{VSR w/wo RM}. Introducing $RM$ effectively removes duplicate predictions and provides more accurate detection results (both labels and coordinates). The colors of semantic labels are: \color{blue}{Figure}, \color{red}{Body}, \color{green}{Figure Caption}.}
\label{figure:result_vis}
\end{figure}
\subsubsection{Effects of relation module.}
\label{e_r_m}
To verify the effectiveness of relation module (\textit{RM}), {we compare two versions of Faster RCNN and VSR in Table~\ref{table:erm}, \ie with \textit{RM} and without \textit{RM}.}
{Since both labels and position coordinates can be refined in \textit{RM}, both unimodal Faster RCNN and VSR show consistent improvements after incorporating relation module, with $5.3\%$ and $2.2\%$ increase respectively.
Visual examples are given in Fig~\ref{figure:result_vis}.
However, for {Figure} component, performance may slightly drop after introducting \textit{RM}.
The reason is that, while removing duplicate predictions, our relation module may also risk removing correct predictions.
But still, we see improvements on overall performances, showing the benefits of introducting relations.
\subsubsection{Limitations.}
As mentioned above, in addition to document images, VSR also requires the positions and contents of texts in the document.
Therefor, the generalization of VSR may be not good enough compared with its unimodal counterparts, which we'll address in the future.
\section{Conclusion}
\label{conclusion}
In this paper, we present a unified framework VSR for multimodal layout analysis combining vision, semantics and relations.
We first introduce semantics of document at character and sentence granularities.
Then, a two-stream convolutional network is used to extract modality-specific visual and semantic features, which are further fused in the adaptive aggregation module.
Finally, given component candidates, a relation module is adopted to model relations between them and output final results.
On three benchmarks, VSR outperforms its unimodal and multimodal single-stream counterparts significantly.
In the future, we will investigate pre-training models with VSR and extend it to other tasks, such as information extraction.
|
2110.05806
|
\section{Introduction}
A number of astrophysical observations provide evidence that the dominant matter component of the Universe is not ordinary matter, but rather non-baryonic dark matter~\cite{Clowe:2006eq,Ade:2015xua}. Theoretically favored dark matter candidates are weakly interacting massive particles~(WIMPs)~\cite{PhysRevLett.39.165,Goodman:1984dc}. Many direct searches for WIMP dark matter in deep underground laboratories have been performed and have yet to find a signal~\cite{PhysRevLett.118.021303,Agnese:2017njq,Aprile:2017iyp,DarkSide:2018bpj,XENON:2018voc,CRESST:2019jnq,Zyla:2020zbs}.
In light of the absence of signal in the WIMP dark matter mass range of GeV/c$^2$ to TeV/c$^2$, there is an increasing interest in low-mass dark matter particles in the sub-GeV/c$^2$ mass range~\cite{Essig:2013lka,XENON:2016jmt,XENON:2019gfn,DAMIC:2019dcn,EDELWEISS:2020fxc,SENSEI:2020dpa,SuperCDMS:2020aus,PandaX-II:2021nsg}.
In this paper, we report on low-mass dark matter searches for WIMP-nuclei interactions by looking for electron recoils induced from secondary radiation via the Migdal process~\cite{Migdal,Ibe:2017yqa} in COSINE-100 data. Data used for COSINE-100 searches has a 1\,keVee analysis threshold~\cite{Adhikari:2020xxj,COSINE-100:2021xqn}, where the unit keVee is the electron recoil-equivalent energy in kiloelectron volts. Since the total energy in the Migdal electron and the nuclear recoil is larger than the deposited energy of typical elastic nuclear recoil, our searches are extended to WIMP masses as low as 200\,MeV/c$^2$.
In the future, this search can be enhanced by lowering the analysis threshold through multivariable analysis or deep machine learning techniques, as discussed in Section~\ref{sec:sens}, where we evaluate sensitivities of the COSINE-200 experiment, which will have lower analysis threshold, reduced internal background by controlled crystal growth~\cite{COSINE:2020egt} and improved light yield using a novel encapsulation method~\cite{Choi:2020qcj}.
\section{Experiment}
The COSINE-100 experiment~\cite{Adhikari:2017esn} is installed in the Yangyang underground laboratory (Y2L) utilizing the space provided by the Yangyang pumped storage power plant in South Korea~\cite{Kims:2005dol,Kim:2012rza}. The laboratory is located at a vertical depth of 700\,m that provides a water-equivalent overburden of 1800\,m~\cite{Prihtiadi:2017inr}. A 2-km-long driveway provides access to the laboratory as well as air ventilation. The laboratory is equipped with a cleanroom and an air-conditioning system providing a low dust level, and constant temperature and humidity of 24.21$\pm$0.06$^\circ$C and 36.7$\pm$1.0\%, respectively~\cite{Adhikari:2017esn,Kim:2021rsj}.
The contamination level of $^{222}$Rn in the room is measured to 36.7$\pm$5.5\,Bq/m$^3$. The readout electronics, high voltages that are applied to photomultiplier tubes~(PMTs), and data acquisition system are also monitored and stably maintained~\cite{Kim:2021rsj}.
The COSINE-100 detector, shown in Fig.~\ref{fig_det}, consists of a 106\,kg array of eight ultra-pure NaI(Tl) crystals each coupled to two PMTs. The crystal array is immersed in an active veto detector comprised of 2,200\,L of linear alkylbenzene~(LAB)-based liquid scintillator~(LS) to attenuate or tag the influence of external or internal radiations~\cite{Park:2017jvs,Adhikari:2020asl}. The LAB-LS is contained within a box made with 1\,cm thick acrylic and 3\,cm thick oxygen-free copper that is surrounded by a 20\,cm thick lead shield. An outer array of plastic scintillation counters is used to tag and veto cosmic-ray muons~\cite{Prihtiadi:2017inr,Prihtiadi:2020yhz}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.49\textwidth]{./cosine_detector_cut.pdf}
\caption{Schematic of the COSINE-100 detector. The NaI(Tl)~(106~kg) detectors are immersed in the 2,200\,L LAB-LS that is surrounded by layers of copper and lead shielding.}
\label{fig_det}
\end{center}
\end{figure}
An event is triggered when coincident single photoelectrons in both PMTs that are coupled to a single crystal are observed within a 200\,ns time window. If at least one crystal satisfies the trigger condition, data from all crystals and the LAB-LS are recorded. The signals from the crystal PMTs are processed by 500\,MHz flash analog-to-digital converters and are 8\,$\mu$s long waveforms that start 2.4\,$\mu$s before the trigger. The LAB-LS and plastic scintillator signals are processed by charge-sensitive flash analog-to-digital converters. Muon events are triggered by coincident signals from at least two plastic scintillators. The LAB-LS signals do not generate triggers, except in the case of energetic muon events that are coincident with one of the muon detector panels. A trigger and clock board read the trigger information from individual boards and generate a global trigger and time synchronizations for all of the modules. Details of the COSINE-100 data acquisition system are described in Ref.~\cite{Adhikari:2018fpo}.
\section{Migdal effect}
Direct detection of WIMP dark matter with mass below sub-GeV/c$^2$ is limited by an energy threshold of the detector in the range about 0.1--1\,keVee. Because the nuclear recoil energies from WIMP-nuclei interactions are quenched (the scintillation signals from nuclear recoils are only an order of 10\,\% of the signals from the same energy deposition of electrons~\cite{Manzur:2009hp,Lee:2015xla,Joo:2018hom,Kimura:2019rdg}), this energy range corresponds to 1--10\,keVnr, where keVnr is kiloelectron volt nuclear recoil energy.
The WIMP-nucleus interaction used in typical direct detection searches assumes that the electron cloud is tightly bound to the nucleus and that the orbit electrons remain in stable states. However, energy transferred to nuclei after collision may lead to excitation or ionization of atomic electrons via the Migdal process~\cite{Migdal,Ibe:2017yqa}.
This process can lead to the production of energetic electron-induced signals that are produced in association with the primary nuclear recoil.
For a WIMP-nucleus interaction, even if the electron equivalent energy implied by the quenching factor is below the energy threshold of the detector, these Midgal-effect secondary electrons can produce electron equivalent energy that are above the threshod, making detectors sensitive to sub-GeV/c$^{2}$ dark matter interactions.
Several experimental groups have already exploited this effect to search for dark matter with sub GeV/c$^2$ masses~\cite{LUX:2018akb,EDELWEISS:2019vjv,CDEX:2019hzn,XENON:2019zpr,GrillidiCortona:2020owp}.
The differential nuclear recoil rate per unit target mass for elastic scattering between WIMPs of mass $m_\chi$ and target nuclei of mass $M$ is~\cite{Savage:2008er},
\begin{eqnarray}
\frac{dR_\mathrm{nr}}{dE_\mathrm{nr}} = \frac{\rho_\chi}{2m_\chi\mu^2}~\sigma(M,~E_\mathrm{nr})
\int_{v>v_\mathrm{min}}d^3v~f(\textbf{v},~t),
\label{eq:recoilrate}
\end{eqnarray}
where $\rho_\chi$ is the local dark matter density, $E_\mathrm{nr}$ is the nuclear recoil energy, $\sigma(M,E_\mathrm{nr})$ is the WIMP-nucleus cross section and $f(\textbf{v},t)$ is the time-dependent WIMP velocity distribution. The reduced mass $\mu$ is defined as $m_\chi M/(m_\chi+M)$ and the minimum WIMP velocity $v_\mathrm{min}$ is $\sqrt{ME_\mathrm{nr}/2\mu^2}$.
The rate of ionization due to the Migdal effect for a nuclear recoil energy $E_\mathrm{nr}$ accompanied by an ionization electron with energy $E_{ee}$ is given by the nuclear recoil in Eq.~\ref{eq:recoilrate} multiplied by the ionization rate~\cite{Ibe:2017yqa},
\begin{equation}
\begin{split}
\frac{dR}{dE_\mathrm{ee}} & = \int dE_\mathrm{nr} dv \frac{d^2R}{dE_\mathrm{nr}dv} \\*
& \times \frac{1}{2\pi}\sum_{n,l} \frac{d}{dE_\mathrm{nr}} p^{c}_{q_e}(n,l \rightarrow E_\mathrm{ee}-E_{n,l}).
\end{split}
\label{eq:migdal}
\end{equation}
Here $p^{c}_{q_e}$ is the probability for an atomic electron with quantum number ($n,l$) and binding energy $E_{n,l}$ to be ejected with a kinetic energy $E_\mathrm{ee}-E_{n,l}$, and $q_e$ is the electron momentum in the nucleus rest frame. When the shell vacancy is refilled, an X-ray or an auger electron with energy $E_{n,l}$ is emitted. This takes into account the fact that the emitted electron may come from an inner orbital and the remaining excited state will release additional energy when returns to its ground state. The differential probability rates for sodium and iodine were calculated in Ref.~\cite{Ibe:2017yqa}. Figure~\ref{fig_signal} shows the differential ionization rates as a function of the electron recoil energy $E_\mathrm{ee}$ for sodium and iodine nuclei considering two different WIMP masses.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.49\textwidth]{./sig_qch.pdf}
\caption{Expected signals without energy resolution from WIMP-nuclei spin-independent (SI) interaction via the Migdal process are presented for WIMP masses of 0.2\,GeV/c$^2$ and 1.0\,GeV/c$^2$ assuming WIMP-nucleon SI cross-section of 1\,pb. Sodium and iodine spectra produced by the Migdal effect are separately presented and compared with the nuclear recoil spectrum of WIMP-sodium interaction for 1.0\,GeV/c$^2$ WIMP mass. Here the nuclear recoil energy is quenched to the electron recoil energy using the measured quenching factor reported in Ref.~\cite{Joo:2018hom}. Although the nuclear recoil deposit is below the 1\,keVee energy threshold, the electron energy from the Migdal effect can produce above-threshold signals. }
\label{fig_signal}
\end{center}
\end{figure}
This description assumes isolated atomic targets that interact with WIMP particles~\cite{Ibe:2017yqa,Kouvaris:2016afs}, which for inner-shell electrons provided a correct estimate of the expected signal rate. For outer electrons, the complicated electronic band structure and crystal form factor can affect the rate for the Migdal effect and significantly improved sensitivity in semiconductors has been reported in Refs.~\cite{Essig:2019xkx,Knapen:2020aky}.
In this analysis, we follow the atomic target approximation because at our analysis threshold, inner shell electrons are the dominant contributors to the Migdal process.
\section{Data analysis}
We use data obtained from October 2016 to July 2018, corresponding to 1.7 years exposure that were used for our first annual modulation search~\cite{Adhikari:2019off} and the model-dependent WIMP dark matter search using the energy spectra~\cite{COSINE-100:2021xqn}. During the 1.7\,years data-taking period, no significant environmental anomalies or unstable detector performance were observed. Six of the eight crystals have a high light yield of approximately 15\,NPE/keVee, where NPE corresponds to the number of photoelectron, and enables an analysis threshold of 1\,keVee. The other two crystals had lower light yields and required higher analysis thresholds~\cite{Adhikari:2017esn,Adhikari:2018ljm}. Since their direct impact on the low-energy signal search is not substantial, we do not include these two crystals in this analysis.
In the offline analysis, muon-induced events are rejected when the crystal hit events and muon candidate events in the muon detector~\cite{Prihtiadi:2017inr,Prihtiadi:2020yhz} are coincident within 30\,ms. Additionally, we require that the leading edges of the trigger pulses start later than 2.0\,$\mu$s after the start of the recording, that the waveforms from the hit crystal contain more than two single photoelectrons and the integral waveform area below the baseline does not exceed a limit. These criteria reject muon-induced phosphor events and electronic interference events. A multiple-hit event is one in which more than one crystal has a signal with more than four photoelectrons in an 8\,$\mu$s time window or, has an LS signal above an 80\,keVee threshold within 4\,$\mu$s of the hit crystal~\cite{Adhikari:2020asl}. A single-hit event is classified as one where only one crystal is hit and none the other detectors meet the above criteria.
\begin{figure*}[!htb]
\begin{center}
\includegraphics[width=0.98\textwidth]{./evtSelectionPar.pdf}
\caption{The distribution of two input parameters, mean-time parameter (left) and the likelihood parameter (center), and output BDT scores (right) are presented separately for the WIMP-search data (top) and $^{60}$Co calibration data (bottom). Details of all parameters are described in~\cite{Adhikari:2020xxj}.}
\label{fig_bdtpar}
\end{center}
\end{figure*}
In the low-energy signal region below 10\,keVee, PMT-induced noise events contribute to the single-hit WIMP-search data.
Although PMT noise involves complex phenomena that are far from being completely understood, we categorize two distinct classes of the PMT-induced noise events.
The first class has a fast decay time of less than 50\,ns, compared with typical NaI(Tl) scintillation of about 250\,ns. This class of noise events may be induced by radioactive decays of U, Th, and K inside the PMT materials. These decays may generate ultraviolet and/or visible photons inside the PMTs. The second class, that occurs less often than the first, is characterized by slow rise times~(about 100\,ns) and decay times~(about 150\,ns), as described in Refs.~\cite{Adhikari:2018ljm,Kang:2019fvz}. This class of noise may be caused by accumulated charge somewhere in the PMT arising from the high voltage and subsequent-discharging that produces a flash inside the PMTs. This class noise events are intermittently produced by certain PMTs. We have developed monitoring tools for data quality verification, including monitoring event rates of the second class of noise. If a crystal has an increased rate due to the second class of noise, the relevant period of data is removed. One crystal detector has this class of noise for the whole data-taking period; for the other five detectors the second-class noise-induced events are absent during more than 95\,\% of the data taking period. The effective data exposure for the five crystals is 97.7\,kg$\cdot$year.
The first class of PMT noise-induced events is efficiently rejected by a boosted decision tree (BDT)-based multivariable analysis techinique~\cite{BDT}. The parameters used in the BDT include the balance of the deposited charge from two PMTs, the ratio of the leading-edge (0--50\,ns) to trailing-edge (100\,ns--600\,ns) charge, a mean-time parameter, which is a logarithm of the amplitude weighted average time of the events, and a likelihood parameter for samples of scintillation-signal events and the fast PMT-induced events~\cite{Adhikari:2020xxj,COSINE-100:2021xqn}. Figure~\ref{fig_bdtpar} shows input parameters of mean-time and likelihood, as well as output BDT scores for calibration samples and WIMP search data.
This procedure reduces the noise contamination level to less than 0.5\% and maintains an 80\% selection efficiency at the lowest energy bin~(1--1.25\,keVee) can be achieved~\cite{Adhikari:2020xxj}.
Geant4~\cite{Agostinelli:2002hh}-based simulations are used to understand the contribution of each background component~\cite{Adhikari:2017gbj,cosinebg,cosinebg2}, as well as to verify the energy scales and resolutions. We classify four categories of the NaI-deposited events based on their energies and detector multiplicities. Single-hit and multiple-hit events are further divided into low-energy events 1--70\,keVee and high-energy events 70--3000\,keVee. The fraction of each background component is determined from a simultaneous fit to the four categorized distributions. For the single-hit events, only 6--3000\,keVee events are used to avoid a bias of the signal in the region of interest (ROI). A detailed description of our modeling of the background with the same data is described in Ref.~\cite{cosinebg2}.
We consider various sources of systematic uncertainties in the background and signal models. Errors associated with the selection efficiency, the energy resolution, the energy scale, and the background modeling technique are accounted for the shapes of the signal and background probability density functions, as well as in rate changes as described in Ref.~\cite{COSINE-100:2021xqn}. These quantities are allowed to vary within their uncertainties as nuisance parameters in the data fit used to extract the signal.
To estimate the dark matter signal enhancement that is provided by the Migdal effect, we generate signals based on Eq.~\ref{eq:migdal} for various interaction models and different masses in the specific context of the standard WIMP galactic halo model~\cite{Lewin:1995rx,Freese:2012xd}. Both a spin-independent (SI) interaction between WIMP and nucleons and a spin-dependent (SD) interaction between WIMP and proton are considered in this analysis. Because both sodium and iodine have non-zero proton spin due to their odd numbers of protons~\cite{Bednyakov:2006ux,Gresham:2014vja}, NaI(Tl) detectors are sensitive to the WIMP-proton SD interactions. Responses that include form factors and proton spin values of the nuclei are implemented from the publicly available {\sc dmdd} package~\cite{dmdd,Gluscevic:2015sqa,Anand:2013yka,Fitzpatrick:2012ix,Gresham:2014vja}. The energy spectra of electron-equivalent energy for the detector are simulated with the energy resolutions of the detectors and the nuclear recoil quenching factors (QFs), where QF is the ratio of the scintillation light yield from sodium or iodine recoil relative to that from electron recoil for the same energy. We used the QF values from recent measurements with monoenergetic neutron beam~\cite{Joo:2018hom}. The measurements were modeled using a modified Lindhard formula~\cite{osti_4701226} and described in Ref.~\cite{Ko:2019enb}. Examples of quenched signal spectra from the Migdal effect of the WIMP-nucleon SI interactions for the WIMP masses of 0.2 and 1.0\,GeV/c$^2$ are presented in Fig.~\ref{fig_signal}. The output events are subjected to the same selection criteria that are applied to the data.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.49\textwidth]{./spectrumFit.pdf}
\caption{An example of the fit for a WIMP mass 1.1\,GeV/c$^2$ using the Migdal effect is presented. The summed energy spectrum for the five crystals (black filled circles) and the best fit (blue solid line) for which no WIMP signals are obtained, are shown together with signal spectra from WIMP masses and SI cross-sections of 0.2\,GeV/c$^2$ and 10$^{7}$pb~(red dashed line), 1.0\,GeV/c$^2$ and 10$^{2}$~\,pb~(green dotted line). The fitted distribution is broken down into cumulative contributions to the background from external sources, the surface of the crystals and nearby materials, internal radionuclide contaminations, and cosmogenic activation, as indicated. The green (yellow) bands are the 68\% (95\%) CL intervals of the systematic uncertainty obtained from the likelihood fit.}
\label{fig_fit}
\end{center}
\end{figure}
A Bayesian approach is adopted to extract the WIMP interaction signals using the Migdal effect from the COSINE-100 data. A likelihood function based on Poisson probability is built including constraints that reflect the known levels of the background components. The likelihood fit is applied to the measured single-hit energy spectra between 1 and 6\,keVee for each WIMP model with various masses. Each crystal is fitted with a crystal-specific background model and a crystal-correlated dark matter signal. The combined fit is constructed by multiplying likelihoods of the five crystals. The systematic uncertainties are included as nuisance parameters with Gaussian constraints. The same machinery was used for the WIMP dark matter searches using the same dataset but with only the energy from the nuclear recoils without the Migdal effect~\cite{COSINE-100:2021xqn}. An example of the fit for the SI interaction for a WIMP mass of 1.1\,GeV/c$^{2}$ is shown in Fig.~\ref{fig_fit}. The summed event spectrum for the five crystals is shown together with the best-fit result. For comparison, the expected signals for WIMP masses (cross-sections) of 0.2 (10$^7$) and 1.0\,GeV/c$^2$ (10$^2$\,pb) are presented together. No significant signal for event excess that could be attributed to WIMP interactions is found and
90\% confidence level (CL) limits are determined from the marginalization of the likelihood function. Figure~\ref{fig_limit} shows the 90\% CL upper limits from the COSINE-100 data for the WIMP-nucleon SI interactions (A) and the WIMP-proton SD interactions (B) compared with the limits from XENON1T~\cite{XENON:2019zpr}, EDELWEISS surface measurement~\cite{EDELWEISS:2019vjv}, and CDEX~\cite{CDEX:2019hzn} using the Migdal effect, as well as CRESST-III~\cite{CRESST:2019jnq}, CRESST LiAlO$_2$G~\cite{CRESST:2020tlq}, DarkSide-50~\cite{Agnes:2018ves}, PICASSO~\cite{Behnke:2016lsk}, CDMSlite~\cite{SuperCDMS:2017nns}, EDELWEISS surface~\cite{EDELWEISS:2019vjv}, Collar~\cite{Collar:2018ydf}, and PICO-60~\cite{PICO:2019vsc} results.
Here, our search extends to the low-mass WIMP of 200\,MeV/c$^2$.
Although our limits do not consider earth shielding effects~\cite{Kouvaris:2015laa,Kavanagh:2017cru,EDELWEISS:2019vjv}, we found no significant impact on the upper limits using the {\sc verne} code~\cite{verne}.
Because of the relatively higher threshold energy and background rates, our search cannot investigate the unexplored parameter space. This will be addressed in the future with improved detectors and analysis methods.
\begin{figure*}[!htb]
\begin{center}
\includegraphics[width=0.98\textwidth]{./limits.pdf}
\caption{Limits on WIMP-nucleon SI interaction (A) and WIMP-proton SD interaction (B) induced by the Migdal effect from the COSINE-100 data (black dots) at 90\% CL are compared with XENON1T~\cite{XENON:2019zpr}, EDELWEISS surface measurement~\cite{EDELWEISS:2019vjv}, and CDEX~\cite{CDEX:2019hzn} with the Migdal effect, and CRESST-III~\cite{CRESST:2019jnq}, CRESST LiAlO$_2$G~\cite{CRESST:2020tlq}, DarkSide-50~\cite{Agnes:2018ves}, PICASSO~\cite{Behnke:2016lsk}, CDMSlite~\cite{SuperCDMS:2017nns}, EDELWEISS surface~\cite{EDELWEISS:2019vjv}, Collar~\cite{Collar:2018ydf}, and PICO-60~\cite{PICO:2019vsc}.}
\label{fig_limit}
\end{center}
\end{figure*}
\section{Sensitivity for COSINE-200}
\label{sec:sens}
An effort to upgrade the on-going COSINE-100 to the next-phase COSINE-200 has resulted in the production of NaI(Tl) crystals with reduced internal background from $^{40}$K and $^{210}$Pb~\cite{Shin:2018ioq,Shin:2020bdq,COSINE:2020egt} as well as an increased light yield of 22\,NPE/keV~\cite{Choi:2020qcj}.
The recrystalization method has achieved chemical purification of the raw NaI powder with sufficient reduction of K and Pb contamination~\cite{Shin:2018ioq,Shin:2020bdq}.
A dedicated Kyropoulos grower for small test crystals has produced low-background NaI(Tl) crystals with reduced $^{40}$K and $^{210}$Pb of less than 20\,ppb and 0.5\,mBq/kg, respectively, corresponding to background rates of less than 1\,counts/day/kg/keVee at the 1--6\,keVee ROI~\cite{COSINE:2020egt}. A full-size Kyropoulos grower has been built for the 100\,kg-size crystal ingot and will provide approximately 200\,kg of low-background NaI(Tl) detectors for the COSINE-200 experiment. The expected background level of those crystals is less than 0.5\,counts/day/kg/keVee in the ROI as shown in Fig.~\ref{fig_exp}. This estimate is based on the measured background levels for small test crystals as discussed in Ref.~\cite{COSINE:2020egt}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.49\textwidth]{./bkgd_cosine200.pdf}
\caption{Expected background spectrum (blue solid line) of the COSINE-200 crystal based on the developed low-background NaI(Tl) crystal~\cite{COSINE:2020egt} is compared with expected signals using the Migdal effect for WIMP-nucleon SI interactions of masses and cross sections of 0.2\,GeV/c$^2$, 100\,pb (red dashed line) and 1.0\,GeV/c$^2$, 1\,pb (green dotted line), respectively. A dominant background contribution is expected from the internal $^{40}$K and $^{210}$Pb remaining after purification.
}
\label{fig_exp}
\end{center}
\end{figure}
A high light yield of the NaI(Tl) crystal is crucial for enabling low-energy thresholds below 1\,keVee. With an optimized concentration of thallium doping in the crystal, we achieved high light yield of 17.1$\pm$0.5\,NPE/keVee, slightly larger than that of the COSINE-100 crystal of approximately 15\,NPE/keVee. Further increase in the light collection efficiency by $\sim$50\% is possible with an improved encapsulation scheme as described in Ref.~\cite{Choi:2020qcj} that directly connects the crystal and PMTs without an intermediate quartz window for which a 22\,NPE/keVee light yield is reported.
The typical trigger requirement of the COSINE-100 experiment is satisfied with coincident photoelectrons in two PMTs attached to each side of the crystal at approximately 0.13\,keVee.
However, the PMT-induced noise events are dominantly triggered below energies of a few keVee.
The multivariable BDT provided a 1\,keVee analysis threshold with less than 0.1\% noise contamination and above 80\% selection efficiency~\cite{Adhikari:2020xxj}.
A key variable in the BDT is the likelihood parameter using the event shapes of the scintillation-like events and the PMT-induced noise-like events.
A further improvement of the low-energy event selection is ongoing with the COSINE-100 data by developing new parameters for the BDT as well as employing a machine learning technique that uses raw waveforms directly.
COSINE-200 targets an analysis threshold of 5\,NPE~(0.2\,keVee), which is similar to the energy threshold that has already been achieved by the COHERENT experiment with CsI(Na) crystals~\cite{COHERENT:2017ipa}.
The COSINE-200 experiment can be realized in a 4$\times$4 array of 12.5\,kg NaI(Tl) modules by replacing crystals inside the COSINE-100 shield~\cite{Adhikari:2017esn}. The COSINE-200 experiment will run at least 3 years for an unambiguous test of the DAMA/LIBRA annual modulation signals~\cite{Adhikari:2015rba}. In addition, this detector can be used for general dark matter searches, especially for low-mass WIMPs via the Migdal process. Assuming the background reduction shown in Fig.~\ref{fig_exp}, a high light yield of 22\,NPE/keVee, and an analysis threshold of 5\,NPE corresponding to 0.2\,keVee, the sensitivities of the COSINE-200 experiment are evaluated below using the Migdal effect.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.49\textwidth]{./rateVSmass}
\caption{The expected signal counts using the Migdal effect for a one year of data assuming 0.2\,keVee analysis threshold are presented as a function of WIMP masses for SI (blue filled points) and SD interactions (green open points) for 1\,kg of sodium (circle points) and 1\,kg of iodine (triangle points). The cross-sections for the SI and SD interactions are assumed to be 1\,pb and 200\,pb, respectively.}
\label{fig_counts}
\end{center}
\end{figure}
\begin{figure*}[!htb]
\begin{center}
\includegraphics[width=0.98\textwidth]{./sensitivity.pdf}
\caption{COSINE-200 expected 90\% CL limits using the Migdal effect on the WIMP-nucleon SI cross-section (A) and the WIMP-proton SD cross-section (B) are presented assuming the background-only hypothesis indicating the 1$\sigma$ and 2$\sigma$ standard deviation probability regions over which the limits have fluctuated. Those limits are compared with the current best limits from the XENON1T Migdal~\cite{XENON:2019zpr}, DarkSide-50~\cite{Agnes:2018ves}, PICASSO~\cite{Behnke:2016lsk}, CDMSlite~\cite{SuperCDMS:2017nns}, Collar~\cite{Collar:2018ydf}, and PICO-60~\cite{PICO:2019vsc} together with the sensitivity limits from future liquid Ar experiment (TEA-LAB)~\cite{GrillidiCortona:2020owp}, Xe~\cite{Essig:2019xkx}, and Si~\cite{Knapen:2020aky} with Migdal.}
\label{fig_sensitivity}
\end{center}
\end{figure*}
We generate the WIMP interaction signals including the Migdal effects as discussed above. Here we assume a one year data exposure with 200\,kg crystals and the aforementioned detector performances. Poisson fluctuations of the measured NPE are considered for the detector resolution. Figure~\ref{fig_counts} presents the expected signal rate as a function of the WIMP mass for 1\,kg of sodium and 1\,kg of iodine for different interactions assuming 0.2\,keVee threshold. As one can see, sodium has an advantage for the low-mass WIMP searches with the Migdal process.
In this scenario, the COSINE-200 experiment can probe the sub-GeV/c$^2$ WIMP with mass down to 20\,MeV/c$^2$.
We use an ensemble of simulated experiments to estimate the sensitivity of the COSINE-200 experiment, expressed as the expected cross-section limits for the WIMP-nucleon SI and WIMP-proton SD interactions using the Migdal effect in the case of no signals. For each experiment, we determine a simulated spectrum for a background-only hypothesis with assumed background from Fig.~\ref{fig_exp}. A gaussian fluctuation of each background component and a poisson fluctuation of each energy bin are considered for each simulated experiment. We then fit the simulated data with a signal-plus-background hypothesis with flat priors for the signal and Gaussian constraints for the backgrounds based on understanding of the NaI(Tl) crystals~\cite{cosinebg2,COSINE:2020egt}. Examples of signal spectra using the Migdal effect for SI interactions with WIMP masses (cross sections) of 0.2\,GeV/c$^2$ (100\,pb) and 1.0\,GeV/c$^2$ (1\,pb) are presented in Fig.~\ref{fig_exp}. The same Bayesian approach is used for the single-hit energy spectra between 5\,NPE (0.2\,keVee) and 130\,NPE (6\,keVee) for each WIMP model for several masses. The 1$\sigma$ and 2$\sigma$ standard-deviation probability regions of the expected 90\% CL limits are calculated from 2000 simulated experiments. Figure~\ref{fig_sensitivity} shows those 1$\sigma$ and 2$\sigma$ regions, for the COSINE-200 experiment using the Migdal effect. The limits on the WIMP-nucleon SI interaction shown in Fig.~\ref{fig_sensitivity} (A) are compared with the current best limit on the low-mass WIMP searches of XENON1T with Migdal, DarkSide-50, and the expected sensitivities from future liquid Ar (TEA-LAB)~\cite{GrillidiCortona:2020owp}, Xe~\cite{Essig:2019xkx}, and Si~\cite{Knapen:2020aky} with Migdal.
We also verify no significant impact on the upper limits using the {\sc verne} code~\cite{verne}.
The COSINE-200 experiment has a potential to probe unexplored cross section values for the WIMP mass below 200\,MeV/c$^2$.
In case of the WIMP-proton SD interactions shown in Fig.~\ref{fig_sensitivity} (B), our projected sensitivities are compared with the current best limits from XENON1T with Migdal, PICO-60, CDMSlite~\cite{SuperCDMS:2017nns}, Collar~\cite{Collar:2018ydf}, and PICASSO. Taking advantages of odd-proton numbers in both iodine and sodium, the projected sensitivities from the COSINE-200 experiments can probe unexplored parameter spaces of WIMP masses below 2\,GeV/c$^2$.
\section{Conclusion}
We consider the Migdal effect to search for the low-mass dark matter using the COSINE-100 data. With 1\,keVee analysis threshold, our search extends down to 200\,MeV/c$^2$ WIMP mass. We have investigated the expected sensitivities of the COSINE-200 experiment with a total mass of 200\,kg, a 1-year period of stable operation, about 0.5 counts/day/keVee/kg background rate, and 0.2\,keVee energy threshold. In this scenario, the COSINE-200 detector can explore low-mass dark matter down to 20\,MeV/c$^2$, with a potential to probe currently unexplored parameter spaces for both SI and SD interactions.
\acknowledgments
We thank the Korea Hydro and Nuclear Power (KHNP) Company for providing underground laboratory space at Yangyang. This work is supported by: the Institute for Basic Science (IBS) under project code IBS-R016-A1 and NRF-2021R1A2C3010989, Republic of Korea; NSF Grants No. PHY-1913742, DGE1122492, WIPAC, the Wisconsin Alumni Research Foundation, United States; STFC Grant ST/N000277/1 and ST/K001337/1, United Kingdom; Grant No. 2017/02952-0 FAPESP, CAPES Finance Code 001, CNPq 131152/2020-3, Brazil.
\bibliographystyle{PRTitle}
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2110.05653
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\section{\label{sec:Intro}Introduction}
The exponential and Gaussian functions are among the most fundamental and important operations, appearing ubiquitously
throughout all areas of science, engineering, and mathematics. Whereas formally, it is well-known that any function,
$f(x)$, may in principle be realized on a quantum
computer \cite{abrams97,zalka98,nielsen,florio04b,kais,preskill18,alexeev19,haner18,sanders20},
in practice present-day algorithms tend to be very expensive.
Currently, one of the best strategies for evaluating general functions---as exemplified by Ref.~\cite{haner18}---divides
the domain, $x$, into a collection of non-intersecting subdomains.
The function, $f(x)$, is then approximated using a separate $d$'th order polynomial for each subdomain,
evaluated through a sequence of $d$ multiplication-accumulation
(addition) operations. The quantum advantage comes from the fact that these polynomial evaluations
can be performed in parallel across all subdomains at once (using conditioned determination of
the coefficients for each subdomain polynomial).
The above parallel quantum strategy is completely general, conceptually elegant, and effective. It can also be
optimized in various ways---e.g., for a given target numerical accuracy, and/or to favor gate complexity (i.e. number of
quantum gates or operations) over space complexity (i.e, number of qubits), or vice-versa. However, in the words
of the Ref. \cite{haner18} authors themselves:
\begin{quotation}
{\em While these methods allow to reduce the Toffoli [gate] and qubit counts significantly, the resulting circuits are still quite expensive, especially in terms of the number of gates that are required.}
\end{quotation}
Part of the reason for the ``significant expense'' is the cost of the requisite quantum
multiplications \cite{draper00,florio04,florio04b,haner17,haner18,sanders20,parent17,gidney19,karatsuba62,kowada06}, each of which---at least for the
most commonly used ``schoolbook'' algorithms \cite{haner17,haner18,sanders20,parent17,gidney19}---requires a sequence of $n$
controlled additions \cite{draper00,cuccaro04,takahashi,takahashi09,parent17}. Here, $n$ is the
number of bits needed to represent the summands using fixed-point arithmetic (which is presumed throughout this work).
Each controlled addition introduces $O(n)$ gate complexity---implying an overall quantum multiplication
gate complexity that scales as $O(n^2)$.
Although alternative multiplication algorithms with asymptotic scaling as low as $O(n^{\log_2 3})$ do
exist \cite{parent17,gidney19,karatsuba62,kowada06},
they do not become competitive until $n$ reaches a few thousand. This is far beyond the values needed for most practical
applications (e.g., those of this work, for which $n=21$--32).
For practical applications, then, there appear to be two strategies that might be relied upon to
significantly improve performance. The first is to wait for better quantum multiplication algorithms to be devised;
this is, after all, an area of active and ongoing development, more so than general function evaluation on quantum
computers. The second is to design entirely new algorithms, customized for \emph{specific} $f(x)$ functions.
The present work is of the latter variety. In particular, we present quantum algorithms designed specifically to evaluate
exponential and Gaussian functions efficiently on quantum computers. These algorithms require a (generally) small
number of multiplications, which represent the overall computational bottleneck. Our general approach is thus
equally applicable to noisy intermediate-scale quantum (NISQ) calculations \cite{preskill18,bharti21}
with non-error-corrected quantum multiplications, as it is in a fault-tolerant context, using error-corrected
multiplications, etc. In all such contexts, we advocate for using the ``total multiplication count'' as the appropriate
gate complexity metric---although the ``Toffoli \cite{nielsen} count'' metric, which is currently quite popular,
will also be used in this paper.
It will be shown that the gate complexity for the present approach is dramatically reduced, when compared with the
state-of-the art competing method by H\"aner et al. \cite{haner18}. For a specific, realistic NISQ application, the Toffoli
count of the exponential function is reduced from 15,690 down to 912, under the most favorable
conditions for each method. For the corresponding Gaussian function comparison, the Toffoli count is reduced
from 19,090 down to 704. Space requirements are also generally reduced, and in any event quite modest---to
the extent that in one case, the above NISQ application can be implemented with as few as
$\sim$70 logical qubits.
Although the range of applications where exponential and Gaussian functions are relevant is virtually limitless,
one particular application area will be singled out for further discussion. Quantum computational chemistry
(QCC) \cite{poplavskii75,feynman82,lloyd96,abrams97,zalka98,lidar99,abrams99,nielsen,aspuru05,kassal08,whitfield11,brown10,christiansen12,georgescu14,kais,huh15,babbush15,kivlichan17,babbush17,babbush18,babbush18b,babbush19,low19,kivlichan19,izmaylov19,parrish19,altman19,cao19,alexeev19,bauer20,aspuru20}---i.e., quantum
chemistry simulations \cite{szabo,jensen,helgaker,RN544} run on quantum
computers---has long been regarded as one of the first important scientific applications where quantum supremacy
will likely be realized \cite{aspuru05,georgescu14,altman19,aspuru20}.
Particularly for ``first-quantized'' or coordinate-grid-based QCC \cite{abrams97,zalka98,lidar99,abrams99,nielsen,aspuru05,kassal08,whitfield11,huh15,babbush15,babbush17,kivlichan17,babbush19,alexeev19,bauer20,aspuru20}, it becomes necessary
to evaluate functions over a (generally) uniformly-distributed set of discrete grid points \cite{aspuru05,babbush18,babbush19,aspuru20,RN553,colbert92,szalay96,light00,poirier02dvrlj,littlejohn02b}---exactly
of the sort that emerges in fixed-point arithmetic, as used here.
Of course, the most natural function to arise in the QCC context is the inverse-square-root
function, $f(x) = x^{-1/2}$, representing Coulombic interactions \cite{szabo,jensen,helgaker,RN544}.
Even for a ``general function evaluator'' code, this specific case poses some special challenges---associated, e.g., with the
singularity at $x=0$---that result in substantially increased computational expense. On the other hand, the
alternative Cartesian-component separated (CCS) approach, as developed recently by the author and
coworkers \cite{jerke15,jerke18,jerke19,mypccp,bittner}, replaces
the inverse square root with a small set of Gaussians. Using the new exponential/Gaussian
evaluator of this work, then, the CCS approach would appear to become a highly competitive contender for
first-quantized QCC.
The remainder of this paper is organized as follows. Mathematical preliminaries are presented in
Sec.~\ref{subsec:prelim}, followed by an exposition of our basic quantum exponentiation
algorithm in Sec.~\ref{subsec:basicquantum}, and its asymptotic scaling in Sec.~\ref{subsec:cost}.
These are the core results, especially for
long-term quantum computing. Secs.~\ref{sec:refined} and~\ref{sec:detailed} then give a detailed explanation
of various algorithmic improvements leading to reduced gate and space complexity, that will be of particular
interest for NISQ computing. In particular, quantum circuits for two specific NISQ implementations are
presented in Sec.\ref{sec:detailed}---one designed to minimize gate complexity (Sec.~\ref{subsec:mult}), and the
other, space complexity (Sec.~\ref{subsec:alternative}). Using the specific ``gate saving'' and ``space saving''
implementations of Sec.~\ref{sec:detailed}, a detailed numerical comparison with Ref.~\cite{haner18} is provided
in Sec.~\ref{sec:analysis}. Finally, concluding remarks are presented in Sec.~\ref{sec:conclusions}.
\section{\label{sec:basic} Basic Method}
\subsection {\label{subsec:prelim} Mathematical preliminaries}
Consider the exponential function,
\eb
f(x') = \exp \of{- \alpha x'}. \label{fexprime}
\ee
We wish to evaluate the
function over the domain interval, $x'_{{\rm min}} \le x' < x'_{{\rm max}}$. Note that
$x'$ and $\alpha$ are presumed to be \emph{real-valued}. If $\alpha$ were pure imaginary, then
\eq{fexprime} would be unitary---i.e., the most well-studied special case in quantum computing \cite{nielsen}.
But this is not the case here.
Without loss of generality, we may restrict consideration to $\alpha > 0$. The negative
$\alpha$ case corresponds to the above, but with $x' \rightarrow -x'$,
$x'_{{\rm min}} \rightarrow - x'_{{\rm max}}$, and $x'_{{\rm max}} \rightarrow - x'_{{\rm min}}$.
Both the domain and the range of \eq{fexprime} are represented discretely,
using a finite number of qubits. For generality, we allow the number of domain qubits, $d$,
to differ from the number of range qubits, $n$. In the first-quantized QCC context, for instance,
the $d \ll n$ case arises very naturally (where `$\ll$' represents perhaps a factor of 3 or 4).
More specifically, something like 100 grid points are needed to accurately
represent each domain degree of freedom---although the function values themselves require
a precision of say, 6--10 digits. Throughout this paper, we mainly focus on the $d \ll n$
case---although the $d=n$ special case is obviously also important, and will also be considered.
The $d$ qubits used to represent the domain correspond to $2^d$ distinct grid points,
distributed uniformly across the $x'$ interval, with grid spacing
$\Delta = 2^{-d} (x'_{{\rm max}} - x'_{{\rm min}})$. Such representations are typical in quantum
arithmetic, and imply fixed-point rather than floating-point implementations \cite{haner18,sanders20}.
Indeed, since fixed-point arithmetic is closely related to integer arithmetic, we find it convenient
to transform $x'$ to the unitless domain variable,
\eb
x = {(x'-x'_{{\rm min}}) \over \Delta }, \label{xeqn}
\ee
such that the $x$ grid points become \emph{integers}, $x = \{0, 1, \ldots, 2^d -1 \}$.
In terms of $x$, the function then becomes
\ea{
f(x) & = & \exp[-\alpha (x'_{{\rm min}} + \Delta x)] \nonumber \\
& = & C A^x, \quad \mbox{where} \label{fCA} \\
C & = & \exp(-\alpha x'_{{\rm min}}) \quad \mbox{and} \quad A = \exp(-\alpha \Delta). \label{CA}
}
Next, we define the following binary decomposition of the $x$ integers, in terms of
the $d$ individual qubits, $x_i$, with $0 \le i < d$ and $x_i = \{0,1\}$:
\eb
x = \sum_{i=0}^{d-1} x_i 2^i \label{xbin}
\ee
Note that increasing $i$ corresponds to \emph{larger} powers of 2; thus, the binary
expansion of the integer $x$ would be $x_{d-1} \cdots x_1 x_0$. Put another way, the lowest
index values correspond to the rightmost, or least significant, digits in the binary expansion.
This convention shall be adopted throughout this work.
Substituting \eq{xbin} into \eq{fCA}, we obtain
\ea{
f(x) & = & C \of{A^{2^0}}^{x_0} \of{A^{2^1}}^{x_1} \cdots \of{A^{2^{d-1}}}^{x_{d-1}} \nonumber \\
& = & C \,A_0^{x_0} \, A_1^{x_1} \, \cdots \, A_{d-1}^{x_{d-1}}, \quad \mbox{where} \label{fCAi} \\
A_i & = & A^{2^i} \label{Ai}
}
In this manner, exponentiation is replaced with a sequence of $d$ multiplications. Note from \eq{Ai}
that $A_0=A$. As additional notation, we find it convenient to introduce the quantities $C_{0 \le i \le d}$,
through the recursion relation $C_{i+1} = C_i A_i^{x_i}$, with $C_0=C$. Thus, $C_d = f(x)$, and
the other $C_{i<d}$ quantities represent partial products in \eq{fCAi}.
\begin{figure*}[htb]
\centering
\includegraphics[height=9cm]{fig1}
\vskip-5pt
\caption{Quantum circuit used to implement basic quantum algorithm for exponentiation,
$f(x') = \exp(-\alpha x')$. All multiplications, $\times$, are presumed to be ``overwriting,'' in the sense
that the second input register is overwritten with the product of the two inputs as output.}
\label{fig1}
\end{figure*}
\subsection{\label{subsec:basicquantum} Basic quantum algorithm}
The exponent of every $A_i$ value in \eq{fCAi}, being the qubit $x_i$, is associated with the two
states or values, 0 and 1. From a quantum computing perspective, therefore, this situation
can be interpreted as an \emph{instruction}:
\begin{itemize}
\item If $x_i = 1$, then multiply by $A_i$.
\item Otherwise, i.e. if $x_i=0$, do nothing.
\end{itemize}
This suggests a simple and straightforward quantum algorithm for exponentiation,
consisting of nothing but a sequence of $d$ \emph{controlled multiplications}, as
indicated in Fig.~\ref{fig1}.
From the figure, each of the $d$ qubits, $x_i$, serves as the control qubit for a separate
target multiplication of $C_i$ by $A_i$, in order to generate the next $C_{i+1}$.
In this basic implementation, each of the $d$ constants, $A_i$, is stored by a separate
bundle of $n$ qubits, initialized prior to the calculation. An additional bundle of $n$
qubits (lowest wire in Fig.\ref{fig1}) is used to represent the value of the function. This output
register is initially assigned the constant value $C$, but through successive controlled multiplications with $A_i$
as described above, ends up taking on the final output value, $C_d=f(x) = \exp(-\alpha x')$.
For the moment, we primarily treat multiplication as an oracle or ``black box'' routine,
whose operational details need not concern us.
However, we note from the above description (and from Fig.\ref{fig1}) that one of the
two input registers gets overwritten with the product value as output, and the other is unaffected.
There are indeed some multiplication algorithms---e.g. those based on the Quantum Fourier
Transform (QFT) \cite{shor94,abrams99,nielsen,draper00,florio04,florio04b}---that behave in this manner.
We call these ``overwriting''
multiplication routines. Other standard multiplication algorithms---e.g., those based on
bit-shifted controlled additions \cite{haner17,haner18,sanders20}---do not have this property.
This issue is revisited again in Sec.~\ref{sec:detailed}.
As discussed, the $C_i$ values are stored in the $n$-qubit output register, whereas the
$A_i$ are stored in $d$ separate $n$-qubit input registers. Since
$0 < A_i < 1$ for all $A_i$, it is convenient to
represent these constants using the following $n$-bit binary expansion:
\eb
y = \sum_{j=0}^{n-1} y_j 2^{-(n-j)} \label{ybin}
\ee
Thus, the binary expansion of $y$ becomes $y=0.y_{n-1} \cdots y_1 y_0$---with the
$y_0$ bit least significant, as discussed. This representation has a resolution of
$2^{-n}$. Likewise, $0 < C_i \le 1$ for all $0 \le i \le d$, provided
$x'_{{\rm min}} \ge 0$ (if not, there are simple remedies that can be applied, although
these are not needed here). We therefore find it convenient
to adopt the \eq{ybin} representation for the $C_i$ as well as the $A_i$ values.
The above describes the basic algorithm for evaluating the exponential function
of \eq{fexprime}. For the Gaussian function, i.e.
\eb
f(x') = \exp(-\alpha x'^2), \label{fexpGauss}
\ee
one proceeds in exactly the same manner, except that it is necessary to perform an additional
multiplication, to obtain $x^2$ from $x$. We note that there are some specialized
quantum squaring algorithms, that shave a bit off of the cost
of a generic multiplication \cite{haner18,sanders20,gidney19}. If $d \ll n$ however, this savings is not significant;
the cost of the extra multiplication itself is much less than the others, since it involves
only $d$ rather than $n$ qubits.
\begin{figure*}[htb]
\centering
\includegraphics[height=6.5cm]{fig2}
\vskip-5pt
\caption{First half of quantum circuit used to implement refined quantum algorithm for exponentiation,
$f(x') = \exp(-\alpha x')$, for specific parameter values, $d=7$, $n=21$, and $A=0.389$.
Overwriting multiplications are presumed.}
\label{fig2}
\end{figure*}
\subsection{\label{subsec:cost} Computational cost and asymptotic scaling}
In terms of memory (i.e., space) usage, the above algorithm requires
$dn + n + d$ qubits in all---not including the ancilla bits needed to actually
implement the multiplications (not shown in Fig.~\ref{fig1}). As mentioned, the
computational cost is simply that of applying $d$ multiplications. Given the tremendous
variety of multiplication algorithms that have been and will be developed---and given
that some will always be better than others in different circumstances---\emph{we feel
it is best to let the number of required multiplications itself serve as the appropriate
gate complexity metric}. Of course, this requires that multiplications comprise the
overall computational bottleneck, as they do here. In similar fashion, the Toffoli
count provides another implementation-independent metric---when
comparing circuits whose bottleneck is the Toffoli gate (Sec.~\ref{sec:analysis}).
If absolute costs are difficult to compare directly between different methods, then
the next best thing to consider is asymptotic scaling---in this case, in terms of the
parameters $n$ and $d$. For our basic exponentiation algorithm, the scaling with
respect to $d$ is clearly linear---both of the space
and gate complexity.
As for the scaling with respect to $n$, this is determined by the multiplication algorithm
itself. At present, the most competitive quantum multiplication algorithm for asymptotically large $n$
in terms of scaling appears to be that of C. Gidney \cite{gidney19}, based on the
recursive Karatsuba scheme \cite{karatsuba62}. The Gidney algorithm requires $O(n)$ space
complexity, so that the overall scaling for our basic exponentiation algorithm
would be $O(nd)$. Likewise, the gate complexity for a single multiplication scales
as $O(n^{\log_2 3})$, implying $O(n^{\log_2 3} d)$ scaling for basic exponentiation.
As mentioned, Gidney does not overtake even
the simplest (i.e. ``schoolbook'') multiplication method until $n$ reaches a few thousand.
It is therefore not practical for NISQ computing. In
Sec.~\ref{sec:analysis}, more precise estimates will be provided for absolute
costs---e.g. in terms of Toffoli counts---presuming multiplication methods that can
be practically applied in a NISQ context (Sec.~\ref{sec:detailed}). We also improve upon the basic
exponentiation algorithm itself---in Sec.~\ref{sec:refined}, where we adopt
a more efficient and refined approach, and in Sec.~\ref{sec:detailed}, where we present
a specific, NISQ implementation.
At this point it is worthwhile to compare the two cases, $d=n$ and $d \ll n$. If the
exponentiation operation is itself part of a more complicated mathematical function network,
with many nested inputs and outputs, then presumably one wants a generic $d=n$ code with $n$
sufficiently large to provide ``machine precision''---i.e., $n\ge 25$ or so for single precision,
or $n \ge 50$ for double precision. The $O(n^2)$ space complexity of our basic exponentiation
algorithm likely places such calculations beyond the current NISQ frontier.
On the other hand, there are situations where $d \ll n$, and where $n$ itself may be
substantially reduced. For first quantized QCC, for example, it is estimated that $d=7$
and $n=21$ may suffice to achieve the so-called ``chemical accuracy'' benchmark \cite{mypccp}.
Such values place the present scheme much closer to the NISQ regime---especially
once the refinements of the next section are introduced.
We conclude this subsection with a reexamination of the true cost of the $d \ll n$
Gaussian function evaluation, within the present basic scheme. Though as stated,
the $x^2$ operation \emph{per se} adds little to the direct cost, it does have the
effect of squaring the size of the domain interval. Thus, if the full resolution
of the domain is to be preserved, this requires $2d$ rather than $d$ qubits---as well
as a commensurate doubling of the gate complexity. On the other hand,
this relative increase can often be largely mitigated by the improvements introduced in the
subsequent sections.
\section{\label{sec:refined} Refined Method}
The basic algorithm can be substantially improved, with respect to both space and gate complexity,
using the refinements described in this section. For definiteness, going forward
we shall generally presume the ``NISQ parameter values,'' $d=7$ and $n=21$, as discussed in
Sec.~\ref{subsec:cost}. However, for comparison and robustness testing, we shall occasionally
use the less spartan parameter values, $d=8$ and $n=32$ (corresponding to ``machine precision'').
In both cases, we find that a NISQ calculation is likely feasible in the near-term future.
There are essentially two distinct ideas presented in this section to improve upon the basic
algorithm---although other possible options certainly also exist. The first idea is to \emph{transform}
the $A_i$ values between successive multiplications, so that only one such constant need be stored
at a time. This will have the effect of reducing the space complexity scaling to $O(n)$, at least for
overwriting multiplications. The second idea reduces the actual number of multiplications that need be
applied.
\subsection{\label{subsec:ref1} Refinement \# 1: reducing space complexity}
The parameters $A_i $ have constant values that can be determined prior to the calculation.
Rather than storing them in $d$ separate registers, it is far less costly in terms of space
to simply transform $A_i \rightarrow A_{i+1}$, prior to each successive multiplication. Such strategies
have been used previously in quantum computing, when constant (unsuperposed) register values
are employed \cite{parent17,haner18,shor94}. If overwriting multiplications are used, it then
becomes necessary to maintain only two such $n$-qubit registers---i.e., one to store all of the
successive $A_i$ values, and the other to store the (conditionally superposed)
$C_i$ values.
The corresponding quantum circuit is presented in Fig.~\ref{fig2}, for $d=7$ and
$n=21$. The upper of the two 21-qubit registers is used to store the $A_i$, with the transformation
gate $X_{i(i+1)}^{\otimes p}$ used to transform $A_i$ into $A_{i+1}$. Similarly, we define
transformation gates $X_{i}^{\otimes p}$ to convert the zero state $0$ into $A_i$ (or vice-versa).
For example, the gate $X_{0}^{\otimes p}$ is used at the start of the circuit to initialize $A_0$ from 0.
Likewise, the lower 21-qubit register is initialized to $C$ from 0, using the transformation gate
$X_{C}^{\otimes p}$. Each successive multiplication operation (conditionally) multiplies this value by
another factor of $A_i$.
In this manner, the total number of qubits is reduced to $2n + d $, or 49 for the present NISQ
example---again, not including the various ancilla bits needed to effect the (overwriting) multiplications
in practice.
The strategy above emphasizes minimal space complexity at the cost of
greater gate depth. Alternatively, using all $d$ $A_i$ registers as in Sec.~\ref{sec:basic},
the multiplications could be performed synchronously and hierarchically, so as to minimize
gate depth, but without any space reduction.
In any event, our analysis in Sec.~\ref{sec:analysis} is all based on
\emph{non}-overwriting multiplications (Sec.~\ref{sec:detailed}), for which the situation is a bit more
complicated.
We next turn our attention to the implementation of the transformation gates.
Since the transformations always correspond to fixed input
and output values, they can easily be implemented as a set of very specific \textsc{NOT} gates,
applied to just those qubits for which the binary expansions of \eq{ybin} differ between
input and output values.
Hence the notation, `$X^{\otimes p}$', to refer to the resultant tensor product of $p \approx n/2$
\textsc{NOT} gates used to effect the transformation.
In Fig.~\ref{fig3}, the specific implementation for $X_{01}^{\otimes p}$ is presented,
corresponding to the specific values, $d=7$, $n=21$, and $A=0.389$. The input qubits
are in an unsuperposed state corresponding to the $n=21$ binary expansion of $A_0=A$,
as expressed in the form of \eq{ybin} (with $y_0$ corresponding to the top wire, etc.) The
output qubits are in a similar state, but corresponding to $A_1 = A^2 = 0.151321$. Generally
speaking, we may expect about half of the qubits to change their values. Indeed, for the
present example with $n=21$, we find $p=10$.
In the refined algorithm as presented in Fig.~\ref{fig2}, we find that there is one transformation
required per multiplication. However, it is clear from Fig.~\ref{fig3} that the gate complexity
of the former is trivial in comparison with that of the latter. In practical terms, therefore, the scheme
of Fig.~\ref{fig2} can be implemented at almost no additional cost beyond that of Fig.~\ref{fig1}---i.e.,
we can continue to use multiplication count as the measure of gate complexity.
\begin{figure}[h]
\centering
\includegraphics[height=14cm]{fig3}
\vskip-5pt
\caption{Quantum circuit used to implement $X_{01}^{\otimes p}$
on a quantum computer, for specific parameter values, $d=7$, $n=21$, and $A_0=A=0.389$.
The $n=21$ binary representation of $A_0$ is $A_0=.011000111001010110000$;
that of $A_1=A^2 = 0.151321$ is $A_1=0.001001101011110011111$. The least significant bit,
i.e. $j=0$, appears at the top of the circuit. For this example, $p=10 \approx n/2$.}
\label{fig3}
\end{figure}
\begin{figure*}[htb]
\centering
\includegraphics[height=9.5cm]{fig4}
\vskip-5pt
\caption{Second half of quantum circuit used to implement refined quantum algorithm for exponentiation,
$f(x') = \exp(-\alpha x')$, for specific parameter values, $d=7$, $n=21$, and $A=0.389$.}
\label{fig4}
\end{figure*}
\subsection{\label{subsec:ref2} Refinement \# 2: reducing gate complexity}
In the initial discussion that follows, it is convenient to reconsider the $d=n$ case.
Note that for both the basic quantum algorithm of Sec.~\ref{sec:basic}, and the refined
version of Sec.~\ref{subsec:ref1}, a total of $n$ multiplications are required---implying
an overall gate complexity that scales asymptotically as $O(n^{1+\log_2 3}) \approx O(n^{2.585})$,
if Karatsuba multiplication is used.
In reality, however, not all $n$ of the multiplications need be applied in practice. In
fact, it will be shown in this subsection that the actual required number of multiplications,
$m$, scales as $\log n$ (for fixed $A$)---thereby implying an asymptotic scaling of gate complexity
no worse than $O(n^{\log_2 3} \log n)$.
The important realization here is that \eq{Ai} implies a \emph{very rapid} reduction in
$A_i$ with increasing $i$---essentially, as the exponential of an exponential.
Consequently, there is no need to apply an explicit multiplication for any $A_i$
whose value is smaller than the smallest value that can be represented numerically
in our fixed-point representation---i.e.,
$2^{-n}$, according to \eq{ybin}. What is needed, therefore, is an expression for $m$
in terms of $A$ and $n$, where $m$ is the smallest $i$ such that $A_i < 2^{-n}$.
For the $d=n$ case, it can easily be shown that
\eb
m = \left \lfloor \log_2 \! \of{{n \over \log_2 (1/A)}} \right \rfloor + 1 . \label{meq}
\ee
For the generic case where $d$ and $n$ are independent, we still never need more
than $d$ multiplications. So \eq{meq} above gets replaced with the general form,
\eb
m = \min \left \{d, \left \lfloor \log_2 \! \of{{n \over \log_2 (1/A)}} \right \rfloor + 1 \right \}. \label{meq2}
\ee
Clearly, $m$ scales asymptotically as either $O(d)$ or $O(\log n)$, rather than $O(n)$, if $A$ is fixed.
This assumes, however, that $d$ and $A$ have no implicit dependence on $n$, which in turn depends
on assumptions about how the $x'$ grid points are increased. If the $x'_{\rm min} \le x' < x_{\rm max}$ domain
interval is expanded keeping the same spacing $\Delta$, \emph{or} if $\Delta$ decreases but $d$ is kept
constant, then the above holds. Otherwise, $A \rightarrow 1$ as $n \rightarrow \infty$, and the
prefactor becomes divergently large, implying a less favorable asymptotic scaling law.
Let us consider the case where $d < n$. Since $m(A)$ as described by \eq{meq} increases
monotonically with $A$, there is in general an interval $0 < A < A_{{\rm max}}$ over which
$m(A) < d$, and so a reduction in the number of multiplications can be realized and $m<d$. Beyond
this point---i.e., for $A_{{\rm max}} \le A < 1$, all $m=d$ basic multiplications must be used.
A bit of algebra reveals the following expression for the transition $A$ value:
\eb
A_{{\rm max}} = 2^{- n/2^{d-1} } \label{Amax}
\ee
As an illustrative example, consider the $d=7$, $n=21$, $A=0.389$ case of
Sec.~\ref{subsec:ref1}. The formula of \eq{meq} predicts that $m=4$ multiplications
will be required, exactly as indicated in Fig.~\ref{fig2}. This represents a significant
reduction versus the $d=7$ multiplications that would otherwise be needed. As
confirmation that $m=4$ is correct, we note that $A_3=0.00052432$, which is
larger than $2^{-21} = 4.768 10^{-7}$. However, $A_4 = 2.749 10^{-7} < 2^{-21}$.
Finally, we can compute $A_{{\rm max}}$ from \eq{Amax}---which, with the above $n$ and $d$
values, is found to be $A_{{\rm max}} = 0.796571$. Thus, one finds a reduction in $m$
down from $d$, over about 80\% of the range of possible $A$ values. Now consider the
\emph{Gaussian} rather than exponential function, for which $d \rightarrow 2d=14$. Here,
we find $A_{{\rm max}} = 0.998225$---implying that there is \emph{almost always}
a reduction in $m$. We will discuss further ramifications in Secs.~\ref{sec:detailed} and
~\ref{sec:analysis}.
\subsection{\label{subsec:second} Second half of refined quantum algorithm}
Although the second refinement of Sec.~\ref{subsec:ref2}, can lead to fewer than $d$
multiplications (depending on the values of $n$, $d$, and $A$), this does not
imply that the refined quantum algorithm simply ends at the right
edge of Fig.~\ref{fig2}. There remains a subsequent computation that must occur, using
the $x_{i\ge m}$ qubits, in order to ensure that the correct final value
for the function is obtained. The multiplication count of the additional computation
is zero, although it does add a cost of $n$ Fredkin gates.
Consider that when $x=2^i$ is a power of two, then all but the $x_i$ binary expansion
coefficients in \eq{xbin} vanish, and the function value becomes simply $f(x) = C A_i$,
according to \eq{fCAi}. This implies that for any $x \ge 2^m$, $f(x) < 2^{-n}$ is smaller than the
minimum non-zero number that can be represented, and so should be replaced with $f(x)=0$.
This situation will occur if \emph{any} of the $(d-m)$ qubits, $x_{i\ge m}$, are in their 1 states.
Otherwise---i.e., if all $(d-m)$ of the $x_{i\ge m}$ are in their 0 states so that $x < 2^m$---then
nothing should happen, as the lowest register is already set to the correct output value, $f(x)=C_m$,
at the right edge of Fig.~\ref{fig2}.
The above can be implemented as follows. First, for the case $A \ge A_{{\rm max}}$,
no additional circuitry is needed; one simply runs the quantum circuit of Fig.~\ref{fig2} as is, except
with explicit controlled multiplications across all $m=d$ of the $x_i$ qubits. For the case $m=d-1$,
then $d-1$ controlled mutiplications are implemented across the lowest $d-1$ qubits,
$x_{i < (d-1)}$. The final qubit, $x_{d-1}$ is then used to conditionally set the lowest
register to zero.
For the last case where $(d-m) \ge 2$, we apply the quantum circuit indicated in Fig.~\ref{fig4}.
This requires first checking if any of the $(d-m)$ $x_{i\ge m}$ qubits are in state 1,
which is implemented using a sequence of $(d-m-1)$ \textsc{OR} gates. The first
is applied to $x_m$ and $x_{m+1}$ to compute $x_m \vee x_{m+1}$. If needed, that
output is then sent to a second \textsc{OR} gate along with $x_{m+2}$, etc. The final
output, which will serve as a control qubit, has value 1 if any of the $x_{i\ge m}$
are in their 1 states; otherwise, it has value 0.
Meanwhile, the upper of the two $n$-qubit registers, which starts out representing
the constant value $A_{m-1}$, is transformed to the value 0, using the transformation
gate, $X_{m-1}^{\otimes p}$. Finally, the upper and lower $n$-qubit registers undergo
a controlled $\textsc{SWAP}^{\otimes n}$, applied in qubit-wise or tensor-product fashion,
across all $n$ qubits of the two registers. If the swap occurs, then the function output
as represented by the lower of the two $n$-qubit registers becomes zero;
otherwise, it is left alone.
We conclude this subsection with a discussion of the reversible quantum \textsc{OR} gate,
used in the quantum circuit of Fig.~\ref{fig4}. Such
a gate can be easily constructed from a single reversible \textsc{NAND} (Toffoli) gate,
together with various \textsc{NOT} gates, as indicated in Fig.~\ref{fig5}. Note
that each such \textsc{OR} gate introduces one new ancilla qubit, initialized to the 1 state.
There are thus no more than $(d-m-1)$ additional ancilla qubits in all that get introduced in
this fashion. The additional costs associated with Fig.~\ref{fig4}, in terms of both gates
and qubits, are thus both very small as compared to those of Fig.~\ref{fig2}, although
they will be included in resource calculations going forward.
\begin{figure}[h]
\centering
\includegraphics[height=1.8cm]{fig5}
\vskip-5pt
\caption{Quantum circuit used to implement reversible \textsc{OR} gate, constructed out of
a single reversible \textsc{NAND} (Toffoli) gate, and various \textsc{NOT} gates.}
\label{fig5}
\end{figure}
\section{\label{sec:detailed} Detailed Implementation Suitable for NISQ Computing}
\subsection{\label{subsec:overview} Overview}
As discussed, there is large variety of quantum multiplication algorithms
on the market currently
\cite{draper00,florio04,florio04b,haner17,haner18,sanders20,parent17,gidney19,karatsuba62,kowada06},
and no doubt many more will follow.
In part for this reason, we prefer to rely on the ``multiplication count'' metric for gate complexity. Indeed,
whereas current multiplication algorithms largely make use of integer or fixed-point
arithmetic, floating-point algorithms---which have very different implementations---are also
of interest going forward, especially for exponentiation. The multiplication
count metric will continue to be relevant for all such innovations.
On the other hand, we are also interested in developing a specific exponentiation circuit
that can be run on NISQ computers for realistic applications. Moreover, we aim
to compare performance against the state-of-the-art competing method by H\"aner and
coworkers \cite{haner18}, for which multiplications are not the only bottleneck.
This constrains us in two important ways.
First, we cannot use the multiplication count metric for accurate comparison; instead, since the
H\"aner algorithm is Toffoli-based (as is our circuit), we use the Toffoli count metric.
Second, to the extent that both exponentiation algorithms do
rely on multiplications, similar multiplication subroutines should be used for both.
Accordingly, we use a modified version of H\"aner's multiplication subroutine, which is itself a
fixed-point version of ``schoolbook'' integer multiplication \cite{haner17,haner18,sanders20,parent17,gidney19},
based on bit shifts and controlled additions. In particular, they exploit truncated additions (that
maintain $n$ fixed bits of precision), together with a highly efficient overwriting, controlled,
ripple-carry addition circuit by
Takahashi \cite{cuccaro04,takahashi,takahashi09,parent17} that minimizes both space and gate complexities.
As it happens, there are some further improvements and simplifications
that arise naturally for our particular exponentiation context, which we also exploit.
All of this is described in detail in Secs.~\ref{subsec:mult} and~\ref{sec:analysis},
wherein we also derive fairly accurate resource estimates for both qubit and Toffoli counts,
respectively.
One disadvantage of H\"aner multiplication is that it does not
overwrite the multiplier input---the way, e.g., that QFT multiplication would \cite{draper00,florio04,florio04b}.
Consequently, each successive multiplication requires additional ancilla bits, unlike what is
presumed in Fig.~\ref{fig2}. Space needs are accordingly greater in this implementation
than what is described in Sec.~\ref{subsec:ref1}---becoming essentially $m n +d$ qubits rather than
$2 n+d$ (without ancilla). For the NISQ applications of interest here, $m$ is still quite small, and so the
increase is generally not too onerous. It is more of a concern for the Gaussian
evaluations, for which $m$ can in principle get twice as large as the corresponding
exponential $d$ value.
Of course, it would be possible to employ QFT-based multiplication in our
exponentiation algorithm---which would require $4n +d$ qubits, with ancilla included.
On the other hand, the QFT approach is not Toffoli-based, and would therefore not lend
itself to direct comparison with H\"aner, vis-\`a-vis
gate complexity. In order to estimate a Toffoli count
for QFT multiplication, one would have to presume some specific implementation for the
Toffoli gate itself (e.g., in terms of \textsc{T} gates), which is not ideal \cite{parent17}.
In any event, Toffoli counts have become a standard gate complexity metric in quantum
computing.
For these reasons, overwriting QFT-based multiplications are not considered further here.
Instead, for cases where the increased space complexity of the non-overwriting multipliers
might pose a problem, we address this situation through the use of a simple alternative
algorithm, describe in Sec.~\ref{subsec:alternative}, that trades increased gate complexity for
reduced space complexity---essentially by uncomputing intermediate results.
In principle, there are any number of ``reversible pebbling strategies'' \cite{parent17,haner18,gidney19,bennett89}
that might also be applied towards this purpose. The particular approach adopted here, though,
is very simple, and appears to be quite effective.
\subsection{\label{subsec:mult} Non-overwriting controlled quantum mutiplication}
\begin{figure*}[htb]
\centering
\includegraphics[height=16cm]{fig6}
\vskip-5pt
\caption{Detailed Implementation of controlled multiplication on a quantum
computer, $\times_0$, for specific parameter values $d=7$ and $n=21$,
fixed multiplier, $A_0 = A = 0.389$, and arbitrary superposed multiplicand, $y$.
The binary representation of $A_0$ is
$A_0=.011000111001010110000$; each bit with value 1 is hard-wired
into the quantum circuit as a distinct controlled addition, $+$.
All operations are controlled by the single domain qubit, $x_0$.}
\label{fig6}
\end{figure*}
As discussed, non-overwriting controlled-addition multiplication subroutines
have three registers. The first is an input register for the multiplier; the second
is another input register for the multiplicand; the third is the output or ``accumulator''
register. The accumulator register is initialized to zero, and therefore serves as
an ancilla register, but
comes to store the product of the multiplier and multiplicand at the end of the calculation.
For integer multiplication, the accumulator register requires $2n$ qubits, assuming
that both input registers are $n$ qubits each. The first register (multiplier) provides the
the control qubits for a cascade of $n$ controlled
additions. The second register (multiplicand) serves as
the first input for each controlled addition. The second input for each controlled
addition is a successively bit-shifted subset of $n+1$ qubits from the accumulator register.
Note that \emph{overwriting} controlled additions are used, so that for each
controlled addition, the second register output is the sum of the two inputs.
In the case of our exponentiation algorithm, we propose a version of the above
basic scheme that is modified in two very important ways. First, the $i$'th multiplication
is \emph{controlled}, via the domain qubit $x_i$ (Fig.~\ref{fig2}). Second, the circuit
exploits the fact that every multiplier has a \emph{fixed} (unsuperposed) value---i.e. the
constant, $A_i$. Adding an overall control to a quantum circuit tends to complicate that circuit---turning
\textsc{CNOT} gates into \textsc{CCNOT} gates, etc. On the other hand, the fixed
multiplier enables substantial simplifications---of the type used in Shor's algorithm for
factoring integers \cite{parent17,shor94}, for instance.
Specifically, we no longer treat the multiplier $A_i$ as an input register---for there is
no longer a need to use the $A_i$ qubits as control bits for the additions.
Instead, the binary expansion of $A_i$ from \eq{ybin} is used to \emph{hard-wire}
what would be a set of \emph{uncontrolled} additions, directly into the quantum circuit---but
only for those binary expansion coefficients equal to 1. In addition to reducing the set of inputs
by one entire $n$-bit register, this modification also reduces the number of additions that must be
performed by a factor of two---since on average, only half of the expansion coefficients have
the value 1.
In addition to the above advantages, fixed-multiplier multiplication reduces circuit complexity
by replacing controlled with uncontrolled additions---effectively converting \textsc{CCNOT}
gates to \textsc{CNOT} gates. Of course, when the $x_i$ qubit control is
thrown back in, to create the requisite \emph{controlled} multiplication subroutine, we find that
the additions become controlled after all---but by $x_i$, rather than $A_i$. In effect, the
control bit for the multiplication is simply passed down to the individual controlled additions
which comprise it.
The above can all be seen in Fig.~\ref{fig6}, our detailed quantum circuit for controlled
multiplication, as implemented for the first multiplication in Fig.~\ref{fig2}, denoted `$\times_0$'
(i.e., multiplication by $A_0$, controlled by the $x_0$ qubit). Note that the individual
multiplication circuits, $\times_i$, differ from each other, due to
the different $A_i$ binary expansions.
Once again, our canonical NISQ parameter values are presumed, i.e.,
$d=7$, $n=21$, and $A=0.389$.
From the figure, another important difference from the basic scheme may be observed:
the accumulator register, $z$, has only $n$ rather than $2n$ qubits.
This is because fixed-point rather than integer arithmetic is being used---as a consequence of
which, it is not necessary to store what would otherwise be the $n$ least significant bits of
the product. This situation provides yet another benefit, which is that each controlled addition
becomes ``truncated'' to an $s$-bit operation---with $s$ increasing with each successive
controlled addition across the range, $2 \le s \le (n-1)$.
Note that the smallest possible addition corresponds to $s=2$ rather than $s=1$.
This is because the first controlled addition can be replaced with a cascade of Toffoli
gates---or controlled bit-copy operations---which is a much more efficient implementation.
This substitution works because the accumulator
register $z$ is set to zero initially. The very first controlled addition thus always (conditionally) adds the
multiplicand register $y$ to zero.
For the example in the figure, the first four binary expansion coefficients for $A$ (from right to left)
are all zero; these bits are simply ignored. The first coefficient equal to one is the $j=4$ or fifth bit. As
indicated in Fig.~\ref{fig6}, this causes the last four bits of the multiplicand register $y$
to be (conditionally) copied into the first four bits of the accumulator register---in what would otherwise
be an $s=4$ controlled addition.
The $j=5$ bit is also equal to one, leading to the first \emph{bona fide} controlled addition
in Fig.~\ref{fig6}, with $s=5$. This pattern continues until the the next-to-last, or $j=19$ bit is reached,
which is the last bit equal to one. This leads to the final controlled addition, with $s=19$.
Although the last ($j=20$) bit is zero, even if it were equal to one,
the corresponding controlled addition gate would extend only up to $y_1$. Thus, the top wire, $y_0$,
or least-significant bit of the multiplicand, is never used. This reflects the fact that both numbers
being multiplied have values less than one, and that $n$ fixed bits of precision are maintained
throughout the calculation. Note also that, as a result, there are never any overflow errors.
The final part of the controlled multiplication circuit is a cascade of $n$ controlled bit-copy operations
(i.e., modified Toffoli gates), which conditionally set the final output of the accumulator register equal to $y$,
when $x_0=0$ (hence the open circles). Otherwise, this register would remain zero. Thus, the ``do nothing''
instruction in Sec.~\ref{subsec:basicquantum} does not literally mean ``do nothing'' when non-overwriting
multiplications are used, as it is still necessary to copy the multiplicand input register to the accumulator output
register.
\subsection{\label{subsec:alternative} Quantum algorithm for exponentiation: space saving alternative}
Now that the specific, controlled quantum multiplication algorithm of Sec.~\ref{subsec:mult} has been
identified, we can determine a precise estimate of space requirements for
our overall exponentiation circuit. (Gate complexity will be discussed in Sec.~\ref{subsec:ours}).
As noted, each of the $m$ multiplications requires a clean $n$-qubit ancilla bundle as input for its
accumulator register, together with the (accumulator) output from the most recent multiplication
as input for its multiplicand register. Thus, for $m$ successive multiplications,
$m+1$ separate registers would be required in all.
However, we can realize significant savings---i.e., one entire register of space, and one
entire controlled multiplication subroutine---by exploiting the fact that the first multiplicand (i.e., $C$)
is a fixed constant. The first controlled multiplication, $\times_0$, is therefore a controlled multiplication of the constant
$C$ by the constant $A_0$. Since both constants are fixed, the controlled multiplication can be
much more efficiently realized using two controlled transformation gates acting on a single
register (i.e., the first two gates shown in Fig.~\ref{fig7}) rather than the controlled multiplication
circuit of Fig.~\ref{fig6}. Note that this controlled $\times_0$ implementation uses only \textsc{CNOT}
gates; thus the Toffoli count is zero.
Since Takahashi addition does not use additional ancilla qubits \cite{takahashi,takahashi09},
the total number of qubits required to implement the $m$ multiplications is just $mn$. In addition to this,
we have the $d$ qubits needed to store the domain register, $x$, that is used to supply the control qubits.
The current qubit count is thus $mn +d$.
However, if $m<d$, then the second half of the refined exponentiation circuit (i.e., Fig.~\ref{fig4}) must
also be executed, which introduces some additional space overhead. To begin with, our current reliance
on non-overwriting multiplications implies that we can no longer generate the requisite zero
ancilla register (i.e., the next-to-last register in the figure) without significant (un)computation.
To avoid this, we instead add a clean new register---at the additional cost of $n$ new qubits.
In addition to this, there are the $(d-m-1)$ ancilla bits used by the \textsc{OR}
gates, as discussed in Sec.~\ref{subsec:second}). Altogether then, the total qubit count becomes:
\eb
q = \left \{ \begin{array}{ll}
dn+d & \mbox{for $m=d$} \\
(m+1)n + 2d -m-1 & \mbox{for $m<d$}
\end{array} \right .
\label{qcost}
\ee
To reduce qubit counts in cases where \eq{qcost} renders a NISQ calculation unfeasible, we have developed
a ``space-saving'' alternative algorithm. The general idea
is to uncompute some of the intermediate quantities, in order to restore some of the ancilla registers to their
initial clean state, so that they can then be reused for subsequent computations. Of course, this requires
additional overhead---i.e., in our case, additional controlled multiplications.
More specifically, our space-saving algorithm reduces space requirements from $O(mn)$ down to
$O(m^{1/2} n)$---a very marked reduction, especially if $m$ is reasonably large. The added cost
in terms of gate complexity, on the other hand, is \emph{never more than double} that of our original
algorithm described above. Thus, $m < m_{{\rm ss}}< 2 m$, with $m_{{\rm ss}}$ the multiplication count for the
space-saving approach.
For values of $m$ that lie in the range
\eb
r(r-1)/2 < m \le r(r+1)/2 \label{rcount}
\ee
(where $r>2$ is an integer), the space-saving method requires a total of $r$ $n$-qubit registers to perform
all multiplications. Note that the $r>2$ restriction implies that the method is only applicable for
$m>3$. However, the $m\le3$ case presents minimal space requirements, and so the space-saving
approach is less likely to be needed. In any event, for all numerical examples considered in Sec.~\ref{subsec:Haner},
(including the worst Gaussian cases), $ 3 \le r \le 5$.
The total qubit count for the space-saving alternative algorithm can be shown to be as follows:
\eb
q_{{\rm ss}} = \left \{ \begin{array}{ll}
r n + d & \mbox{for $m=d$} \\
(r+1)n + 2d -m-1 & \mbox{for $m=r(r+1)/2<d$} \\
rn + 2d -m-1 & \mbox{otherwise}
\end{array} \right . \label{qsscost}
\ee
Note that unlike our original non-space-saving or ``gate-saving'' algorithm,
a zero $n$-qubit ancilla register can always be made
available for the final controlled $\textsc{SWAP}^{\otimes n}$ operation of Fig.~\ref{fig4}---\emph{except}
when $m=r(r+1)/2$, which thus has an additional qubit cost. (See technical note at the
end of this subsection).
The space-saving algorithm itself proceeds as follows.
First, apply the first $r$ multiplications, exactly as for the earlier gate-saving
algorithm. This leaves the $r$ registers in the states, $C_1$, $C_2$, \ldots, $C_r$.
Then, uncompute all but the most recent multiplication (i.e., the one that provided $C_r$). The first $(r-1)$ registers
are thereby restored to zero, but the final register remains in the $C_r$ state. It is then possible to perform
$(r-1)$ additional multiplications, before once again running out of registers. All but the last of these is
then uncomputed, allowing $(r-2)$ more multiplications to be performed, and so on.
The space-saving quantum circuit used for $d=7$ and $m=$4--6 is presented in Fig.~\ref{fig7},
corresponding to $r=3$ registers.
For the first wave, there are three clean registers, allowing for three successive multiplications,
$\times_0$, $\times_1$, and $\times_2$ (provided $\times_0$ is implemented as discussed above).
This is followed by two uncompute multiplications for the first two multiplications, denoted $\times_1^{-1}$ and
$\times_0^{-1}$ (the latter, again with the new implementation).
%
In the second wave, we apply $\times_3$ and $\times_4$, generating $C_4$ and $C_5$,
respectively. This suffices for $m=4$ and $m=5$, respectively. However, if $m=6$, we
must undergo a third and final wave, as indicated in the figure.
As is clear from the above description, and from Fig.~\ref{fig7}, the number of uncompute
multiplications, $m_{{\rm un}}$ , is always less than $ m$. Thus, $m_{{\rm ss}} = m + m_{{\rm un}} < 2m$,
as claimed.
Precise values can be found as follows. Let $l$ be the largest integer such that
\eb
l(l+1)/2 \le r(r+1)/2 - m. \label{lcond}
\ee
Then,
\eb
m_{{\rm un}} = r(r-1)/2 - l(l+1)/2.
\ee
Table~\ref{mtab} indicates specific values for all $m\le36$. Note that the $m_{{\rm ss}}$ multiplication
count includes
both $\times_0$ and $\times_0^{-1}$; thus, the total \emph{actual} number of controlled multiplication
subroutines that must be executed is $(m_{{\rm ss}}-2)$, as indicated in the final column. From the table,
also, it may be observed that greater space savings are usually associated with increased
multiplication counts, and vice-versa.
\emph{Technical note:}
For $m < d$ space-saving calculations, a zero ancilla register is automatically available
at the end of the Fig.~\ref{fig7} circuit (to be used in the subsequent Fig.~\ref{fig4} circuit),
whenever \eq{lcond} is a true inequality. When \eq{lcond} is an \emph{equality}, then the $l=0$ case
requires the addition of a new zero ancilla register (as discussed), but for $l>0$, a zero register can be easily
created from an existing non-zero register. This is done by applying the single uncompute multiplication,
$\times_{m-2}^{-1}$. As an example, the case $m=5$ corresponds to $l=1$ and $r=3$,
thus satisfying \eq{lcond} as an equality, with both sides equal to one. The necessary
uncompute multiplication can be seen in Fig.~\ref{fig7}, just to the right of the vertical dashed line
marked `$m=5$'. Note that the Toffoli count associated with such $i \approx m$ multiplications is
greatly reduced in comparison with the other multiplications, as will be discussed in Sec.~\ref{subsec:ours}.
Moreover, this event is fairly rarely realized in practice, including the examples given in the present work.
Nevertheless, the small additional cost required in such cases is included in the Toffoli count formulas presented
in Sec.~\ref{subsec:ours}.
\begin{table}
\centering
\caption{Number of $n$-qubit registers $r$, uncompute multiplications $m_{{\rm un}}$ (including
$\times_0^{-1}$), and total actual multiplications $(m_{{\rm ss}}-2)$, as a function of number of
compute multiplications $m$ (including $\times_0$), for the
space saving alternative quantum exponentiation algorithm of Sec.~\ref{subsec:alternative} .}
\begin{tabular}{ p{1cm} p{1.25cm} p{1.25cm} p{1.3cm} p{1.25cm} p{1.25cm}}
\hline
\hline
$m$ & $r$ & $m_{{\rm un}}$ & $(m_{{\rm ss}}-2)$ \\
\hline
4 & 3 & 2 & 4 \\
5 & 3 & 2 & 5 \\
6 & 3 & 3 & 7 \\
7 & 4 & 3 & 8 \\
8 & 4 & 5 & 11 \\
9 & 4 & 5 & 12 \\
10 & 4 & 6 & 14 \\
11 & 5 & 7 & 16 \\
12 & 5 & 7 & 17 \\
13 & 5 & 9 & 20 \\
14 & 5 & 9 & 21 \\
15 & 5 & 10 & 23 \\
16 & 6 & 12 & 26 \\
17 & 6 & 12 & 27 \\
18 & 6 & 12 & 28 \\
19 & 6 & 14 & 31 \\
20 & 6 & 14 & 32 \\
21 & 6 & 15 & 34 \\
22 & 7 & 15 & 35 \\
23 & 7 & 18 & 39 \\
24 & 7 & 18 & 40 \\
25 & 7 & 18 & 41 \\
26 & 7 & 20 & 44 \\
27 & 7 & 20 & 45 \\
28 & 7 & 21 & 47 \\
29 & 8 & 22 & 49 \\
30 & 8 & 22 & 50 \\
31 & 8 & 25 & 54 \\
32 & 8 & 25 & 55 \\
33 & 8 & 25 & 56 \\
34 & 8 & 27 & 59 \\
35 & 8 & 27 & 60 \\
36 & 8 & 28 & 62 \\
\hline
\end{tabular}
\label{mtab}\\
\end{table}
\begin{figure*}[htb]
\centering
\includegraphics[height=17cm]{fig7}
\vskip-5pt
\caption{Quantum circuit used to implement space-saving alternative quantum algorithm for exponentiation,
$f(x') = \exp(-\alpha x')$, for specific parameter values, $d=7$ and $m=$4--6, corresponding to $r=3$.
Replaces first half of earlier refined circuit, i.e. Fig. 2. Bold face and dashed vertical lines indicate
final circuit outputs for $m=4$, 5, and 6, respectively. All controlled multiplications are
implemented via the overwriting quantum circuit of Fig. 6.}
\label{fig7}
\end{figure*}
\section{\label{sec:analysis} Analysis: Toffoli counts}
\subsection{\label{subsec:ours} Present methods}
To a rough approximation, the total Toffoli count for the proposed exponentiation algorithm
[or for the space-saving alternative] is simply $(m-1)$ [or $(m_{{\rm ss}}-2)$]
times the Toffoli count needed to execute a single controlled multiplication subroutine.
Before working out the latter, however, we first describe another reduction of effort that in
practical terms, converts the total cost to that of only $(m-8/3)$ [or $(m_{{\rm ss}}-16/3)$]
controlled multiplications. This additional savings is fully realized
whenever $m<d$, which in practice occurs much of the time, if the domain interval is
realistically large.
The rationale is as follows. When $m <d$, the $A_i$ values \ span the
entire range from 1 down to $2^{-n}$. This implies that for $i \approx m$,
the corresponding $A_i$ have many leading zeroes. Consequently, these later
multiplications can be performed using fewer than $n$ binary digits, leading to
significant computational savings.
Note that the \emph{worst-case} scenario vis-\`a-vis the aforementioned savings---i.e.,
that for which the $A_{i\approx m}$ have the \emph{fewest} leading zeros---corresponds to
$A_m \rightarrow 2^{-n}$ from below.
Now, in general, the approximate number of leading zeros for the binary expansion
of $0 < y <1$ is given by $-\log_2 y$. Thus, for $y=A_m$, we find $\sim \!\!n$ leading
zeros, as expected. More generally, \eq{Ai} in the worst-case scenario leads to
\eb
\mbox{leading zeros}(A_i) \approx n\, 2^{-(m-i)} = n \, 2^{-k},
\ee
where $k = (m-i)$.
The number of binary digits needed for the $A_{m-k}$ controlled multiplication is thus
$n_k = n - n \, 2^{-k}$.
Going forward, we shall for simplicity presume the asymptotic limit, $n \rightarrow \infty$.
In this limit, the Toffoli count per multiplication scales as $O(n_k^2)$. The Toffoli
\emph{savings} (i.e., reduction in the Toffoli count relative to multiplication with
$n$ digits) is therefore
\eb
s_k \propto (n^2 - n_k^2) = n^2 (2 \, 2^{-k} - 2^{-2k}) \label{savings}
\ee
Summing \eq{savings} from $k=1$ to $\infty$ then yields a total savings of $5/3$
multiplications.
In practice---i.e., for finite $n$---the series is truncated, and so the actual
savings is less than $5/3$ multiplications. In the worst case (of the worst case),
only the $s_1$ term contributes to the sum, resulting in a lower bound of $3/4$
multiplications. On the other hand, a small increase in $A_m$, such that the
new value is slightly greater than $2^{-n}$, will increment the value of $m$---thus,
effectively increasing the savings by one whole additional multiplication. On
balance, we therefore take our $5/3$
``best case of the worst case'' value as a reasonable middle-ground estimate.
Next, we move on to a calculation of the Toffoli cost of each controlled multiplication.
As discussed in Sec.~\ref{subsec:mult}, these are implemented using a sequence
of controlled additions, with from $s=2$ qubits up to $s=(n-1)$ qubits. Note that
the Toffoli cost of the highly efficient overwriting, controlled, ripple-carry addition
circuit of Takahashi \cite{takahashi,takahashi09,haner18} with $s$
qubits is $3s+3$. Thus, if \emph{every} $2\le s \le (n-1)$ required a controlled
addition, the total contribution to the Toffoli cost of multiplication would be
$(3/2)n^2 + (3/2) n -9$. However, since only \emph{half} of these multiplications
are realized on average, in practice, the actual cost per multiplication is half of this.
The total contribution to the cost of the exponentiation circuit is then this value, multiplied
by the \emph{effective} number of multiplications, i.e. $(m-8/3)$.
Now, in addition, each controlled multiplication in the Fig.~\ref{fig6} circuit also
begins and ends with a cascade of additional Toffoli gates. The initial cascade
can be easily shown to consist of two Toffoli gates, on average. The final cascade,
is always $n$ Toffoli gates, even when fewer than $n$ qubits are needed to execute
the main part of the multiplication circuit (i.e., the controlled additions). Note that
both Toffoli cascades are required in every \emph{actual} multiplication. The
total contribution to the Toffoli cost of the exponentiation circuit is thus $(m-1)(n+2)$.
Finally, there are the additional costs associated with the second half of the (refined)
exponentiation circuit, as presented in Fig.~\ref{fig4}, presuming $m <d$. Since
each Fredkin gate can be implemented using a single Toffoli gate, the Toffoli cost of the
final $\textsc{SWAP}^{\otimes n}$ operation is $n$. Likewise, each \textsc{OR} gate
requires one Toffoli gate, for a total Toffoli count of $(d-m-1)$. Altogether, we
wind up with the following expression for the total Toffoli cost for the entire
gate-saving exponentiation circuit:
\eb
T = \left \{ \begin{array}{ll}
\of{{3d \over 4}\!-\!2} n^2 + \of{{7d \over 4}\!-\!3} n - {5 \over 2} d + 10 & \mbox{for $m=d$} \\ [2mm]
\of{{3m \over 4}\!-\!2} n^2 + \of{{7m \over 4}\!-\!2} n - {7 \over 2} m + d +9 & \mbox{for $m<d$}
\end{array} \right .
\label{Tcost}
\ee
Things are a bit more complicated in the space-saving algorithm case. In particular, there are
three cases instead of two. In addition to $m=d$, there are two different $m<d$ cases, i.e. one
corresponding \eq{lcond} being an equality, and one to the inequality case, as discussed in the
Technical note at the end of Sec.~\ref{subsec:alternative}. Note also that the uncompute
multiplications that are \emph{not} included in $m_{{ \rm un}}$ (as compared to $m$) are in
fact the $i \approx m$ multiplications, that do not cost as much. Consequently, the effective
number of uncompute multiplications is reduced relative to the actual number, by an amount
\emph{less than} 5/3 multiplications. A more accurate estimate of the uncompute savings is given by
\eb
S_{\Delta m} = \sum_{k=1+\Delta m}^{\infty} s_k\! = \!n^2\of{ 2\, 2^{-\Delta m} - 2^{-2 \Delta m}/3},
\ee
where $\Delta m = (m - m_{{\rm un}})$.
Taking all of the above into account, we obtain the following expression for the Toffoli count of
the space-saving exponentiation algorithm:
\eb
T_{{\rm ss}} = \left \{ \begin{array}{ll}
\of{m_{{\rm ss}}\! -\! {11 \over 3} -S_{\Delta m}}\of{{3 \over 4}n^2 \!+ \!{3 \over 4} n \!-\! {9 \over 2}}
+ (m_{{\rm ss}} \!-\!2)(n\!+\!2) \\
\hskip 6.0cm \mbox{for $m=d$} \\ [2mm]
\mexp{\mbox{above}} + n+d-m-1 \hskip 2.2cm \mbox{for $m<d $} \\ [4mm]
\mexp{\mbox{above}} + {9 \over 16}\of{{3 \over 4}n^2 \!+ \!{3 \over 4} n \!-\! {9 \over 2}} + (n\!+\!2) \\ [0.5mm]
\hskip 0.1cm \mbox{for $m\!<\!d$ and $l\!>\!0$ and $l(l\!+\!1)/2 = r(r\!+\!1)/2\! -\! m$} \\
\end{array} \right .
\label{Tsscost}
\ee
Note that in the final case above, the (worst-case) cost of the additional $\times_{m-2}^{-1}$
uncompute multiplication is obtained from $s_2$ in \eq{savings} to be $9/16$ that of a regular
multiplication---at least insofar as the controlled addition contribution is concerned.
\subsection{\label{subsec:Haner} Explicit numerical comparison with H\"aner approach}
In the approach by H\"aner et al. \cite{haner18}, arbitrary functions are evaluated via a decomposition
of the $x'$ domain into $M$ non-intersecting subdomain intervals, as discussed in Sec. ~\ref{sec:Intro}.
A given function is then approximated
using a separate $d$'th order polynomial in each subdomain. Both the polynomial coefficients,
and the subdomain intervals themselves, are optimized for a given target accuracy, using the
Remez algorithm \cite{remez34}.
Once the optimized parameters have been determined
for a given function $f(x')$, domain interval $x'_{{\rm min}} \le x' <x'_{{\rm max}}$, and (H\"aner) $d$ value,
the quantum algorithm is then implemented as follows.
First, polynomials are evaluated using a sequence of $d$ multiplication-accumulation
(addition) operations. On a quantum platform, these can be performed in parallel, across
all $M$ subdomains at once, using conditioned determination of the coefficients for each
subdomain. The multiplication count
would thus be $d$, \emph{irrespective} of $M$. Also, since non-overwriting multiplications are used,
the qubit count is $O(nd)$.
Note that generally speaking, lower $d$ corresponds to greater $M$, and
vice-versa. Thus, were the above multiplication-accumulation operations the only significant computational cost,
one would simply choose a very small value such as $d=1$. However, there is additional
space and gate complexity overhead associated with managing and assigning the $M$ sets of polynomial
coefficients. These costs do increase with $M$ (although in a manner that is naturally
measured in Toffoli gates rather than multiplications). There is thus a competition between $M$ and $d$,
with minimal Toffoli counts resulting when $d=4$ or 5---at least for the numerical examples from
\Reff{haner18} that are considered here.
The minimal-Toffoli choice of $d$ can thus be thought of as a ``gate-saving'' H\"aner
implementation.
Note that a rudimentary ``space-saving'' alternative may also be obtained, simply by reducing the value of $d$. That
said, \Reff{haner18} also discusses the use of much more sophisticated pebbling strategies.
However, such strategies are not actually implemented for
the numerical results presented in \Reff{haner18} that are used for comparison with the present
results.
Instead, H\"aner and coworkers perform calculations for different functions, and for different
target accuracies, across a range of different $d$ values---providing total qubit and Toffoli
counts for each. In particular, they consider both the exponential and Gaussian functions,
with $x'_{{\rm min}}=0$ and $\alpha=1$. It thus becomes mostly possible to provide a direct
comparison between the H\"aner approach and our
methods, with respect to these metrics. Such a comparison is provided in Table~\ref{restab}.
One slight difficulty arises from the fact that \emph{no $x'_{{\rm max}}$ value is provided in \Reff{haner18}};
moreover, the authors have not been available for clarification.
We thus present results for our methods using two very different $x'_{{\rm max}}$ values---i.e., 10
and 100. Although for most purposes, even the former interval is wide enough to capture the main
function features, the latter interval is actually more realistic for certain applications such as QCC
(as discussed in greater detail in Sec.~\ref{sec:conclusions}). In any case, it should be mentioned
that in the large $x'_{{\rm max}}$ limit, our methods become less expensive, whereas the H\"aner approach
becomes more expensive---owing to the increased $M$ values needed to achieve a given
level of accuracy.
The issue of target range accuracy merits further discussion. In H\"aner, calculations were performed
to an accuracy of $10^{-7}$, and also $10^{-9}$. The corresponding number of bits needed to resolve the
range to these thresholds are 23.25 and 29.89, respectively. Note that these values are quite close to the $n=21$
and $n=32$ values considered in our examples thus far---in one case a bit high, in the other a bit low. Of
course, a few extra bits might also be needed to compensate for round-off error in the fixed-point arithmetic.
These are relatively small effects however; in particular, they are likely no larger than those associated with the unknown
H\"aner $x_{{\rm max}}$ value.
For our purposes, therefore, we take these $n$ values as reasonable
estimates, at least for the exponential function evaluations. For the Gaussian case, we do go ahead and use
$n=24$ and $n=30$, as the closest integers larger than the threshold values listed above. As for the
corresponding $d$ values, roughly speaking, these would be double those from the exponential
calculation---except that we exploit symmetry of the domain to reduce these by one bit each.
Thus, $d=13 = 2 \times7 -1 $ and $d= 15=2 \times 8 -1 $, respectively, for $n=24$ and $n=30$.
In Table~\ref{restab}, we present Toffoli and qubit counts for both function evaluations (i.e., exponential and Gaussian),
for both target accuracy thresholds (i.e. $10^{-7}$ and $10^{-9}$), for both of our methods (i.e. gate-saving
and space-saving), and for both domain intervals (i.e. $0 \le x' < 10$ and $0 \le x' < 100$). For each
function, the minimal Toffoli count is given in bold face. For each function and accuracy
threshold, we also present results for the full set of H\"aner calculations, as obtained
from \Reff{haner18}. Here too, the minimal Toffoli count is highlighted in bold face.
In all cases, our method requires \emph{far} fewer Toffoli gates than the H\"aner approach.
In comparing minimal-Toffoli calculations for the exponential function, the Toffoli count is reduced from
15,690 using H\"aner, down to just 912 using our approach. For Gaussian function evaluation, the
Toffoli count comparison
is even more stark---i.e., 19,090 vs. just 704. Generally speaking, our methods also require fewer qubits
than H\"aner. This is especially true for the space-saving alternative, which in one instance requires
as few as 71 qubits (and 1409 Toffoli gates)---a NISQ calculation, certainly, by any standard.
\begin{table*}
\centering
\caption{Toffoli and qubit counts for exponential [$\exp (-x')$] and Gaussian [$\exp (-{x'}^2)$] function
evaluations, using three different methods: gate-saving (ours); space-saving (ours); H\"aner.
For our methods, two different domain intervals are used: $0 \le x' < 10$ and $0 \le x' < 100$.
Two different target accuracies are considered:
$10^{-7}$ (Columns IV--VI) and $10^{-9}$ (Columns VII--IX).
For the former, bold face indicates minimal Toffoli count from among a given set of calculations,
i.e. ours vs. H\"aner.}
\begin{tabular}{ccc|rrr|rrr}
\hline
\hline
Function & Method & Domain & \multicolumn{3}{|c}{$10^{-7}$ Accuracy} & \multicolumn{3}{|c}{$10^{-9}$ Accuracy} \\
\cline{4-9}
& & interval & $\quad (n,d,m)$ & $\quad$ Toffolis & $\quad$ qubits & $\quad(n,d,m)$ & $\quad$ Toffolis & $\quad$ qubits \\
\hline
$\exp(-x')$ & $\quad$ gate saving $\quad$ & $0\le x' \le 10$ & $(21,7,7)$ & 1620 & 154 & $(32,8,8)$ & 4438 & 264 \\
& & $\quad 0\le x' \le 100\quad$ & $(21,7,5)$ & {\bf 912} & 134 & $(32,8,6)$ & 2828 & 233 \\
& space saving & $0\le x' \le 10$ &$(21,7,7)$ & 2308 & 91 & $(32,8,8)$ & 7531 & 136 \\
& & $\quad 0\le x' \le 100\quad$ & $(21,7,5)$ & 1409 & 71 & $(32,8,6)$ & 4278 & 105 \\
& H\"aner & & & 17304 & 149 & & 45012 & 175 \\
& & & & {\bf 15690} & 184 & & 28302 & 216 \\
& & & & 16956 & 220 & & 25721 & 257 \\
& & & & 18662 & 255 & & 26452 & 298 \\
\hline
$\exp(-{x'}^2)$ & $\quad$ gate saving $\quad$ & $0\le x' \le 10$ &$\quad (24,13,12)$ & 4468 & 325 & $\quad(30,15,14)$ & 8300 & 465 \\
& & $\quad 0\le x' \le 100\quad$ & $(24,13,4)$ & {\bf 704} & 141 & $(30,15,7)$ & 3232 & 262 \\
& space saving & $0\le x' \le 10$ &$(24,13,12)$ & 7546 & 133 & $(30,15,14)$ & 14479 & 165 \\
& & $\quad 0\le x' \le 100\quad$ & $(24,13,4)$ & 962 & 93 & $(30,15,7)$ & 5018 & 142 \\
& H\"aner & & & 20504 & 161 & & 49032 & 187 \\
& & & & {\bf 19090} & 199 & & 32305 & 231 \\
& & & & 21180 & 238 & & 30234 & 275 \\
& & & & 23254 & 276 & & 31595 & 319 \\
\hline
\end{tabular}
\label{restab}\\
\end{table*}
\section{\label{sec:conclusions}Summary and Conclusions}
After the various refinements and NISQ-oriented details as presented
in the latter 2/3 of this paper, it might be easy to lose sight of the main point,
which is simply this: \emph{the method presented here allows the exponential
function to be evaluated on quantum computers for the cost of a few multiplications.}
This basic conclusion will continue to hold true, regardless of the many quantum
hardware and software innovations that will come on the scene in ensuing
decades.
In particular, there is a plethora of multiplication algorithms available, both
overwriting (e.g. QFT-based) and non-overwriting (e.g. controlled addition),
and for both integer and fixed-point arithmetic---with new strategies for floating-point arithmetic,
quantum error correction, etc., an area of ongoing development. Given
this milieu, we propose the implementation-independent ``multiplication count''
as the most sensible gate complexity metric, for any quantum algorithm whose
dominant cost can be expressed in terms of multiplications. The present algorithms
are certainly of this type.
For our exponentiation strategy, the (controlled) multiplication count $m$ will
indeed be rather small in practice---at least for the applications envisioned.
To begin with, in a great many simulation contexts,
the domain resolution as expressed in total qubits $d$, is far less than the range
resolution $n$---with $m \le d$. For QCC, for instance, the $d=7$ and $d=8$ values
considered throughout this work are likely to suffice in practice \cite{mypccp}.
Conversely, we also consider the asymptotically large $n=d$ limit, in which it can
be shown [in \eq{meq}] that $m=O(\log n)$ for fixed $A$. In this limit, Karatsuba
multiplication provides better asymptotic scaling. Using the Gidney implementation,
the Toffoli and qubit counts for exponentiation scale as $O(n^{\log_2 3} m)$ and
$O(nm)$, respectively.
In the latter part of this paper, we present two specific, NISQ
implementations of our general exponentiation strategy, in order that detailed resource
estimates can be assessed, and compared with competing methods. When compared
with the method of H\"aner and coworkers, our implementations are found to
reduce Toffoli counts by an order of magnitude or more. Qubit
counts are also (generally) substantially reduced. Note that our two
implementations are complementary, with one designed to favor gate and the other
space resource needs. Together, they may provide the flexibility needed to actually
implement exponentiation on NISQ architectures---which
could serve as the focus of a future project.
Finally, we assess the present exponentiation algorithms within the context in which
they were originally conceived---i.e., quantum computational chemistry (QCC).
The long-awaited ``(QCC) revolution''\cite{poplavskii75,feynman82,lloyd96,abrams97,zalka98,lidar99,abrams99,nielsen,aspuru05,kassal08,whitfield11,brown10,christiansen12,georgescu14,kais,huh15,babbush15,kivlichan17,babbush17,babbush18,babbush18b,babbush19,low19,kivlichan19,izmaylov19,parrish19,altman19,cao19,alexeev19,bauer20,aspuru20}''
may be nearly upon us, although achieving full quantum supremacy will likely require quantum
platforms that can accommodate first-quantized methods. On classical computers, the
Cartesian-component separated (CCS) approach, as developed by the
author \cite{jerke15,jerke18,jerke19,mypccp,bittner}
offers a highly competitive first-quantized strategy.
On quantum computers, the question
appears to boil down to the relative costs of the exponential function vs. the inverse
square root \cite{babbushcomm}. Toffoli count estimates for the former appear in Table~\ref{restab}.
Note that the larger-domain-interval calculations---i.e., those with lower Toffoli counts---are the
more realistic in this context. This is because the QCC CCS
implementation requires multiple exponentiations with different $\alpha$ values to be performed,
across the same grid domain interval---which must accordingly be large enough to
accommodate all of them. Our exponentiation cost of 704 Toffoli gates should thus be
compared to the cost of the inverse-square-root function, which---again, according to the highly optimized
method of H\"aner and coworkers---is estimated to be 134,302 Toffoli gates.
\section*{\label{sec:acknw}Acknowledgement}
The author gratefully acknowledges support from a grant from the Robert A. Welch Foundation (D-1523).
|
1603.05722
|
\section{Introduction}
In recent years, computational electrocardiology has attracted the attention of mathematicians, engineers and clinicians. Computational methods have been continuously refined to match clinical applications.
Just to mention one example,the ideal lesion pattern with minimal burn in a cardiac ablation therapy has been studied in \cite{Krueger2013} by electrocardiological modeling.
However, to provide reliable and efficient simulations of cardiac electrical activity is not easy. Differential models in electrocardiology depend on several parameters typically coming from empirical constitutive laws, so their quantification for a specific patient is difficult.
In particular, these models
are strongly sensitive to the cardiac conductivity parameter \cite{johnston2011sensitivity}. While experimental data generally disagree on the values of conductivities, mathematically sound estimation methods---generally based on the solution of an inverse problem---have been considered only recently in \cite{graham2010estimation} and in our previous work \cite{HH2015IP}.
In addition, electrocardiological modeling for clinical application is computationally intensive. This is even more true for the inverse conductivity problem as we carried out in \cite{HH2015IP}, since high computational cost arises in many ``queries'' of forward simulations with different conductivity guesses.
Model-order-reduction techniques have been investigated in the literature, but their application to cardiac conductivity estimation is challenging, due to the nonlinearity of the models and the specific features like wave-front propagation of the solutions. In this paper, we apply model reduction techniques to dramatically decrease the computational cost of solving the inverse conductivity problem.
A numerical solution to the forward problem by Galerkin projection can be represented, in general terms, as an expansion ${u_h}(\boldsymbol{x},t)=\sum\limits_{j=1}^{n} u_j(t)\phi_j(\boldsymbol{x})$. In finite elements, the basis $\{ \phi_j \}_{j=1}^n$ is selected to be a set of piecewise polynomials. This basis is of ``general purpose'' as it does not have any specific clue of the problem to solve.
Model reduction techniques aim at cheaply solving the forward equations in a low-dimensional space still by a Galerkin projection process.
To this aim they construct a rather small set of basis functions (known as {\it reduced basis}), which we call ``educated'' basis as it includes features of solutions to the forward problem considered. This allows to an efficient low-dimensional yet accurate representation of the solution.
Among various techniques for the reduced basis construction (see the recent review\cite{Benner2015}), typical ones are the {\it Proper Orthogonal Decomposition} (POD) approach \cite{Kunisch2002, frangos2010surrogate}, the {\it Greedy Reduced Basis} (GRB) approach \cite{quarteroni2011certified, maday}, and their combination \cite{Nguyen2010}.
In these approaches, the reduced basis is constructed from a set of parameter-dependent solutions of the full-order model.
The POD approach has drawn widespread attention for its optimal ability to approximate the snapshots of solution with minimized error by the selection of the most important modes, and for its easy-to-use feature in practice. The GRB approach selects the snapshots following a greedy process according to a rule controlled by an {\it a posteriori} error estimator.
The requirement of a rigorous error estimator currently limits its application to electrocardiological models due to the model complexity.
In this paper we resort to the POD approach as a starting point, the development of an {\it a posteriori} error estimator for the GRB method is a part of ongoing work.
The POD approach has been used in numerous fields of science and engineering such as fluid-structure interaction \cite{BertagnaVeneziani} and aerodynamics \cite{Thanh2004}, but its practical application in electrocardiology only starts from 2011 \cite{Boulakia2011POD, boulakia2012, corrado2015identification}. In these references, the POD method allows reasonable estimation of cardiac ionic model parameters, however, no systematic study on the improvement of efficiency of solving the full nonlinear electrocardiological model is available.
In fact, one critical aspect when reducing a nonlinear problem by projecting onto a low-dimensional space is to approximate the projected nonlinear terms in a way independent of the full-order model size. This point was not thoroughly addressed in current electrocardiology publications.
Several techniques are available in general to reduce the cost of evaluating nonlinear terms, such as the {\it trajectory piecewise-linear} (TPWL \cite{TPWL2003}) approach, the {\it Best Point Interpolation Method} (BPIM \cite{NguyenBestPoint}), the {\it Empirical Interpolation Method} (EIM \cite{Barrault2004667}) and its discrete variant Discrete EIM (DEIM \cite{thesisDEIM}). Although TPWL was successfully applied to some practical problems, it may not be effective or efficient for systems with high order of nonlinearity. The BPIM and EIM approaches are similar \cite{Galbally2010}, both select a small set of spatial interpolation points to avoid the expensive calculation of inner products and use the points for nonlinear approximation. BPIM is optimal in the point selection and gains a little improvement on accuracy, but it is more computational expensive.
Here, we resort to EIM in its discrete variant DEIM.
Precisely, we apply the POD-DEIM for the first time to the conductivity estimation problem.
It is worth mentioning that the conductivity parameter to be estimated considered in this work is more troublesome than other ionic model parameters, since it dominates the speed and direction of fast transient of electrical potential through the cardiac tissue, which is an intrinsic feature of the forward electrocardiology model. This fact thus prevents a successful model reduction via a
classical POD procedure.
Model reduction procedures need to be specifically customized for the problem, and in particular the construction of the educated basis is a delicate step.
Nevertheless, we show here how an appropriate sampling for the basis computation actually leads to significant reduction of the full-order computational costs with a great level of accuracy. We address the sampling required for basis construction based on the novel concept of ``Domain of Effectiveness'' in the parameter space.
A rather small set of samples is obtained by sampling the parameter space based on polar coordinates, with refinement in the ``small angle--short arc'' zone of the sample space utilizing Gaussian nodes.
In this way, we manage to use the POD-DEIM reduced-order model with a computational reduction
of at least 95\% of the full-order conductivity estimation.
The present work relies on but largely improved the study presented in \cite{HHthesis}, to which we refer for more details.
We consider specifically the Monodomain problem. Notwithstanding that some authors consider this model reliable enough for many clinical applications, the extension of the present work to the more accurate
Bidomain system is a follow-up of the present work.
The outline is as follows. After a brief statement of the full-order {\it Monodomain inverse conductivity problem} (MICP) in a discrete form (Sec.~\ref{MICPfull}), the POD-DEIM approach is introduced (Sec.~\ref{DEIMintr}) and applied to solve a reduced MICP (Sec.~\ref{DEIM-use}) by derivative-based optimization. The reduced-order model is tested in Sec.~\ref{DEIMresult}: both the efficiency and accuracy of the POD-DEIM approach in conductivity estimation are investigated; we report both pitfalls and successful examples.
\section{The full-order Monodomain inverse conductivity problem}\label{MICPfull}
\subsection*{The cardiac Bidomain and Monodomain models}
The {\it Bidomain model} is considered as the most physiologically founded description for the dynamics of cardiac electric potentials---the transmembrane potential $u$ and the extracellular potential $u_{\rm e}$---at the level of cardiac tissue.
Its parabolic-elliptic form (see e.g. \cite{pullan2005mathematically}) reads
\begin{equation} \label{stateequations}
\left\{
\begin{array}{ll}
\beta C_{\rm m} \dfrac{\partial u}{\partial t}
-\nabla \cdot \left( \boldsymbol{\boldsymbol{\sigma}}_{\rm i} \nabla u \right)-\nabla \cdot \left( \boldsymbol{\sigma}_{\rm i}\nabla u_{\rm e} \right) + \beta I_{\rm ion}(u,w) =I_{\rm si} &\hspace{0.05cm}\mbox{ in } \Omega\times[0,T] \\[0.3cm]
- \nabla \cdot \left( \boldsymbol{\sigma}_{\rm i} \nabla u \right)-\nabla \cdot \left(\boldsymbol{\sigma}_{\rm i}+\boldsymbol{\sigma}_{\rm e}\right)\nabla u_{\rm e} =I_{\rm si}-I_{\rm se} & \hspace{0.05cm}\mbox{ in } \Omega\times[0,T]
\end{array} \right.
\end{equation}
with initial condition $u(\boldsymbol{x},0)=u_0(\boldsymbol{x})$. Typically, homogeneous Neumann boundary conditions are prescribed to model an isolated tissue.
Here $\Omega \subset \mathbb{R}^3$ is a spatial domain denoting the portion of cardiac tissue of interest and $[0, T]$ is a fixed time interval.
The symbol $C_{\rm m}$ is the membrane capacitance per unit area with $\beta = 2000 \mbox{ cm}^{-1}$ being the surface-to-volume ratio of the membrane; $\boldsymbol{\sigma}_{\rm i}$ ($ \boldsymbol{\sigma}_{\rm e}$) is the intracellular (extracellular) conductivity tensor; $I_{\rm si}$ ($I_{\rm se}$) represents the intracellular (extracellular) stimulation current. An explicit form of the total ionic current $I_{\rm ion}$ is described by an ionic model in which the {\it gating variable} $w$ is used to control the depolarization and repolarization phases of the cardiac action potential. The time evolution of the gating variable $w$ is generally modeled in the form
$$\dfrac{d w}{d t} + g(u, w) = 0 \quad\mbox{ in } \Omega\times[0,T]$$
with initial condition $w(\boldsymbol{x},0)=w_0(\boldsymbol{x})$.
We can represent the conductivity tensors as
$\boldsymbol{\sigma}_{\rm k}(\boldsymbol{x})=
{\sigma}_{\rm kl}\mathbf{a_l}(\boldsymbol{x})\mathbf{a_l}(\boldsymbol{x})^T + {\sigma}_{\rm kt}\mathbf{a_t}(\boldsymbol{x})\mathbf{a_t}(\boldsymbol{x})^T
+ {\sigma}_{\rm kn}\mathbf{a_n}(\boldsymbol{x})\mathbf{a_n}(\boldsymbol{x})^T, $
where k stands for i or e, $ (\mathbf{a_l}, \mathbf{a_t}, \mathbf{a_n} )$ are orthonormal vectors related to the structure of the myocardium with $\mathbf{a_l}$ parallel to the fibre direction of the myocardial tissue. We further assume that the tissue is axial isotropic (i.e.~$\sigma_{\rm kn} = \sigma_{\rm kt}$) and postulate the conductivity $[\sigma_{\rm kl}, \sigma_{\rm kt}]$ to be constant, as has been done by several groups \cite{Clerc1976,Roberts2} for conductivity estimation in experiments.
The Bidomain model has been widely used due to its ability to reproduce cardiac phenomena \cite{Trayanova}. However, its numerical solution for clinical application requires high computational cost, since it is a degenerate system of PDEs and the mesh and time constraints are significant for simulating fast potential variation.
The {\it Monodomain model} as a heuristic approximation of the Bidomain model has been proposed to provide computational improvements. Its derivation \cite{nielsen2007optimal} is based upon a proportionality assumption
$\boldsymbol{\sigma}_{\rm e} = \lambda \boldsymbol{\sigma}_{\rm i}$, where $\lambda$ is a constant. A formulation of the Monodomain model is then obtained, by denoting $\boldsymbol{\sigma}_{\rm m} = \frac{\lambda}{1+\lambda}\boldsymbol{\sigma}_{\rm i}$ and $I_{\rm app}=\frac{\lambda}{1+\lambda}I_{\rm si}+\frac{1}{1+\lambda}I_{\rm se}$, as
\begin{equation} \label{monoDerive}
\begin{array}{ll}
\beta C_{\rm m} \dfrac{\partial u}{\partial t}-\nabla \cdot (\boldsymbol{\boldsymbol{\sigma}}_{\rm m} \nabla u) + \beta I_{\rm ion} =I_{\rm app} &\quad\mbox{ in } \Omega\times[0,T].
\end{array}
\end{equation}
Although the assumption on its derivation lacks physiological foundation, the Monodomain model has been intensively used in clinic-oriented simulations \cite{Villongco2014305, Barros2015} since it requires significantly less computational efforts than the Bidomain model. More importantly, a comparison between the Bidomain and Monodomain models in \cite{Bourgault2010} concluded that the discrepancy between the models at the continuous level may be quite small: of order 1\% or even below in terms of activation time relative error.
We resort to the Monodomain model in this paper for the cardiac conductivity estimation, following the ``potential oriented'' line. Namely, we speculate that an appropriate estimate of the conductivity tensor $\bsb{\sigma}_{\rm m}$ based on our variational procedure can still lead to an accurate reconstruction of the potential propagation. This is demonstrated by the numerical result shown in Fig.~\ref{uAtPt}, where the Bidomain solution $u$ computed on a slab mesh with $[\sigma_{\rm il}, \sigma_{\rm el}, \sigma_{\rm it}, \sigma_{\rm et}] = [3.5, 3, 0.3, 1.8]$ was used as synthetic measurement data to estimate the conductivity $\bsb{\sigma}_{\rm m}$ in the Monodomain solver. The reconstruction of potential by the Monodomain solver with the estimated conductivity $[\sigma_{\rm ml}, \sigma_{\rm mt}] = [1.704, 0.3551]$ gives an excellent matching with the Bidomain solution (for more details, see \cite{HHthesis}).
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{uAtPt.pdf}
\caption{The reconstruction of potential by the Monodomain solver, computed with the estimated conductivity $[\sigma_{\rm ml}, \sigma_{\rm mt}] = [1.704, 0.3551]$, gives an excellent matching with the Bidomain solution, which is computed with $[\sigma_{\rm il}, \sigma_{\rm el}, \sigma_{\rm it}, \sigma_{\rm et}] = [3.5, 3, 0.3, 1.8]$.}
\label{uAtPt}
\end{center}
\end{figure}
In the following subsections, we describe the time and space discretization schemes for the full-order Monodomain model, and state the Monodomain inverse conductivity problem in a discrete version.
As a starting point of studies on reduced-order modeling for conductivity estimation, here we simply use the classical Rogers--McCulloch \cite{rogers} ionic model, given by
\begin{equation}
\begin{array}{l}
I_{\rm ion}(u,w) = C_{\rm m}[\beta_1(u-V_{\rm r})(u-V_{\rm th})(u-V_{\rm p})+c_2(u-V_{\rm r})w ]\label{RMmodel} \\%[0.3cm]
g(u,w) = -\beta_2(u-V_{\rm r})+bdw
\end{array}
\end{equation}
with $\beta_1 = \frac{c_1}{(V_{\rm p}-V_{\rm r})^2}, ~ \beta_2=\frac{b}{V_{\rm p}-V_{\rm r}}.$
The values of ionic model parameters are taken from \cite{colli2004}: $C_{\rm m} = 1~\mu\mbox{Fcm}^{-2}$, $V_{\rm r}=-85$ mV, $V_{\rm th}=-72$ mV, $V_{\rm p}=15$ mV, $c_1=11.54 \mbox{ ms}^{-1}$, $c_2=4.4 \mbox{ ms}^{-1}$, $b = 0.012 \mbox{ ms}^{-1}$, $d = 1$.
\subsection*{Time and space discretization}
To improve the computational efficiency, we split the Monodomain problem into a PDE and an ODE representing the Rogers--McCulloch model. The ODE gating variables are integrated with an implicit backward Euler method. The PDE is solved by a semi-implicit method based on a backward differentiation formula (BDF). The nonlinear terms are tackled by an explicit second-order time extrapolation \cite{HH2015IP}.
Let $\Delta t$ be the time step, we define $L=T/\Delta t, ~t^l=l\Delta t$. Hereafter we use superscripts $l$ and $l+1$ for those variables at time $t^l$ and $t^{l+1}$, respectively.
The finite element method (FEM) is used for space discretization. Let $\{\phi_j\}_{j=1}^n$ be the finite element basis, we denote by $\mathbf{M}$ the mass matrix
and by $ \mathbf{S_{\rm l}}$ and $ \mathbf{S_{\rm t}}$ the stiffness matrices with entries
$$[\mathbf{S_{\rm l}}]_{jk}=\int_\Omega \mathbf{a}_{\rm l}\mathbf{a}_{\rm l}^T\nabla\phi_k\cdot\nabla\phi_j d\boldsymbol{x}, \quad [\mathbf{S_{\rm t}}]_{jk}=\int_\Omega (\mathbf{I}-\mathbf{a}_{\rm l}\mathbf{a}_{\rm l}^T)\nabla\phi_k\cdot\nabla\phi_j d\boldsymbol{x}.$$
The bold symbol $\mathbf{u}^l$ will denote the vector representation of $u(x,t^l)$ in the finite element space. We adopt a similar notation for the other variables.
At time $t=t^{l+1}$, the gating variable $\mathbf{w}^{l+1}$ in the ionic model is updated by
\begin{equation}\label{ion-algebra}
\dfrac{\mathbf{w}^{l+1}-\mathbf{w}^l}{\Delta t}=-g(\tilde{\mathbf{u}}^{l+1}, \mathbf{w}^{l+1})
\end{equation}
with $\tilde{\mathbf{u}}^{l+1} = 2\mathbf{u}^{l}-\mathbf{u}^{l-1}$ ($\tilde{\mathbf{u}}^{1}=\mathbf{u}^{0}$ in particular) being the second-order time extrapolation of $\mathbf{u}^{l+1}$. Here $g(\cdot)$ is evaluated component-wise, its $i$-th entry is computed as $g([\tilde{\mathbf{u}}^{l+1}]_i, [\mathbf{w}^{l+1}]_i)$.
For the sake of computational efficiency (see Sec.~\ref{pod-DEIM-resultInv}: The measures) and for the convenience of model reduction (Sec.~\ref{DEIM}: (\ref{fappx})), we approximate the nonlinear function $I_{\rm ion}$ and the stimulus $I_{\rm app}$ piecewise linearly. Under this assumption, the discretized Monodomain system reads
\begin{equation}\label{mono-algebra}
\mathbf{A_{\rm m}} \mathbf{u}^{l+1} = \mathbf{b}^{l+1}
\end{equation}
where $\mathbf{A_{\rm m}} = \beta C_{\rm m}\frac{\alpha_0}{\Delta t}\mathbf{M}+\sigma_{\rm ml}\mathbf{S_{\rm l}} + \sigma_{\rm mt}\mathbf{S_{\rm t}}$ and the right-hand side is evaluated as
\begin{equation}\label{mono-b-ptws}
\mathbf{b}^{l+1} = \mathbf{M}(\mathbf{I}_{\rm app}^{l+1} - \beta I_{\rm ion}(\tilde{\mathbf{u}}^{l+1}, \mathbf{w}^{l+1})) + \beta C_{\rm m}\mathbf{M} \sum\limits_{i=1}^{2}\frac{\alpha_i}{\Delta t}\mathbf{u}^{l+1-i}.
\end{equation}
Here $\alpha_i$'s are the coefficients of the BDF (for a BDF of order two we have $\alpha_0 = 3/2,~\alpha_1 = 2,~\alpha_2 = -1/2$). In (\ref{mono-b-ptws}), $I_{\rm ion}(\tilde{\mathbf{u}}^{l+1}, \mathbf{w}^{l+1})$ is a component-wise evaluation. As we will see in Sec.~\ref{pod-DEIM-resultInv}, this component-wise approximation improves computational efficiency without significant
loss of accuracy as compared with the exact finite element approximation.
The linear system (\ref{mono-algebra}) can be solved by an ILU preconditioned conjugate gradient method implemented for instance in the Trilinos package ({\tt{www.trilinos.org}}).
A mass lumping technique (\cite{Zienkiewicz2005}, Sec.~16.2.4) is employed for the sake of computational efficiency and
stability.
Specifically, we apply a diagonal scaling with the factor being the total mass $\mathbf{M}_{\rm tot}$ ($=\sum_{i,j}\mathbf{M}_{ij}$) divided by the trace, i.e.~the diagonal lumped mass matrix $\mathbf{M}_{\rm L}$ satisfies $[\mathbf{M}_{\rm L}]_{ii} = \dfrac{\mathbf{M}_{\rm tot}}{\mbox{Tr}(\mathbf{M})}\mathbf{M}_{ii}$.
For simplicity of notation, we will use the same symbol $\mathbf{M}$ for the lumped mass matrix as for the mass.
We solve the Monodomain and Bidomain models using LifeV, an object oriented C++ finite element library developed by different groups worldwide \cite{lifev}.
\subsection*{The inverse conductivity problem}
The {\it Monodomain inverse conductivity problem} (MICP) reads: find $\boldsymbol{\sigma} = [\sigma_{\rm ml}, \sigma_{\rm mt}] \in \mathcal{C}_{\rm ad} \subset \mathbb{R}^2$ that minimizes
\begin{equation}\label{}
\mathcal{J}_{\rm m}(\boldsymbol{\sigma}) = \dfrac{1}{2}\sum\limits_{l=1}^L(\mathbf{u}^l-\mathbf{u}_{\rm meas}^l)^T \mathbb{X}_{\rm site} (\mathbf{u}^l-\mathbf{u}_{\rm meas}^l) \chi^l_{\rm snap} + \dfrac{\alpha}{2}\mathcal{R}(\boldsymbol{\sigma})
\end{equation}
subject to the Monodomain system (\ref{mono-algebra}) coupled with (\ref{ion-algebra}). Here $\mathcal{C}_{\rm ad}$ is an admissible domain given by inequality constraints on the conductivity,
in the general form $\mathbf{h}(\boldsymbol{\sigma})\geq \boldsymbol{0}$. The term $\mathbf{u}_{\rm meas}^l$ denotes experimental data at time $t^l$.
The data can be obtained in vitro using voltage optical mapping \cite{fentonMeasure}, or in vivo by back-mapping body surface potentials \cite{Cluitmans2013} or possibly by potential reconstruction from
electrocardiogram phase analysis of standard gated SPECT \cite{Chen2008JNC}.
We assume that the measurement sites are always grid points and
$\mathbb{X}_{\rm site} \in \mathbb{R}^{n\times n}$ is the matrix recording observation sites.
The off-diagonal entries in $\mathbb{X}_{\rm site}$ are zeros; in the diagonal, $[\mathbb{X}_{\rm site}]_{ii} = 1$ if the spatial grid $x_i$ is an observation site and $0$ otherwise. We also introduce the {\it snapshot marker} $\chi^l_{\rm snap}$ which equals 1 if $t^l$ is an observation moment and $0$ otherwise.
The MICP can be solved by the BFGS optimization \cite{nocedal2006} approach as done in \cite{HH2015IP}. However, the computation is intensive since each ``query'' of the forward system or its adjoint counterpart is performed in the full-order $n$ by solving large algebraic systems.
Moreover, in practice one may find also pathological tissues where scars inside have different (anomalous) values of conductivities. In this case the number of ``queries'' will substantially increase due to the increase of the total number of conductivities (see \cite{HH2015IP, HHthesis}).
Altogether, in the current scenario, the MICP is too computationally expensive to be applied in clinic. This motivates us to work on a model reduction investigation with the aim of reducing significantly the computational cost.
\section{Model order reduction for nonlinear systems}\label{DEIMintr}
In this section, we describe the method in general by considering the discrete system of a parameterized nonlinear differential equation
\begin{equation}\label{generalSys}
\mathbf{A}\mathbf{y}(\tau) = \mathbf{F}(\mathbf{y}(\tau);\tau)
\end{equation}
where $\tau$ denotes the model parameter of interest in an inverse problem and is contained in a closed bounded domain $\mathcal{D}$.
We assume that the matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$ has affine dependence on $\tau$ and omit the explicit dependency $\mathbf{A}(\tau)$ for simplicity; $\mathbf{F}(\cdot)$ is a nonlinear function evaluated at the solution $\mathbf{y}(\tau) = [y_1(\tau), \cdots, y_n(\tau)]^T$ component-wise, i.e.
$\mathbf{F}(\mathbf{y}(\tau);\tau) = [F(y_1(\tau);\tau), \cdots, F(y_n(\tau);\tau)]^T$
with $F(\cdot)$ being a nonlinear scalar-valued function. In a finite element discretization, this is equivalent to a piece-wise linear approximation.
Our goal is to construct a small set of basis functions $\{\bsb{\varphi}_i \}_{i=1}^N$ in $\mathbb{R}^n$
such that the solution $\mathbf{y}(\tau)$ can be well approximated in the space $\spn\{\bsb{\varphi}_i\}$ (called {\it reduced space}). The functions $\{\bsb{\varphi}_i \}_{i=1}^N$ form the so-called {\it reduced basis} (RB).
Let $\mathbb{Z}_{\rm y} = [\bsb{\varphi}_1, \cdots, \bsb{\varphi}_N] \in \mathbb{R}^{n\times N}$, we represent the solution in the reduced space as
$
\mathbf{y}(\tau) = \mathbb{Z}_{\rm y}\mathbf{y_r}(\tau)
$
with $\mathbf{y_r}(\tau)$ being the vector of coordinates in the reduced space.
A {\it reduced-order model} (ROM) of (\ref{generalSys}) is then obtained by Galerkin projection
\begin{equation}\label{generalReduced}
\underbrace{\mathbb{Z}_{\rm y}^T\mathbf{A}\mathbb{Z}_{\rm y}}_{\mathbf{A_r}}\mathbf{y_r}(\tau) = \mathbb{Z}_{\rm y}^T\mathbf{F}(\mathbb{Z}_{\rm y}\mathbf{y_r}(\tau); \tau).
\end{equation}
Notice that $\mathbf{A_r} = \mathbb{Z}_{\rm y}^T\mathbf{A}\mathbb{Z}_{\rm y} \in \mathbb{R}^{N\times N}$ is a dense matrix but in general it features a very small size, hence the linear system (\ref{generalReduced}) can be tackled with a direct solver.
The RB is constructed from the {\it full-order model} (FOM) (\ref{generalSys}) which is of large scale, thus the computation is usually expensive and performed offline.
In the online phase, the ROM (\ref{generalReduced}) is solved many times for different parameter values with remarkably lower computational costs than the FOM.
Techniques for RB construction may rely on a sampling on the parameter $\tau$. We introduce here a sample $S = \{\tau_1, \cdots, \tau_s \}$
consisting of $s$ distinct parameter points in $\mathcal{D}$.
A RB is constructed so to guarantee that for each $\tau_i \in S$ the error of approximating $\mathbf{y}(\tau_i)$ in the reduced space is bounded by a desired tolerance. Hereafter we follow the POD approach, which usually constructs an ``optimal" reduced basis as specified below.
\subsection{Proper Orthogonal Decomposition (POD)}\label{pod}
For the sake of completeness we briefly recall basic features of POD. More details can be found e.g.~in \cite{Kunisch2002}.
Given the parameter sample $S$, we solve the FOM (\ref{generalSys}) for each parameter value in $S$. The solutions are called {\it snapshots} and denoted by $\{\mathbf{y}_i \}_{i=1}^m$ ($m=s$ in current setting). The Proper Orthogonal Decomposition (POD) approach seeks an orthonormal {\it POD basis} $\{\boldsymbol{\varphi}_1, \cdots, \boldsymbol{\varphi}_N \}$ (also known as a set of {\it POD modes}) in $\mathbb{R}^n$ of a given rank $N~(N\ll m)$ that can best approximate the training space $X^{trn} = \spn\{\mathbf{y}_i \}_{i=1}^m$. Here ``best'' means the POD basis solves
\begin{equation}\label{minPOD}
\min_{\{\boldsymbol{\psi}_i\}} \sum_{j=1}^m ||\mathbf{y}_j-\sum_{i=1}^N\langle \mathbf{y}_j,\boldsymbol{\psi}_i \rangle\boldsymbol{\psi}_i ||^2
\qquad\mbox{ s.t. } \langle\boldsymbol{\psi}_i,\boldsymbol{\psi}_j\rangle = \delta_{ij}.
\end{equation}
We gather the snapshots into the so called {\it snapshot matrix} $\mathbf{Y}=[\mathbf{y}_1, \cdots, \mathbf{y}_m] \in \mathbb{R}^{n\times m}$. The POD modes are given by the $N$ left singular vectors of $\mathbf{Y}$ associated with the $N$ largest singular values \cite{Kunisch2002}. Without loss of generality, we assume that the snapshot mean is zero. In fact we can replace each $\mathbf{y}_i$ with $\mathbf{y}_i-\frac{1}{m}\sum\limits_{j=1}^m\mathbf{y}_j$.
An efficient way for computing the POD modes through snapshot matrix $\mathbf{Y}$
is to first compute the {\it thin QR factorization} of $\mathbf{Y}$ as $\mathbf{Y} = \mathbf{QR}$ , and then compute the singular value decomposition of matrix $\mathbf{R}\in\mathbb{R}^{m\times m}$ as $\mathbf{R}=\mathbf{U}_{\rm R}\mathbf{S}_{\rm R}\mathbf{V}_{\rm R}^T$. The POD modes can be extracted in order as the first $N$ columns of $\mathbf{QU}_{\rm R}$.
\subsubsection*{Snapshots selection / Sampling}
The effectiveness of model reduction is clearly related to the representativity of the snapshots.
Standard schemes of sampling in the parameter space, which determines snapshots selection, include uniform sampling, random sampling, the Latin hypercube sampling (LHS, \cite{McKay1979}), and the centroidal voronoi tessellation (CVT) sampling \cite{Du1999}. When sampling a high-dimensional parameter space, the greedy sampling method could be used \cite{quarteroni2011certified, Nguyen2010}. The key feature of greedy sampling is to adaptively select a parameter at which the estimate of the solution error in the ROM is maximal. In this way,
we select the most effective parameter for controlling the error of the ROM.
However, the application of this method is limited to problems where sharp error estimators are available.
For optimal control applications, several online adaptive sampling procedures have been proposed to let the sampling procedure take into account the optimization trajectory. The Trust Region POD \cite{Arian00trust} approach constructs successively improved POD bases according to parameter values updated during optimization. The updating procedure was embedded with the trust region method which determines whether after an optimization step the POD basis should be updated.
The Compact POD \cite{Carlberg2008} uses snapshots of the full-order model solutions as well as their derivatives (known as {\it sensitivities}) with respect to the model parameters of interest.
\subsection{Discrete Empirical Interpolation Method (DEIM)}\label{DEIM}
In the reduced system (\ref{generalReduced}) obtained by the POD projection, we have to evaluate the nonlinear term
\begin{equation}\label{nonlinear-n}
\mathbf{n}(\tau) = \underbrace{\mathbb{Z}_{\rm y}^T}_{N\times n}\underbrace{\mathbf{F}(\mathbb{Z}_{\rm y}\mathbf{y_r}(\tau);\tau)}_{n\times 1}.
\end{equation}
This evaluation has computational complexity depending on the size $n$ of the FOM (\ref{generalSys}), which is possibly in the magnitude of hundred thousand. Therefore, solving the ROM (\ref{generalReduced}) without an appropriate methodology may be as expensive as solving the full one.
The complexity of evaluating (\ref{nonlinear-n}) can be made independent of the full order $n$ by using the Discrete Empirical Interpolation Method (DEIM \cite{thesisDEIM}). The DEIM provides an interpolation approximation for the nonlinear term $\mathbf{F}(\mathbb{Z}_{\rm y}\mathbf{y_r}(\tau);\tau)$ (simply denoted as $\mathbf{f}(\tau)$ in the sequel) by a projection onto a low-dimensional subspace. For this purpose, we introduce an $M$ dimensional space ($M \ll n$) where we look for an approximation of $\mathbf{f}(\tau)$ for values $\tau$ of interest. In particular, we can sample $\tau$, take snapshots of $\mathbf{f}(\tau)$ computed from the FOM with those samples, and then apply the POD on the snapshots to extract a projection basis $\{\mathbf{z}_1,\cdots,\mathbf{z}_M \}$.
Let $\mathbb{Z}_{\rm f} = [\mathbf{z}_1,\cdots,\mathbf{z}_M] \in \mathbb{R}^{n\times M}$, the DEIM approximation of $\mathbf{f}$ is in the form
$\mathbf{f} \approx \hat{\mathbf{f}} =\mathbb{Z}_{\rm f} \mathbf{c}.$
To determine the coefficient vector $\mathbf{c}$, the DEIM optimally extracts $M$ distinct rows from the over-determined system $\mathbf{f}=\mathbb{Z}_{\rm f} \mathbf{c}$. Specifically, the DEIM selects row indices $p_1,\cdots, p_M$ in $\{1, ... ,n\}$ and requires:
$[\mathbf{f}]_{p_i} = [\hat{\mathbf{f}}]_{p_i}.$
If we denote ${\rm P} = [\mathbf{e}_{p_1}, \cdots, \mathbf{e}_{p_M}] \in \mathbb{R}^{n\times M} $ with $\mathbf{e}_{p_i}$ being the $p_i\mbox{-th}$ unit vector in $\mathbb{R}^n$, the coefficient vector $\mathbf{c}$ is solved
from ${\rm P}^T\mathbf{f}=({\rm P}^T\mathbb{Z}_{\rm f})\mathbf{c}$.
Finally the approximation of $\mathbf{f}$ writes
\begin{equation}\label{fappx}
\mathbf{f} \approx \hat{\mathbf{f}} = \mathbb{Z}_{\rm f} \mathbf{c} = \mathbb{Z}_{\rm f}({\rm P}^T \mathbb{Z}_{\rm f})^{-1} {\rm P}^T \mathbf{f} = \mathbb{Z}_{\rm f}({\rm P}^T \mathbb{Z}_{\rm f})^{-1}\mathbf{F}({\rm P}^T\mathbb{Z}_{\rm y}\mathbf{y_r}(\tau);\tau).
\end{equation}
The last equality in (\ref{fappx}) follows from the assumption that the function $\mathbf{F}(\cdot)$ evaluates component-wise at its input vector. The nonlinear term (\ref{nonlinear-n}) can then be efficiently computed through
\begin{equation}
\mathbf{n}(\tau) \approx \underbrace{\mathbb{Z}_{\rm y}^T\mathbb{Z}_{\rm f}({\rm P}^T \mathbb{Z}_{\rm f})^{-1}}_{N\times M}\underbrace{\mathbf{F}({\rm P}^T\mathbb{Z}_{\rm y}\mathbf{y_r}(\tau);\tau)}_{M \times 1}.
\end{equation}
Notice that the matrices $\mathbb{Z}_{\rm y}^T\mathbb{Z}_{\rm f}({\rm P}^T \mathbb{Z}_{\rm f})^{-1} \in \mathbb{R}^{N\times M}$ and ${\rm P}^T\mathbb{Z}_{\rm y} \in \mathbb{R}^{M\times N}$
can be precomputed so that the computational complexity of evaluating $\mathbf{n}(\tau)$ is only $\mathcal{O}(MN)$.
The interpolation indices ${p_1}, \cdots, {p_M}$ are selected inductively from the projection basis $\{\mathbf{z}_1 , \cdots, \mathbf{z}_M \}$ by the DEIM algorithm. For the sake of completeness,
we recall in Algorithm \ref{DEIM_alg} the DEIM described in \cite{thesisDEIM}. At each iteration, an interpolation index is selected to limit growth of the error bound of the approximation $\hat{\mathbf{f}}$. In particular, the first index $p_1$ is the index on which $\mathbf{z}_1$ has the largest magnitude; each of the remaining indices $p_l$ is the index on which the residual of approximating $\mathbf{z}_l$ by the first $l-1$ basis vectors $\{\mathbf{z}_1,\cdots,\mathbf{z}_{l-1} \}$ has the largest magnitude.
It is demonstrated that the DEIM algorithm is well-defined (\cite{thesisDEIM}, lem.~2.2.2). In fact, ${\rm P}^T\mathbf{Z}_{\rm f}$ is non-singular in each iteration of Algorithm \ref{DEIM_alg} and the interpolation indices are not repeated.
\begin{algorithm}[tp]
\caption{DEIM \cite{thesisDEIM}}\label{DEIM_alg}
\begin{algorithmic}[1]
\Input $\{\mathbf{z}_l\}_{l=1}^M \subseteq \mathbb{R}^n $ linear independent
\Output indices $\mathbf{p} = [p_1, \cdots, p_M]^T \in \mathbb{R}^M$, $\mathbf{M}$ as the inverse of ${\rm P}^T\mathbb{Z}_{\rm f}$
\State $[|\rho|, p_1] = \max\{|\mathbf{z}_1|\} $
\State $\mathbb{Z}_{\rm f}\leftarrow[\mathbf{z}_1],~ {\rm P}\leftarrow[\mathbf{e}_{p_1}], ~\mathbf{p} \leftarrow [p_1]
\For{$l=2, \cdots, M$}
\State Solve $ ({\rm P}^T\mathbb{Z}_{\rm f})\mathbf{c} = {\rm P}^T\mathbf{z}_l $
\State $\mathbf{r}=\mathbf{z}_l - \mathbb{Z}_{\rm f}\mathbf{c}$
\State $[|\rho|, p_l] = \max\{|\mathbf{r}|\} $
\State $\mathbf{a}^T = \mathbf{e}_{p_l}^T \mathbb{Z}_{\rm f}$
\State $$\mathbf{M} \leftarrow
\left[
\begin{array}{cc}
\mathbf{I} & -\mathbf{c} \\
\boldsymbol{0} & 1
\end{array}
\right]
\left[
\begin{array}{cc}
\mathbf{M} & \boldsymbol{0} \\
-\rho^{-1}\mathbf{a}^T \mathbf{M} & \rho^{-1}
\end{array}
\right] $$
\State $\mathbb{Z}_{\rm f}\leftarrow [\mathbb{Z}_{\rm f} ~\mathbf{z}_l],~ {\rm P}\leftarrow [{\rm P} ~\mathbf{e}_{p_l}], ~\mathbf{p} \leftarrow [\mathbf{p}^T ~p_l]^T$
\EndFor
\end{algorithmic}
\end{algorithm}
\section{The reduced-order Monodomain inverse conductivity problem}\label{DEIM-use}
The combination of POD and DEIM as described can be applied to the MICP. However, the real accuracy (and also efficiency) of the procedure depends on the nature of the problem and ultimately on the snapshots selection. In this section we present the standard tools to apply POD-DEIM to the MICP. The performance and the specific customization needed to make the procedure effective are discussed in Sec. \ref{DEIMresult}.
\subsection*{The POD basis for the Monodomain model}
In constructing a reduced basis for the parameterized Monodomain model, two ways can be followed as mentioned in \cite{Boulakia2011POD}. One is to store a set of solutions of the full-order model computed at different instants with a set of different parameter values in a given sample space, then collect all these solutions to build a unique snapshot matrix, upon which the POD basis is finally built. In other words, we treat both the conductivity tensor $\boldsymbol{\sigma}$ and also the time $t$ as parameters of the model. Another way for the reduced basis construction is to build multiple POD bases instead of a unique one. Each POD basis is constructed from the snapshots of solutions computed with a particular conductivity parameter, which we call {\it generating parameter} and denote by $\bsb{\sigma}_{\rm gen}$.
The idea is that for a given value of the tensor $\bsb{\sigma}$ we select the POD basis obtained with the closest generating parameter (among all available) to approximate the Monodomain problem.
The dynamic of the transmembrane potential $u$ is not quite smooth in time, due to the wavefront propagation or the upstroke (depolarization) spreading. This has an important impact on POD procedures. In fact, the singular values of the snapshot matrix of $u$ do not decay fast in general. We confirm this by a numerical experiment shown in Fig.~\ref{svd-onePara-multi}~(left). In this test, the time step $\Delta t = 0.05$~ms and the full-order model dimension is 24272. We collected snapshots of the transmembrane potential $u$ (and the ionic current $I_{\rm ion}$) computed with a fixed conductivity parameter $\boldsymbol{\sigma}=[3, 1]$ for 500 time steps. Slow decay shown in Fig.~\ref{svd-onePara-multi}~(left) is apparent especially when compared with other problems, in which singular values of a snapshot matrix usually decrease fast hence few POD modes are enough to give an accurate approximation of the solution considered (see e.g.~\cite{BertagnaVeneziani} for a FLuid-Structure Interaction problem).
The slow decay is detrimental to the actual model reduction as many modes need to be considered in the ROM construction.
\begin{figure}[!tp]
\begin{center}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.35]{u+ion-svd-onePara-dash.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.35]{u+ion-svd-multiPara-dash.pdf}
\end{minipage}
\caption{Singular values of snapshot matrices of $u$ and $I_{\rm ion}$. Left: snapshots generated by one parameter value. Right: snapshots generated by four parameter values.}
\label{svd-onePara-multi}
\end{center}
\end{figure}
Fig.~\ref{svd-onePara-multi} (left) shows that the singular value ${s_i}$ of snapshots of the transmembrane potential relative to the leading singular value ${s_1}$ decays to $10^{-3}$ (cross markers
in Fig.~\ref{svd-onePara-multi}) when $i$ is 49 (113 for the ionic current).
It suggests that $u$ ($I_{\rm ion}$) may still be well approximated by a POD basis of dimension less than 50 (120), provided that the model parameter is close enough to the generating parameter of the POD basis. We also infer that the ionic current $I_{\rm ion}$ features even more complex nonlinearity than the transmembrane potential $u$, since more POD modes are needed for approximating the ionic current with the same accuracy of the transmembrane potential (circular markers in Fig.~\ref{svd-onePara-multi})
A consequence of this nature of the problem is that building a unique POD basis actually provides worthless results. This can be identified from Fig.~\ref{svd-onePara-multi} right, in which the leading 500 singular values of snapshots generated by four different conductivity parameters are reported.
The relative singular value $s_i/s_1$ of snapshots of the transmembrane $u$ decays to $10^{-3}$ when $i$ is 136 (399 for the ionic current). Compared with the one-parameter case shown in Fig.~\ref{svd-onePara-multi} left, a combination of snapshots from different parameter values does not reduce the number of POD modes necessary for accurate POD approximation.
This feature demands for specific customization of the POD procedure based on an educated selection of the snapshots as detailed in Sec.~\ref{DEIMresult}.
\subsection*{Sensitivity equations}
One possible way to improve our model reduction is to add snapshots of the sensitivity of the Monodomain solution to the conductivity tensor. To this aim, we explicitly report the sensitivity equations.
Applying differentiation with respect to $\sigma_{\rm mk}$ on the Monodomain model ((\ref{mono-algebra}) coupled with (\ref{ion-algebra})), at time $t^{l+1}$ we can solve $\frac{\partial \mathbf{u}^{l+1}}{\partial \sigma_{\rm mk}}$ from
\begin{equation}
\frac{\partial\mathbf{A_{\rm m}}}{\partial \sigma_{\rm mk}} \mathbf{u}^{l+1} + \mathbf{A_{\rm m}}\frac{\partial \mathbf{u}^{l+1}}{\partial \sigma_{\rm mk}} = \frac{\partial \mathbf{b}^{l+1}}{\partial \sigma_{\rm mk}}.
\end{equation}
The sensitivity of the ionic current, stated in the continuous form without loss of generality, can be evaluated through
$$ \frac{\partial I_{\rm ion}}{\partial \sigma_{\rm mk}} = \partial_u I_{\rm ion}\frac{\partial u}{\partial \sigma_{\rm mk}} + \partial_w I_{\rm ion}\frac{\partial w}{\partial \sigma_{\rm mk}}$$
where ${\rm k}$ stands for either ${\rm l}$ or ${\rm t}$.
\subsection*{Application of DEIM on the Monodomain solver}
Let us denote the reduced bases for the transmembrane potential $u$ and the ionic current $I_{\rm ion}$ (evaluated in the degrees of freedom) by $\mathbb{Z}_{\rm u}$ and $\mathbb{Z}_{\rm ion}$ respectively. By projecting the discrete Monodomain system (\ref{mono-algebra}, \ref{mono-b-ptws}) onto the reduced space $\mathbb{Z}_{\rm u}$ , we obtain
\begin{equation}\label{reducedMono}
\left\{
\begin{array}{l}
\mathbb{Z}_{\rm u}^T \mathbf{A_{\rm m}}\mathbb{Z}_{\rm u}\mathbf{u}_\mathbf{r}^{l+1} = \mathbf{b}_\mathbf{r}^{l+1}
\quad\quad l\in\{0, 1, \cdots, L-1 \} \cr
\mbox{with }\quad \mathbf{b}_\mathbf{r}^{l+1} = \mathbb{Z}_{\rm u}^T\mathbf{b}^{l+1}
= \underbrace{\mathbb{Z}_{\rm u}^T\mathbf{M}\mathbf{I}_{\rm app}^{l+1}}_{N\times 1}
+ \beta C_{\rm m}\underbrace{\mathbb{Z}_{\rm u}^T\mathbf{M}\mathbb{Z}_{\rm u}}_{N\times N}\sum\limits_{i=1}^{2}\frac{\alpha_i}{\Delta t}\underbrace{\mathbf{u}_\mathbf{r}^{l+1-i}}_{N\times 1} \cr
\hspace{4cm} - \beta\underbrace{\mathbb{Z}_{\rm u}^T\mathbf{M}}_{N\times n}\underbrace{I_{\rm ion}(\mathbb{Z}_{\rm u}\widetilde{\mathbf{u}}_\mathbf{r}^{l+1}, \mathbf{w}^{l+1})}_{n\times 1}
\end{array}
\right.
\end{equation}
where $\mathbf{u}_\mathbf{r}^{l+1}$ is the solution at time $t=t^{l+1}$ in the reduced space and
$\widetilde{\mathbf{u}}_\mathbf{r}^{l+1} = 2\mathbf{u}_\mathbf{r}^{l} - \mathbf{u}_\mathbf{r}^{l-1}$ is its second-order extrapolation.
As discussed in Sec.~\ref{DEIM}, the complexity in computing the nonlinear term $I_{\rm ion}$ in the right hand side is $\mathcal{O}(n)$. This mainly results from the fact that $I_{\rm ion}(\mathbb{Z}_{\rm u}\widetilde{\mathbf{u}}_\mathbf{r}^{l+1}, \mathbf{w}^{l+1})$ cannot be precomputed, since it depends nonlinearly on the full-order vector $\mathbb{Z}_{\rm u}\widetilde{\mathbf{u}}_\mathbf{r}^{l+1}$ and $\mathbf{w}^{l+1}$. We will apply the DEIM approximation to $I_{\rm ion}(\mathbb{Z}_{\rm u}\widetilde{\mathbf{u}}_\mathbf{r}^{l+1}, \mathbf{w}^{l+1})$.
We use the POD basis $\mathbb{Z}_{\rm ion} \in \mathbb{R}^{n\times M}$ of snapshots of $I_{\rm ion}$ as an input basis for the DEIM algorithm, where $M$ is the number of POD modes. The DEIM algorithm generates interpolation indices $\mathbf{p} = [p_1, \cdots, p_M]^T $ for constructing the extraction matrix ${\rm P}$. The DEIM approximation reads then
\begin{equation}
I_{\rm ion}(\mathbb{Z}_{\rm u}\widetilde{\mathbf{u}}_\mathbf{r}^{l+1}, \mathbf{w}^{l+1})
\approx \underbrace{\mathbb{Z}_{\rm ion}({\rm P}^T\mathbb{Z}_{\rm ion})^{-1}}_{n\times M}I_{\rm ion}(\underbrace{{\rm P}^T\mathbb{Z}_{u}\tilde{\mathbf{u}}_\mathbf{r}^{l+1}}_{M\times 1},~ \underbrace{{\rm P}^T\mathbf{w}^{l+1}}_{M \times 1}).
\end{equation}
If we set
\begin{align*}
&\mathbf{A_r} = \mathbb{Z}_{\rm u}^T\mathbf{A_{\rm m}}\mathbb{Z}_{\rm u} \in \mathbb{R}^{N\times N},\\
&\mathbf{M}_{\rm u} = \mathbb{Z}_{\rm u}^T\mathbf{M}\mathbb{Z}_{\rm u} \in \mathbb{R}^{N\times N},
~\mathbf{M}_{\rm iu} = \underbrace{\mathbb{Z}_{\rm u}^T\mathbf{M}}_{N\times n}\underbrace{\mathbb{Z}_{\rm ion}({\rm P}^T\mathbb{Z}_{\rm ion})^{-1}}_{n\times M} \in \mathbb{R}^{N\times M}, \\
&\mathbf{I}_\mathbf{r}^{l+1} = \mathbb{Z}_{\rm u}^T\mathbf{M}\mathbf{I}_{\rm app}^{l+1} \in \mathbb{R}^{N\times 1} ,
~\mathbf{w}_\mathbf{r}^{l+1} = {\rm P}^T\mathbf{w}^{l+1} \in \mathbb{R}^{M\times 1},
~{\rm U} = {\rm P}^T\mathbb{Z}_{\rm u} \in \mathbb{R}^{M\times N} ,
\end{align*}
the reduced Monodomain system is then formulated as
\begin{equation}\label{rbDEIM_mono}
\left\{
\begin{array}{l}
\mathbf{A_r}\mathbf{u}_\mathbf{r}^{l+1}
= \mathbf{I}_\mathbf{r}^{l+1}
+ \beta C_{\rm m}\mathbf{M}_{\rm u}\sum\limits_{i=1}^{2}\frac{\alpha_i}{\Delta t}\mathbf{u}_\mathbf{r}^{l+1-i}
- \beta \mathbf{M}_{\rm iu} I_{\rm ion}({\rm U}\widetilde{\mathbf{u}}_\mathbf{r}^{l+1},~\mathbf{w}_\mathbf{r}^{l+1}) \\%[.3cm]
\dfrac{\mathbf{w}_\mathbf{r}^{l+1}-\mathbf{w}_\mathbf{r}^{l}}{\Delta t} = -g({\rm U}\widetilde{\mathbf{u}}_\mathbf{r}^{l+1}, \mathbf{w}_\mathbf{r}^{l+1})
\end{array}
\right..
\end{equation}
\subsection*{The reduced Monodomain inverse conductivity problem}
We are in position of formulating the reduced MICP: find $\boldsymbol{\sigma} = [\sigma_{\rm ml}, \sigma_{\rm mt}]$ that minimizes
\begin{equation}\label{}
\mathcal{J}_{\rm r}(\boldsymbol{\sigma}) = \dfrac{1}{2}\sum\limits_{l=1}^L(\mathbb{Z}_{\rm u}\mathbf{u}_\mathbf{r}^l-\mathbf{u}_{\rm meas}^l)^T \mathbb{X}_{\rm site} (\mathbb{Z}_{\rm u}\mathbf{u}_\mathbf{r}^l-\mathbf{u}_{\rm meas}^l) \chi^l_{\rm snap} + \dfrac{\alpha}{2}\mathcal{R}(\boldsymbol{\sigma})
\end{equation}
subject to the reduced Monodomain system (\ref{rbDEIM_mono}) and the inequality constraint $\mathbf{h}(\boldsymbol{\sigma})\geq \boldsymbol{0}$.
In particular, we restrict the admissible domain of the conductivities as
$$\big\{ \sigma_{\rm ml}/\sigma_{\rm mt}\geq 1,~\sigma_{\rm ml}/\sigma_{\rm mt}\leq 100, ~\sigma_{\rm mt}\geq 0.05, ~\sigma_{\rm ml}\leq 7 \big\},$$
inspired from the range of conductivity measures listed in Table 1 of \cite{HH2015IP}. We will denote the components of $\mathbf{h}(\boldsymbol{\sigma})$ as $h_i(\boldsymbol{\sigma})$ in
the algorithm description.
In order to obtain linear algebra operations of complexity $\mathcal{O}(N)$ only, in computing the cost function $\mathcal{J}_{\rm r}$, we can rewrite
\begin{equation}\label{}
\mathcal{J}_{\rm r}(\boldsymbol{\sigma}) = \dfrac{1}{2}\sum\limits_{l=1}^L
( (\mathbf{u}_\mathbf{r}^l)^T \mathbf{X}_{\rm u} \mathbf{u}_\mathbf{r}^l
-2(\widehat{\mathbf{u}}_{\rm meas}^l)^T\mathbf{u}_\mathbf{r}^l
+ ||\widehat{\vphantom{\rule{1pt}{6.5pt}}\smash{\widehat{\mathbf{u}}}}_{\rm meas}^l||^2
) \chi^l_{\rm snap} + \dfrac{\alpha}{2}\mathcal{R}(\boldsymbol{\sigma})
\end{equation}
where
$\mathbf{X}_{\rm u} = \mathbb{Z}_{\rm u}^T\mathbb{X}_{\rm site}\mathbb{Z}_{\rm u} \in \mathbb{R}^{N\times N}$.
The projected measurements
$\widehat{\mathbf{u}}_{\rm meas}^l= \mathbb{Z}_{\rm u}^T\mathbb{X}_{\rm site}\mathbf{u}_{\rm meas}^l \in \mathbb{R}^{N\times 1} \mbox{ and } \widehat{\vphantom{\rule{1pt}{6.5pt}}\smash{\widehat{\mathbf{u}}}}_{\rm meas}^l= \mathbb{X}_{\rm site}\mathbf{u}_{\rm meas}^l \in \mathbb{R}^{n\times 1}$
can be precomputed.
As we did in \cite{HH2015IP} for the full-order problem, we introduce the Lagrange multipliers $\{\mathbf{q}_\mathbf{r}^1,\cdots,\mathbf{q}_\mathbf{r}^L, \mathbf{r}_\mathbf{r}^1,\cdots,\mathbf{r}_\mathbf{r}^L\}$
and the Lagrangian functional
\begin{equation}
\begin{split}
&\mathcal{L}_{\rm r}(\mathbf{u}_\mathbf{r}^1,\cdots,\mathbf{u}_\mathbf{r}^L, \mathbf{w}_\mathbf{r}^1,\cdots,\mathbf{w}_\mathbf{r}^L, \boldsymbol{\sigma}, \mathbf{q}_\mathbf{r}^1,\cdots,\mathbf{q}_\mathbf{r}^L, \mathbf{r}_\mathbf{r}^1,\cdots,\mathbf{r}_\mathbf{r}^L) = \mathcal{J}_{\rm r}(\boldsymbol{\sigma}) \\
&\qquad\qquad-\sum\limits_{l=1}^L (\mathbf{q}_\mathbf{r}^{l})^T(\mathbf{A_r}\mathbf{u}_\mathbf{r}^{l} - \mathbf{I}_\mathbf{r}^{l} - \beta C_{\rm m}\mathbf{M}_{\rm u}\sum\limits_{i=1}^{2}\frac{\alpha_i}{\Delta t}\mathbf{u}_\mathbf{r}^{l-i} + \beta \mathbf{M}_{\rm iu} I_{\rm ion}({\rm U}\widetilde{\mathbf{u}}_\mathbf{r}^{l},~\mathbf{w}_\mathbf{r}^{l})) \\
&\qquad\qquad-\sum\limits_{l=1}^L (\mathbf{r}_\mathbf{r}^{l})^T(\dfrac{\mathbf{w}_\mathbf{r}^{l}-\mathbf{w}_\mathbf{r}^{l-1}}{\Delta t} + g({\rm U}\widetilde{\mathbf{u}}_\mathbf{r}^{l}, \mathbf{w}_\mathbf{r}^{l}) ) .
\end{split}
\end{equation}
The adjoint form of the reduced discretized Monodomain system can be constructed by setting $\dfrac{\partial \mathcal{L}_{\rm r}}{\partial \mathbf{u}_\mathbf{r}^{l}} = 0$ for $l = 1, \cdots, L$. It reads
\begin{equation}\label{}
\mathbb{Z}_{\rm u}^T \mathbf{A_{\rm m}}\mathbb{Z}_{\rm u} \mathbf{q}_\mathbf{r}^{l} =
\mathbf{d}_\mathbf{r}^{l} \quad\quad l\in\{1, \cdots, L \}
\end{equation}
with
\begin{align*}
& \mathbf{d}_\mathbf{r}^{l} =
\beta{\rm U}^T \{\mathbf{M}_{\rm iu}^T\mathbf{q}_\mathbf{r}^{l+2} \}\circ \partial_u I_{\rm ion}({\rm U}\tilde{\mathbf{u}}_\mathbf{r}^{l+2}, \mathbf{w}_\mathbf{r}^{l+2})
-2 \beta{\rm U}^T \{ \mathbf{M}_{\rm iu}^T\mathbf{q}_\mathbf{r}^{l+1} \}\circ \partial_u I_{\rm ion}({\rm U}\tilde{\mathbf{u}}_\mathbf{r}^{l+1}, \mathbf{w}_\mathbf{r}^{l+1}) \\
&\qquad -\partial_ug{\rm U}^T\tilde{\mathbf{r}}_\mathbf{r}^l
+(\mathbf{X}_{\rm u}\mathbf{u}_\mathbf{r}^l - \hat{\mathbf{u}}_{\rm meas}^l ) \chi_{\rm snap}^l
+ \beta C_{\rm m}\mathbf{M}_{\rm u}\sum\limits_{i=1}^{2}\frac{\alpha_i}{\Delta t}\mathbf{q}_\mathbf{r}^{l+i}.
\end{align*}
Here the operation $\circ$ means entry-wise product.
The dual gating variable $\mathbf{r}_\mathbf{r}^l$ is updated by the equation below derived from setting $\dfrac{\partial \mathcal{L}_{\rm r}}{\partial \mathbf{w}_\mathbf{r}^{l}} = 0$ for $l = 1, \cdots, L$:
\begin{equation}
\frac{\mathbf{r}_\mathbf{r}^{l+1}-\mathbf{r}_\mathbf{r}^{l}}{\Delta t} =
\partial_wg\mathbf{r}_\mathbf{r}^l + \beta \{\mathbf{M}_{\rm iu}^T\mathbf{q}_\mathbf{r}^{l} \}\circ \partial_{w}I_{\rm ion}({\rm U}\tilde{\mathbf{u}}_\mathbf{r}^l, \mathbf{w}_\mathbf{r}^{l}).
\end{equation}
For superscripts exceeding $L$, we set $\mathbf{q}_\mathbf{r}^{L+1} = \bsb{0} = \mathbf{q}_\mathbf{r}^{L+2}$ and $\mathbf{r}_\mathbf{r}^{L+1} = \bsb{0} = \mathbf{r}_\mathbf{r}^{L+2}$.
Based on the adjoint equations we then get the derivatives of $\mathcal{J}_{\rm r}$
\begin{eqnarray}
& \dfrac{{\mathcal D} \mathcal{J}_{\rm r}}{{\mathcal D} \sigma_{\rm mk}} =
-\sum\limits_{l=1}^L(\mathbf{q}_\mathbf{r}^{l})^T\mathbf{S}_{\rm ku} \mathbf{u}_\mathbf{r}^{l} +
\dfrac{\alpha}{2}\dfrac{\partial\mathcal{R}}{\partial\sigma_{\rm mk}} \label{DJ-rb1}
\end{eqnarray}
where
$\mathbf{S}_{\rm ku} = \mathbb{Z}_{\rm u}^T \mathbf{S}_{\rm k}\mathbb{Z}_{\rm u} \in \mathbb{R}^{N\times N}$ and ${\rm k}$ stands for ${\rm l}$ and ${\rm t}$. Notice that
$\mathbf{A_r} = \mathbb{Z}_{\rm u}^T \mathbf{A_{\rm m}}\mathbb{Z}_{\rm u} = \beta C_{\rm m}\frac{\alpha_0}{\Delta t}\mathbf{M}_{\rm u} + \sigma_{\rm ml}\mathbf{S}_{\rm lu} + \sigma_{\rm mt}\mathbf{S}_{\rm tu}.$
Eventually, the reduced inverse conductivity problem can be solved by a line search or trust region interior-point method \cite{nocedal2006}, for instance the primal-dual method or the logarithmic barrier method with a quasi-Newton update on the Hessian computation. For simplicity, we describe in Algorithm \ref{optRB-alg} only the line search barrier method. In particular, the norm $||\cdot||$ in step \ref{searchNorm} of Algorithm \ref{optRB-alg} could be customized. For instance, we can use the Euclidean norm of the polar coordinates to introduce a priori information weighting different values as we propose in Sec.~\ref{pod-DEIM-resultInv}. The parameters $\mu$ and $\varsigma$ are related to the enforcement of the unilateral constraints on the solution.
The optimization iteration updating in our simulation is based on the Optizelle package\footnote{http://www.optimojoe.com/products/optizelle/}, an open source library for nonlinear optimization.
If one wants to avoid off-line reduced bases construction, as an alternative the conductivity parameter can be estimated by an online POD-DEIM framework with adaptivity. This is explained by Algorithm \ref{adaptivePOD-alg}.
Specifically, one can solve the full-order Monodomain system at the initial guess, build the POD bases from
snapshots and solve the reduced inverse solver, then use the reduced-order solution as a new initial guess for the next computing cycle. In particular, for the inverse conductivity problem, instead of storing a unique basis by merging previous POD modes we typically store multiple reduced bases online. In the reduced-order computation, we still use polar coordinates to choose a single appropriate basis (see step \ref{searchNorm} of Algorithm \ref{optRB-alg}).
\begin{algorithm}[tp]
\caption{Optimization-POD-DEIM:~ \texttt{opt-reduced($\mathbf{\boldsymbol{\sigma}}^0$, $\{\mathbb{Z}_{\rm u}^i\}_{i=1}^s$, $\{\mathbb{Z}_{\rm ion}^i\}_{i=1}^s$, $\{\boldsymbol{\sigma}_{\rm gen}^i \}_{i=1}^s$, $\varsigma$)}}\label{optRB-alg}
\begin{algorithmic}[1]
\Input initial guess $\mathbf{\boldsymbol{\sigma}}^0\in\mathcal{C}_{\rm ad}$, POD basis $\{\mathbb{Z}_{\rm u}^i\}_{i=1}^s$ and $\{\mathbb{Z}_{\rm ion}^i\}_{i=1}^s$, POD bases generating parameters $\{\boldsymbol{\sigma}_{\rm gen}^i \}_{i=1}^s$, factor $\varsigma\in(0, 1)$
\Output estimated conductivity values $\mathbf{\boldsymbol{\sigma}}$
\State $\boldsymbol{\sigma}\leftarrow \boldsymbol{\sigma}^0, \mu \leftarrow 1, k\leftarrow 0, i_0\leftarrow -1$
\While{$k<k_{\rm max}$ and not converged}
\While{stopping tolerance not reached}
\State Search $i_* = arg\min\limits_{1\leq j\leq s}||\boldsymbol{\sigma}-\boldsymbol{\sigma}_{\rm gen}^j || $ \label{searchNorm}
\If{($i_*\neq i_0 )$}
\State Import the POD basis $\mathbb{Z}_{\rm u}^{i_*}$ and $\mathbb{Z}_{\rm ion}^{i_*}$
\State $i_0 \leftarrow i_*$
\EndIf
\State Solve $\mathbf{u_r}^{1\cdots L}$ with $\boldsymbol{\sigma}$ from $t^1$ to $t^L$, using bases $\mathbb{Z}_{\rm u}^{i_*}$ and $\mathbb{Z}_{\rm ion}^{i_*}$
\State Compute the perturbed cost function value $\mathcal{J}_{\rm r}(\boldsymbol{\sigma}) - \mu \sum\limits_i\log \big(h_i(\boldsymbol{\sigma})\big)$
\State Solve $\mathbf{q_r}^{1\cdots L}$ with $\boldsymbol{\sigma}$ from $t^L$ to $t^1$, using bases $\mathbb{Z}_{\rm u}^{i_*}$ and $\mathbb{Z}_{\rm ion}^{i_*}$
\State Compute the gradient $\nabla\mathcal{J}_{\rm r}(\boldsymbol{\sigma}) - \mu\sum\limits_i\frac{\nabla h_i(\boldsymbol{\sigma})}{h_i(\boldsymbol{\sigma})}$, using (\ref{DJ-rb1})
\State Update the inverse Hessian approximation and compute the search direction $\boldsymbol{v}^k$ (BFGS \cite{nocedal2006})
\State Set $\boldsymbol{\sigma} = \boldsymbol{\sigma} + \gamma_k \boldsymbol{v}^k$ where $\gamma_k \in (0, \infty)$ is computed from a line search
\State $k\leftarrow k+1$
\EndWhile
\State $\mu \leftarrow \varsigma\mu$
\EndWhile
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[tp]
\caption{Adaptive POD-DEIM Optimization}\label{adaptivePOD-alg}
\begin{algorithmic}[1]
\Input initial guess $\boldsymbol{\sigma}^0$, factor $\varsigma\in(0, 1)$
\Output estimated conductivity value $\boldsymbol{\sigma}^0$
\State $R_{\rm u} = [~]$, $R_{\rm ion} = [~]$, $\Sigma_{\rm gen} = [~]$
\While{stopping criterion not fulfilled}
\State Create snapshots: solve full-order $u(\boldsymbol{\sigma}^0)$ and $I_{\rm ion}(\boldsymbol{\sigma}^0)$\label{creSnap}
\State Build POD bases $\mathbb{Z}_{\rm u}(\boldsymbol{\sigma}^0)$ and $\mathbb{Z}_{\rm ion}(\boldsymbol{\sigma}^0)$ using above snapshots
\State Include new POD bases: $R_{\rm u} = [R_{\rm u}, \mathbb{Z}_{\rm u}(\boldsymbol{\sigma}^0)]$, $R_{\rm ion} = [R_{\rm ion}, \mathbb{Z}_{\rm ion}(\boldsymbol{\sigma}^0)]$, $\Sigma_{\rm gen} = [\Sigma_{\rm gen}, \boldsymbol{\sigma}^0]$
\State Solve the reduced-order problem $\min\limits_{\boldsymbol{\sigma}}\mathcal{J}_{\rm r}(\boldsymbol{\sigma})$:~ \texttt{opt-reduced($\boldsymbol{\sigma}^0$, $R_{\rm u}$, $R_{\rm ion}$, $\Sigma_{\rm gen}$, $\varsigma$)}
\State Set $\boldsymbol{\sigma}^0 \leftarrow \mbox{arg}\min\limits_{\boldsymbol{\sigma}}\mathcal{J}_{\rm r}(\boldsymbol{\sigma})$
\EndWhile
\end{algorithmic}
\end{algorithm}
\section{Model reduction in action: pitfalls and success}\label{DEIMresult}
In the beginning of this section, we will focus only on the offline-online procedure of POD-DEIM approximation. That is, we would like to obtain a set of reduced bases offline so that it can be use at any moment and for many times to solve an inverse conductivity problem. The observation on adaptive POD-DEIM will be provided at the last.
\subsection{POD-DEIM on the forward problem}\label{resultFWD}
To investigate the performance of POD-DEIM model reduction technique on the Monodomain system, we preliminarily solved the ROM on a slab $\Omega=[0, 5]\times[0, 5]\times[0, 0.5] \subseteq \mathbb{R}^3$ with 24272 mesh nodes. In each simulation, five stimuli of $ I_{\rm app} = 10^5 ~\mu \mbox{A/cm}^{3}$ were applied with four at the corners and one at the center of the domain for a duration of $1$~ms. We set the myocardium fibers to be constantly along the $x$-axis. The snapshots for POD basis construction were taken every 0.05 ms with a duration of 25 ms.
\begin{figure}[!tp]
\begin{minipage}{0.30\textwidth}
\centering{POD mode 1}
\includegraphics[scale=0.2]{3-2-podMode0.png}
\end{minipage}
\begin{minipage}{0.30\textwidth}
\centering{POD mode 2}
\includegraphics[scale=0.2]{3-2-podMode1.png}
\end{minipage}
\begin{minipage}{0.35\textwidth}
\centering{POD mode 3\hspace{1cm} \par}
\includegraphics[scale=0.35]{3-2-podMode2.pdf}
\end{minipage}
\begin{minipage}{0.30\textwidth}
\centering{POD mode 1}
\includegraphics[scale=0.2]{4-01-podMode0.png}
\end{minipage}
\begin{minipage}{0.30\textwidth}
\centering{POD mode 2}
\includegraphics[scale=0.2]{4-01-podMode1.png}
\end{minipage}
\begin{minipage}{0.35\textwidth}
\centering{POD mode 3\hspace{1cm} \par}
\includegraphics[scale=0.35]{4-01-podMode2.pdf}
\end{minipage}
\caption{POD modes of the transmembrane potential constructed with $\boldsymbol{\sigma}_{\rm gen} = [3, 2]$ (first row) and $\boldsymbol{\sigma}_{\rm gen} = [4, 0.1]$ (second row).}
\label{pod-u1-u2}
\end{figure}
In Fig.~\ref{pod-u1-u2} (first row), we plot the first three POD modes of the transmembrane potential constructed from snapshots of $u$ that were computed with conductivity parameter $\boldsymbol{\sigma}_{\rm gen} = [3, 2]$. The parameter chosen has $\sigma_{\rm ml}/\sigma_{\rm mt}$ close to one, which makes the tissue almost isotropic. This explains why the wave front propagated from the center of the slab tissue is almost circular. We also plot in the second row the leading POD modes of the transmembrane potential constructed with $\boldsymbol{\sigma}_{\rm gen} = [4, 0.1]$. The wave front in this case is elliptic since the ratio $\sigma_{\rm ml}/\sigma_{\rm mt} \gg 1$.
This comparison suggests that the POD modes constructed with different conductivity values are very weakly correlated. Therefore, the approach of extracting POD modes from combined snapshots computed with different parameter values can not achieve enough dimension reduction, as already pointed out in Fig.~\ref{svd-onePara-multi} (right).
\begin{figure}[!tp]
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.25]{errDiffRBsize_good_dot.pdf
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.258]{errDiffSigma_dot16x16.pdf}
\end{minipage}
\caption{Errors of $u$ by POD-DEIM approximation. Left: errors w.r.t.~the dimension of the POD basis, with $\dim(\mathbb{Z}_{\rm u})$ ranging from 34 to 46 and $\dim(\mathbb{Z}_{\rm ion})$ ranging from 82 to 106 ; Right: errors on $16\times 16$ different conductivity parameters, with fixed POD-basis dimensions: $\dim(\mathbb{Z}_{\rm u}) = 45$ and $\dim(\mathbb{Z}_{\rm ion}) = 100$.}
\label{err-pod-DEIM}
\end{figure}
To check the performance of reduced-order modeling, we took a uniform sampling on the conductivity parameter: 25 samples were generated over the domain $[1, 5]\times [0, 2]$. For each value of the parameter, snapshots were computed to construct offline a POD basis of the transmembrane potential (ionic current). In this way, 25 POD bases were available for importing during online computation of the reduced-order Monodomain model (Algorithm \ref{optRB-alg}).
As a preliminary assessment of the impact of model reduction, we computed the transmembrane potential with the POD-DEIM procedure. Fig.~\ref{err-pod-DEIM} reports the corresponding relative error. The errors in Fig.~\ref{err-pod-DEIM} (left) are with respect to the dimension of the POD basis, with $\dim(\mathbb{Z}_{\rm u})$ ranging from 34 to 46 and $\dim(\mathbb{Z}_{\rm ion})$ ranging from 82 to 106. The test conductivity parameter is $\boldsymbol{\sigma} = [2.6, 1.05]$. We observe that the POD-DEIM method provides stable and accurate approximation when the POD basis dimension is around 40 for $u$ and 90 for $I_{\rm ion}$.
In Fig.~\ref{err-pod-DEIM} (right) we plot the errors on $16\times 16$ different conductivity parameters, with fixed POD-basis dimensions: $\dim(\mathbb{Z}_{\rm u}) = 45$ and $\dim(\mathbb{Z}_{\rm ion}) = 100$. We observe that 60.55\% of the parameters in this test feature errors below 0.005, 21.48\% of them lead errors greater than 0.01 and they are mainly associated with conductivities at the boundary of the sample space. These errors indicate that a {\it uniform sampling} with 25 parameter values would be generally inappropriate for its use in the inverse problem of conductivity estimation.
\subsection{Domain of effectiveness (DOE) of the reduced basis}\label{DOEresult}
From previous results we argue that a single reduced basis may be effective only for problems with parameter values close to the generator one. To quantify the role of this basis, we introduce the concept of {\it ``domain of effectiveness''} (DOE). This is the region of the parameter space where a reduced-order solution provide accurate results as we specify later on.
We carry out a supportive study on the domains of effectiveness of different reduced bases, when each basis has a unique generating parameter $\boldsymbol{\sigma}_{\rm gen}$ that differs from others in the conductivity ratio.
\begin{figure}[!tp]
\begin{center}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{errDiffSigma_oneRB31dot3_impactArea2nd.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{errDiffSigma_oneRB1dot480dot7_impactArea2nd.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{errDiffSigma_oneRB30dot35_impactArea2nd.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{errDiffSigma_oneRB1dot480dot2_impactArea2nd.pdf}
\end{minipage}
\caption{Domains of effectiveness of different reduced bases. The relative errors (denoted by $e$) of the ROM solutions are indicated in different colors. Black: $e \leq 0.002$, Cyan(Gray): $0.002 < e \leq 0.005$, White: $e > 0.005$. The parameter space is partitioned by the red dash lines using an equi-spaced partition on $\arctan(\frac{\sigma_{\rm mt}}{\sigma_{\rm ml}})$. The POD basis generating parameter $\boldsymbol{\sigma}_{\rm gen}$ in each picture is indicated by a red triangle. }
\label{err-oneRB-noSensi}
\end{center}
\end{figure}
\begin{figure}[!tp]
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{errorDiffSigma+SensiSnap_30dot35_scaleAt3dot81dot32_2nd.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{errorDiffSigma+SensiSnap_30dot35_scaleAt4dot20dot48_2nd.pdf}
\end{minipage}
\caption{Domains of effectiveness of sensitivity-based reduced bases both generated by $\bsb{\sigma}_{\rm gen} = [3, 0.35]$. The left one is based on $\bsb{\sigma}_{\rm gen} + [\delta_{\rm l}, \delta_{\rm t}] = [3.8, 1.32]$ (marked by the red circle) and the right one takes $\bsb{\sigma}_{\rm gen} + [\delta_{\rm l}, \delta_{\rm t}] = [4.2, 0.48]$ (marked by the red circle).}
\label{err-oneRB-Sensi}
\end{figure}
Given a POD basis $\mathbb{Z}_{\rm u}$ and a test parameter $\boldsymbol{\sigma}$, we measure the effectiveness of $\mathbb{Z}_{\rm u}$ at $\boldsymbol{\sigma}$ by the relative error of reduced-order solution
$e(\boldsymbol{\sigma}) = {\sum\limits_{l=1}^L || \mathbb{Z}_{\rm u}\mathbf{u}^l_\mathbf{r}-\mathbf{u}^l ||^2}/{\sum\limits_{l=1}^L ||\mathbf{u}^l ||^2}$
where $\mathbf{u}^l$ is the full-order solution at time $t^l$ given $\boldsymbol{\sigma}$ and $\mathbf{u}^l_\mathbf{r}$ is the corresponding reduced-order solution solved with the reduced basis $\mathbb{Z}_{\rm u}$. We study the domains of effectiveness of different reduced bases and plot them in Fig.~\ref{err-oneRB-noSensi}. In each picture, a unique generating parameter $\boldsymbol{\sigma}_{\rm gen}$ (indicated by the red triangle) was used for the construction of the reduced basis. The reduced-order simulation errors are less than 0.002 at black points, between 0.002 and 0.005 at cyan/gray points, and greater than 0.005 at white points.
We define the DOE associated to the reduced basis as the region collecting black and cyan points, where the reconstruction error is less than 0.005.
In Fig.~\ref{err-oneRB-noSensi}, the parameter space is partitioned by the red dash lines using an equi-spaced partition on the range of $\arctan ({\sigma_{\rm mt}}/{\sigma_{\rm ml}})$.
It is interesting to see that the DOE of a reduced basis is apparently confined to an angular region of its generating parameter. The region is wide when the arc and the angle of $\boldsymbol{\sigma}_{\rm gen}$ are large (as in Fig.~\ref{err-oneRB-noSensi} upper left), and is relatively narrow while the arc or the angle is relatively small (as in Fig.~\ref{err-oneRB-noSensi} upper right, lower left and lower right).
This study confirms that the transmembrane potentials solved with the Monodomain model are strongly sensitive to the conductivity ratio and amplitude \cite{johnston2011sensitivity}.
As we mentioned earlier, a natural idea to enlarge the domain of effectiveness of a reduced basis is to include extra sensitivity snapshots in the POD basis construction. Following the idea in Compact POD \cite{Carlberg2008}, we took snapshots for the transmembrane potential $u$ as
\begin{equation*}
\bigg[ \mathbf{u}^1, \delta_{\rm l}\frac{\partial\mathbf{u}^1}{\partial \sigma_{\rm ml}}, \delta_{\rm t}\frac{\partial\mathbf{u}^1}{\partial \sigma_{\rm mt}}, \cdots , \mathbf{u}^L, \delta_{\rm l}\frac{\partial\mathbf{u}^L}{\partial \sigma_{\rm ml}}, \delta_{\rm t}\frac{\partial\mathbf{u}^L}{\partial \sigma_{\rm mt}} \bigg],
\end{equation*}
where $\delta_{\rm l}$ and $\delta_{\rm t}$ are scaling factors applied on the sensitivity snapshots.
We also took snapshots for the ionic current $I_{\rm ion}$ in a similar way. To investigate the feasibility of this concept, we chose a generating parameter $\bsb{\sigma}_{\rm gen} = [3, 0.35]$ and compare the DOE of the sensitivity-based RB with the older one shown in Fig.~\ref{err-oneRB-noSensi} lower left. Two sets of scaling factors were chosen in a way such that
$$\bsb{\sigma}_{\rm gen} + [\delta_{\rm l}, \delta_{\rm t}] = [3.8, 1.32]\qquad\bsb{\sigma}_{\rm gen} + [\delta_{\rm l}, \delta_{\rm t}] = [4.2, 0.48].$$
Two reduced bases were generated accordingly and their DOE are plotted in Fig.~\ref{err-oneRB-Sensi}. The ROM in this test took 100 POD modes for $u$ and 250 for $I_{\rm ion}$ which are appropriate numbers based on our experience. From this test, we notice that adding sensitivity snapshots is not significantly effective for widening the DOE. In addition, the largely increased necessary number of POD modes makes this idea not feasible for application in inverse problem such as conductivity estimation.
\subsection{POD-DEIM for the inverse problem}\label{pod-DEIM-resultInv}
\subsubsection*{The measures}
Considering the loss of accuracy in using the reduced-order model, we may need to acquire enough measures to maintain the stability of the inverse conductivity solver. In the following tests, we used $100\times 100$ measurement sites on the tissue surface, which is achievable in experiments\footnote{F. Fenton (GA Tech), personal communication.}.
We point out that when reducing the number of sites (down to 1000), we experienced just modest (expected) slowing of the iterative procedure and slightly worse estimation in most test cases.
We took snapshots every $dt_{\rm snap} = 2$ ms for a duration of $T = 30$ ms.
\begin{figure}[!tp]
\begin{center}
\includegraphics[scale=0.2]{76kMono_t28_bar.png}
\begin{minipage}{0.32\textwidth}
\includegraphics[scale=0.2]{76kMono_u_t28_noNoise_3D.png}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[scale=0.2]{76kMono_uPW_t28_noNoise_3D.png}
\end{minipage}
\begin{minipage}{0.32\textwidth}
\includegraphics[scale=0.25]{76kMono_u_t28_15Noise_3D.png}
\end{minipage}
\caption{Left: $u$ computed with exact finite element approximation on the nonlinear term $I_{\rm ion}$; middle: $u$ computed with component-wise evaluation on the nonlinear term $I_{\rm ion}$; right: synthetic measure of $u$ generated from the left by adding 15\% noise uniformly.}
\label{uPW+noise}
\end{center}
\end{figure}
The DEIM approximation for nonlinearity was applied under the assumption that the nonlinear term $I_{\rm ion}$ is evaluated component-wise in the FOM. This assumption greatly improves the computational efficiency of the full-order system while keeping enough accuracy when the mesh is fine. This is verified by a numerical experiment shown in Fig.~\ref{uPW+noise}. In this test the mesh contains 76832 nodes. As one can see, the component-wise computation on $I_{\rm ion}$ (Fig.~\ref{uPW+noise} middle) presents small error as compared with the exact finite element approximation (Fig.~\ref{uPW+noise} left). More importantly, the computational cost is reduced from 759.284 seconds to 53.6954 seconds (more than 95\% reduction).
Fig.~\ref{uPW+noise} (right) displays the synthetic measures generated by adding 15\% noise to the potential $u$ computed with exact evaluation of $I_{\rm ion}$. This typically represents the way we generate measures in our subsequent numerical tests.
\subsubsection*{Sampling}
As we have reported in Sec.~\ref{resultFWD}, a uniform sampling on the conductivity parameter to construct POD bases is not a viable approach, since it needs too many sample points and the corresponding ROM approximation still lacks accuracy. However, based on the DOE study in Sec.~\ref{DOEresult}, a nonuniform sampling with refinement in the ``small angle--short arc'' zone could substantially decrease the number of sample points while preserving the accuracy of the ROM approximation. This can be achieved by a sampling on polar coordinates, so to take advantage of the angular pattern of the DOE.
We illustrate an example in Fig.~\ref{10smpNewandTst}, where nine Gaussian nodes were generated to cover nonuniformly the parameter space and one extra node is added in the left corner to slightly extend the coverage of those samples. Specifically, the Gaussian points were obtained in the polar coordinates $[\rho, \theta]$:
\begin{eqnarray}
&\theta \leftarrow \frac{\theta_i+\theta_{i+1}}{2}, \quad\rho \leftarrow \frac{\rho_j^i+\rho_{j+1}^i}{2}\\
&\theta_i = \theta_{\rm max}-(\theta_{\rm max} - \theta_{\rm min})\cos\big(\frac{(i-1)\pi}{2(n_\theta-1)} \big) , \quad i=1,\cdots n_\theta \\
&\rho_j^i = \rho_{\rm max}-(\rho_{\rm max} - \rho_{\rm min})\cos\big(\frac{(j-1)\pi}{2(n_\rho^i-1)} \big), \quad j=1,\cdots n_\rho^i.
\end{eqnarray}
For the example above, we typically take $\theta_{\rm max}=\arctan(1.2)$, ~$\theta_{\rm min}=\arctan(1/14)$, ~$n_\theta=4$,~ $n_{\rho}^i=\max(6-i,3)$, ~$\rho_{\rm max}=6.5$, ~$\rho_{\rm min}=1.5$. The extra point in the left corner was given as $[\rho, \theta] \leftarrow [1.3, \frac{\theta_{\rm min}+\theta_{\rm max}}{2}]$.
\begin{figure}[tp]
\begin{center}
\includegraphics[scale=0.5]{10smpCustomized_new_addTestPt.pdf}
\caption{Ten samples (stars in red) generated by a nonuniform sampling on the polar coordinates of conductivity values. The ``small angle--short arc'' zone of the sample space is refined by the use of Gaussian nodes. The six blue dots will be used as test points.}
\label{10smpNewandTst}
\end{center}
\end{figure}
\subsubsection*{Conductivity estimation}
We conduct a group of tests on the performance of the POD-DEIM approach applying to the inverse conductivity problem, using the ten samples plotted in Fig.~\ref{10smpNewandTst}. For the sake of reliability of our results, six test points (blue dots in Fig.~\ref{10smpNewandTst}) were carefully chosen not too close to the sample points. With each test point, we solved the full-order Monodomain equation and then added 15\% noise to generate the synthetic measures.
We performed the numerical experiments on a 76832-node mesh with a simulation time step $\Delta t = 0.025$ ms. In each run of the reduced inverse conductivity solver, 35 POD modes were taken for $u$ and 80 for $I_{\rm ion}$, the norm $||\cdot||$ in step \ref{searchNorm} of Algorithm \ref{optRB-alg} was taken as the Euclidean norm of the polar coordinates. The optimization iteration started from a prescribed initial guess $\boldsymbol{\sigma}_{\rm initial}=[1.5, 1]$ and was constrained by a maximum iteration number 40.
Each simulation was performed on a processor having an Intel(R) Core(TM) i7-3740QM CPU @ 2.70GHz.
\begin{table}[tp]
\begin{center}
\caption{Conductivity estimation on a slab mesh with DOF = 76832. $T=~30$ ms, $\Delta t=~0.025$ ms, $dt_{\rm snap}=~2$ ms, noise = 15\%, $\boldsymbol{\sigma}_{\rm initial}=[1.5, 1]$. }
{\footnotesize
\begin{tabular}{|c||c|c|c|c|c|}
\hline\hline
& $\boldsymbol{\sigma}_{\rm exact}$ & $\boldsymbol{\sigma}_{\rm estimated}$ & \# fwd $|$ bwd & Total exe.~time & Time perc\\
\hline
Full Order & [3.2, 0.5] & [3.237, 0.625] & 51 $|$ 12 & 2720 s& 100\%\\
POD+DEIM & [3.2, 0.5] & [3.084 0.4277] & 103 $|$ 40 & 123.7 s & 4.5\%\\
\hline
Full Order & [4.5, 1] & [4.535, 1.099] & 58 $|$ 17 & 5409 s& 100\%\\
POD+DEIM & [4.5, 1] & [4.482 0.9846] & 161 $|$ 25 & 72.62 s & 1.3\%\\
\hline
Full Order & [5.5, 3] & [5.538, 3.059] & 109 $|$ 36 & 8699 s & 100\%\\
POD+DEIM & [5.5, 3] & [5.293 3.099] & 156 $|$ 24 & 58.25 s & 0.6\%\\
\hline
Full Order & [4, 2] & [4.045, 2.076] & 71 $|$ 23 & 5388 s & 100\%\\
POD+DEIM & [4, 2] & [3.762 1.965] & 94 $|$ 40 & 97.44 s & 1.8\%\\
\hline
Full Order & [3, 2] & [3.055, 2.077] & 59 $|$ 19 & 5626 s & 100\%\\
POD+DEIM & [3, 2] & [2.741, 2.148] & 83 $|$ 11 & 49.52 s & 0.9\%\\
\hline
Full Order & [6, 5] & [6.038, 5.049] & 50 $|$ 16 & 3366 s & 100\%\\
POD+DEIM & [6, 5] & [6.035, 4.910] & 54 $|$ 16 & 31.31 s & 0.9\%\\
\hline
\end{tabular}
}
\label{76k6testTable}
\end{center}
\end{table}
\begin{figure}[!h]
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{3d20d5_itrFunDJ_reRun.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{4d51_itrFunDJ_reRun.pdf}
\end{minipage}
\vspace{1.2cm}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{5d53_itrFunDJ_reRun.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{42_itrFunDJ_reRun.pdf}
\end{minipage}
\vspace{1.2cm}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.375]{32_itrFunDJ_reRun.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.375]{65_itrFunDJ_reRun.pdf}
\end{minipage}
\caption{Optimization iterations corresponding to Table \ref{76k6testTable}}
\label{6testPt-itrFunDJ}
\end{figure}
\begin{figure}[!h]
\begin{center}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{2016102432_itrPlane.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{2016102502_itrPlane.pdf}
\end{minipage}
\vspace{1cm}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{2016102434_itrPlane.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{2016102504_itrPlane.pdf}
\end{minipage}
\vspace{1cm}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{20161024362_itrPlane.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.38]{2016102506_itrPlane.pdf}
\end{minipage}
\caption{Optimization trajectory corresponding to Table \ref{76k6testTable} for test points: [4.5, 1], [4, 2], [6, 5]. The left column corresponds to the full-order inverse solver, while the right column corresponds to the reduced inverse solver.}
\label{3testPt-itrPlane}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{minipage}{0.45\textwidth}
\includegraphics[scale=0.32]{meas276k_PW_3d20d5_15noise_t262nd_cut.png}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\includegraphics[scale=0.32]{meas276k_PW_3d20d5Estim_t262nd_cut.png}
\end{minipage}
\caption{ Screenshots of the transmembrane potential (in mV) at $t = 26$ ms, computed on a real left ventricular
geometry reconstructed from SPECT images. The white arrows represent myocardial fiber orientation used in the simulation. Left: synthetic measure of the potential created by simulating with $\bsb{\sigma}_{\rm exact} = [3.2, 0.5]$ and adding 15\% uniform noise. Right: reconstruction of the potential computed with the estimated conductivity $\bsb{\sigma}_{\rm estimated} = [3.07, 0.425]$.}
\label{snapOnVentricle}
\end{center}
\end{figure}
The estimated conductivities are listed in Table \ref{76k6testTable}, from which we infer the following conclusion. (a) On average each solve of the reduced Monodomain system (or its dual), including reduced basis importing, takes about only 1 ms as compared with 60 ms for the full-order model. The computational cost reduction on the forward problem is of about two orders of magnitude. (b) For conductivity estimation, the reduced inverse solver returns slightly worse estimates than the full-order inverse solver, due to the loss of accuracy of the ROM. However, the total execution time can be reduced by at least 95\% and the estimation results using ROM are very satisfying with all the test points. Reasonable and similar results were obtained with the coarse 24272-node mesh, which we do not report for the sake of brevity. In summary, the message we retrieve from these results is that even with the 24272-node mesh, notwithstanding the obvious difference between the potential computed with the exact finite element approximation on nonlinearity and the potential solved by component-wise evaluation on nonlinearity, the conductivity estimations using ROM are still vey good.
The iterations corresponding to the 76832-node case are plotted in Fig.~\ref{6testPt-itrFunDJ}. Some of them are also shown in the plane as displayed in Fig.~\ref{3testPt-itrPlane}. We can see that the optimal pathway in the reduced inverse solver is similar to that in the full-order inverse solver.
\subsubsection*{Simulations on the left ventricle}
To demonstrate the independence of the method on geometries, we also performed simulations on a real left ventricular geometry reconstructed from SPECT images (see Figure \ref{snapOnVentricle}).
The main features of the cardiac fiber field (shown as white arrows in Figure \ref{snapOnVentricle}) were modeled by an analytical representation of the fiber orientation, which was originally proposed in \cite{colli2004} eqn.~(6.2) for an ellipsoid domain and properly adapted to a real domain retrieved from SPECT images (as done in \cite{gerardo2009model}). Other computational models of cardiac fibers, such as \cite{bayer2012}, are also available.
In this test, we still took measurements every $dt_{\rm snap} = 2$ ms for a duration of $T = 30$ ms.
The synthetic measure of the transmembrane potential at time $t=26$ ms is displayed in the left of Figure \ref{snapOnVentricle}, which was created by simulating with $\bsb{\sigma}_{\rm exact} = [3.2, 0.5]$ and adding 15\% uniform noise.
With the same sample points of the conductivity parameter mentioned before, we created the corresponding set of reduced bases and employ them in the POD-DEIM procedure. The main difference between current and previous tests lies in the geometric complexity of the fiber orientation. To capture enough variety of the propagation pattern on the ventricle, we need to increase the number of POD modes in the reduced space. A typical example is at the first test point $\boldsymbol{\sigma}_{\rm exact} = [3.2, 0.5]$. When 35 POD modes were taken for $u$ and 80 for $I_{\rm ion}$ we obtain $\boldsymbol{\sigma}_{\rm estimated} = [2.491, 0.6]$. After increasing the number of POD modes to 45 and 90 respectively, the estimation is clearly improved to $\bsb{\sigma}_{\rm estimated} = [3.07, 0.425]$.
For further tests we refer to Table \ref{22kTable}. On the ventricle, the reduced inverse solver lost certain accuracy as compared to the slab tissue, but still returned acceptable estimations of the conductivity. We reconstruct the transmembrane potential corresponding to the first test point and show the snapshot at time $t=26$ ms in the right of Figure \ref{snapOnVentricle}. As demonstrated by the figure, the reconstruction matches the synthetic measure on the wave front.
\begin{table}[tp]
\begin{center}
\caption{Conductivity estimation by the reduced inverse solver on a left ventricular mesh: $\boldsymbol{\sigma}_{\rm initial}=[1.5, 1]$. }
\label{22kTable}
{\footnotesize
\begin{tabular}{c|cccccc}
\toprule
$\boldsymbol{\sigma}_{\rm exact}$ & [3.2, 0.5] & [4.5, 1] & [5.5, 3] & [4, 2] & [3, 2] & [6, 5] \\ %
\midrule
$\boldsymbol{\sigma}_{\rm estimated}$ & [3.07, 0.425] & [4.757, 0.94] & [5.356, 3.151] & [3.933 1.782] & [2.8, 2.07] & [6.186, 5.488]\\
\bottomrule
\end{tabular}
}
\label{22k6testTable}
\end{center}
\end{table}
\subsubsection*{Adaptive POD-DEIM}
We finally test the viability of adaptation by an online procedure. As described in Sec.~\ref{pod}, POD adaptivity has been studied in many publications, such as \cite{zahr2015progressive}. The main purpose is to let the sampling procedure take into account the optimization trajectory. Since we have observed that merging snapshots from different conductivity values doesn't reduced the necessary number of POD modes for accurate potential approximation, we still used multiple reduced bases in online reduced-order computation.
The following experiments were conducted with the same six test values as before, on the slab mesh having 76832 nodes. The measurement data were still generated synthetically with extra 15\% noise. We execute Algorithm \ref{adaptivePOD-alg} given initial guess $\bsb{\sigma}_{\rm initial} = [1.5, 1]$ and particularly control the reduced-order minimization problem inside the loop by a maximum iteration number 20. Each single online basis $\mathbb{Z}_{\rm u}$ has size 35 and $\mathbb{Z}_{\rm ion}$ has size 80. Accordingly to our experience, five cycles of reduced-order optimizations are generally enough to obtain stable convergence. We list the estimated conductivities in Table \ref{76k6testOnlineTable}. As one can see, the accuracy of conductivity estimation is promising, and the execution time is reduced to 10.2\%--38.8\%. Although the online computational reduction is not as intense as the offline-online procedure, this is an effective alternative use of model reduction especially when we have no offline reduced basis at hand.
To have an insight on the optimization trajectory, we report in Fig.~\ref{onlinePOD4d51} (resp.~Fig.~\ref{onlinePOD65}) the five cycles of iterations correspondingly to $\bsb{\sigma}_{\rm exact} = [4.5, 1]$ (resp.~$\bsb{\sigma}_{\rm exact} = [6, 5]$).
\begin{table}[tp]
\begin{center}
\caption{Conductivity estimation by online adaptive POD-DEIM. }
{\footnotesize
\begin{tabular}{|c||c|c|c|c|c|}
\hline\hline
& $\boldsymbol{\sigma}_{\rm exact}$ & $\boldsymbol{\sigma}_{\rm estimated}$ & \# fwd $|$ bwd & Total exe.~time & Time perc\\
\hline
Full Order & [3.2, 0.5] & [3.237, 0.625] & 51 $|$ 12 & 2720 s & 100\% \\%sigma2016102431 ok
Adaptive POD+DEIM & [3.2, 0.5] & [3.223, 0.6635] & 544 $|$ 89 & 1055 s & 38.8\%\\
\hline
Full Order & [4.5, 1] & [4.535, 1.099] & 58 $|$ 17 & 5409 s & 100\% \\
Adaptive POD+DEIM & [4.5, 1] & [4.522, 0.9839] & 507 $|$ 87 & 733.5 s & 13.6\%\\
\hline
Full Order & [5.5, 3] & [5.538, 3.059] & 109 $|$ 36 & 8699 s& 100\% \\
Adaptive POD+DEIM & [5.5, 3] & [5.294, 3.212] & 519 $|$ 93 & 889.4 s & 10.2\%\\
\hline
Full Order & [4, 2] & [4.045, 2.076] & 71 $|$ 23 & 5388 s& 100\% \\
Adaptive POD+DEIM & [4, 2] & [3.9, 2.084] & 318 $|$ 81 & 812.6 s & 15.1\%\\
\hline
Full Order & [3, 2] & [3.055, 2.077] & 59 $|$ 19 & 5626 s & 100\% \\
Adaptive POD+DEIM & [3, 2] & [3.237, 1.784] & 366 $|$ 80 & 995.9 s & 17.7\%\\
\hline
Full Order & [6, 5] & [6.038, 5.049] & 50 $|$ 16 & 3366 s & 100\%\\
Adaptive POD+DEIM & [6, 5] & [5.686, 5.064] & 425 $|$ 95 & 709 s & 21\%\\
\hline
\end{tabular}
}
\label{76k6testOnlineTable}
\end{center}
\end{table}
\begin{figure}[tp]
\begin{center}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{4d51_itrFunDJ_online2016102672.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{2016102672_itrPlane.pdf}
\end{minipage}
\caption{Optimization with adaptive POD-DEIM corresponding to Table \ref{76k6testOnlineTable} for test point $\bsb{\sigma}_{\rm exact} = [4.5, 1]$. Left: Optimization iterations. The vertical dash lines denote the position where a new POD basis was generated. Each optimization cycle (five in total) is constrained by a maximum iteration number 20. Right: Optimization trajectory. The points for online reduced basis generation are marked by the black dots.}
\label{onlinePOD4d51}
\end{center}
\end{figure}
\begin{figure}[tp]
\begin{center}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{65_itrFunDJ_online2016102676.pdf}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[scale=0.45]{2016102676_itrPlane.pdf}
\end{minipage}
\caption{Optimization with adaptive POD-DEIM corresponding to Table \ref{76k6testOnlineTable} for test point $\bsb{\sigma}_{\rm exact} = [6, 5]$. Left: Optimization iterations. The vertical dash lines denote the position where a new POD basis was generated. Each optimization cycle (five in total) is constrained by a maximum iteration number 20. Right: Optimization trajectory. The points for online reduced basis generation are marked by the black dots.}
\label{onlinePOD65}
\end{center}
\end{figure}
\section{Conclusions}
Cardiac conductivity estimation is a critical step for bringing computational electrocardiology in clinical settings. Currently there are no established procedures, available methodologies are computationally demanding. Model reduction is generally challenged by the mathematical features of the Monodomain problem and this prevents ``classical'' approaches like POD (coupled with DEIM) to be promptly applied. This paper gives a first contribution to model reduction applying to the inverse conductivity problem.
As a matter of fact, the main challenge of model reduction lies in the POD basis construction. To quantify this effect, we have introduced the concept of DOE. We have observed that the DOE of a POD basis based on a single generating parameter is narrow, especially when the amplitude of the conductivity value is small. This phenomena has also been shown in \cite{Boulakia2011POD} where POD was applied to estimate ionic model parameters. The situation in our case is even worse, considering the fact that we can not group snapshots for basis construction from different generating parameters, due to the strong sensitivity of the transmembrane potential to the conductivities. In other words, a unique POD basis is inadequate for the inverse conductivity problem, where the simulations are performed with various conductivity parameters.
Another challenge that deserves to mention is the failure of adding sensitivity snapshots, which is expected to highly improve the effectiveness of the POD basis in other problems \cite{Carlberg2008}. We find that putting extra sensitivity snapshots into the snapshot matrix of state variables does not significantly enlarge the DOE of the constructed RB as much as we desire. Even if it did to some extent, the required number of POD modes is almost tripled, hence it is not appropriate for online application.
Nevertheless, there are still some interesting aspects that finally lead to a success. We detect that the DOE of a reduced basis is confined to an angular region of its generating parameter, and the region would be enlarged if the parameter amplitude gets larger. Based on this, we sample the parameter space utilizing the polar coordinates and the Gaussian nodes. A sample set of size ten is then obtained.
The usage of multiple POD bases, each generated with a sampled parameter, provides satisfactory results. Overall, by utilizing this POD-DEIM reduced-order model, the computational effort can be reduced by at least 95\% in conductivity estimation.
This work opens several interesting challenges to be investigated in future works. In particular, we would like to theoretically quantify the conductivity estimation error caused by reduced-order modeling. An error analysis for some particular optimal control problems has been studied in \cite{Gubisch2016A-pos-33371,kammann2013posteriori}, but the work on conductivity estimation is still open sine the control parameter appears in the differential core of the system.
We also plan to extend the sampling strategy proposed here to 3D Bidomain inverse conductivity problem.
\section*{Acknowledgements}
This work has been supported by the NSF Project DMS 1412973/1413037 ``Novel Data Assimilation Techniques in Mathematical Cardiology''. We thank the referees for extremely valuable suggestions and remarks in improving this paper.
\bibliographystyle{model1b-num-names}
|
1711.02259
|
\section{Introduction}\label{sec1}
Based on the AdS/CFT correspondence \cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}, one may study bulk gravity dynamics in a rather precisely defined setup.
Numerous aspects of the correspondence have been checked for many examples in the zero temperature limit and well understood by now.
In this non thermal regime,
there are no conceptual issues such as non-unitarity of description. On the other hand,
a precise understanding of the gravity description of thermal field theories is still lacking, which is relevant to the problem of the black hole
information paradox \cite{Hawking:1976ra}.
In Ref.~\cite{Bak:2017xla}, rather general perturbations of BTZ black hole \cite{Banados:1992wn} are constructed
which are dual to state
deformations of the thermofield double theory \cite{Takahasi:1974zn}.
This construction may be viewed as
a realization of micro thermofield deformations of the BTZ geometry and may serve as an ideal setup to clarify the issues of black hole dynamics in AdS spacetime.
The perturbation makes one-point function of operator nonvanishing initially. This one-point function decays exponentially in general, which is
contradicting with the unitarity of the CFT defined on $R \times S^{d-1}$.
Furthermore the future horizon area of this classical geometry grows in time in general.
Interpretation of the corresponding geometric entropy is the key issue which we would like to address in this note. We would like to clarify the nature of gravity description
by investigating the degrees relevant to the growing of gravitational entropy.
In short, the gravity description of thermal field theories
naturally involves a coarse-graining of certain degrees. We identify the coarse-grained degrees as nonclassical degrees such as Hawking radiation
quanta, which cannot be seen in the classical gravity description.
The interaction between
the classical gravity degrees with these coarse-grained nonclassical degrees is tiny but nonvanishing and hence they work as a dark sector from the view point of
the classical gravity description. In this note, we shall describe the development of entanglement between the two and transfer of
information from the classical gravity to the coarse-grained degrees.
We shall also discuss how one can regain the unitarity from the view points of both
the gravity and the boundary field theory.
In section \ref{sec2}, we review the general perturbations of BTZ background studied in Ref.~\cite{Bak:2017xla} and the corresponding AdS/CFT
correspondence \cite{Maldacena:2001kr}. The basic puzzles of entropy and unitarity of the gravity description are described in detail. We clarify the degrees responsible for coarse-graining of bulk gravity description.
In section \ref{sec3}, we present a field theory model to explain the entropy growth of the perturbation as forming entanglement between the classical gravity
and the coarse-grained degrees. We also discuss how one can regain unitarity in this model.
In section \ref{sec4}, we explain what happens from the bulk view point. This represents the quantum version of ER = EPR \cite{
Einstein:1935tc,Einstein:1935rr,
Maldacena:2013xja} where ER bridges are formed
between the classical and nonclassical degrees. The latter degrees cannot be seen from the view point of the classical gravity description.
\section{Eternal black holes and perturbation of states}\label{sec2}
The eternal black hole in AdS spacetime is dual to the thermofield double of a CFT, which is maximally entangled but non interacting \cite{Maldacena:2001kr}.
The Hamiltonian of the total system is given by\footnote{The left and the right systems do not have to be the same in general,
which leads to non maximal entanglement of the left and the right CFT's \cite{Bak:2007jm}. }
\begin{eqnarray}
H_{tfd}= H_L + H_R =H \otimes 1 + 1 \otimes H
\end{eqnarray}
where there are absolutely no interactions between the left and the right CFT's\footnote{Interactions between the two CFT's can be introduced by a double trace deformation in \cite{Gao:2016bin}, which leads to a traversable wormhole solution.}. We denote the CFT Hamiltonian by
$H$.
The initial unperturbed thermal vacuum state is given by a particularly prepared entangled state
\begin{eqnarray}
|\Psi(0) \rangle =\frac{1}{\sqrt{Z}} \sum_{n,n'} \, \langle n | U | n' \rangle \, |n' \rangle \otimes | n \rangle =\frac{1}{\sqrt{Z}}
\sum_{n} \, e^{-\frac{\beta}{2} E_n }\, |n\rangle \otimes | n \rangle
\label{initial}
\end{eqnarray}
with a Euclidean evolution operator $U=U_0 = e^{-\frac{\beta}{2} H}$ and $Z$ denoting the normalization constant. The entanglement here is maximal for a given temperature $T=1/\beta$.
In Figure \ref{figbtz}, we illustrate the Penrose diagram of the corresponding BTZ black hole in
three dimensions where two boundary spacetimes connected though an ER bridge representing the left-right entanglement. Here two copies of a 2d CFT live
on the left and the right boundary spacetimes respectively.
\begin{figure}[th!]
\centering
\includegraphics[width=4.5cm,clip]{btzete.pdf} \hskip2cm
\includegraphics[width=4.5cm,clip]{erbtz.pdf}
\caption{\small On the left, the Penrose diagram of the BTZ black hole is depicted. Two copies of a CFT live in the left and the right boundaries denoted by
$L$ and $R$ respectively. On the right, we draw the ER bridge connecting the two boundaries. The left-right entanglement equals to the ER bridge of BTZ black hole connecting the left and the right boundaries.
}
\label{figbtz}
\end{figure}
This BTZ initial state can be deformed by introducing a mid-point insertion of operators as
\begin{eqnarray}
U = e^{-\frac{\beta}{4} H} e^{-\sum_k c_k O_k} e^{-\frac{\beta}{4} H}
\label{initialnew}
\end{eqnarray}
where $O_k$ denote general operators of the underlying CFT \cite{Bak:2017xla}. Then we evolve the system in time
in a standard manner
as
\begin{eqnarray}
|\Psi(t) \rangle = e^{-i H_{tfd} \, t} \, |\Psi(0) \rangle
\end{eqnarray}
The one-point function
\begin{eqnarray}
\langle O_R \rangle = \langle \Psi(t) | 1\otimes O |\Psi(t) \rangle
\end{eqnarray}
was computed in \cite{Bak:2017xla} to the leading order in its coefficient $c_k$ using the conformal perturbation theory and the two-point function \cite{KeskiVakkuri:1998nw, Maldacena:2001kr}
\begin{eqnarray}
\frac{1}{Z} {\rm tr} O(t, \varphi) O(t', \varphi') e^{-\beta H }
=
\sum^\infty_{m=-\infty}\frac{{\cal N}_\Delta}{
\left[\cosh \frac{2\pi}{\beta}( t - t') - \cosh\frac{2\pi \ell}{\beta} (\varphi-\varphi'+2\pi m)- i \epsilon\right]^{\Delta}}
\label{twopoint}
\end{eqnarray}
where $\Delta$ is the dimension of the scalar primary operator $O$ and ${\cal N}_\Delta$ \cite{Bak:2017rpp} is an appropriate normalization. Here for the sake of
an illustration, we
consider 2d CFT on $R \times S^1$ where we use coordinates $(t, \varphi)$ with $\varphi \sim \varphi+ 2\pi$. (Of course, our presentation can be generalized to
other dimensions in a straightforward manner.)
Then the field theory result agrees with the holographic computation of one-point function precisely \cite{Bak:2017xla}. One finds that the resulting expression
decays exponentially
in time, which contradicts with the quantum Poincare recurrence theorem \cite{Dyson:2002pf}. Indeed the above expression of the two-point function is not exact but
involves
a large $N \, (\sim 1/\sqrt{G\ell^{1-d}})$ approximation\footnote{Here $\ell$ denotes the AdS radius and we shall consider the boundary $d$ dimensional CFT on $R \times S^{d-1}$ where the radius of the sphere is set to be $\ell$. For the simplicity of our presentation, $\ell$ is frequently set to be unity.
One can be more precise about the large $N$ limit for the well known AdS$_5$/CFT$_4$ correspondence \cite{Maldacena:1997re} where the boundary CFT is given by ${\cal N}=4$ SU(N) super
Yang-Mills theories for which $N$ is identified as the number of colors.}. To understand this classical gravity approximation, we need to clarify the nature of the above large $N$ limit in
the dual field theories.
Next we discuss the issue of entropy involved with the above deformations.
Let us first
introduce the
reduced density matrix
\begin{eqnarray}
\rho_R (t)= {\rm tr}_L \, |\Psi(t) \rangle \langle \Psi(t) |
\end{eqnarray}
and then the corresponding von Neumann entropy
\begin{eqnarray}
S_R = -{\rm tr}_{R\,} \rho_R (t) \log \rho_R (t)
\label{vonr}
\end{eqnarray}
gives us an entanglement measure
of the left-right system. The perturbation in the above makes this left-right entanglement non-maximal initially.
One finds this L-R entanglement is time independent since one may undo the similarity transformation of
the time evolution of the reduced density matrix inside the trace. Since there are no interactions between the left right system, any net information of one side
cannot be transferred to the other side. This is consistent with the causal structure of the eternal black hole spacetime where
the left and the right boundaries are causally
disconnected from each other.
In Ref.\,\,\cite{Bak:2017xla}, it was also shown that, for any perturbation of initial state in (\ref{initial}),
there exists a corresponding one-to-one deformation of the eternal black hole spacetime. See Figure \ref{figtdbh} for a Penrose diagram of perturbed
BTZ black hole.
These geometries show two general
features. Any initial perturbations
decay exponentially in time with the time scale of so called relaxation time scale
$\beta$. In the gravity side, the perturbation outside horizon generically falls into the horizon and the corresponding horizon
perturbations decay away exponentially in the time scale of the relaxation time scale. This is generic in thermal AdS spacetime. With black holes,
the decay is explained by the physics of quasi-normal modes which makes any perturbations of the black horizon decay away exponentially. This is consistent with
the field theory result using the large $N$ approximation in (\ref{twopoint}).
\begin{figure}[hb!]
\vskip0.5cm
\centering
\includegraphics[width=4.5cm,clip]{pennew.pdf}
\caption{\small Penrose diagram of perturbed BTZ black hole is depicted here. The Hamiltonian of the system is not perturbed but only initial state
is perturbed in a non maximally entangled manner. The diagram is elongated horizontally in such a way that the two sides are causally disconnected.
}
\label{figtdbh}
\end{figure}
The second is regarding entropy whose precise nature
is our primary concern in this note. In geometric side, the leading order gravitational entropy is given by the horizon area divided by $4 G$. For instance we
take the view point of the bulk observer outside of the future horizon from the right. The future horizon is nondecreasing monotonically in time
reaching its asymptotic value after scrambling of the initial perturbation where the change of area $\Delta A$ is given by
\begin{eqnarray}
\Delta A = A(\infty)-A(0)
\end{eqnarray}
which is finite
and the function of the coefficients $c_k$.
In the corresponding gravitational entropy is given by
\begin{eqnarray}
S_{grav}(t)= \frac{A(t)}{4 G} + S_\delta (t)
\label{grentropy}
\end{eqnarray}
where $S_\delta$ denotes the contribution of order $G^0$ such as bulk entanglement entropy \cite{Bombelli:1986rw} which may be ignored in our discussion.
In the late time, the area approaches
its final value $A(\infty)$ exponentially in time again with
the relaxation time scale. This leading term is the classical
gravity contribution.
It is rather clear that this entropy
cannot be identified with the L-R entanglement entropy
introduced in the above. The L-R entanglement entropy is time independent
whereas the above is time dependent.
Understanding these two aspects will be our primary concern of this note. As was already noted in \cite{Bak:2007jm}, these two at least indicate that
the classical gravity description cannot be fully fine-grained. If it were fully fine-grained, the exponential decay of perturbation
is not possible
since it is contradicting with the quantum Poincare recurrence theorem \cite{Dyson:2002pf}. It is also contradicting with the time independence of the L-R entanglement entropy
which is dictated by the unitarity of the underlying system. Therefore it is rather clear that the classical gravity description cannot be fully fine-grained. This certainly
implies that the classical gravity description is involving an inevitable coarse-graining. In this note, we would like to clarify the nature of gravity description in this respect.
Of course the gravity description involves a large $N$ (or large central charge) approximation but the main question is which and how degrees are
coarse-grained in the gravity description. We shall not be fully general here and our main focus is on the perturbations around thermofield double state.
The unperturbed case corresponds to the BTZ spacetime, which involves a maximal entanglement between the left and right system leading to an
equilibrium finite-temperature thermal system from the view point of one-sided
observer. The perturbations then make the system non-maximally entangled.
We divide each-sided system by $B$ and $C$ where $B$ is for the degrees responsible for the bulk classical gravity
and $C$ the system of the remaining degrees that are coarse-grained by the classical gravity description.
Let us choose the right-side CFT whose
Hamiltonian is given by $H$ in the above.
The part $C$ is the complement of $B$ with $R=B+C$ and forms an environment
of $B$ in some sense. There has to be nonvanishing interactions between $B$ and $C$. If there were no interactions, then the entropy would be time
independent after integrating out those degrees of $C$. The interaction has to be as weak as $O(1/N)$.
Otherwise it should be visible from the view point of
the classical
gravity.
The part $C$ here is describing the quantum gravity degrees such as Hawking radiation of quanta whose existence is invisible from the view point of
the classical gravity.
These include any of nonclassical degrees which are produced by
$O(1/N)$ interactions.
Namely for instance one
cannot see any classical signal of such quanta from the BTZ background though their mass may be included into the definition of the energy of the
dual field theory.
Then $B$ is representing the part described by the classical gravity
fluctuations. In this context, we shall refer the degrees of $C$ as nonclassical cloud (NCC) degrees\footnote{Here ``non-classicality" or ``quantum" refers
to $1/N$ quantum gravity effects in the bulk that basically correspond to joining and splitting of strings in the string theory.},
which are coarse-grained from the view point of classical gravity
description. These degrees will be in thermal equilibrium with the black hole states forming a quantum cloud around the black hole since there is no net radiation
for the case of BTZ black hole or the large AdS Schwarzschild black hole of other dimensions in general.
Below
we shall present such a model of coarse-graining degrees and explain the above two characteristics of the eternal black hole dynamics.
\section{Field theory model}\label{sec3}
In this section, we model the above phenomena observed in the gravity side. As we described already, $B$ is representing
the part described by the bulk fluctuation of the classical gravity
that is one-to-one correspondence with the
boundary deformation of the thermofield initial state with mid-point insertion of CFT
operators. These operators are basically dual to those bulk fields, which are basic elements of the gravity description.
We shall represent the corresponding field-theory fluctuation above the thermofield vacuum by
\begin{eqnarray}
|\psi_B \rangle =\sum_i \alpha_i |i \rangle
\end{eqnarray}
where $ |i \rangle $ is the eigenstate of $H_B$ with eigenvalue $ \epsilon_i$ with $i=1,2,\cdots$.
The coarse-grained degrees in $C$ are responsible for the Poincare recurrence to happen in the full system $R=B+ C$. These degrees are excited in the black
hole phase nonclassically forming the Hawking radiation cloud as we described previously. In the zero temperature limit, basically one may ignore these degrees
since their occupation numbers are practically zero. Hence the effect of $C$ disappears
in the zero temperature limit.
It is not directly to do with
higher derivative corrections since those higher derivative corrections are classical, which are well controlled in the AdS/CFT correspondence.
Thus these higher derivative corrections
will be included into the part $B$ of the classical gravity.
Our assumption is rather mild here. First of all, the interaction between
$B$ and $C$ has to be extremely weak.
Hence the degrees in $E=L+C$ should work as a dark sector from the view
point of the gravity system $B$ where the left side $L$ is completely dark. In the zero temperature limit of non thermal field theories, the occupation number of NCC degrees vanishes
and their effect can be
ignored completely.
This is why the gravity description in the zero temperature limit is unitary and fully fine-grained.
On the other hand, at finite temperature,
the number of excited NCC degrees in $C$ can be estimated as of order $N^0$ due to their semi-classical nature, which turns
out to be still large enough\footnote{In the black hole phase, their number
denoted by $N_s$
can be estimated as $N_s \sim f(\lambda, Tr) {V_{S^{d-1}}}(T r)^{d-1}$ where $r$ is the radius of the sphere $S^{d-1}$, ${V_{S^{d-1}}}$ the volume of a unit
$d-1$ sphere and $\lambda$ denotes some moduli
parameter such as 't Hooft coupling in 4d ${\cal N}=4$ SYM theories. $f(\lambda, Tr)$ is counting the bulk fields whose Hawking radiation quanta are significantly excited.
Since $rT \gg 1$ and $f(\lambda, Tr)$ is large in the black hole phase, the number of excited $NCC$ degrees has to be large.}.
Thus many such NCC degrees are excited which we label by $I=1, 2, \cdots, N_s $.
The number of relevant states for the full environment of $E=L+C$ is of order $M^{N_s} \sim e^{\alpha N^2}$ where
we assume
there are $M$ states for each NCC degree.
Then, for the $I$-th
NCC degree, the relevant state is given by
\begin{eqnarray}
|\psi \rangle_I = \sum^M_{m=1} c^I_m \, | m \rangle_I
\end{eqnarray}
where we use the basis defined by $H_I | m \rangle_I =\frac{1}{M} e_m | m \rangle_I$ where $M$ is as large as $M \sim e^{\alpha N^2/N_s}$.
The full relevant Hilbert space will be described by the tensor product
\begin{eqnarray}
| \psi_E \rangle = \prod^{N_s}_{I=1} |\psi \rangle_I
\end{eqnarray}
We take the initial state of $E$ as
\begin{eqnarray}
|\psi \rangle_I (0)=\frac{1}{\sqrt{M}} \sum^M_{m=1} e^{i \varphi^I_m} \, | m \rangle_I
\end{eqnarray}
with random phases $\varphi^I_m$ where all eigenstates are equally probable.
The interaction Hamiltonian is given by \footnote{We assume here the interaction Hamiltonian is diagonalized by the basis $|i \rangle \otimes |m\rangle_I$. This assumption is not necessary and just for the simplicity of our presentation.}
\begin{eqnarray}
H_{int} = \frac{g_0}{N'} H_B \otimes \sum_I H_I
\end{eqnarray}
where we assume that $N'$ goes to infinity as $N_s$ tends to infinity.
We shall assume that $H_B$ responsible of interactions is non-degenerate.
Thus we begin with an initial state
\begin{eqnarray}
\sum_i \alpha_i\, |i\rangle \otimes |\psi_E \rangle (0)
\end{eqnarray}
The entanglement entropy between $B$ and $E$ is zero since the reduced density matrix tracing over $E$ is
still pure. Then the time evolution of the system is given by
\begin{eqnarray}
\sum_i \alpha_i \,|i\rangle \otimes |\psi^i_E \rangle (t)
\end{eqnarray}
where one has
\begin{eqnarray}
| \psi^i_E \rangle (t) = \prod^N_{I=1} |\psi^i \rangle_I (t)
\end{eqnarray}
with
\begin{eqnarray}
|\psi^i \rangle_I (t)=\frac{1}{\sqrt{M}} \sum^M_{m=1} e^{i \varphi^I_m-i \frac{g_0 \epsilon_i e_m}{N' M}t} \, | m \rangle_I
\label{sistate}
\end{eqnarray}
Then the overlap can be evaluated as
\begin{eqnarray}
\langle \psi^i |\psi^{j} \rangle_I =\frac{1}{M} \sum^M_{m=1} e^{i g_0 \frac{ \epsilon_i -\epsilon_j}{N'} \frac{e_m}{M}t}
\end{eqnarray}
Since $M$ is large enough, we can turn this to an integral
\begin{eqnarray}
\langle \psi^i |\psi^{j} \rangle_I =\int^{\infty}_{-\infty} dx \, D(x) \, e^{i g_0 \frac{ \epsilon_i -\epsilon_{j}}{N'}x t }
\end{eqnarray}
where we introduce the normalized density of state $D(x)$ by
\begin{eqnarray}
D(x)= \frac{1}{M} \sum^M_{m=1} \delta \Big(x-\frac{e_m}{M}\Big)
\end{eqnarray}
which satisfies the normalization
\begin{eqnarray}
\int^{\infty}_{-\infty} dx\, D(x) =1
\end{eqnarray}
For instance choosing $D (x)=D_0(x)=\frac{1}{2 \pi} \Theta(\pi-x)\Theta(x+\pi)$ leads to
\begin{eqnarray}
\langle \psi^i |\psi^{j} \rangle_I = \frac{\sin \pi \frac{ \epsilon_i -\epsilon_{j}}{N'}g_0 t }{\pi \frac{ \epsilon_i -\epsilon_{j}}{N'} g_0t}
\end{eqnarray}
This leads to the overlap
\begin{eqnarray}
\langle \psi_E^i |\psi_E^{j} \rangle = e^{-\frac{\pi^2}{6} (\epsilon_i -\epsilon_{j})^2 g_0^2 t^2 }
\end{eqnarray}
where we assume $ N_s={N'}^2$ and take the large $N_s$ limit.
This decay rate is too fast since in the gravity side we have just exponential decay rate in the late time.
Hence this example is not consistent with our bulk gravity dynamics described in section \ref{sec2}.
In this example,
let us take
$M$ and $N'$ to be finite and further assume $M$ is even integer. By taking $e_m =m -M/2$ and $\epsilon_i = i$, one finds $D(x)=D_0(x)$ in the large $M$ limit.
With this choice of $e_m$ and $\epsilon_i$, the state in (\ref{sistate}) will return to its initial state as
\begin{eqnarray}
|\psi^i \rangle_I (0) =|\psi^i \rangle_I (t_{rec})
\end{eqnarray}
where $t^E_{rec}= \pi N'M/g_0$
with $t^E_{rec}$ denoting
the Poincare
recurrence time of the system. If one allows a slight random variation of multiplicity $M$ over $I$ with its mean
value $\langle M \rangle =e^{\alpha N^2/N_s}$ and assumes $N'=N_s$,
the recurrence time scale
becomes $t^E_{rec} \sim e^{N_s\log N_s} e^{ \alpha N^2 } \sim e^{S_E} $.
\begin{figure}[th!]
\centering
\includegraphics[height=3.5cm]{bcfnew.pdf}
\caption{\small $B$ and $C$ represent respectively the classical gravity degrees and the remaining nonclassical degrees in $R$. EPR pairs between
$B$ and $C$ are formed by entanglement, which are represented by the dotted lines. The interaction is small and the location of degrees in $C$ are random and $C$
is embedded in $E$.
}
\label{figbcfield}
\end{figure}
To consider a more realistic case, we choose
\begin{eqnarray}
D(x) =\frac{1}{\pi}\frac{a_0}{x^2+ a_0^2}
\end{eqnarray}
together with $N'=N_s$.
Then one has
\begin{eqnarray}
\langle \psi^i |\psi^{j} \rangle_I =e^{-| \frac{ \epsilon_i -\epsilon_{j}}{N'}a_0 g_0t|}
\end{eqnarray}
This leads to the overlap
\begin{eqnarray}
\langle \psi_B^i |\psi_B^{j} \rangle = e^{-|(\epsilon_i -\epsilon_{j})a_0g_0 t | }
\end{eqnarray}
This is describing a typical decoherence \cite{Zurek:2003zz}
and consistent with the late time behavior of the perturbations in \cite{Bak:2017xla}. We take $a_0$ to be an order of typical thermal energy scale as $a_0 = b_0 T$.
For example consider the case
\begin{eqnarray}
|\psi_B \rangle = \frac{1}{\sqrt{2}}\big( |0 \rangle + |1 \rangle \big)
\end{eqnarray}
Then the reduced density matrix is obtained as
\begin{eqnarray}
\rho_B =\frac{1}{2} \Big[ |0 \rangle \langle 0 | + |1 \rangle \langle 1 | + \chi (t) ( |0 \rangle \langle 1 |+ |1 \rangle \langle 0 | ) \Big]
\end{eqnarray}
where $\chi(t) = e^{-k_0 \frac{t}{\beta}}$ with $k_0 =|( \epsilon_1 -\epsilon_{0})b_0 g_0| $ for $t >0$.
The corresponding entanglement entropy is given by
\begin{eqnarray}
\
\Delta S_B(t) = \log 2 -\frac{1}{2} (1+ \chi(t)) \log (1+ \chi(t)) -\frac{1}{2} (1- \chi(t)) \log (1- \chi(t))
\end{eqnarray}
with $\Delta S_B(t) =S_B(t)-S_B(0)$. (See below for the definition of $S_B$ in the full thermofield double system.)
One finds initially there is no entanglement as $\Delta S_B(0)=0$. Then the information is leaking out from the system $B$ to the dark cloud environment $C$
though the interaction between $B$ and $C$ is negligibly small. The final entanglement entropy approaches the maximal value $\log 2$ exponentially
as a time scale of typical relaxation time scale whose precise value depends on the properties of system $B$. This feature is true for any $|\psi_B\rangle (0)$.
At the initial moment of perturbation, there is no entanglement between $B$ and $C$. Then through the interactions which is extremely small, the information
leaks out from $B$ to $C$ by forming EPR pairs as described in Figure \ref{figbcfield}.
Since the system $E=B+C$ is huge, it is practically impossible to collect this entanglement back to the original
non entangled state by performing simple operations. Thus in the gravity description, there is an effective loss of information through the entanglement
between $B$ and $C$. This is the story well before the Poincare recurrence time scale.
The typical recurrence time scale becomes $t^E_{rec}\sim e^{ S_E}\sim e^{S_{grav}}$.
It is clear that the initial information of system $B$ can be recovered if one waits
for the order of the recurrence time scale. In this sense there is certainly no loss of information by the entanglement transfer since
the transferred information will be regained if one waits for enough time that is order of the recurrence time scale.
\section{Bulk interpretation} \label{sec4}
In this section we shall describe how the above transfer of information from $B$ to $C$ looks like from the view point of the bulk
side.
The expansion in the gravity side is organized as follows. There are saddle-point solutions each of which
is weighted
by the probability $\frac{1}{Z} e^{-I}$ where $I$ is the on-shell action and $Z$ is the full partition function\footnote{Of course each saddle point receives
quantum gravity corrections of $1/N$ expansions}.
Hence the entropy for instance is given by
\begin{eqnarray}
S_{grav}=\langle \hat{S}_{grav} \rangle = \frac{1}{Z}\sum_\alpha e^{-I_\alpha} S^{grav}_\alpha
\end{eqnarray}
where $\alpha$ is labeling each saddle-point and $S^{grav}_\alpha$ is the entropy in (\ref{grentropy}) associated with the saddle-point.
This entropy is well defined at least for the static case in which the system is in thermal equilibrium. As we showed already, the leading contribution of the
above entropy is from that of the deformation of BTZ spacetime if the temperature is larger than the Hawking-Page transition temperature
$T_{HP}= \frac{1}{2\pi \ell}$ \cite{Hawking:1982dh}. Then the area and the entropy grows in time, which implies that the corresponding entropy in field theory
side cannot be identified with the L-R entanglement entropy $S_R$ in (\ref{vonr}) obtained by tracing over the CFT on the left side. As
we said before, this R-L entanglement is time-independent since there is no interaction between the left and the right sides. This then should be identified with
the entanglement entropy where we further trace over the degrees in $C$ which is invisible from the view point of gravity description. We introduce
further reduced density matrix by
\begin{eqnarray}
\rho_B = {\rm tr}_C \, \rho_R
\end{eqnarray}
Hence the entanglement entropy between $B$ and $E=C+L$ (with $R= B+C$)
\begin{eqnarray}
S_{B}= -{\rm tr}_B \rho_B \log \rho_B
\end{eqnarray}
can be identified with the gravitational entropy $S_{grav}$ in (\ref{grentropy}).
But there is a nonvanishing interaction between $B$ and $C$ as emphasized previously. Thus there is a leak of information from $B$
to $C$, which leads to the increase of the entropy $S_{grav}$. This leak happens in the form of developing entanglement between
$B$ and $C$ as described in the previous section.
\begin{figure}[th!]
\centering
\includegraphics[height=4cm]{thads.pdf}
\caption{\small In the thermal AdS phase the left and right sides are classically disconnected. But there is a contribution from graviton exchange between them
semiclassically which is responsible for the entanglement between the left and the right sides. This is depicted in b) where the dotted line represents the graviton
exchange.
}
\label{figthads}
\end{figure}
Hence according to the ER = EPR conjecture \cite{Maldacena:2013xja},
the corresponding ER bridge connecting $B$ to $C$ should develop in the gravity description. However
our solutions found in \cite{Bak:2017xla} do not show any signature of the development of ER bridge. This ER bridge ought to be nonclassical and
cannot be seen in the classical gravity description.
A similar behavior may be found in other examples. For instance consider the thermal AdS geometry that gives a dominant contribution to the partition function or the sum over geometries for $T < T_{HP}$,
which is basically the Hawking-Page transition \cite{Hawking:1982dh}.
There are two thermal AdS geometries for the left and the right CFT's separately. More precisely, as described in
\cite{Maldacena:2001kr}, two Lorentzian thermal AdS spaces are connected by a half of Euclidean thermal AdS solution in order to provide the
initial state in (\ref{initial}). Thus this geometry is still dual to the thermofield double dynamics. It is clear that there is a nonvanishing L-R entanglement
that is given by (\ref{vonr}) once the temperature is nonzero. The corresponding entanglement entropy is of higher order in $G$
and hence cannot be seen in the classical
gravity
description.
Indeed the two Lorentzian geometries of the left and the right sides are completely disconnected as depicted in Figure \ref{figthads}. Hence at the level of solution, the ER
bridge does not appear, which is of course not a contradiction. There are many other examples of this type: The connected string solution in Figure\,\,1c of \cite{Bak:2007fk}
appears as two completely separate strings at the level of classical description.
\begin{figure}[th!]
\centering
\includegraphics[height=3.7cm]{bcsystem.pdf}
\caption{\small On the boundaries, the green dots represent the degrees of $C$ not covered by the classical gravity system. The rest represent the degrees in $B$ of the classical gravity including blue
dots. The left-right entanglement is reduced to a non maximal one by the perturbations.
The bulk degrees in $B$ (denoted by blue dots) is connected by the ER connecting $C$ (denoted by green dots) which is nonclassical. This effect is of order $1/N$ and cannot be seen in the
classical gravity solutions.
}
\label{figbc}
\end{figure}
Similarly in our example, the ER bridge is not seen at the level of classical description.
The relevant degrees in $B$ fall gradually into the horizon that is responsible for the growth of the horizon area. Thus the entanglement is between
these behind-horizon degrees and the corresponding degrees in $C$. The degrees in $C$ is not accessible from the classical gravity description.
The connection is the quantum gravity fluctuation
which is the semiclassical picture of the relevant ER bridge.
We illustrate this in Figure \ref{figbc}.
Finally let us comment on the issue of the recurrence in the bulk.
In the classical gravity description, the large $N$ limit
is already taken, so there is basically no way to see this even in principle. But through the higher order effect in $1/N$ such as Hawking radiation,
regaining of the transferred entanglement is
allowed in principle, which can be a possible resolution of the black hole information paradox.
But here we are dealing with an eternal black hole. Unlike the case of evaporating black hole in the flat space,
the regaining time scale will be as large as $t_{rec}\sim e^{\alpha N^2}\sim e^{S_{grav}}$.
At the final stage of recurrence, the entanglement between $B$ and $C$ is reduced and the system is back to the original state.
In the time dependent perturbation of black hole in \cite{Bak:2017xla}, this return actually happens when $t < 0$, in which the entanglement between $B$ and $C$ is decreasing through
interactions. Not that there is a failure of description at $t= \pm \infty$, which appears as the orbifold singularity in the three dimensional BTZ
black hole. We view this as a simple failure of description, so in the bulk the recurrence will occur if one waits for long enough time.
\section*{Acknowledgement}
This work was
supported in part by
NRF Grant 2017R1A2B4003095.
|
1711.02060
|
\section*{Methods}
In this work the effective nonlinear coupling was $d_{\text{eff}} = 2.5 \times 10^{-12}$ V/m, the effective mode area was $S = 1 \times 1$ $\mu$m$^2$, the waveguide length was 50.638 mm for $\kappa z = 4$, and the pump power was 20 dBm. The RF signal tones were $\Omega_1$ = 9 MHz and $\Omega_2$ = 11 MHz. The optical carrier was $\omega_c$ = 200.1 THz and the pump frequency was $\omega_p$ = 201.2 THz. All results in this work were computed by expanding the template Eqn. \ref{eq:template} into a set accounting for all 11 frequencies shown in Fig. \ref{fig:fig3}b with the sum- and difference-frequency coupling accounted for by the Dirac delta functions. The resulting set of coupled equations was solved with Newton's method and a Crank-Nicolson finite differencing scheme \cite{crank_practical_1996}.
\section*{Acknowledgments}
This work was supported by the Packard Fellowships for Science and Engineering, the National Science Foundation (NSF) (EFMA-1641069), and the US Office of Naval Research (ONR) (N00014-16-1-2687).
|
1506.01008
|
\section{Introduction}
Finding the ground state of strongly-correlated quantum many-body systems poses one of the main challenges of contemporary condensed matter physics. The physical properties of these systems, however, are not determined by the ground state but rather by the low-lying, elementary excitations relative to this ground state. In contrast to the strongly-correlated ground state, these elementary excitations are of a particularly simple character: in most cases they can be treated as a collection of independent, weakly-interacting particles living on non-trivial vacuum state \cite{Landau1941, Anderson1963}.
\par In condensed matter theory these ``quasi-particles'' are typically defined starting from some non-interacting limit. In Fermi liquid theory for interacting electron systems \cite{Landau1956, *Landau1957, Nozieres1964, *Pines1966} -- the most prominent example of this approach -- the quasi-particles are defined in the free-electron system, but remain well-defined modes when turning on the interactions. The effect of a finite lifetime and quasi-particle interactions can be treated in perturbation theory. In strongly-correlated lattice systems, however, there is typically no obvious way to start from a non-interacting theory to define the quasi-particles that determine the system's properties (notable counter-examples in one dimension include integrable systems \cite{Essler2004a} and continuous unitary transformations \cite{Knetter2003a}). The variational approach, which we will advocate in this paper, is orthogonal to the perturbative approach by starting from the strongly-correlated ground state and finding the low-lying excitations of the interacting system variationally. As exact eigenstates, these excitations have an infinite lifetime, but a priori it is not clear that they should have a local, particle-like nature.
\par In relativistic quantum field theory, a picture of localized elementary excitations on top of a strongly-correlated vacuum has been formulated in a rigorous fashion. Haag-Ruelle scattering theory \cite{Haag1958, *Ruelle1962, *Haag1996} does indeed construct a many-particle Fock space by acting with local operators on the vacuum and even defines an S matrix describing the interactions between these particles. Because this formalism depends heavily on Lorentz invariance, there is a priori no straightforward translation to lattice systems. Indeed, on the lattice there are fewer restrictions on the spectrum: different elementary excitation branches typically have different characteristic velocities and are not bound to be stable in the whole Brillouin zone -- a typical spectrum is shown in Fig.~\ref{fig:typicalspectrum}.
\par Recently though, it was realized that by using Lieb-Robinson bounds \cite{Lieb1972} as the soft lattice analog of strict causality in relativistic QFT, the locality of elementary excitations can be established in a rigorous way. More specifically, it was shown in Ref.~\onlinecite{Haegeman2013a} that an elementary excitation that lives on an isolated branch in the energy-momentum spectrum and has a finite overlap with an arbitrary local operator, can be created out of the ground state by the action of a momentum superposition of a local operator (to an exponential precision in the size of the support of this operator). In Ref.~\onlinecite{Bachmann2014} the scattering problem of these particle excitations was formulated by translating the Haag-Ruelle formalism to the lattice setting.
\par These theoretical developments provide a motivation for the variational approach towards a particle picture of the low-energy physics of lattice systems. Indeed, by making use of the fact that gapped excitations should be local, it might prove possible to describe them with only a small number of variational parameters. This is the idea of the single-mode approximation, or Feynman-Bijl ansatz, pioneered by Feynman in his study on liquid helium \cite{Feynman1953a, *Feynman1954, *Feynman1956, Girvin1986} and later successfully applied to quantum spin systems \cite{Arovas1988, Takahashi1988, Sorensen1994}. Although providing qualitative insight into the nature of the elementary excitations, the single-mode approximation is often too crude as a variational ansatz to obtain quantitative results on the low-lying spectrum of generic spin chains.
\par Indeed, constructing a variational ansatz for excitations with quantitative accuracy requires both an accurate parametrization of the ground state and a systematic way to change this ground state locally. For one-dimensional systems, the framework of matrix product states \cite{Schollwock2011a, Verstraete2008a} (MPS) has proven to meet both requirements. The ground state of one-dimensional quantum spin systems can indeed be efficiently parametrized by the class of MPS \cite{Verstraete2006, Hastings2007}; the success of the density matrix renormalization group \cite{White1992} is based on MPS serving as the class of states over which it optimizes \cite{Ostlund1995a, *Rommer1997a, Dukelsky1997}. The defining characteristic of MPS -- or tensor network states in general -- is the presence of a ``virtual'' level that carries the (quantum) correlations in the many-body wave function. By acting both on the physical and the virtual level, a variational ansatz for elementary excitations on an MPS ground state was introduced in Refs.~\onlinecite{Haegeman2012a} and \onlinecite{Pirvu2012}. The ansatz was used for calculating dispersion relations and dynamical correlation functions of quantum spin chains \cite{Haegeman2013b, Zauner2015, Keim2015}, quantum field theories \cite{Draxler2013, Milsted2013} and gauge theories \cite{Buyens2014}.
\par A more general understanding of these efforts is obtained by realizing that the low-energy dynamics correspond to small variations around the variational ground state and are therefore not necessarily contained within the variational class itself. Indeed, for the smooth manifold of MPS \cite{Haegeman2014} it is the tangent space constructed around the MPS ground state that provides a natural parametrization of the low-energy dynamics. For example, the best approximation to time evolution within the MPS manifold can be obtained by projecting the Schr\"{o}dinger equation into the MPS tangent bundle, according to the time-dependent variational principle (TDVP) \cite{Haegeman2011d,Haegeman2014a}. Similarly, the ansatz for an elementary excitation corresponds exactly to a vector in the tangent space around the MPS ground state. These ideas can be grouped under the concept of \emph{post-MPS} methods \cite{Haegeman2013b} as an alternative for the standard MPS algorithms for tackling the low-energy dynamics around an MPS ground state.
\par The crucial next step in this approach -- after the construction of single-particle excitations -- consists of studying the interactions between these particles and, more specifically, computing the two-particle S matrix \cite{Vanderstraeten2014}. This information can then be used as the input for the ``approximate Bethe ansatz'' \cite{Krauth1991, Kiwata1994a, Okunishi1999a} (i.e. neglecting all three-particle scattering processes) in order to provide a first-quantized description of a finite density of excitations on top of the strongly-correlated vacuum.
\par These developments should eventually lead to the ab initio construction of an effective second-quantized Hamiltonian, acting in a Fock space of interacting particles. In contrast to standard effective field theory constructions, the variational approach would automatically incorporate all symmetries and correlations of the vacuum state on which these particles live without relying on phenomenological considerations.
\par In this paper we further elaborate on the framework that we introduced in Ref.~\onlinecite{Vanderstraeten2014}. In Sec.~\ref{sec:section2} we show how to construct one- and two-particle excitations on an MPS vacuum state. We formulate a definition of the S matrix based on the form of the two-particle wave function and prove that it is unitary. Finally, we construct the projector on the one- and two-particle subspace which shows up in the spectral representation of dynamical correlation functions. In Sec.~\ref{sec:section3} we take a step back and show that the S matrix as defined in Sec.~\ref{sec:section2} corresponds to the one that shows up in standard dynamical scattering theory. Next we elaborate on the approximate Bethe ansatz as a way of dealing with a finite density of excitations in a first-quantized many-particle formalism. In Sec.~\ref{sec:section4} we apply our variational method to study the Heisenberg antiferromagnetic two-leg ladder. We compute the elementary excitation spectrum, the two-particle S matrix and one- and two-particle contributions to dynamical correlation functions. Afterwards, we apply the approximate Bethe ansatz to the magnetization process, at zero and finite temperature, and compute both thermodynamic properties and correlation functions of the magnetized ladder. In the last section, we provide an overview of some interesting extensions of our framework and give an outlook towards the construction of effective field theories in second quantization.
\begin{figure}
\includegraphics[width=\columnwidth]{./spectrum.pdf}
\caption{A typical momentum-energy excitation spectrum of a one-dimensional lattice system. We have depicted three elementary (one-particle) excitations (full lines) and the many-particle continuum (grey). Both $\alpha_2$ and $\alpha_3$ are stable in the whole Brillouin zone; the latter remains stable even inside the continuum, possibly because it cannot decay in a two-particle state through symmetry constraints. Particle $\alpha_1$ becomes unstable upon entering the continuum, so that it ceases to be a one-particle excitation (cannot be created by a local operator).}
\label{fig:typicalspectrum}
\end{figure}
\section{Constructing scattering states}
\label{sec:section2}
In this section we construct variational one- and two-particle states on an MPS background \footnote{Note that our approach was inspired by the works of Kohn \cite{Kohn1948} and Feynman \cite{Feynman1972}}. Based on the form of these wave functions, we define the S matrix and introduce the projectors on the one- and two-particle subspaces (i.e. the low-energy subspace) which can be used to compute the low-energy part of dynamical correlation functions. Note that, while the complete framework is presented in the main body, technical details and long equations are taken up in the appendix. A short remark on notation is also in order. Vectors of any length will be denoted in boldface, whereas matrices will use a sans serif font. Vector entries will be referred to using a superscript (in which case the boldface will be dropped), whereas subscripts of a boldface vector typically refer to a label of a set of vectors, such as a basis. The only exception to these rules is that physical states are denoted using Dirac's bra-ket notation and the matrices appearing in the definition of the matrix product state (which can also be interpreted as rank three tensors) are typeset using the normal serif type (italic).
\subsection{Ground state}
Consider a one-dimensional quantum spin system with local dimension $d$ in the thermodynamic limit, described by a local and translation invariant Hamiltonian. While in no way crucial, we restrict to nearest-neighbour Hamiltonians, i.e. $\ensuremath{\hat{H}}=\sum_{n\in\ensuremath{\mathbb{Z}}}\hat{h}_{n,n+1}$, for reasons of simplicity. We furthermore assume that the translation invariant ground state of this system (we restrict to a unique ground state, see Sec.~\ref{sec:section5} for extensions) can be accurately described by an injective uniform matrix product state (uMPS) \cite{Fannes1992,Vidal2007b,Haegeman2011d}
\begin{equation} \label{eq:mps}
\ket{\Psi[A]} = \sum_{\{s\}=1}^d \ensuremath{\vect{v_L^\dagger}} \left[ \prod_{m} A^{s_m} \right] \ensuremath{\vect{v_R}} \ensuremath{\ket{\{s\}}},
\end{equation}
where $A^s$ is a set of $D\times D$ matrices for $s=1,\dots,d$, or, equivalently, $A$ can be interpreted as a $D\times d \times D$ tensor; $\ensuremath{\vect{v_L^\dagger}}$ and $\ensuremath{\vect{v_R}}$ are $D$-dimensional boundary vectors. In the thermodynamic limit, all physical observables are independent of these boundary vectors \cite{Haegeman2014}, so that the tensor $A$ provides a complete description of the ground state $\ket{\Psi[A]}$.
\begin{figure}
\subfigure{\includegraphics[width=0.4\columnwidth]{./tensor.pdf}}\\
\subfigure{\includegraphics[width=0.85\columnwidth]{./mps.pdf}}
\caption{Graphical representation of an MPS ground state. The circles represent the ($D\times d \times D$)-dimensional tensor $A$: every outgoing leg corresponds to a tensor index. Whenever two legs are connected, this corresponds to a contraction of the two indices. In the MPS all virtual indices are contracted, while the physical indices correspond to the physical degrees of freedom in the MPS wave function \eqref{eq:mps}. The matrix product structure contains the (quantum) correlations of this ground state.}
\end{figure}
\par The set of injective MPS of a certain bond dimension constitute a complex manifold \cite{Haegeman2014}. Finding the best approximation of the ground state of a certain Hamiltonian within this manifold can be achieved using different algorithms \cite{Schollwock2011a} -- in our simulations we will always use the TDVP algorithm\cite{Haegeman2011d,Haegeman2014a}.
\subsection{One-particle excitations}
This ground state serves as our vacuum, on top of which we will build localized, particle-like excitations. A first guess for the wave function of an elementary excitation with momentum $\kappa$ is the single-mode approximation
\begin{equation} \label{eq:sma}
\ket{\Phi_\text{SMA}(\kappa)} = \sum_n \ensuremath{\mathrm{e}}^{i\kappa n} \hat{O}_n \ket{\Psi[A]},
\end{equation}
where $\hat{O}_n$ is an operator acting at site $n$. The choice of operator can be inspired by physical intuition \cite{Feynman1954,Girvin1986,Arovas1988,Takahashi1988, Sorensen1994, Talstra1997} or determined by numerical optimization \cite{Chung2010a}. Though providing some qualitative insight into elementary excitation spectra, this ansatz is typically not a good quantitative approximation for the true wave function of the excitation. Systematically improving on this would ask for the introduction of bigger local operators $\hat{O}_n$. It was indeed proven \cite{Haegeman2013a} that, in the case of an isolated excitation branch, the exact wave function can be arbitrary well approximated in this way. More specifically, it was shown that the localized nature of an excitation depends on the gap to the nearest eigenvalue of the Hamiltonian in the same momentum sector.
\begin{figure}
\includegraphics[width=\columnwidth]{./ansatz.pdf}
\caption{Graphical representation of the one-particle excitation ansatz. The ground state tensor $A$ is changed at site $n$ into a new tensor $B$ (square) and a momentum superposition is taken. The matrix product structure allows that the tensor $B$ can change the ground state over a finite distance.}
\label{fig:ansatz}
\end{figure}
\par Within the framework of matrix product states, it is possible to construct a variational ansatz that is able to capture the localized nature of the excitation by directly modifying the local tensors. Indeed, instead of only operating on the physical level, we can change one ground state tensor $A^s$ with a new tensor $B^s$ and take a momentum superposition \cite{Ostlund1995a, *Rommer1997a, Haegeman2012a,Haegeman2013b}
\begin{multline} \label{oneparticle}
\ket{\Phi_\kappa[B]} = \sum_n \ensuremath{\mathrm{e}}^{i\kappa n} \sum_{\{s\}} \ensuremath{\vect{v_L^\dagger}} \left[ \prod_{m<n} A^{s_m} \right] \\
\times B^{s_n} \left[ \prod_{m>n} A^{s_m} \right] \ensuremath{\vect{v_R}} \ensuremath{\ket{\{s\}}}.
\end{multline}
Through the virtual level of the MPS, this ansatz is able to perturb the ground state over a finite length determined by the bond dimension $D$. All variational freedom of this ansatz is contained within the tensor $B^s$. As the parametrization of the state \eqref{oneparticle} is linear in the elements of $B^s$, variationally optimizing amounts to solving the Rayleigh-Ritz problem
\begin{equation} \label{eig1p}
\mathsf{H}_\text{eff,1p}(\kappa) \vect{u} = \lambda \vect{u}
\end{equation}
with $\mathsf{H}_\text{eff,1p}(\kappa)$ the momentum dependent effective one-particle Hamiltonian and vector $\vect{u}$ containing the coefficients $u^i$ to expand tensor $B$ in the state \eqref{oneparticle} with respect to a suitably chosen basis $\{B_{(i)}, i=1,\ldots,(d-1)D^2\}$. We refer to Appendix \ref{sec:1pA} for details on how to calculate $\mathsf{H}_\text{eff,1p}$.
\par Upon solving the eigenvalue problem in Eq.~\eqref{eig1p}, we obtain a set of $(d-1)D^2$ eigenvalues for every momentum $\kappa$. Some of those eigenvalues $\Delta_{\alpha}(\kappa)$ offer a good approximation to the exact dispersion relations of the elementary excitations, i.e.\ the isolated branches in the spectrum of the Hamiltonian. Moreover, from the corresponding eigenvectors $\vect{u}_{\alpha}(\kappa)$ we obtain an explicit expression for the wave function of the elementary excitations by inserting the tensors $B_{\alpha}(\kappa)=\sum_i u_{\alpha}^i(\kappa) B_{(i)}$ in Eq.~\eqref{oneparticle}. This expression can be used to calculate the spectral weights of the excitations and, consequently, the one-particle contribution to dynamical correlation functions.
\par Other eigenvalues obtained from Eq.~\eqref{eig1p} will fall in the continuous part of the spectrum of the Hamiltonian, i.e.\ in the set of scattering states. Scattering states cannot be described by a single local perturbation, so we expect the ansatz \eqref{oneparticle} to fail. In fact, instead of a scattering state, the variational optimization will create a localized wave packet of two-particle states within some energy range. Obviously, the variational eigenstates of the form in Eq.~\eqref{oneparticle} will not provide a good approximation to the exact scattering eigenstates of the full Hamiltonian. A more appropriate variational ansatz for two-particle scattering states is discussed in the remainder of this section.
\par Remark that we have so far not discussed the case of bound states. When defining (quasi-) particles along a path of Hamiltonians using e.g. perturbation theory or continuous unitary transformations \cite{Knetter2003a}, bound states can be identified with isolated eigenstates emerging from a multi-particle continuum along the path. In our variational framework, we consider one particular Hamiltonian which is not necessarily related to a one-parameter family. All isolated branches in the spectrum are equally elementary (see Ref.~\onlinecite{Zimmermann1958} for the analogous result in QFT). While there might be quantum numbers that indicate the ``history'' of an elementary excitation along a path of Hamiltonians, there is typically no particle number symmetry to make the interpretation of bound states unambiguous. On a related note, elementary excitations are by this definition exact eigenstates of the Hamiltonian and therefore have an infinite life time. We cannot and do not target resonances within the continuous part of the spectrum. Therefore, the Hamiltonian does not contain interactions that link the one-particle sector with higher particle states.
\par As mentioned previously, the spectrum of general quantum spin chains can be very complex. Within certain regions of the Brillouin zone, the energy of elementary excitations can fall within the continuum (this typically requires a quantum number that protects them against decay) or there might be no elementary excitations at all (e.g. around momentum zero in the spin-1 Heisenberg antiferromagnet). We therefore need a way to determine which variational eigenvalues of Eq.~\eqref{eig1p} correspond to elementary excitations and therefore offer a good approximation to actual eigenstates of the Hamiltonian. Upon enlarging the variational one-particle space, e.g. by increasing the bond dimension or the spatial support of the local perturbation, eigenvalues that correspond to elementary excitations will converge quickly (related to the gap to the nearest eigenvalue) and remain at a fixed position. Eigenvalues in the continuous part of the exact spectrum, on the other hand, will not really converge and several new eigenvalues will appear in those regions. A more quantitative way to assess how well an exact eigenstate is approximated consists of calculating the variance of the Hamiltonian \cite{Davison1968}, i.e.\ $\bra{\Phi_\kappa[B]}(\ensuremath{\hat{H}}-\Delta(\kappa))^2 \ket{\Phi_\kappa[B]}$. For elementary excitations, these variances should be small (see Sec.~\ref{sec:ladder1p} for numerical values). For the other solutions of the one-particle problem \eqref{eig1p}, which correspond to scattering states, the variance should be larger. For a typical gapped system, the difference will be some orders of magnitude. Consequently, this quantity allows for the identification of one-particle states, even within higher-particle bands and without exploiting symmetries.
\par Note finally that, without Galilean invariance on the lattice, the tensor $B_{\alpha}(\kappa)$, which describes the particle $\alpha$ on a dispersion branch $\Delta_\alpha(\kappa)$, is momentum dependent. On the other hand, we expect that for a well-defined particle in a certain momentum range this momentum dependence is not too strong. Indeed, it turns out that by a suitable choice of the basis tensors $\{B_{(i)}, i=1,\ldots,(d-1)D^2\}$, we can fully capture $B_{\alpha}(\kappa)$ for all elementary excitations $\alpha$ and for all momenta $\kappa$ in the span of just a small number $\ell\ll (d-1)D^2$ basis vectors $\{B_{(i)},i=1,\ldots,\ell\}$. Although more sophisticated optimization strategies should be possible, we construct this reduced basis from a number of $B$'s at different momenta. This reduced basis will be important for solving the scattering problem in the next sections.
\subsection{Variational ansatz for two-particle states}
In the previous section it became clear that we need another ansatz to capture the delocalized nature of a two-particle state. We will start from a one-particle spectrum consisting of a number of different types of particles, labelled by $\alpha$, with dispersion relations $\Delta_\alpha(\kappa)$. In the thermodynamic limit, constructing the two-particle spectrum is trivial: the momentum and energy are the sum of the individual momenta and energies of the two particles \cite{Anderson1963}. The two-particle wave function, however, depends on the particle interaction. The interactions, which depend on both the Hamiltonian and the ground state correlations, are reflected in the wave function in two ways: (i) the asymptotic wave function has different terms, with the S matrix elements as the relative coefficients, and (ii) the local part of the wave function.
\par In order to capture both we introduce the following ansatz for describing states with two localized, particle-like excitations with total momentum $K$
\begin{align} \label{eq:ansatz}
\ket{\Upsilon(K)} = \sum_{n=0}^{+\infty} \sum_{j=1}^{L_n} c^j(n) \ket{\chi_{K,j}(n)}
\end{align}
where the basis states are
\begin{widetext}
\begin{align}
& \ket{\chi_{K,j}(n=0)} = \sum_{n_1=-\infty}^{+\infty} \ensuremath{\mathrm{e}}^{i K n_1} \sum_{\{s\}=1}^d \ensuremath{\vect{v_L^\dagger}} \left[ \prod_{m<n_1} A^{s_m} \right] B_{(j)}^{s_{n_1}} \left[ \prod_{m>n_1} A^{s_m} \right] \ensuremath{\vect{v_R}} \ensuremath{\ket{\{s\}}} \label{local} \\
& \ket{\chi_{K,(j_1,j_2)}(n>0)} = \sum_{n_1=-\infty}^{+\infty} \ensuremath{\mathrm{e}}^{i K n_1} \sum_{\{s\}=1}^d \ensuremath{\vect{v_L^\dagger}} \left[ \prod_{m<n_1} A^{s_m} \right] B_{(j_1)}^{s_{n_1}} \left[ \prod_{n_1<m<n_1+n} A^{s_m} \right] B_{(j_2)}^{s_{n_1+n}} \left[ \prod_{m>n_1+n} A^{s_m} \right] \ensuremath{\vect{v_R}} \ensuremath{\ket{\{s\}}}. \label{nonLocal}
\end{align}
\end{widetext}
\begin{figure*}
\centering
\includegraphics[width=0.7\columnwidth]{./twoP.pdf}
\caption{Graphical representation of the basis states \eqref{nonLocal}. The ground state is changed at two sites at a distance of $n$ sites and a momentum superposition is taken (with the total momentum $K$).}
\end{figure*}
We collect the variational coefficients either in one half-infinite vector $\vect{C}$ with $C^{j,n} = c^j(n)$ or using the finite vectors $\vect{c}(n)$ with entries $\{c^j(n),j=1,\ldots,L_n\}$ for every $n=0,1,\ldots$. Here, we have $L_0 = (d-1) D^2$ and $L_{n>0} = [(d-1)D^2]^2$. Note that the sum in Eq.\eqref{eq:ansatz} only runs over values $n\geq0$, because a sum over all integers would result in an overcomplete basis.
\par Already at this point, we will reduce the number of variational parameters to keep the problem tractable. The terms with $n=0$ (corresponding to the basis vectors in Eq.~\eqref{local}) are designed to capture the situation where the two particles are close together. No information on how this part should look like is a priori available, so we keep all variational parameters $c^j(0)$, $j=1,\dots,L_0=D^2(d-1)$. The terms with $n>0$ corresponding to the basis vectors in Eq.~\eqref{nonLocal} represent the situation where the particles are separated. We know that, as $n\rightarrow\infty$, the particles decouple and we should obtain a combination of one-particle solutions. With this in mind, we restrict the range of $j_1$ and $j_2$ to the first $\ell$ basis tensors $\{B_{(i)}, i=1,\ldots,\ell\}$, which were chosen so as to capture the momentum dependent solutions of the one-particle problem. Consequently, the number of basis states of Eq.~\eqref{nonLocal} for $n>0$ satisfies $L_n=\ell^2$, which we will henceforth denote as just $L$.
\par This might seem like a big approximation for $n$ small: when the two particles approach the wave functions might begin to deform, so that the $B$ tensors of the one-particle problem no longer apply. Note, however, that the local ($n=0$) and non-local ($n>0$) part are not orthogonal, so that the local part is able to correct for the part of the non-local wave function where the one-particle description is no longer valid.
\par As the state \eqref{eq:ansatz} is linear in its variational parameters $\vect{C}$, optimizing the energy amounts to solving a generalized eigenvalue problem
\begin{equation} \label{eig}
\mathsf{H}_\text{eff} \vect{C} = \omega \mathsf{N}_{\text{eff}} \vect{C}
\end{equation}
with $\omega$ the total energy of the state and
\begin{align}
& (\mathsf{H}_\text{eff})_{n'j',nj} = \bra{\chi_{j',K}(n')} \hat{H} \ket{\chi_{j,K}(n)} \label{Heff} \\
& (\mathsf{N}_\text{eff})_{n'j',nj} = \braket{\chi_{j',K}(n') | \chi_{j,K}(n)} \label{Neff}
\end{align}
two half-infinite matrices. They have a block matrix structure, where the submatrices are labelled by $(n',n)$ and are of size $L_{n'} \times L_n$. The computation of the matrix elements is quite involved and technical, so we refer to the appendix for the explicit formulas.
\par Since the eigenvalue problem is still infinite, it cannot be diagonalized straightforwardly. Since we actually know the possible energies $\omega$ for a scattering state with total momentum $K$, we can also interpret Eq.~\eqref{eig} as an overdetermined system of linear equations for the coefficients $C^{j,n}=c^j(n)$. In the next two sections we will show how to reduce this problem to a finite linear equation.
\subsection{Asymptotic regime}
First we solve the problem in the asymptotic regime, where the two particles are completely decoupled. This regime corresponds to the limit $n',n\to\infty$, where the effective norm and Hamiltonian matrices, consisting of blocks of size $L\times L$, take on a simple form. Indeed, if we properly normalize the basis states, the asymptotic form of the effective norm matrix reduces to the identity, while the effective Hamiltonian matrix is a repeating row of block matrices centred around the diagonal
\begin{equation} \label{Am}
(\mathsf{H}_\text{eff})_{n',n}\rightarrow \mathsf{A}_{n-n'}, \qquad n,n' \rightarrow \infty.
\end{equation}
The blocks decrease exponentially as we go further from the diagonal, so we can, in order to solve the problem, consider them to be zero if $|n-n'|>M$ for some suitably chosen integer $M$. In this approximation, the coefficients $\vect{c}(n)$ obey
\begin{equation} \label{recurrence}
\sum_{m=-M}^M \mathsf{A}_m \vect{c}(n+m) = \omega \vect{c}(n), \qquad n\rightarrow\infty.
\end{equation}
We can reformulate this as a recurrence relation for the $\vect{c}(n)$ vectors and therefore look for elementary solutions of the form $\vect{c}(n) = \mu^n \vect{v}$. For fixed $\omega$, the solutions $\mu$ and $\vect{v}$ are now determined by the polynomial eigenvalue equation
\begin{equation} \label{polynomial}
\sum_{m=-M}^M \mathsf{A}_m \mu^m \vect{v} = \omega \vect{v} .
\end{equation}
From the special structure of the blocks $\mathsf{A}_m$ (see Appendix \ref{sec:asymptotic}) and their relation to the one-particle effective Hamiltonian $\mathsf{H}_\text{eff,1p}$, we already know a number of solutions to Eq.~\eqref{polynomial}. Indeed, if we can find $\Gamma$ combinations of two types of particles $(\alpha,\beta)$ with individual momenta $(\kappa_1,\kappa_2)$ such that $K=\kappa_1+\kappa_2$ and $\omega=\Delta_\alpha(\kappa_1)+\Delta_\beta(\kappa_2)$, then the polynomial eigenvalue problem will have $2\Gamma$ solutions $\mu$ on the unit circle. These solutions take the form $\mu=\ensuremath{\mathrm{e}}^{i\kappa_2}$ and the corresponding eigenvector is given by
\begin{equation} \label{asModes}
\vect{v} = \vect{u}_{\alpha}(\kappa_1) \otimes \vect{u}_{\beta}(\kappa_2)
\end{equation}
(in the case of degenerate eigenvalues we can take linear combinations of these eigenvectors that no longer have this product structure). Every combination is counted twice, because we can have particle $\alpha$ on the left and particle $\beta$ on the right, and vice versa.
\par Moreover, since $\mathsf{A}_m\dag=\mathsf{A}_{-m}$, the number of eigenvalues within and outside the unit circle should be equal. This allows for a classification of the eigenvalues $\mu$ as
\begin{align*}
& \left|\mu_i\right| <1 \qquad \text{for} \qquad i=1,\dots,LM-\Gamma\\
& \left|\mu_i\right| =1 \qquad \text{for} \qquad i= LM-\Gamma+1,\dots,LM+\Gamma\\
& \left|\mu_i\right| >1 \qquad \text{for} \qquad i=LM+\Gamma+1,\dots,2LM.
\end{align*}
The last eigenvalues with modulus bigger than one are not physical (because the corresponding $\vect{c}(n)\sim \mu_i^n \vect{v}_i$ yiels a non-normalizable state) and should be discarded. The $2\Gamma$ eigenvalues with modulus 1 are the oscillating modes discussed above; we will henceforth label them with $\gamma=1,\ldots,2\Gamma$ such that $\mu = \ensuremath{\mathrm{e}}^{i \kappa_{\gamma}}$ ($\kappa_\gamma$ being the momentum of the particle of the right) and the corresponding eigenvector is given by
\begin{equation*}
\vect{v}_{\gamma} = \vect{u}_{\alpha_\gamma}(K-\kappa_\gamma) \otimes \vect{u}_{\beta_\gamma}(\kappa_\gamma).
\end{equation*}
Finally, the first eigenvalues are exponentially decreasing and represent corrections when the excitations are close to each other. We henceforth denote them as $e^{-\lambda_i}$ with $\Re(\lambda_i)>0$ for $i=1,\ldots,LM-\Gamma$ and denote the corresponding eigenvectors as $\vect{w}_{i}$.
\par With these solutions, we can represent the general asymptotic solution as
\begin{equation} \label{asymptotic}
\vect{c}(n) \rightarrow \sum_{i=1}^{LM-\Gamma} p^i \ensuremath{\mathrm{e}}^{-\lambda_i n} \vect{w}_i + \sum_{\gamma=1}^{2\Gamma} q^\gamma \ensuremath{\mathrm{e}}^{i\kappa_\gamma n} \vect{v}_\gamma.
\end{equation}
Of course, we still have to determine the coefficients $\{p^i,q^\gamma\}$ by solving the local problem.
\subsection{Solving the full eigenvalue equation}
Since the energy $\omega$ was fixed when constructing the asymptotic solution, the generalized eigenvalue equation is reduced to the linear equation
\begin{equation*}
(\mathsf{H}_\text{eff}-\omega \mathsf{N}_\text{eff}) \vect{C} = 0.
\end{equation*}
We know that in the asymptotic regime this equation is fulfilled if and only if $\vect{c}(n)$ is of the form of Eq.~\eqref{asymptotic}. We will introduce the approximation that the elements for the effective Hamiltonian matrix [Eq.~\eqref{Heff}] and norm matrix [Eq.~\eqref{Neff}] have reached their asymptotic values when either $n>M+N$ or $n'>M+N$, where $N$ is a finite value and can be chosen sufficiently large. This implies that we can safely insert the asymptotic form for $n>N$ in the wave function, which we can implement by rewriting the wave function as
\begin{equation} \label{Qx}
\vect{C}=\mathsf{Z}\times \vect{x}
\end{equation}
where
\begin{align*}
\mathsf{Z} = \begin{pmatrix} \mathds{1}_\text{local} & & \\ & \{ \ensuremath{\mathrm{e}}^{-\lambda_i n}\vect{w}_i \} & \{ \ensuremath{\mathrm{e}}^{-i\kappa_\gamma n} \vect{v}_{\gamma} \} \end{pmatrix}.
\end{align*}
The $\{ \ensuremath{\mathrm{e}}^{-\lambda_i n}\vect{w}_i \}$ and $\{ \ensuremath{\mathrm{e}}^{-i\kappa_\gamma n} \vect{v}_{\gamma} \}$ are the vectors corresponding to the damped, resp. oscillating modes, while the identity matrix is inserted to leave open all parameters in $\vect{c}(n)$ for $n\leq N$. The number of parameters in $x$ is reduced to the finite value of $D^2(d-1)+NL+LM+\Gamma$.
\par Since the equation is automatically fulfilled after $M+N$ rows, we can reduce $\mathsf{H}_\text{eff}$ and $\mathsf{N}_\text{eff}$ to the first rows, so we end up with the following linear equation
\begin{equation} \label{scatEq}
[\mathsf{H}-\omega \mathsf{N}]_\text{red}\times \mathsf{Z} \times \vect{x} = 0
\end{equation}
with
\begin{equation*}
\left[\mathsf{H}-\omega \mathsf{N}\right]_\text{red} =
\left(\begin{array}{c|cccc}
& 0 & 0 & \dots & 0 \\
& \vdots & \vdots & \ddots & \vdots \\
& 0 & 0 & \dots & 0 \\
(\mathsf{H}-\omega \mathsf{N})_\text{ex} & \mathsf{A}_M & 0 & \dots & 0 \\
& \mathsf{A}_{M-1} & \mathsf{A}_M & \dots & 0 \\
& \vdots & \vdots & \ddots & \vdots \\
& \mathsf{A}_1 & \mathsf{A}_2 & \dots & \mathsf{A}_M \end{array} \right).
\end{equation*}
This ``effective scattering matrix'' consists of the first $(M+N)\times(M+N)$ blocks of the exact effective Hamiltonian and norm matrix and the $\mathsf{A}$ matrices of the asymptotic part [Eq.~\eqref{Am}] to make sure that these matrices remain the truncated versions of a hermitian problem. This matrix has $D^2(d-1)+(N+M)L$ rows, which implies that the linear equation \eqref{scatEq} has $\Gamma$ exact solutions, which is precisely the number of scattering states we expect to find. Every solution consists of a local part ($D^2(d-1) + NL$ elements), the $LM-\Gamma$ coefficients $\vect{p}$ of the decaying modes and the $2\Gamma$ coefficients $\vect{q}$ of the asymptotic modes.
\subsection{S matrix and normalization}
\label{sec:norm}
After having shown how to find the solutions of the scattering problem, we can now elaborate on the structure of the asymptotic wave function and define the S matrix.
\par We start from $\Gamma$ linearly independent scattering eigenstates $\ket{\Upsilon_i(K,\omega)}$ ($i=1,\ldots,\Gamma$) at total momentum $K$ and energy $\omega$ with asymptotic coefficients $\vect{q}_i(K,\omega)$. The asymptotic form of these eigenstates is thus a linear combination of all possible non-decaying solutions of the asymptotic problem:
\begin{multline} \label{asForm}
\ket{\Upsilon_i(K,\omega)} = \sum_{\gamma=1}^{2\Gamma} q_i^\gamma(K,\omega) \\ \times \sum_{n>N}\sum_{j} \ensuremath{\mathrm{e}}^{i\kappa_\gamma n}v_\gamma^j(\kappa_\gamma) \ket{\chi_{j,K}(n)}
\end{multline}
where the coefficients are obtained from solving the local problem. The number of eigenstates equals half the number of oscillating modes that appear in the linear combination. With every oscillating mode $\gamma$ we can associate a function $\omega_\gamma(\kappa)$ giving the energy of this mode as a function of the momentum $\kappa_\gamma$ of the second particle at a fixed total momentum $K$. If $\gamma$ corresponds to the two-particle mode with particles $\alpha_\gamma$ and $\beta_\gamma$, this function is given by $\omega_{\gamma}(\kappa) = \Delta_{\alpha_\gamma}(K-\kappa) + \Delta_{\beta_\gamma}(\kappa)$. The derivative of this function, which will prove of crucial importance, is $\omega'_{\gamma}(\kappa) = \Delta'_{\beta_\gamma}(\kappa) - \Delta'_{\alpha_\gamma}(K-\kappa)$. It can be interpreted as the difference in group velocity between the two particles, i.e. the relative group velocity in the center of mass frame.
\par Much like the proof of conservation of particle current in one-particle quantum mechanics, it can be shown that (see Appendix \ref{proof}), if \eqref{asForm} is to be the asymptotic form of an eigenstate, the coefficients $q_i^\gamma(K,\omega)$ should obey
\begin{equation} \label{conservation}
\sum_{\gamma} \left|q^\gamma_i(K,\omega) \right|^2 \left(\frac{\d\omega_\gamma}{\d\kappa}(\kappa_\gamma)\right) = 0.
\end{equation}
This equation can indeed be read as a form of conservation of particle current, with $\omega_\gamma'(\kappa_\gamma)$ playing the role of the (relative) group velocity of the asymptotic mode $\gamma$. As any linear combination of eigenstates with the same energy $\omega$ is again an eigenstate, this relation can be extended to
\begin{equation*}
\sum_{\gamma} \overline{q^\gamma_j(K,\omega)} q^\gamma_i(K,\omega) \left(\frac{\d\omega_\gamma}{\d\kappa}(\kappa_\gamma)\right) = 0.
\end{equation*}
With this equation satisfied, we can define the two-particle S matrix $S(K,\omega)$. Firstly, the different modes are classified according to the sign of the derivative: the incoming modes have $\frac{\d\omega}{\d\kappa}>0$ (two particles moving towards each other), the outgoing modes have $\frac{\d\omega}{\d\kappa}<0$ (two particles moving away from each other), so that we have
\begin{multline*}
\sum_{\gamma\in\Gamma_{\text{in}}} \overline{q_j^\gamma(K,\omega)} q_i^\gamma(K,\omega) \left|\frac{\d\omega_\gamma}{\d\kappa}(\kappa_\gamma)\right| \\ = \sum_{\gamma\in\Gamma_{\text{out}}} \overline{q_j^\gamma(K,\omega)} q_i^\gamma(K,\omega) \left|\frac{\d\omega_\gamma}{\d\kappa}(\kappa_\gamma)\right|.
\end{multline*}
If we group the coefficients of all solutions in (square) matrices $Q_\text{in}(K,\omega)$ and $Q_\text{out}(K,\omega)$, so that the $i$'th column is a vector with the coefficients $q^\gamma_i$ for the in- and outgoing modes of the $i$'th solution, we can rewrite this equation as
\begin{multline*}
Q_\text{in}(K,\omega)\dag V^2_\text{in}(K,\omega) Q_\text{in}(K,\omega) \\= Q_\text{out}(K,\omega)\dag V^2_\text{out}(K,\omega) Q_\text{out}(K,\omega),
\end{multline*}
with $V_\text{in,out}(K,\omega)_{ij}= \delta_{ij} \left|\frac{\d\omega_\gamma}{\d\kappa}(\kappa_\gamma)\right|^{1/2}$ a diagonal matrix. As $Q_\text{in}(K,\omega)$ and $Q_\text{out}(K,\omega)$ should be connected linearly, we can define a unitary matrix $S(K,\omega)$ as
\begin{multline*}
V_\text{out}(K,\omega) Q_\text{out}(K,\omega) = S(K,\omega) V_\text{in}(K,\omega) Q_\text{in}(K,\omega).
\end{multline*}
In Sec.~\ref{sec:moller} we will show that this definition corresponds to the standard S matrix. Note, however, that $S(K,\omega)$ is only defined up to a set of phases. Indeed, since the vectors $\vect{v}_\gamma$ can only be determined up to a phase, the coefficient matrices $C_\text{in}$ and $C_\text{out}$ are only defined up to a diagonal matrix of phase factors. These arbitrary phase factors show up in the S matrix as well. We will show how to fix them in the case of the elastic scattering of two identical particles (Sec. \ref{sec:oneType}); in the case where we have different outgoing channels only the square of the magnitude of the S matrix elements is physically well-defined (see Sec. \ref{sec:moller}).
\par This formalism allows to calculate the norm of the scattering states in an easy way. Indeed, the general overlap between two scattering states is given by
\begin{widetext}
\begin{align*}
\braket{\Upsilon_{i'}(K',\omega')|\Upsilon_i(K,\omega)} &= 2\pi\delta(K-K') \left( \sum_{\gamma,\gamma'} \ol{q_{i'}^\ensuremath{{\gamma'}}(K',\omega')} q_i^\gamma(K,\omega) \vect{v}_{\gamma'}\dag \vect{v}_\gamma \sum_{n,n'>N} \ensuremath{\mathrm{e}}^{i(\kappa_\gamma-\kappa'_{\gamma'} ) n} + \text{finite} \right) \\
&= 2\pi\delta(K-K') \left( \sum_{\gamma,\gamma'} \ol{q_{i'}^{\gamma'}(K',\omega')} q_i^\gamma(K,\omega)\vect{v}_{\gamma'}\dag \vect{v}_\gamma \pi\delta(\kappa_\gamma(\omega)-\kappa'_{\gamma'}(\omega')) + \text{finite} \right).
\end{align*}
The $\delta$ factor for the momenta $\kappa_\gamma$ is obviously only satisfied if $\omega=\omega'$, so we can transform this to a $\delta(\omega-\omega')$. Moreover, if $\kappa_\gamma(\omega) = \kappa'_{\gamma'}(\omega')$ for $\gamma\neq\gamma'$, then necessarily $\vect{v}_{\gamma'}\dag \vect{v}_\gamma=0$, so we can reduce the double sum in $\gamma,\gamma'$ to a single one. If we omit all finite parts, we have
\begin{align*}
\braket{\Upsilon_{i'}(K',\omega')|\Upsilon_i(K,\omega)} &= 2\pi\delta(K-K') \pi \delta(\omega-\omega') \sum_{\gamma} \ol{q_{i'}^{\gamma}(K',\omega')} q_i^\gamma(K,\omega) \left|\frac{\d\omega_\gamma}{\d\kappa}(\kappa_\gamma)\right|.
\end{align*}
With the $Q_\text{in/out}$ as defined above the overlap reduces to
\begin{align*}
\braket{\Upsilon_{i'}(K',\omega')|\Upsilon_i(K,\omega)} &= 2\pi\delta(K-K') 2\pi \delta(\omega-\omega') \left[Q_\text{in}(K,\omega)\right]_{i'} \dag V^2_\text{in}(K,\omega) \left[Q_\text{in}(K,\omega)\right]_i \\
&= 2\pi\delta(K-K') 2\pi \delta(\omega-\omega') \left[Q_\text{out}(K,\omega)\right]_{i'} \dag V^2_\text{out}(K,\omega) \left[Q_\text{out}(K,\omega)\right]_i.
\end{align*}
\end{widetext}
\subsection{One type of particle}
\label{sec:oneType}
Let us make things more concrete by working out the case where the one-particle spectrum consists of just one type of particle with dispersion relation $\Delta(\kappa)$. Suppose we have only one combination of momenta $\kappa_1$ and $\kappa_2$ such that they add up to total momentum $K=\kappa_1+\kappa_2$ and total energy $\omega=\Delta(\kappa_1)+\Delta(\kappa_2)$. There are two asymptotic modes -- one mode with $\kappa_1$ on the left and $\kappa_2$ on the right, and one mode with the momenta interchanged -- that combine into one scattering state with the asymptotic form
\begin{equation*}
\vect{c}(n) \rightarrow q^1 \ensuremath{\mathrm{e}}^{i\kappa_2 n} \vect{v_1} + q^2 \ensuremath{\mathrm{e}}^{i\kappa_1n} \vect{v_2}.
\end{equation*}
The conservation equation that was derived in the previous section takes on the simple form
\begin{equation*}
\left|q^1\right|^2 = \left|q^2\right|^2
\end{equation*}
because $\omega'(\kappa_1) = -\omega'(\kappa_2)$. As we mentioned above in the general case, the relative phase of the two vectors $\vect{v_1}$ and $\vect{v_2}$ can be chosen arbitrarily. However, since the two modes correspond to the interchanging of two identical particles, it makes sense to fix the phase such that $\vect{v_2}\dag\vect{v_1}>0$. Due to the momentum dependence of the one-particle solutions, this overlap will be slightly smaller than one.
\par The S matrix reduces to a phase factor and is defined as
\begin{equation*}
S(\kappa_1,\kappa_2) = S(K,\omega) = \frac{q^2}{q^1}.
\end{equation*}
The asymptotic wave function takes the form
\begin{multline} \label{oneType}
\ket{\Upsilon(\kappa_1,\kappa_2)} \rightarrow \sum_{n_1<_2} \ensuremath{\mathrm{e}}^{i(\kappa_1n_1+\kappa_2n_2)} \left[ \text{$B_{\kappa_1}$ at $n_1$,$B_{\kappa_2}$ at $n_2$} \right] \\ + S(\kappa_1,\kappa_2) \ensuremath{\mathrm{e}}^{i(\kappa_2n_1+\kappa_1n_2)} \left[ \text{$B_{\kappa_2}$ at $n_1$,$B_{\kappa_1}$ at $n_2$} \right].
\end{multline}
From simple arguments \cite{Sachdev2011} one can argue that in one dimension the S matrix should have the universal limiting value for low-energy scattering \cite{Sachdev1997,Damle1997}
\begin{equation*}
S(\kappa_1,\kappa_2) \rightarrow -1 \quad \text{as} \quad |\kappa_1-\kappa_2|\rightarrow 0.
\end{equation*}
We define the scattering phase $\theta$ as the phase shift of the S matrix relative to its universal low-energy value $S(\kappa_1,\kappa_2)=-\ensuremath{\mathrm{e}}^{i\theta(\kappa_1,\kappa_2)}$.
\subsection{Spectral functions}
With the variational wave functions of one- and two-particle states, we can now calculate the low-energy part of spectral functions at zero temperature. We consider the following function
\begin{equation*}
S(\kappa,\omega) = \sum_n \int \d t \, \ensuremath{\mathrm{e}}^{i(\omega t - \kappa n)} \bra{\Psi_0} O_n\dag(t) O_0(0) \ket{\Psi_0}
\end{equation*}
with $O_n(t)$ an operator at site $n$ in the Heisenberg picture. In order to approximate the low-energy part, we insert a projector on the one- and two-particle subspaces
\begin{multline*}
P_\text{1p,2p} = \int \frac{\d\kappa}{2\pi} \sum_{\alpha\in\Gamma_1(\kappa)} \ket{\Phi_\alpha(\kappa)}\bra{\Phi_\alpha(\kappa)} \\
+ \int \frac{\d K}{2\pi} \int \frac{\d\omega}{2\pi} \sum_{\gamma\in\Gamma_2(K,\omega)}\ket{\Upsilon_\gamma(K,\omega)}\bra{\Upsilon_\gamma(K,\omega)}
\end{multline*}
where $\Gamma_{1}$ ($\Gamma_{2}$) is the set of all types of one-particle (two-particle) states at that momentum (momentum-energy). The states are orthonormalized as
\begin{align*}
& \braket{\Phi_{\gamma'}(\kappa')|\Phi_\gamma(\kappa)} = 2\pi\delta(\kappa'-\kappa) \delta_{\gamma\gamma'} \\
& \braket{\Upsilon_{\gamma'}(K',\omega')|\Upsilon_\gamma(K,\omega)} = 4\pi^2\delta(K'-K) \delta(\omega'-\omega) \delta_{\gamma\gamma'}
\end{align*}
so that we obtain the Lehmann representation \cite{Lehmann1954} for the spectral function up to two-particle contributions
\begin{align*}
S(\kappa,\omega) &= \sum_{\alpha\in\Gamma_1(\kappa)} 2\pi\delta(\Delta_\alpha(\kappa)-\omega) \left|\bra{\Phi_\alpha(\kappa)} \hat{O}_0 \ket{\Psi_0} \right|^2 \\
& \qquad + \sum_{\gamma\in\Gamma_2(\kappa,\omega)} \left| \bra{\Upsilon_\gamma(\kappa,\omega)}\hat{O}_0 \ket{\Psi_0} \right|^2 \\
& + ...
\end{align*}
In gapped systems, the one- and two-particle contributions saturate the full spectral function below the three-particle threshold, while contributions from higher-particle excitations might become important for higher energies. Yet, it appears that typically the one- and two-particle sectors already contain the largest portion of the spectral function, see e.g. Ref.~\onlinecite{Caux2008}. The one- and two-particle form factors appearing in the above expression are calculated explicitly in Appendix \ref{sec:form1p}.
\par To get a quantitative estimate of how well the spectral function is approximated, we look at the zeroth and first frequency moment at a certain momentum, which are defined as
\begin{equation*}
s_0(\kappa) = \int \frac{\d \omega}{2\pi} S(\kappa,\omega) \quad \text{and} \quad s_1(\kappa) = \int \frac{\d \omega}{2\pi} \omega S(\kappa,\omega).
\end{equation*}
These quantities follow the sum rules \cite{Hohenberg1974}
\begin{equation*} \begin{split}
s_0(\kappa) &= \int \frac{\d \omega}{2\pi} \bra{\Psi_0} O_{-\kappa}\dag 2\pi\delta(\omega-\hat{H}) O_0(0) \ket{\Psi_0} \\
&= \bra{\Psi_0} O_{-\kappa}\dag O_0(0) \ket{\Psi_0}
\end{split} \end{equation*}
and
\begin{align*}
s_1(\kappa) &= \int \frac{\d \omega}{2\pi} \omega \bra{\Psi_0} O_{-\kappa}\dag 2\pi\delta(\omega-\hat{H}) O_0(0) \ket{\Psi_0} \\
&= \bra{\Psi_0} O_{-\kappa}\dag \hat{H} O_0(0) \ket{\Psi_0}.
\end{align*}
If the ground state is taken to be an MPS, these quantities can be calculated exactly. Note that $s_0$ is just the static correlation function and that the ratio of the two is equal to the single mode approximation for the dispersion relation \cite{Arovas1992}
\begin{align*}
\Delta_\text{SMA}(\kappa) = \frac{s_1(\kappa)}{s_0(\kappa)} = \frac{\bra{\Psi_0} O_{-\kappa}\dag \hat{H} O_0(0) \ket{\Psi_0}}{\bra{\Psi_0} O_{-\kappa}\dag O_0(0) \ket{\Psi_0}}.
\end{align*}
By comparing the one- and two-particle contributions for $s_0$ and $s_1$ to the exact values, we can get an idea of how well these eigenstates capture the effect of the operators working on the ground state and, consequently, how well the spectral function is approximated by only looking at these contributions.
\section{Two-particle S matrix and approximate Bethe ansatz}
\label{sec:section3}
We now discuss how the variational formulation of scattering theory using matrix product states, as developed in the previous section, relates to standard scattering theory. We then discuss how we can use the information provided by the scattering matrix to build an effective description of the low-energy behaviour of the spin chain using the approximate Bethe ansatz.
\subsection{Stationary scattering states and the S matrix in one dimension}
\label{sec:moller}
In standard scattering theory the S matrix is typically defined from a dynamical point of view: its elements are the overlaps of asymptotically free in and out states with respect to the full time-evolution operator. Although it is a priori not clear that this definition corresponds to the one that was presented in the previous sections, we can show that this is indeed the case.
\par Appendix \ref{sec:mollerA} provides a summary of the standard scattering formalism of single-particle quantum mechanics \cite{Taylor1972}, which we have adapted to the one-dimensional setting with general Hamiltonians (e.g. potentials which are not diagonal in real space) and arbitrary dispersion relations (non-quadratic eigenvalue spectrum of the ``free'' Hamiltonian $\ensuremath{\hat{H}}_0$). More specifically we have shown how the S matrix elements $f(q_\beta\leftarrow p_\alpha)$ show up in the asymptotic form of the scattering eigenstates $\ket{p_\alpha\pm}$ of the full Hamiltonian $\ensuremath{\hat{H}}$.
\par To make the connection to the variational scattering states of Sec.~\ref{sec:section2}, we have to make a few modifications. First of all, we can reformulate the two-particle scattering problem as a one-particle problem by factoring out the conservation of total momentum and only focus on the matrix elements between different relative momenta. At every value of the total momentum, we can define relative momentum states $\ket{p_\gamma}$ with dispersions $\omega(p_\gamma)$, which are solutions of the free Hamiltonian $\hat{H}_0$. This free Hamiltonian corresponds to the effective two-particle Hamiltonian matrix in the asymptotic regime \eqref{Am} and the states $\ket{p_\gamma}$ are the asymptotic modes \eqref{asModes}.
\par Secondly, our ``one-particle'' Hilbert space is only defined on a half-infinite line, because the particles are essentially bosonic. The way around this consists of artificially assigning particle labels and distinguishing the situation where particle 1 (2) is on the left (right), and the opposite situation; the relative coordinate $n=n_2-n_1$ now ranges over the positive and negative integers. Alternatively, one could add to the free Hamiltonian $\ensuremath{\hat{H}}_0$ a potential $\hat{V}$ which is infinite everywhere on the negative real line, making this a forbidden region. Scattering theory would still work (existence of the M\"{o}ller operators etc.), provided that we restrict the ``in'' states to momenta for which $\frac{\d \omega}{\d p}(p) <0$ and the out states to momenta for which $\frac{\d \omega}{\d p}(p)>0$. This corresponds exactly to how we defined the incoming and outgoing modes in Sec.~\ref{sec:norm}.
\par Translating the expression for the asymptotic wave function of the scattering states $\ket{p_\alpha+}$ to the framework of Sec.~\ref{sec:norm} amounts to the following form for the wave function $\vect{c}_\alpha(n)$
\begin{multline*}
\vect{c}_\alpha(n) \rightarrow \left| \frac{\d\omega}{\d\kappa}(\kappa_\alpha) \right|^{-1/2} \vect{v}_\alpha e^{i p_\alpha n} \\
+ \sum_{\gamma\in A^{+}(\kappa_\alpha)} f(\kappa_\gamma\leftarrow\kappa_\alpha) \left| \frac{\d\omega}{\d\kappa}(\kappa_\gamma) \right|^{-1/2} \vect{v}_\gamma e^{i\kappa_\gamma n}
\end{multline*}
for every incoming mode $\alpha=1,\dots,\Gamma$. In this representation, we choose one incoming mode $\alpha$ that couples only to all outgoing modes $\{\gamma\in A^{+}(\kappa_\alpha)\}$. The coefficient matrix for the incoming modes that was defined earlier takes on the form
\begin{equation*}
(Q_\text{in})_{\gamma,\alpha} = \delta_{\gamma\alpha} \left| \frac{\d\omega}{\d\kappa}(\kappa_\alpha) \right|^{-1/2}
\end{equation*}
while the coefficients for the outgoing modes are given by
\begin{equation*}
(Q_\text{out})_{\gamma,\alpha} = \left| \frac{\d\omega}{\d\kappa}(\kappa_\gamma) \right|^{-1/2} f(\kappa_\gamma\leftarrow\kappa_\alpha).
\end{equation*}
The S matrix $S(K,\omega)$ that was defined takes on the simple form (as $V_\text{in}Q_\text{in}=\mathds{1}$ in this representation)
\begin{align*}
S(K,\omega) &= V_\text{out}Q_\text{out} \\
&= f(\kappa_\gamma\leftarrow \kappa_\alpha).
\end{align*}
Through this identification the unitariness of the S matrix $S(K,\omega)$ that was proven in the previous section is indeed equivalent to the unitary S matrix defined through the M{\o}ller operators as $S=\Omega_-\dag\Omega_+$.
\subsection{Scattering length and bound states}
\label{sec:scatBound}
Suppose we have the scattering process of two identical particles in the limit of vanishing relative momentum. We expect that the equation for the relative wave function $\psi(x)$ should obey the zero energy and zero potential Schr\"odinger equation
\begin{equation*}
\frac{\d^2\psi(x)}{\d x^2} = 0
\end{equation*}
in the region $x>x_0$ where $x_0$ is the length of the interaction. The solutions are of the form $\psi(x) \propto x - a$ for large $x$ which matches the asymptotic form of Sec.~\ref{sec:oneType} if the phase of the S matrix reduces to
\begin{equation} \label{linearTheta}
\theta(\kappa_1,\kappa_2) \approx - a (\kappa_1-\kappa_2)
\end{equation}
in the limit for $\kappa_1-\kappa_2\rightarrow0$. The slope $a$ will be called the scattering length and still depends on the total momentum $\kappa_1+\kappa_2$.
\par Suppose now the existence of a bound state with very low binding energy $-\epsilon$. The wave function of this bound state should look like $\psi(x)=\ensuremath{\mathrm{e}}^{-\kappa x}\approx 1-\kappa x$ with $\omega(i\kappa)= -\epsilon \rightarrow0$. If we want the formation of this bound state to follow smoothly from a scattering state with vanishing energy, the scattering length should diverge. This means that the formation of a bound state out of a scattering continuum at a certain momentum should be accompanied by a diverging scattering length.
\subsection{Approximate Bethe Ansatz}
\label{sec:aba}
In this section we will develop a method to describe a finite density of excitations based on the coordinate Bethe ansatz. For simplicity, we will for the remainder of this section restrict to the case of one type of particle -- making the consistency conditions for factorized scattering (Yang-Baxter equation) trivial -- but the framework can be extended to multicomponent situations \cite{Sutherland2004}. We will interpret the strongly correlated MPS ground state as a vacuum state on which we can build $N$-particle states, described by a $N$-particle wave function $\Psi(x_1,\dots,x_N)$. Although in general we have no particle conservation in the system, we will argue that the first-quantized approach gives a good approximation at low densities. Indeed, particle-number violating processes involve three or more particles and can be neglected at low densities. In Sec.~\ref{sec:section5} we will discuss how to develop a second-quantization approach.
\par We start with one particle. We can link the one-particle excitation $\ket{\Phi_\kappa[B]}$ with dispersion $\Delta(\kappa)$ in an obvious way with a one-particle wave function $\Psi_1(x)$ in first quantization as
\begin{equation*}
\Psi_1(x) = \ensuremath{\mathrm{e}}^{i\kappa x}.
\end{equation*}
Adding a second particle can be done by only taking account of the asymptotic part of the two-particle wave function [Eq.~\eqref{oneType}] ($x_1<x_2$)
\begin{align*}
\Psi_2(x_1,x_2) = \ensuremath{\mathrm{e}}^{i(\kappa_1x_1 + \kappa_2x_2)} + S(\kappa_1,\kappa_2) \ensuremath{\mathrm{e}}^{i(\kappa_2x_1 + \kappa_1x_2)}.
\end{align*}
As we are working with identical particles, the wave function in the other sector ($x_1>x_2$) has to be determined from the statistics of the particles. On the level of the spin system, the addition of a particle is a local operation, so we will work with bosonic many-particle wave functions.
\par The addition of a third particle can only be done approximately. Indeed, a three-particle wave function has the general form \cite{Sutherland2004}
\begin{align}
& \Psi_3(x_1,x_2,x_3) = \ensuremath{\mathrm{e}}^{i(\kappa_1x_1 + \kappa_2x_2 + \kappa_3x_3)} \nonumber\\
& \qquad + S(\kappa_1,\kappa_2) \ensuremath{\mathrm{e}}^{i(\kappa_2x_1 + \kappa_1x_2+\kappa_3x_3)} \nonumber \\
& \qquad + \dots \nonumber \\
& \qquad + \iiint \d\kappa_1'\d\kappa_2'\d\kappa_3' \; S(\kappa_1'\kappa_2'\kappa_3')\; \ensuremath{\mathrm{e}}^{i(\kappa_1'x_1 + \kappa_2'x_2 + \kappa_3'x_3)} \nonumber \\
& \qquad + \text{other particle numbers} .\label{three}
\end{align}
The first terms represent a sum over all six permutations of the three momenta, with the S matrices for all possible two-particle scattering processes as prefactors. The next term is the diffractive part, which accounts for the three-particle scattering. For these scattering processes, the two conservation laws are not enough to preserve individual momenta and we can generate a whole continuum of other momenta. The last term accounts for the non-particle preserving scattering processes, which can generate two- or four-particle states as well. As a result, it is no longer possible to assign a set of individual momenta $\{\kappa_1,\kappa_2,\kappa_3\}$ (or even a particle number) to this wave function, because they are completely mixed up with all other possible sets of momenta that are compatible with conservation of total energy and momentum.
\par The crucial approximation of our approach is that we neglect the two last terms in Eq.~\eqref{three}: every many-particle scattering event can be decomposed into two-particle scatterings that preserve particle number and individual momenta. This implies that three-particle eigenstates can be labeled by three individual momenta and that the three-particle wave function is given by the permutation terms only. The absence of diffractive scattering is the hallmark of integrability \cite{Sutherland2004}, so we are essentially assuming that our many-particle system is integrable \cite{Krauth1991, Kiwata1994a, Okunishi1999a}.
\par If this approximation proves to be valid, we can apply the Bethe ansatz machinery\cite{Bethe1931,Sutherland2004,Korepin1997}. The first-quantized wave function of an integrable $N$-particle system, unambiguously defined by a set of momenta $\{\lambda_1,\dots,\lambda_M\}$, is a sum of plane waves with all possible permutations of the momenta
\begin{equation} \label{bethe}
\Psi(x_1,\dots,x_N) = \sum_{\ensuremath{{\mathcal{P}}}} A(\ensuremath{{\mathcal{P}}}) \ensuremath{\mathrm{e}}^{i(\lambda_{\ensuremath{{\mathcal{P}}} 1}x_1 + \dots+\lambda_{\ensuremath{{\mathcal{P}}} N}x_N)}
\end{equation}
where $A(\ensuremath{{\mathcal{P}}})/A(\ensuremath{{\mathcal{P}}}') = S(\lambda_i,\lambda_j)$ if the permutations $\ensuremath{{\mathcal{P}}}$ and $\ensuremath{{\mathcal{P}}}'$ differ by the interchange of the momenta $\lambda_i$ and $\lambda_j$.
\par By imposing periodic boundary conditions on the Bethe wave function in the thermodynamic limit, we arrive at a description of the ground state as a Fermi sea of ``pseudo-momenta'' filled up to a certain Fermi level $q$. In contrast to the free-fermion case, the density of occupied modes is not constant but given by the function $\rho(\lambda)$ such that $\rho(\lambda)=0$ for $|\lambda|>q$. The energy of the modes $\epsilon(\lambda)$ can be determined from the integral equation
\begin{equation} \label{liebEq1}
\epsilon(\lambda) -\frac{1}{2\pi} \int_{-q}^{q} K(\lambda,\mu) \epsilon(\mu) \d\mu = \epsilon_0(\lambda)
\end{equation}
where $\epsilon_0(\lambda)$ is the ``bare energy'' of the particle, i.e. the energy an isolated particle with momentum $\lambda$ would have in an infinite system. The kernel of the integral equation is given by the derivative of the scattering phase $K(\lambda,\mu)=\partial_\lambda\theta(\lambda,\mu)$. The value of the Fermi level is computed self-consistently from this equation and the requirement that $\epsilon(\pm q)=0$. Once $q$ has been determined, the density $\rho(\lambda)$ is the solution of a similar integral equation \cite{Lieb1963a, *Lieb1963b}
\begin{equation} \label{liebEq2}
\rho(\lambda) -\frac{1}{2\pi} \int_{-q}^{q} K(\lambda,\mu) \rho(\mu) \d\mu = \frac{1}{2\pi}.
\end{equation}
The total density and energy density are given by
\begin{align} \label{density}
D = \int_{-q}^{q} \rho(\lambda)\d\lambda \qquad \text{and} \qquad E = \frac{1}{2\pi} \int_{-q}^q \epsilon(\lambda)\d\lambda .
\end{align}
\par The excitation spectrum is easily characterized in terms of the pseudo-particles of the Bethe ansatz. We can construct two types of elementary excitations: either we take one particle with momentum $|\lambda|<q$ out of the Fermi sea (hole excitation) or we add one particle with momentum $|\lambda|>q$ (particle excitation). These elementary particle and hole excitations have a topological nature \cite{Korepin1997}, so that the physical excitations -- the ones having a finite overlap with a local operator -- consist of an even number of particles and holes \cite{Lieb1963b}. This gives rise to the physical excitation spectrum as shown in Fig.~\ref{fig:sea}.
\begin{figure}
\subfigure[]{\includegraphics[width=0.7\columnwidth]{./sea.pdf}}
\subfigure[]{\includegraphics[width=0.7\columnwidth]{./spectrumSea.pdf}\label{fig:sea}}
\caption{(a) The Fermi sea of pseudo momenta, filled up to the Fermi level $q$. Physical excitations can be pictured as particle-hole excitations close to the Fermi-level. (b) The physical excitation spectrum, the grey area represents a continuum of states. Because of the fact that physical excitations always come in pairs, the spectrum has its minima at momentum 0 and $2k_F$. The slope of the dispersion relation at these momenta is the Fermi velocity $u$.}
\end{figure}
\par This critical one-dimensional bose gas can be described as a Luttinger liquid (LL) \cite{*Haldane1981b, Giamarchi2004}. A first important quantity is the Fermi momentum $k_F$, the physical momentum of the gapless particle and hole excitations. It is given by the dressed momentum of the Fermi level and is directly related to the density as (see appendix)
\begin{align} \label{eq:luttingerTheorem}
k_F = \pi D.
\end{align}
Since we have gapless excitations at $0$ and $\pm 2k_F$, correlation functions will have their oscillation periods at these values. The slope of the dispersion relation is the Fermi velocity $u$ and can be calculated from the Bethe ansatz. The third important characteristic quantity is the LL parameter $K$ which determines the power-law decay of correlation functions. In order to calculate it, we define the function $S_R(\lambda)$ as ($h$ is a chemical potential for the particles)
\begin{equation*}
S_R(\lambda) = - \frac{\partial \epsilon(\lambda)}{\partial h}
\end{equation*}
which (from Eq.~\eqref{liebEq1}) follows the integral equation
\begin{equation}
S_R(\lambda) - \frac{1}{2\pi} \int_{-q}^q K(\lambda,\mu) S_R(\mu) \d\mu = 1.
\end{equation}
In the context of a dilute gas of magnons (see Sec.~\ref{sec:magnProcess}) $S_R(q)$ can be interpreted as the renormalized spin of the magnon close to the Fermi surface. With the low-energy excitations just above the Fermi sea behaving as free fermions \cite{Lesage1997} (i.e. their S matrix is -1), one can show that the LL parameter $K$ is related to $S_R(q)$ as \cite{Konik2002}
\begin{equation} \label{eq:LuttK}
K = S_R(q)^2.
\end{equation}
By thus making the connection between the approximate Bethe ansatz and the LL description, we can infer information on the critical correlations in a system where a finite density of excitations forms on top of a strongly-correlated vacuum state. More specifically, we can infer the long-range behaviour of one-particle and pair correlation functions as \cite{Haldane1981a, Cazalilla2011a}
\begin{equation} \label{eq:correlators} \begin{split}
& g_1(x) = A_0 \frac{1}{x^{1/2K}} - A_1 \frac{\cos(2\pi Dx)}{x^{2K+1/2K}} + \dots \\
& D_2(x) = D^2 - \frac{K}{2\pi^2x^2} + B_1 \frac{\cos(2\pi Dx)}{x^{2K}} + \dots
\end{split} \end{equation}
where $D$ is the density, $A_0$, $A_1$, and $B_1$ are non-universal constants and the dots denote higher order terms. Depending on whether the operator targets a particle or a pair, the corresponding correlation functions will decay according to one of these two forms.
\subsection{Limiting cases}
\label{sec:limiting}
The Bethe ansatz equations of the previous section can be greatly simplified if we assume that we work at very low densities. Indeed, assuming that only the lowest pseudo-momentum states are occupied, we can approximate the full dispersion relation by its quadratic form $\epsilon_0(\lambda)\approx c\lambda^2-h$, and the full two-particle S matrix by its limiting value of $S(\theta,\mu)\approx-1$. With the kernel of the integral equation zero, we find easily the density and the (physical) Fermi momentum
\begin{align*}
D=\frac{1}{\pi}\sqrt\frac{h}{c}, \qquad k_F = \sqrt\frac{h}{c}
\end{align*}
and the LL parameters
\begin{align*}
u = 2\pi c D, \qquad K = 1.
\end{align*}
\par Upon increasing the density, the limiting value of the S matrix will no longer apply. From Sec.~\ref{sec:scatBound} we know that the first order correction to the scattering phase is given by the scattering length, so we can insert the form \eqref{linearTheta} into the Bethe equations, while still assuming a quadratic dispersion relation. The first order correction to the Fermi level is linear in the scattering length
\begin{equation}
q = q_\text{FF} + \delta q = q_\text{FF} - \frac{ah}{3\pi c},
\end{equation}
so that the correction to the density is given by
\begin{align} \label{densityScatLength}
D = \frac{1}{\pi} \sqrt{\frac{h}{c}} - \frac{4ha}{3\pi^2c} + \mathcal{O}(a^2).
\end{align}
This result coincides with the one in Ref.~\onlinecite{Lou2000}. The LL parameters in first order in $a$ are given by \cite{Affleck2005, Affleck2004}
\begin{align*}
u = 2 c \sqrt{\frac{h}{c}} + \frac{4ah}{3\pi} + \mathcal{O}(a^2)
\end{align*}
and
\begin{align*}
K = 1 - 2 a D + \mathcal{O}(a^2).
\end{align*}
\subsection{Thermodynamic Bethe ansatz}
At zero temperature, the coordinate Bethe ansatz describes an integrable system in its ground state by filling up a Fermi sea of quasi-momentum states; its excitations are holes and particles above this Fermi sea. When a finite temperature $T$ is applied, these particles and holes will have finite distribution densities. By associating an entropy to these distributions and minimizing the free energy, one arrives at the celebrated Yang-Yang equation \cite{Yang1969}
\begin{multline*}
\epsilon(\lambda) = \epsilon_0(\lambda) \\ -\frac{T}{2\pi} \int_{-\infty}^{+\infty} K(\lambda,\mu) \ln\left( 1+\ensuremath{\mathrm{e}}^{-\epsilon(\mu)/T} \right) \d\mu,
\end{multline*}
a non-linear integral equation for the dressed energy $\epsilon(\lambda)$ of the quasi momentum states; the equation can be solved by iteration \cite{Takahashi2005}. The density of occupied vacancies $\rho(\lambda)$ is given by
\begin{equation*}
\theta(\lambda)=\frac{\rho(\lambda)}{\rho_v(\lambda)} = \frac{1}{1+\ensuremath{\mathrm{e}}^{\epsilon(\lambda)/T}}
\end{equation*}
with $\rho_v(\lambda)$ the density of all (occupied and empty) vacancies. Through this equation the density of occupied vacancies satisfies the integral equation
\begin{equation}
\rho(\lambda) = \frac{\theta(\lambda)}{2\pi} \left( 1+ \int_{-\infty}^{+\infty} K(\lambda,\mu) \rho(\mu)\d\mu \right),
\end{equation}
such that the total density can be calculated as
\begin{equation}
D = \int_{-\infty}^{+\infty} \rho(\lambda) \d\lambda.
\end{equation}
\subsection{Effective integrable field theories}
Another way of dealing with a finite density of excitations, based on information on the one-particle dispersion and the two-particle S matrix, consists of mapping the system to an effective integrable field theory. The parameters in this effective theory should be tuned to fit the variational information as good as possible. This approach has the advantage that integrability is exact for the effective model, but the mapping is typically only valid in some small region (e.g. low density and/or low temperature).
\par One possible field theory is obtained by making the approximation that the particles interact through a contact potential \cite{Okunishi1999, Okunishi1999a}, so that we end up with a Lieb-Liniger model \cite{Lieb1963a, *Lieb1963b}. The first-quantized Hamiltonian for a collection of $N$ bosons is given by
\begin{equation}
H = -\frac{1}{2m} \sum_{j=1}^N \frac{\partial^2}{\partial x_j^2} + 2c \sum_{j<k=1}^N \delta(x_j-x_k)
\end{equation}
with the mass $m$ of the bosons and the interaction strength $c$ as the two tunable parameters. The two-boson S matrix is given by $S(\lambda_1,\lambda_2) = -\ensuremath{\mathrm{e}}^{i\theta(\lambda_1-\lambda_2)}$ with
\begin{equation}
\theta(\lambda) = 2\arctan\left(\frac{\lambda}{c}\right),
\end{equation}
so that the scattering length for a $\delta$ potential is $a_\delta=-2/c$. The boson dispersion relation is just quadratic, i.e. $\Delta(\lambda) = \lambda^2/(2m)$. By variationally calculating the dispersion relation and the scattering length of the relevant excitations, we can fix the two parameters and map the density of excitations to a Lieb-Liniger model. At low densities, we expect that this mapping is quantitatively correct.
\par Another possibility is the non-linear sigma model, which has proven to capture the qualitative behaviour of Haldane-gapped spin chains such as the spin-1 Heisenberg model \cite{Haldane1983a} or two-leg spin-1/2 ladders. In contrast to the Lieb-Liniger model, however, we can not tune any parameters to fit the exactly known \cite{Zamolodchikov1979} two-particle S matrix. The universal behaviour of e.g. the magnon condensation of a gapped spin chain in a magnetic field \cite{Konik2002}, can nonetheless be captured with this mapping.
\section{Application to spin ladders}
\label{sec:section4}
We will study the spin-1/2 Heisenberg antiferromagnetic (HAF) two-leg ladder in a magnetic field, defined by the Hamiltonian
\begin{equation}
H = \sum_{i,l} \vec{S}_{i,l} \cdot \vec{S}_{i+1,l} + \gamma \sum_{i} \vec{S}_{i,1} \cdot \vec{S}_{i,2} - h \sum_{i,l}S^z_{i,l}
\end{equation}
where $l=1,2$ denote the two legs of the ladder and $\vec{S}_{i,l}$ denotes the spin operator at site $i$ in the $l$'th leg (see Fig.~\ref{fig:ladder}).
\begin{figure}
\includegraphics[width=0.7\columnwidth]{./ladder.pdf}
\caption{The ladder geometry with $J_\parallel$ and $J_\perp$ the couplings along the leg, resp. rung. We will always put $J_\parallel=1$ and define the coupling ratio $\gamma=J_\perp/J_\parallel$.}
\label{fig:ladder}
\end{figure}
\par The two-leg HAF ladder and its excitation spectrum have been studied intensively for many reasons. First of all, it is the first step to study the transition from one-dimensional systems to higher-dimensional versions. Secondly, the excitation spectrum has a lot of interesting features, such as the presence of a gap \cite{Haldane1983} and the existence of bound states, and can be studied with a variety of methods depending on the parameter regime. These features can also be observed experimentally \cite{Masuda2006, Notbohm2007, Shapiro2007, Schmidiger2013a}, so that ladders provide an ideal test for these theoretical methods \cite{Bouillot2011, Schmidiger2013}. Finally, the experimental realization of magnetized spin ladders provides an ideal quantitative test of the Luttinger liquid model \cite{Giamarchi2008,Klanjsek2008,Ruegg2008}.
\par In this section we will test our variational method on the two-leg ladder. An MPS approximation for the ground state can be found by first blocking two spins on a rung into one four-level system and applying an MPS optimization algorithm (we have used the TDVP algorithm \cite{Haegeman2011d,Haegeman2014a}). In this representation (to every rung there corresponds one MPS tensor $A$) we find a ground state that is invariant under translations over one site in the leg direction; all momenta in the following subsections are defined with respect to this translation operator. The Hamiltonian and the ground state are invariant under the reflection operator $\mathcal{P}$ which flips the two legs of the ladder. We impose no additional symmetries (e.g. SU(2) invariance) on the MPS, but our variational solution will of course have the right symmetries to high precision.
\par In the first three subsections we will investigate the low-lying spectrum of the ladder without magnetic field. In the following two subsections we will apply the approximate Bethe ansatz to the magnetization process, at zero and finite temperature.
\subsection{One-particle excitations: elementary spectrum and bound states}
\label{sec:ladder1p}
The nature of the elementary excitations in the ladder can be understood starting from different limits.
\par At zero coupling ($\gamma\rightarrow0$), we have two independent spin-1/2 Heisenberg chains where the elementary excitations are spinons (carrying spin 1/2). These spinons are topologically non-trivial excitations and can only be created in pairs by the action of a local operator. Upon coupling the chains, the spinons are confined into magnons carrying integer spin. This picture has been studied with bosonization techniques \cite{Shelton1996}, showing that the interchain coupling opens up a gap to a triplet of massive magnons (triplons) and a higher-up singlet.
\par At infinite coupling ($\gamma\rightarrow\infty$) we have a collection of independent rungs with antiferromagnetic interaction. In the ground state all rungs are in a singlet state and an elementary excitation is constructed by promoting one rung to a triplet state. When the leg coupling $J_\perp$ is turned on, this triplet obtains a kinetic energy and we get a non-trivial dispersion. This qualitative picture survives for intermediate couplings: through perturbative continuous unitary transformations an effective particle picture can be established and very accurate results on e.g. the elementary dispersion relation and bound states can be obtained \cite{Trebst2000, *Knetter2001, *Schmidt2001, *Knetter2003, *Coester2014, Schmidt2005}.
\par In Fig.~\ref{fig:gap} we have plotted the gap in function of the interchain coupling. One can observe that the gap goes to zero in the weak-coupling limit, while it grows to the constant value that one expects from a strong-coupling expansion. Our variational results smoothly interpolates between these two limits.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./gap.pdf}
\caption{The rescaled gap $\frac{\Delta}{\sqrt{1+\gamma^2}}$ in function of the interchain coupling $\gamma$. The blue dashed lines are the first order correction from the strong-coupling limit ($\gamma\rightarrow\infty$) and results from bosonization in the weak-coupling limit ($\gamma\rightarrow0$) \cite{Shelton1996, Larochelle2004}.}
\label{fig:gap}
\end{figure}
\par A typical excitation spectrum in the intermediate region ($\gamma=2$) is shown in Fig.~\ref{fig:spectrum}. The lowest-energy state is an elementary triplet excitation (magnon) with a minimum at momentum $\pi$. The magnon has odd parity under the reflection operator $\mathcal{P}$. The lowest-energy state around momentum zero is a two-magnon scattering state and has even parity. Because the one- and two-magnon state have different parity, the elementary magnon cannot decay and is stable in the whole Brillouin zone. From Fig.~\ref{fig:variance}, where we have plotted the variance of the excitation ansatz, we can indeed see that the magnon is a bona fide particle excitation for all momenta. Note that under a parity-breaking interaction the stability of the magnon inside the continuum breaks down \cite{Fischer2010} and it might prove an interesting question whether we can capture its decay within our framework.
\par The elementary excitation spectrum at $\gamma=2$ has two more elementary particle excitations, a singlet and a triplet, which are stable in a limited region around momentum $\pi$. Both are even under the parity operator $\mathcal{P}$. From the strong-coupling expansion, we can interpret them as two-magnon bound states \cite{Trebst2000}, hence the even parity (without a well-defined particle number, we cannot make this interpretation, so we regard these branches as elementary particles). The variance of the bound states is sufficiently small in the stable region, but it grows larger as the momentum approaches the continuum. From Ref.~\onlinecite{Haegeman2013a} we know that the localized nature of an elementary excitation is related to the gap below and above the excitation branch, so we expect the bound state to become wider as the gap to the continuum closes. This explains the increasing variance of the bound states in Fig.~\ref{fig:variance}. Upon entering the continuum, the bound state has become completely delocalized and no longer exists as a stationary eigenstate of the Hamiltonian.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./spectrum2.pdf}
\caption{The one-particle spectrum consists of a triplet (magnon) which is stable over the whole Brillouin zone (lowest lying blue curve), an singlet (bound state) which is stable for momenta between $\kappa_\text{BS1}\approx0.39\pi$ and $\pi$ (second blue curve), and a triplet (bound state) which is stable for momenta between $\kappa_\text{BS2}\approx0.46\pi$ and $\pi$ (third blue curve). Note that the determination of $\kappa_\text{BS1}$ and $\kappa_\text{BS2}$ is not very precise because the one-particle ansatz is not accurate near the transition. The red region is the two-magnon continuum, the green region is the three-magnon continuum; the other continua (e.g. triplet-singlet continuum) are not shown. }
\label{fig:spectrum}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{./variance.pdf}
\caption{The ($\log_{10}$ of the modulus of the) variance of the one-particle excitations; dots, resp. crosses are positive, resp. negative variances (see Appendix \ref{sec:variance} for the meaning of a negative variance). The magnon (green) is clearly a well-defined particle excitation in the whole Brillouin zone. The singlet (red) and triplet (blue) get larger variances as they come closer to the two-particle band, until they actually dive in and are no longer stable. Calculations were done at $\gamma=2$ with bond dimension $D=30$; the ground state variance density is $2.27\times10^{-8}$ at that bond dimension.}
\label{fig:variance}
\end{figure}
\par As a last illustration of the one-particle ansatz we have included Table~\ref{table:variance} with excitation energies and variances in the weak-coupling region, showing the elementary triplet and singlet excitations that we expect from a bosonization calculation. We observe that the variances are some orders of magnitude larger in this weak-coupling region. Since the gaps above and below these excitations are a lot smaller at small $\gamma$, this is not unexpected. Note that both the energies and the variances have the right degeneracies, even though we never imposed the corresponding symmetries explicitly.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
energy & variance \\
\hline
0.081841224772803 & -0.000178252361115 \\
\hline
0.081841224779434 & -0.000178252351941 \\
\hline
0.081841224792513 & -0.000178252347304 \\
\hline
0.331378942771407 & 0.000337897356458 \\
\hline
0.367322866763615 & 0.029803975299627 \\
\hline
0.410460620351393 & 0.044970779553592 \\
\hline
\dots & \dots \\
\hline
0.513408977989184 & 0.014052233372105 \\
\hline
0.513408978649963 & 0.014052233100514 \\
\hline
0.513408978939573 & 0.014052232922150 \\
\hline
\dots & \dots \\
\hline
\end{tabular}
\caption{Excitation energy and variance of the first 6 solutions of the one-particle problem for the HAF ($\gamma=0.2$) at momentum $\pi$ with bond dimension $D=108$. The variance density of the ground state is $9.28.10^{-6}$. The first triplet has negative variance, which shows that this excitation is closer to an exact eigenstate locally than the ground state (see Appendix \ref{sec:variance}). The fourth solution is also a true one-particle (singlet) excitation. All other solutions have a considerably larger variance and correspond to artificial two-particle states. Further up in the continuum, however, we have another triplet with quite small variance, although it is difficult to say whether this corresponds to a true bound state.}
\label{table:variance}
\end{table}
\subsection{Two-particle S matrix}
In this section we will look at the two-magnon S matrix; the scattering of, e.g., an elementary magnon with a bound state will not be considered. The S matrix was defined in Secs.~\ref{sec:norm} and \ref{sec:moller}; in our setting we have three types of particles (the three components of the magnon triplet) and they all have the same dispersion relation. This implies that, for every combination of total momentum $K$ and total energy $\omega$ within the two-magnon continuum, we can build 9 scattering states. The relative coefficients of the asymptotic modes in these scattering states give rise to a ($9\times9$) unitary S matrix (the group velocities will factor out, as all particles have the same dispersion). Furthermore, instead of labeling these scattering states with momentum and energy $(K,\omega)$, we can equally well label them with total and relative momentum $(K,\kappa_1-\kappa_2)$ where $\kappa_1$ and $\kappa_2$ are the two momenta that show up in the asymptotic modes (there is still an ambiguity in the ordering of the momenta, we will always take the convention that $\kappa_1>\kappa_2$, i.e. positive relative momentum).
\par We can simplify the S matrix by making use of SU(2) invariance. Indeed, if we make linear combinations of the asymptotic modes that diagonalize the total spin $S_T^2$ and its projection $S_T^z$, the S matrix should be diagonal. Moreover, since the magnon interactions are SU(2) invariant (both Hamiltonian and ground state are), the S matrix elements should be constant within every sector of total spin. This means that the general expression for the magnon-magnon S matrix in this representation should reduce to
\begin{equation*}
S = \begin{pmatrix} -\ensuremath{\mathrm{e}}^{i\theta_0} \times \mathds{1}_{1\times1} & & \\ & -\ensuremath{\mathrm{e}}^{i\theta_1} \times \mathds{1}_{3\times3} & \\ & & -\ensuremath{\mathrm{e}}^{i\theta_2} \times \mathds{1}_{5\times5} \end{pmatrix},
\end{equation*}
i.e. the S matrix reduces to three phases for every sector of total spin. In our simulations, we always found this reduced form to high precision, so in the following we can restrict to plotting these three phases.
\begin{figure}
\includegraphics[width=\columnwidth]{./Smatrix.pdf}
\caption{The S matrix in function of relative momentum $\kappa_1-\kappa_2$ at total momentum $K=0$. Plotted are the phases of the S matrix in the $S=0$ (red), $S=1$ (blue) and $S=2$ (green) sector. Calculations were done at $\gamma=2$ and with bond dimension $D=32$.}
\label{fig:Smatrix}
\end{figure}
\par In Fig.~\ref{fig:Smatrix} we have plotted the S matrix in function of the relative momentum $\kappa_1-\kappa_2$ for total momentum $K=0$. One can observe (i) the limit $S=-\mathds{1}$ for the relative momentum going to zero, and (ii) the linear region around this limit (the slope is the scattering length). The sign of the phase is positive for all three sectors (although this does not have to be the case, see Figs.~\ref{fig:SmatrixDispersion} and \ref{fig:scatLengthFull}).
\par In Fig.~\ref{fig:SmatrixDispersion} we have plotted the S matrix in the $S=2$ sector for different values of the total momentum. We observe that the S matrix depends strongly on $K$ in a non-trivial way, but there seems to be a small region around $K=0$ where it is quasi-constant. This points to the presence of a region around the minimum of the dispersion relation where the interaction is Galilean invariant (note that the dispersion should be quadratic in this region). At larger momenta, this Galilean invariance is broken, as one expects in a lattice system.
\begin{figure}
\includegraphics[width=\columnwidth]{./SmatrixDispersion.pdf}
\caption{The scattering phase in the $S=2$ sector for 8 equally spaced values of the total momentum between $K=0$ (upper line) and $K=\pi/3$ (lower line). Around $K=0$ there is a region where the S matrix is independent of total momentum, which points to some Galilean invariance around the minimum of the dispersion relation. Calculations were done at $\gamma=2$ and with bond dimension $D=32$}
\label{fig:SmatrixDispersion}
\end{figure}
\par Even more spectacular things can happen when we vary the total momentum, such as the formation of a bound state. In Fig.~\ref{fig:scatLengthFull} we have plotted the scattering lengths in all three sectors in function of the total momentum. We can see that the scattering lengths in the $S=0$ and $S=1$ sectors diverge, signalling the formation of the singlet and triplet bound states (in agreement with the discussion in Sec.~\ref{sec:scatBound}).
\begin{figure}
\includegraphics[width=\columnwidth]{./scatLength.pdf}
\caption{The scattering lengths $a_0$ (red), $a_1$ (blue) and $a_2$ (green) in function of the total momentum $K$. In the $S=2$ sector nothing spectacular happens, although it does change sign. In the other sectors we see a divergence at the momentum where a bound state forms. The plotted range does not show all data points around the divergences, the full lines are a guide to the eye and give an indication on where the other points are situated. Calculations were done at $\gamma=2$ and bond dimension $D=32$.}
\label{fig:scatLengthFull}
\end{figure}
\subsection{Spectral function}
\label{sec:spectral}
Since we have a two-leg ladder system, we can look at spectral functions with transversal momentum $q$ equal to $0$ or $\pi$. We define the two rung operators (defined on rung $i$)
\begin{align}
& (S^z_0)_i = S^z_{i,1} + S^z_{i,2} \label{evenP} \\
& (S^z_\pi)_i = S^z_{i,1} - S^z_{i,2} . \label{oddP}
\end{align}
These operators have even, resp. odd parity under the action of the reflection operator $\mathcal{P}$. We will look at spectral functions $S_{0/\pi}(\kappa,\omega)$ with respect to these two operators,
\begin{multline}
S_{0/\pi}(\kappa,\omega) = \sum_{n} \int\d t \; \ensuremath{\mathrm{e}}^{i(\omega t-\kappa n)} \\ \times \bra{\Psi_0} \ensuremath{\mathrm{e}}^{-iHt} (S^z_{0/\pi})_n\dag \ensuremath{\mathrm{e}}^{iHt} (S^z_{0/\pi})_0 \ket{\Psi_0}
\end{multline}
where $\sum_n$ represents a sum over rungs.
\par Let us first look at the one-particle contributions. Since the elementary magnon is odd under $\mathcal{P}$, it can only carry spectral weight with respect to the odd operator. From SU(2) symmetry we know that the singlet bound state does not carry any spectral weight with respect to both operators (they are both spin-1 operators). Lastly, the triplet bound state is even under $\mathcal{P}$, so it only contributes to the even operator spectral function $S_0(\kappa,\omega)$. These considerations lead to the picture in Fig.~\ref{fig:spectralOne}. One can see that the spectral weight of the bound state goes to zero as it approaches the continuum.
\begin{figure}
\includegraphics[width=\columnwidth]{./spectralOne.pdf}
\caption{The one-particle spectral weights; these appear in the spectral functions $S_{0/\pi}(\kappa,\omega)$ as the prefactor of the $2\pi\delta(\omega-\Delta(\kappa))$ function (where $\Delta(\kappa)$ is the dispersion relation of the particle). We have plotted the magnon weights w.r.t. to the odd operator (green) and the weight of the triplet bound state w.r.t. to the even operator (blue). All the other one-particle spectral weights are identically zero. These results are in accordance with Ref. \onlinecite{Schmidt2005}. Note that the one-particle description of the bound state gets worse when coming closer to the continuum, so that the calculation of its spectral weight loses accuracy in this region. It is nevertheless clear that the spectral weight goes to zero as the bound state loses stability.}
\label{fig:spectralOne}
\end{figure}
\par Next we look at the two-magnon contribution, which has only overlap with the even parity operator. In Fig.~\ref{fig:spectralTwo} we have plotted different momentum slices of the spectral function. At momentum zero, the spectral function is identically zero (the ground state is a singlet) and grows for small momenta as $\propto\kappa^2$ (cfr. Ref.~\onlinecite{Affleck1992}). For larger momenta, we see that the spectral function gets strongly peaked at some value for $\kappa$, after which the peak again disappears. The origin of this resonance is of course the formation of the bound state: before it is stable, the bound state is already visible in the spectral function as a resonance.
\par To further confirm this picture, we have plotted the maximum of the peak in function of the momentum in Fig.~\ref{fig:specMax}. One can see the resonance clearly diverging at the point where the bound state reaches stability: from that point on the stable bound state contributes a delta peak to the spectral function.
\par We have also plotted the integrated spectral function in Fig.~\ref{fig:integrated}. Before the formation of the bound state, we see that the sum rules are completely satisfied (up to numerical errors), which shows that the one- and two-particle sectors indeed capture the full spectral function, at least in this momentum range (see also Ref. \onlinecite{Schmidt2005}). Again, we clearly see the $\propto\kappa^2$ dependence at small momenta. After the bound state has formed, however, the two-magnon part loses increasing spectral weight to the bound state.
\begin{figure}
\includegraphics[width=\columnwidth]{./spectralTwo.pdf}
\caption{The two-particle contribution to the spectral function $S_0(\kappa,\omega)$ for equally spaced values of the momentum between $\kappa=0$ and $\kappa=\pi/2$. The $\kappa=0$ curve is not shown as it is equal to zero everywhere. Calculations were done at $\gamma=2$ with bond dimension $D=32$.}
\label{fig:spectralTwo}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{./specMax.pdf}
\caption{The maximum of the two-particle contribution to the spectral function $S_{0}(\kappa,\omega)$ for different momentum slices. The full line is a guide to the eye. In the inset we show a close-up of the small momentum region, the full line is quadratic fit. Calculations were done at $\gamma=2$ with bond dimension $D=32$.}
\label{fig:specMax}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{./integrated.pdf}
\caption{The integrated spectral function $\int\d\omega/2\pi S(\kappa,\omega)$ in function of the momentum $\kappa$ (red dots) compared with the momentum space correlation function $s_0(\kappa)$ (blue line). In the inset we plot the (log10 of the) difference between the two; values below $10^{-2}$ are not shown. Calculations were done at $\gamma=2$ with bond dimension $D=32$.}
\label{fig:integrated}
\end{figure}
\subsection{Magnetization process}
\label{sec:magnProcess}
Let us now turn on the magnetic field. For SU(2) invariant systems, this perturbation does not affect the singlet ground state and induces a Zeeman splitting of the elementary magnon excitation. When the magnetic field reaches the value of the gap, one of the components of the triplet forms a pseudo-condensate (no real condensate can form in one dimension); the system undergoes a continuous phase transition from a commensurate phase with zero magnetization to an incommensurate phase with non-zero magnetization \cite{Schulz1980}.
\par The physical picture of this condensation can be understood from the approximate Bethe ansatz that was developed in Sec.~\ref{sec:aba}. Indeed, once it crosses the gap, the magnetic field serves as a chemical potential for the +1 component of the magnon triplet (the other components remain gapped, so we will not consider them in our calculations). The information on the magnon dispersion relation and the magnon-magnon S matrix we have gathered in the previous sections will allow us to compute both thermodynamic properties and correlation functions for the magnetized chain.
\par We start very close to the phase transition, where only the momenta around the minimum will be occupied, so that we can approximate them as free fermions. If we introduce a characteristic velocity $v$ for the magnon dispersion around its minimum as
\begin{equation*}
\Delta(\kappa) = \Delta + \frac{v^2}{2\Delta} (\kappa-\kappa_\text{min})^2,
\end{equation*}
the magnetization (i.e. the density of condensed magnons) will be given by \cite{Tsvelik1990, Affleck1991, Sorensen1993}
\begin{equation} \label{freeFermion}
m(h) = \frac{\sqrt{2\Delta}}{\pi v} \sqrt{(h-h_c)}.
\end{equation}
When more pseudo-momentum levels are filled up, the two-particle S matrix will deviate from its limiting value of $-1$ and the free-fermion approximation will no longer hold. As a first order correction, we can assume a linear scattering phase with the scattering length $a$ as the slope (and still a quadratic dispersion). From Eq.~\eqref{densityScatLength} it follows that the correction to the magnetization curve is given by
\begin{equation} \label{mScatLength}
m(h) = \frac{\sqrt{2\Delta}}{\pi v} \sqrt{(h-h_c)} - \frac{8\Delta a}{3\pi^2v^2} (h-h_c),
\end{equation}
a result which was obtained in Ref.~\onlinecite{Lou2000} by a similar reasoning.
\par When even higher momenta are occupied these approximations (quadratic dispersion relation, linear scattering phase and Galilean invariance) will get worse and only a full Bethe ansatz calculation will give the correct magnetization curve. In Fig.~\ref{fig:magnetization} we have plotted this.
\begin{figure}
\includegraphics[width=\columnwidth]{./magnetization.pdf}
\caption{The magnetization of the ladder ($\gamma=2$) in function of the applied magnetic field $h$. The dots are calculated with a direct MPS optimization (using an adapted version of Ref.~\onlinecite{Haegeman2011d}), the red line is the free-fermion result [Eq.~\eqref{freeFermion}], the green one is with the scattering length correction [Eq.~\eqref{mScatLength}], and the blue line is a full approximate Bethe ansatz calculation. }
\label{fig:magnetization}
\end{figure}
\par Next we look at correlation functions of the magnetized ladder. With our methods, we have no direct access to these correlation functions, but we can infer their form by combining the Luttinger liquid formalism with the thermodynamic properties computed from the approximate Bethe ansatz. Indeed, since we have seen in Sec.~\ref{sec:spectral} that the $S^x_\pi$ operator essentially creates a magnon out of the vacuum at momentum $\pi$ and the $S^z_0$ operator creates a two-magnon state at momentum $0$, we can translate the expressions for the Bose gas correlators [Eq.~\eqref{eq:correlators}] to the magnetized ladder as
\begin{align}
& \braket{(S_\pi^x)_{i'}(S_\pi^x)_{i}} = A_x \frac{(-1)^{i-i'}}{|i-i'|^{1/2K}} \nonumber \\ & \hspace{2.5cm} - B_x (-1)^{i-i'} \frac{\cos(2\pi m (i-i'))}{|i-i'|^{2K+1/2K}} \label{xx} \\
& \braket{(S_0^z)_{i'}(S_0^z)_{i}} = m^2 - \frac{K}{2\pi^2|i-i'|^2} \nonumber \\ & \hspace{2.5cm} + A_z \frac{\cos(2\pi m (i-i'))}{|i-i'|^{2K}}, \label{zz}
\end{align}
in accordance with Ref.~\onlinecite{Hikihara2001}. The power-law decay of these correlation functions is controlled by the LL parameter $K$. In Fig.~\ref{fig:luttinger} we have plotted $K$ in function of the magnetization $m$ for the ladder at different values of $\gamma$. At very low magnetization $m\rightarrow0$ the LL parameter reaches the universal value of 1, but it appears that, beyond this limiting value, $K(m)$ changes qualitatively as we vary $\gamma$. The same behaviour was observed in Ref.~\onlinecite{Hikihara2001} by fitting the analytic form of the correlation functions \eqref{xx} and \eqref{zz} with numerical calculations.
\begin{figure}
\includegraphics[width=\columnwidth]{./luttinger.pdf}
\caption{The LL parameter in function of the magnetization for $\gamma=5$ (blue), $\gamma=2$ (red), $\gamma=1$ (green) and $\gamma=1/2$ (magenta).}
\label{fig:luttinger}
\end{figure}
\par This behaviour can again be explained by starting with the free-fermion limit at very low densities. In Sec.~\ref{sec:limiting} we have shown that the LL parameter equals $K=1$ in this case. The first order correction on this value is determined by the magnon-magnon scattering length; in first order in $m$ the LL parameters is given by \cite{Affleck2005}
\begin{equation} \label{LLa}
K(m) = 1 - 2am.
\end{equation}
In Fig.~\ref{fig:scatLengthRange} we have plotted the scattering length in function of the interchain coupling $\gamma$. Based on Eq.~\eqref{LLa}, the change of the sign of $a$ confirms the varying qualitative behaviour of $K(m)$ as observed in Fig.~\ref{fig:luttinger} and in Ref.~\onlinecite{Hikihara2001}.
\begin{figure}
\includegraphics[width=\columnwidth]{./scatLengthRange.pdf}
\caption{The scattering length for different values of the interchain coupling $\gamma$.}
\label{fig:scatLengthRange}
\end{figure}
\par Finally, we can study the magnetization process at finite temperatures using the thermodynamic Bethe ansatz. In Fig.~\ref{fig:tba} we have plotted the magnetization curve for different temperatures, showing that the zero-temperature square-root dependence around the phase transition is smoothed out at finite temperature. Note that we have included the other components of the magnon triplet -- they are thermally excited as well -- in a decoupled fashion. In a more correct analysis we would have to solve the fully coupled Bethe equations for the three components, but this falls outside the scope of this paper.
\begin{figure}
\includegraphics[width=\columnwidth]{./tba.pdf}
\caption{The magnetization in function of the magnetic field $h$ for three values of the temperature: $T=.01\Delta$ (blue), $T=.045\Delta$ (green) and $T=.08\Delta$ (red).}
\label{fig:tba}
\end{figure}
\section{Future directions}
\label{sec:section5}
In the previous sections we have shown how to variationally determine all properties of one- and two-particle excitations of generic quantum spin chains. In this last section we show how our framework can be extended to study domain wall excitations and bound states and how to compute spectral functions at finite temperature. Since we believe that our work provides a crucial step towards the construction of an effective Fock space of interacting, particle-like excitations, we provide some further steps in this direction. Lastly, we reflect shortly on the application of our methods to two dimensional systems.
\subsection{Topological excitations and bound states}
In the previous sections we have restricted our framework to the case where we have a unique ground state. We can easily extend the framework, however, to situations where we have symmetry breaking and the elementary excitations are domain walls rather than localized particles.
\par Suppose we have a doubly degenerate ground state, approximated by two MPS $\ket{\Psi[A_1]}$ and $\ket{\Psi[A_2]}$. The obvious ansatz for a domain wall excitation is
\begin{multline} \label{domainWall}
\ket{\Phi_\kappa[B]} = \sum_n \ensuremath{\mathrm{e}}^{i\kappa n} \sum_{\{s\}} \ensuremath{\vect{v_L^\dagger}} \left[ \prod_{m<n} A_1^{s_m} \right] \\
\times B^{s_n} \left[ \prod_{m>n} A_2^{s_m} \right] \ensuremath{\vect{v_R}} \ensuremath{\ket{\{s\}}},
\end{multline}
i.e. the domain wall interpolates between the two ground states. The ansatz has been successfully applied to the gapped XXZ model in Ref.~\onlinecite{Haegeman2012a}, where the elementary excitations are spinons, and to the Lieb-Liniger model in Ref.~\onlinecite{Draxler2013}, where topological excitations are elementary.
\par Strictly speaking, however, the momentum of the ansatz [Eq.~\eqref{domainWall}] is not well defined: multiplying the tensor $A_2$ with an arbitrary phase factor $A_2\leftarrow A_2\ensuremath{\mathrm{e}}^{i\phi}$ shifts the momentum with $\kappa\leftarrow\kappa+\phi$. The origin of this ambiguity is the fact that one domain wall cannot be properly defined when using periodic boundary conditions.
\par Physically, however, domain walls should come in pairs. The procedure for constructing a scattering state of two domain walls is completely analogous as in Sec.~\ref{sec:section2}. For these states the total momentum is well-defined, although the individual momenta can be arbitrarily transferred between the two domain walls. Scattering states of two domain walls are especially relevant as they are the first excitations that carry any spectral weight. Consequently, a first non-trivial contribution to dynamical correlation functions asks for a solution of the scattering problem.
\par A second extension of the scattering formalism is towards the study of bound states. As we explained above, a bound state should be interpreted as a one-particle excitation and described by a one-particle ansatz. Yet, in the case where the bound state becomes very wide -- e.g. when it is close to a two-particle continuum -- the one-particle ansatz is not able to capture its delocalized nature. One possible extension consists of working on multiple MPS tensors at once, leading to the ansatz \cite{Haegeman2013a,Haegeman2013b}
\begin{multline}
\ket{\Phi_\kappa[B]} = \sum_n \ensuremath{\mathrm{e}}^{i\kappa n} \sum_{\{s\}} \ensuremath{\vect{v_L^\dagger}} \left[ \prod_{m<n} A^{s_m} \right] \\
\times B^{s_n,s_{n+1},\dots , s_{n+N}} \left[ \prod_{m>N+n} A^{s_m} \right] \ensuremath{\vect{v_R}} \ensuremath{\ket{\{s\}}}.
\end{multline}
The number of the variational parameters in the big $B$ tensor grows exponentially in the number of sites, so that we cannot systematically grow the block as the bound state gets wider.
\begin{figure*}
\includegraphics[width=1.3\columnwidth]{./block.pdf}
\caption{Graphical representation of the bound state ansatz. The $B$ tensor of the one-particle ansatz in Fig.~\ref{fig:ansatz} is spread over more than one site.}
\label{fig:block}
\end{figure*}
\par As a more systematic way to study wide bound states, we should use the two-particle ansatz \eqref{eq:ansatz} to describe them. In contrast to a scattering state the energy of a bound state is not known from the one-particle dispersions, so that we will have to scan a certain energy range in search of bound state solutions -- of course, with the one-particle ansatz we can get a pretty good idea where to look. A bound state corresponds to solutions for the eigenvalue equation \eqref{eig} with only decaying modes in the asymptotic regime. In principle we should even be able to find bound state solutions within a continuum of scattering states (i.e. a stationary bound-state, not a resonance within the continuum) by the presence of additional localized solutions for the scattering problem.
\subsection{Spectral functions at finite temperature}
At finite temperatures, the thermally excited density of excitations already present in the thermal state destroys the perfect coherence of one-particle contributions to spectral functions: the delta peaks at zero temperature will get smeared out in finite temperature spectral functions. It appears that this thermal broadening depends heavily on the interactions between the particles \cite{Essler2008,Tennant2012}, so that a full quantum mechanical treatment is needed to accurately resolve it.
\par At zero temperature the spectral function $S(\kappa,\omega)$ can be expressed in terms of the spectral weights of the low-energy excitations of the system. At finite temperatures, this is no longer true as we generally need form factors corresponding to states with arbitrarily high energies. In gapped integrable systems -- where the higher energy states can be labelled with a particle number $n$ and have an energy of the order $n\Delta$ -- the higher-energy form factors are suppressed with a Boltzmann factor $\mathcal{O}\left(\ensuremath{\mathrm{e}}^{-n\Delta/T}\right)$, so one can restrict to low-particle form factors at low enough temperatures (compared to the gap) \cite{Konik2003, Essler2008, Tennant2012}.
\par In this paper we have shown that, even in non-integrable systems, the particle picture remains valid at low densities (low temperatures), which makes the low-temperature expansion in $\mathcal{O}\left(\ensuremath{\mathrm{e}}^{-\Delta/T}\right)$ possible for the non-integrable case as well (see also Ref.~\onlinecite{Fauseweh2014} for a similar expansion for non-integrable systems). So we can associate a particle number to higher excitations and we can write down the finite temperature expression for the spectral function in the Lehmann representation as
\begin{multline} \label{LehmannT}
S(\kappa,\omega) = \frac{1}{Z} \sum_{mn} \sum_{\left\{\alpha\right\}\left\{\beta\right\}} 2\pi\delta\Big(E(\{\alpha\}) - E(\{\beta\}) - \omega\Big) \\ 2\pi\delta\Big(K(\{\alpha\}) - K (\{\beta\}) -\kappa\Big) \\ \ensuremath{\mathrm{e}}^{-\beta E(\{\alpha\})} \left|\bra{m,\{\alpha_m\}} O \ket{n,\{\beta_n\}} \right|^2
\end{multline}
where $\sum_{mn}$ is a double sum over particle numbers ranging to $\infty$ and $\{\alpha_m\}$ is a set of $m$ particle types: the states $\ket{m,\{\alpha_m\}}$ can then be identified with the multi-particle states in the approximate Bethe ansatz picture of Sec.~\ref{sec:aba}. We can see that, for gapped systems, the Boltzmann factor provides a small parameter, so that excitations with many particles only play a limited role at low temperatures. In the thermodynamic limit, two difficulties remain: (i) when coming close to the one-particle dispersion curve (where the zero-temperature spectral function has its $\delta$ peak divergence) we have to perform a resummation in order to take into account an infinite number of terms, and (ii) the form factors appearing in Eq.~\eqref{LehmannT} can be divergent in the thermodynamic limit. A careful analysis shows that both difficulties can be overcome in the case of integrable (free and interacting) massive field theories \cite{Essler2009}. Within our framework, it should prove possible to calculate finite-temperature spectral functions for generic spin chains (non-integrable) and go beyond the perturbative approaches of Refs.~\onlinecite{James2008} and \onlinecite{Goetze2010a}.
\subsection{Effective field theory}
Whereas the approximate Bethe ansatz provides a way to construct an effective first-quantized wave function for a finite density of excitations, a systematic construction of an interacting many-particle model should be formulated in second quantization \cite{Haegeman2013b, Vanderstraeten2014, Keim2015}. We introduce momentum space creation and annihilation operators that act on the ground state as
\begin{equation*} \begin{split}
& c_\alpha\dag(\kappa) \ket{\Psi[A]} = \ket{\Phi_\alpha(\kappa)} \\
& c_\alpha(\kappa) \ket{\Psi[A]} = 0
\end{split} \end{equation*}
and write down an effective interacting theory
\begin{multline} \label{eq:effective}
H = \sum_\alpha\int\frac{\d\kappa}{2\pi} \Delta_\alpha(\kappa) c_\alpha\dag(\kappa) c_\alpha(\kappa) \\ + \sum_{\alpha'\beta'\alpha\beta} \int \frac{\d\kappa}{2\pi}\frac{\d\kappa_1}{2\pi}\frac{\d\kappa_2}{2\pi} V_{\alpha'\beta',\alpha\beta}(\kappa,\kappa_1,\kappa_2) \\ \times c_{\alpha'}\dag(\kappa_1+\kappa_2-\kappa) c_{\beta'}\dag(\kappa) c_\beta(\kappa_2) c_\alpha(\kappa_1).
\end{multline}
Since we only have explicit access to the operator acting on the ground state and not the operator itself, it is a priori not clear how to determine the $c_\alpha\dag(\kappa)$ and $c_\alpha(\kappa)$ in a unique way. Moreover, there seems to be no trivial way for imposing the correct commutation relations. Thirdly, because these operators will be momentum-dependent, the transition to a local, real-space representation of the Fock operators might not be well-defined. The construction of Wannier states out of the momentum eigenstates might provide a good starting point \cite{Keim2015}, although it is still not clear how to find the unique real-space operators that are essential for computing the interaction term in Eq.~\eqref{eq:effective}.
\par A different approach can be taken by starting from a free theory of particles with generalized statistics that match the two-particle S matrix. The following effective Hamiltonian
\begin{equation*}
H_0 = \sum_\alpha\int\frac{\d\kappa}{2\pi} \Delta_\alpha(\kappa) Z_\alpha\dag(\kappa) Z_\alpha(\kappa)
\end{equation*}
indeed captures the low-lying spectrum of the original Hamiltonian if the $Z_\alpha$ and $Z_\alpha\dag$ are the so-called Faddeev-Zamolodchikov (FZ) operators obeying the following commutation relations
\begin{align*}
& Z_\alpha(\kappa_1)Z_\beta(\kappa_2) = S_{\alpha\beta}^{\gamma\delta}(\kappa_1,\kappa_2) Z_\delta(\kappa_2)Z_\gamma(\kappa_1) \\
& Z_\alpha(\kappa_1)\dag Z_\beta(\kappa_2)\dag = S_{\alpha\beta}^{\gamma\delta}(\kappa_1,\kappa_2) Z_\delta(\kappa_2)\dag Z_\gamma(\kappa_1)\dag \\
& Z_\alpha(\kappa_1) Z_\beta(\kappa_2)\dag = 2\pi\delta(\kappa_1-\kappa_2)\delta_{\alpha\beta} \\ & \hspace{3cm} + S_{\beta\gamma}^{\delta\alpha}(\kappa_1,\kappa_2) Z_\delta(\kappa_2)\dag Z_\gamma(\kappa_1).
\end{align*}
The idea is to look at perturbations of $H_0$ and express them in terms of these FZ operators. Indeed, when applying a non-commuting perturbation, we could have a new Hamiltonian of the form
\begin{multline} \label{perturbationFZ}
H = H_0+\sum_{\alpha\beta} \int\frac{\d\kappa}{2\pi} \left( M^{\alpha\beta}_p Z\dag_\alpha(\kappa)Z_\beta(\kappa) \right. \\ + \left. M_n^{\alpha\beta} Z_\alpha(-\kappa)Z_\beta(\kappa) + h.c. \right)
\end{multline}
where
\begin{align*}
& M_p^{\alpha\beta}(\kappa) = \bra{\Phi_\alpha(\kappa)} \hat{M} \ket{\Phi_\beta(\kappa)} \\
& M_n^{\alpha\beta}(\kappa) = \bra{\Psi[A]} \hat{M} \ket{\Upsilon_{\beta\alpha}(\kappa,-\kappa)}
\end{align*}
are the particle preserving, resp. particle non-preserving parts of the perturbation. For small perturbations, we can assume that only small momentum states will be occupied and that the S matrix is approximately $-\mathds{1}$. In that case, the FZ operators reduce to fermion creation and annihilation operators and we can diagonalize the Hamiltonian [Eq.~\eqref{perturbationFZ}] with a Bogoliubov rotation. In general, this proves not to be possible \cite{Sotiriadis2012} and a more sophisticated strategy will have to be developed.
\par When studying the time evolution of integrable systems, the occupation numbers $n_\alpha(\kappa) = Z_\alpha\dag(\kappa)Z_\alpha(\kappa)$ corresponding to the FZ operators are integrals of motion \cite{Essler2014}. For non-integrable systems this is no longer the case, although the observation of so-called prethermalization plateaus might point to the fact that they are almost preserved. Indeed, the mode occupation numbers $n_\alpha(\kappa)$ provide a way to distinguish a thermal Gibbs ensemble from a generalized Gibbs ensemble \cite{Essler2014a,*Bertini2015a}. Consequently, by finding an explicit (real-space) representation of the FZ operators we could follow the occupation numbers $n_\alpha(\kappa)$ through time, also when starting from an interacting theory.
\subsection{Breaking of integrability and Yang-Baxter equation}
Integrable systems possess a number of interrelated properties -- diffractionless scattering, local conservation laws, etc. -- that makes them amenable to a number of analytical techniques. Once the integrability is broken, these techniques are no longer applicable. An important question is to what extent the different manifestations of integrability survive in an approximate way close to an integrable point.
\par One simple consistency condition for integrability is the Yang-Baxter equation \cite{Yang1967,*Yang1968,Baxter1982}, expressing that three-particle scattering should be indepedent of the order in which it is decomposed into consecutive two-particle processes. As such it is a condition on the two-particle S matrix. Our methods provide a way to test this condition for non-integrable systems, and, more specifically, to study the breaking of the Yang-Baxter equation for systems close to integrable points \cite{preparation}.
\subsection{Higher dimensions}
Matrix product states have a higher-dimensional generalization called projected entangled-pair states (PEPS) \cite{Verstraete2004b}. Just as in one dimension, it has been shown that PEPS are able to capture the ground state properties of generic two-dimensional quantum spin systems \cite{Jordan2008}, so it should be able to straightforwardly generalize the one-particle ansatz of Eq.~\eqref{oneparticle} to the PEPS formalism. Compared to the MPS setting, however, the computation of the effective one-particle Hamiltonian is a lot more involved, because of the fact that the environment in a PEPS contraction is a one-dimensional object itself (compared to the zero-dimensional environment in MPS).
\par In Refs.~\onlinecite{Zauner2015} and \onlinecite{Haegeman2014b} elementary particle excitations in two dimensions were studied by looking at the spectrum of the transfer matrix. The next step, i.e. a full variational calculation of the effective Hamiltonian matrix, should lead to quantitative estimates of the gap and full dispersion relations of generic two-dimensional spin systems \cite{Vanderstraeten2015b}.
\begin{acknowledgments}
The authors would like to thank Sven Bachmann, Fabian Essler, Paul Fendley, Tobias Osborne, and Didier Poilblanc for inspiring discussions. Research supported by the Research Foundation Flanders (LV,JH), the Austrian FWF SFB grants FoQuS and ViCoM, and the European grants SIQS and QUTE (FV).
\end{acknowledgments}
|
1506.00739
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\begin{document}
\title{\textbf{Compactness of conformal metrics with constant $Q$-curvature. I}
\bigskip}
\author{\medskip YanYan Li\footnote{Supported in part by NSF grants DMS-1065971 and DMS-1203961.} \ \ and \ \
Jingang Xiong\footnote{Supported in part by Beijing Municipal Commission of Education for the Supervisor of Excellent Doctoral Dissertation (20131002701).}}
\date{}
\fancyhead{}
\fancyhead[CO]{Compactness of conformal metrics with constant $Q$-curvature}
\fancyhead[CE]{Y. Y. Li \& J. Xiong}
\fancyfoot{}
\fancyfoot[CO, CE]{\thepage}
\renewcommand{\headrule}{}
\maketitle
\begin{abstract}
We establish compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are
universally bounded. If the $Q$-curvature is $1$, assuming that Paneitz operator's kernel is trivial and its Green function is positive, we establish universal energy bounds on manifolds which are either locally conformally flat (LCF) or of dimension $\le 9$. By assuming a positive mass type theorem for the Paneitz operator, we prove compactness in $C^4$. Positive mass type theorems have been verified recently on LCF manifolds or manifolds of dimension $\le 7$, when the Yamabe invariant is positive. We also prove that, for dimension $\ge 8$, the Weyl tensor has to vanish at possible blow up points of a sequence of solutions. This implies the compactness result in dimension $\ge 8$ when the Weyl tensor does not vanish anywhere.
To overcome difficulties stemming from fourth order elliptic equations, we develop a blow up analysis procedure via integral equations.
\end{abstract}
\tableofcontents
\section{Introduction}
On a compact smooth Riemannian manifold $(M,g)$ of dimension $\ge 3$, the Yamabe problem, which concerns the existence of constant scalar curvature metrics in the conformal class of $g$, was solved through the works of Yamabe \cite{Y}, Trudinger \cite{Tr}, Aubin \cite{Aubin} and Schoen \cite{Schoen84}. Different proofs of the Yamabe problem in the case $n\le 5$ and in the case $(M,g)$ is locally conformally flat are given by Bahri and Brezis \cite{BB} and Bahri \cite{B}.
The problem is equivalent to solving the Yamabe equation
\begin{equation}} \newcommand{\ee}{\end{equation} \label{Yamabe equation}
-L_g u= Sign(\lda_1) u^{\frac{n+2}{n-2}}, \quad u>0 \quad \mbox{on }M,
\ee
where $L_g:=\Delta_g -\frac{(n-2)}{4
(n-1)}R_g$, $\Delta_g$ is the Laplace-Beltrami operator associated with $g$, $R_g$ is the scalar curvature, and $Sign(\lda_1)$ denotes the sign of the first eigenvalue $\lda_1$ of the conformal Laplacian $-L_g$. The sign of $\lda_1$ is conformally invariant, i.e., it is the same for every metric in the conformal class of $g$.
If $\lda_1<0$, there exists a unique solution of \eqref{Yamabe equation}. If $\lda_1=0$, the equation is linear and solutions are unique up to multiplication by a positive constant. If $\lda_1>0$, non-uniqueness has been established; see Schoen \cite{Schoen89} and Pollack \cite{P}. If $(M,g)$ is the standard unit sphere, all solutions are classified by Obata \cite{Obata} and there is no uniform $L^\infty$ bound for them. Schoen \cite{Schoen91} established a uniform $C^2$ bound for all solutions if $M$ is locally conformally flat but not conformal to the sphere. The uniform $C^2$ bound was established in dimensions $n\le 7$ by Li-Zhang \cite{Li-Zhang05} and Marques \cite{Marques} independently. For $n = 3, 4, 5$, see works of Li-Zhu \cite{Li-Zhu99}, Druet \cite{Druet03, Druet04} and Li-Zhang \cite{Li-Zhang04}.
For $8\le n\le 24$, the answer is positive provided
that the positive mass theorem holds in these dimensions; see Li-Zhang \cite{Li-Zhang05, Li-Zhang06}
for $8\le n\le 11$, and Khuri-Marques-Schoen \cite{KMS} for $12\le n\le 24$. On the other
hand, the answer is negative in dimension $n \ge 25$; see Brendle \cite{Brendle} for
$n \ge 52$, and Brendle-Marques \cite{BM} for $25\le n \le 51$.
In this paper, we are interested in a fourth order analogue of the Yamabe problem. Namely, the constant $Q$-curvature problem. Let us recall the conformally invariant Paneitz operator and the corresponding \emph{$Q$-curvature}, which are defined as \footnote{If $n=4$, $\frac12 Q_g$ is defined as the $Q$-curvature in some papers.}
\begin{align}
\label{Paneitz operator}
P_g&= \Delta_g^2 -\mathrm{div}_g(a_n R_g g+b_nRic_g)d+\frac{n-4}{2}Q_g \\
\label{Q-curvature}
Q_g&=-\frac{1}{2(n-1)} \Delta_g R_g+\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2} R_g^2-\frac{2}{(n-2)^2} |Ric_g|^2,
\end{align} where $R_g$ and $Ric_g$ denote the scalar curvature and Ricci tensor of $g$ respectively, and $
a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}.$ The self-adjoint operator $P_g$ was discovered by Paneitz \cite{Pan83} in 1983, and $Q_g$ was introduced later by Branson \cite{Bra85}. Paneitz operator is conformally invariant in the sense that
\begin{itemize}
\item If $n=4$, for any conformal metric $\hat g=e^{2w} g$, $w\in C^\infty(M)$, there holds
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:conformal change2}
P_{\hat g}=e^{-4w}P_g \quad \mbox{and} \quad P_g w+Q_g=Q_{\hat g} e^{4w}.
\ee
\item If $n=3$ or $n\ge 5$, for any conformal metric $\hat g=u^{\frac{4}{n-4}} g$, $0<u\in C^\infty(M)$, there holds
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:conformal change1}
P_{\hat g}(\phi)=u^{-\frac{n+4}{n-4}}P_g(u\phi)\quad \forall~ \phi\in C^\infty(M).
\ee
\end{itemize}
Hence, finding constant $Q$-curvature in the conformal class of $g$ is equivalent to solving
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Q-4d}
P_g w+Q_g=\lda e^{4w} \quad \mbox{on }M
\ee if $n=4$, and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:4th Yamabe}
P_g u=\lda u^{\frac{n+4}{n-4}}, \quad u>0 \quad \mbox{on }M,
\ee
if $n=3$ or $n\ge 5$, where $\lda$ is a constant.
When $n=4$, there is a Chern-Gauss-Bonnet type formula involving the $Q$-curvature; see Chang-Yang \cite{CY99}. The constant $Q$-curvature problem has been studied by Chang-Yang \cite{CY95}, Djadli-Malchiodi \cite{DM}, Li-Li-Liu \cite{LLL} and references therein. Bubbling analysis and compactness for solutions have been studied by Druet-Robert \cite{DR}, Malchiodi \cite{Mal}, Weinstein-Zhang \cite{WZhang} among others.
When $n\ge 5$, the constant $Q$-curvature problem is a natural extension of the Yamabe problem. However, the lack of maximum principle for fourth order elliptic equations makes the problem much harder. The first eigenvalues of fourth order self-adjoint elliptic operators are not necessarily simple and the associated eigenfunctions may change signs. We might not be able to divide the study of \eqref{eq:4th Yamabe} into three mutually exclusive cases by linking the constant $\lda$ to the sign of the first eigenvalue of the Paneitz operator. Up to now, the existence of solutions has been obtained with $\lda=1$, roughly speaking, under the following three types of assumptions. The first one is on the equation. Assuming, among others, the coefficients of the Paneitz operator are constants, Djadli-Hebey-Ledoux \cite{DHL} proved some existence results, where they decompose the operator as a product of two second order elliptic operators and use the maximum principle of second order elliptic equations. This assumption is fulfilled, for instance, when the background metric is Einstein. The second one is on the geometry and topology of the manifolds. Assuming that the Poincar\'e exponent is less than $(n-4)/2$, Qing-Raske \cite{QR0, QR} proved the existence result on locally conformally flat manifolds of positive scalar curvature. The last one is purely geometric. Assuming that there exists a conformal metric of nonnegative scalar curvature and semi-positive $Q$-curvature, Gursky-Malchiodi \cite{GM} recently proved the existence result for $n\ge 5$. By their condition, the scalar curvature was proved to be positive. In a very recent preprint, Hang-Yang \cite{HY14b} replaced the positive scalar curvature condition by the positive Yamabe invariant (which is equivalent to $\lda_1>0$). More precisely, \eqref{eq:4th Yamabe} admits a solution with $\lda=1$ if
\begin{equation}} \newcommand{\ee}{\end{equation}\label{condition:main}
\lda_1(-L_{g})>0, \quad Q_{g}\ge 0 \mbox{ and } Q_{g}>0 \mbox{ somewhere on }M,
\ee
where $\lda_1(-L_{g})$ is the first eigenvalue of $-L_g$ defined above. See also Hang-Yang \cite{HY14a} for $n=3$.
Each of the above three types of assumptions implies that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{condition:main2} \mathrm{Ker} P_g=\{0\} \mbox{ and the Green's function }G_g \mbox{ of $P_g$ is positive}.
\ee
In fact, $P_g$ is coercive in Djadli-Hebey-Ledoux \cite{DHL}, Qing-Raske \cite{QR0, QR} and Gursky-Malchiodi \cite{GM}. We refer to the latest paper Gursky-Hang-Lin \cite{GHL} for further discussions on these conditions.
If \eqref{condition:main2} holds and $\lda_1>0$, there exists a positive mass type theorem for $G_g$, provided $M$ is locally conformally flat or $n=5,6,7$, but not conformal to the standard sphere; see Humbert-Raulot \cite{HuR}, Gursky-Malchiodi \cite{GM} and Hang-Yang \cite{HY14c}.
Starting from this paper, we study the compactness of solutions of the constant $Q$-curvature equation for $n\ge 5$.
For positive constant $Q$-curvature problem, there are non-compact examples. If $(M,g)$ is a sphere, the constant $Q$-curvature metrics are not compact in $C^4$ due to the non-compactness of the conformal diffeomorphism group of the sphere. Recently, Wei-Zhao \cite{WZ} produced non-compact examples on manifolds of dimension $n\ge 25$ not conformal to the standard sphere.
\begin{thm} \label{thm:main theorem} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$, but not conformal to the standard sphere.
Assume (\ref{condition:main2}).
For $1< p\le \frac{n+4}{n-4}$, let $0<u\in C^4(M)$ be a solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:main1}
P_g u=c(n)u^{p} \quad \mbox{on }M,
\ee where $c(n)=n(n+2)(n-2)(n-4)$. Suppose that one of the following conditions is also satisfied:
\begin{itemize}
\item[i)] $\lda_1(-L_g)>0$ and $(M,g)$ is locally conformally flat,
\item[ii)] $\lda_1(-L_g)>0$ and $n=5,6,7$,
\item[iii)] $(M,g)$ is locally conformally flat or $5\le n\le 9$, and the positive mass type theorem holds for the Paneitz operator,
\item[iv)] The Weyl tensor of $g$ does not vanish anywhere, i.e., $|W_g|^2>0$ on $M$.
\end{itemize} Then there exists a constant $C>0$, depending only on $M,g$, and
a lower bound of $p-1$, such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:C4 estimate}
\|u\|_{C^4(M)} +\|1/u\|_{C^4(M)}\le C.
\ee
\end{thm}
The assumption (\ref{condition:main2})
in the theorem can be
replaced by \eqref{condition:main},
as explained above. The positive mass type theorem for Paneitz operator in dimension $8,9$ is understood as in Remark \ref{rem:positive mass}. The case $5\le n\le 9$ for positive constant $Q$-curvature equation shows some similarity to $3\le n\le 7$ for the Yamabe equation with positive scalar curvature.
The following situations, included in
Theorem \ref{thm:main theorem}, were proved before.
If $M$ is locally conformally flat and $p=\frac{n+4}{n-4}$, \eqref{eq:C4 estimate} was established
by Qing-Raske \cite{QR0, QR} with the assumptions that $\lda_1>0$ and the Poincar\'e exponent
is less than $(n-4)/2$, and by Hebey-Robert \cite{HR, HR11} with $C$ depending on the $H^2$ norm of $u$,
where they assumed that $P_g$ is coercive.
Neither $\lda_1(-L_g)>0$ nor the positive mass type theorem for Paneitz operator is assumed, we have an energy bound of solutions.
\begin{thm}\label{thm:energy} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$.
Assume (\ref{condition:main2}), and assume that either $n\le 9$ or $(M,g)$ is
locally conformally flat.
Let $0<u\in C^4(M)$ be a solution of \eqref{eq:main1}. Then
\[
\|u\|_{H^2(M)}\le C,
\]
where $C>0$ depends only on $M,g$, and
a lower bound of $p-1$.
\end{thm}
Next, we establish Weyl tensor vanishing results.
\begin{thm}\label{thm:main-b} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 8$.
Assume (\ref{condition:main2}).
Let $u_i$ be a sequence of positive solutions of
\[
P_g u_i=c(n)u_i^{p_i},
\] where $p_i\le \frac{n+4}{n-4}$,
$p_i\to \frac{n+4}{n-4}$ as $i\to \infty$. Suppose that there is a sequence of $X_i\to \bar X\in M$ such that $u_i(X_i)\to \infty$. Then the Weyl tensor has to vanish at $\bar X$, i.e., $W_g(\bar X)=0$.
Furthermore, if $n=8,9$, there exists
$X_i'\to \bar X$
such that, for all $i$,
\[
|W_g(X_i')|^2\le C\begin{cases} (\log u_i(X_i'))^{-1},& \quad \mbox{if }n=8,\\
u_i(X_i')^{-\frac{2}{n-4}}, &\quad \mbox{if }n=9,
\end{cases}
\]
where $C>0$ depends only on $M$ and $g$.
\end{thm}
\begin{thm}\label{thm:main-c}
In addition to the assumptions in Theorem \ref{thm:main-b} with $n\ge 10$, we assume that
there exist a neighborhood $\Omega} \newcommand{\pa}{\partial$ of $\bar X$ and a constant $\bar b>0$
such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:technical condition 12}
u_i(X)\le \bar b \cdot dist_g(X, X_i)^{ -\frac 4{p_i-1}} \quad \forall~ X\in \Omega} \newcommand{\pa}{\partial,
\ee
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:technical condition}
X_i\ \mbox{is a local maximum point of}\
u_i,
\quad \sup_{\Omega} \newcommand{\pa}{\partial}u_i\le \bar bu_i(X_i).
\ee
Then, for sufficiently large $i$,
\[
|W_g(X_i)|^2\le C\begin{cases} u_i(X_i)^{-\frac{4}{n-4}}\log u_i(X_i), &\quad \mbox{if }n=10,\\
u_i(X_i)^{-\frac{4}{n-4}}, &\quad \mbox{if }n\ge 11,
\end{cases}
\]
where $C>0$ depends only on $M,g, dist_g(\bar X,\pa \Omega} \newcommand{\pa}{\partial)$ and $\bar b$.
\end{thm}
The rates of decay of $|W_g(X_i)|$ in
Theorem \ref{thm:main-b} and
Theorem \ref{thm:main-c} correspond to the Yamabe
problem case $n=6,7$ and $n\ge 8$ respectively;
see theorem 1.3 and theorem 1.2 in
\cite{Li-Zhang05}.
Condition (\ref{eq:technical condition 12}) and (\ref{eq:technical condition})
can often be reduced to, by some elementary consideration,
in applications.
In a subsequent paper, we will establish compactness results analogous to those established in $8\le n\le 24$ for the Yamabe equation by Li-Zhang \cite{Li-Zhang05,Li-Zhang06} and Khuri-Marques-Schoen \cite{KMS}. The present paper provides analysis foundations.
For the negative constant $Q$-curvature equation, we have
\begin{thm}\label{thm:compact1} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. Then for any $1<p<\infty$, there exists a positive constant $C$, depending only on $M,g$ and $p$, such that every nonnegative $C^4$ solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:-Q}
P_g(u)=-u^{p} \quad \mbox{on }M
\ee
satisfies
\[
\|u\|_{C^4(M)} \le C.
\]
\end{thm}
The proofs of Theorems \ref{thm:main theorem}, Theorem \ref{thm:main-b} and Theorem \ref{thm:main-c}
make use of important ideas for the proof of compactness of positive solutions of the Yamabe equation, which were outlined first by Schoen \cite{Schoen89, Schoen89b, Schoen91}, as well as methods developed through the work Li \cite{Li95}, Li-Zhu \cite{Li-Zhu99}, Li-Zhang \cite{Li-Zhang04,Li-Zhang05,Li-Zhang06},
and Marques \cite{Marques}.
Our main difficulty now stems from the fourth order equation, which we explain in details. To understand the profile of possible blow up solutions, it is natural to scale the solutions in local coordinates centered at local maximum points. By the Liouville theorem in Lin \cite{Lin}, one can conclude that these solutions are close to some standard bubbles in small geodesic balls, whose sizes become smaller and smaller as solutions blowing up; see e.g., Proposition \ref{prop:reduction}. Then we need to answer two questions:
\begin{itemize}
\item[(i)] Do these blow up points accumulate?
\item [(ii)] If not, how do these solutions behave in geodesic balls with some fixed size?
\end{itemize}
For the first one, we may scale possible blow up points apart and look at them individually. It turns out that we end up with the situation of question (ii). After scaling we need to carry out local analysis.
In the Yamabe case, properties of second order elliptic equations, which include the maximum principle, comparison principle, Harnack inequality and B\^ocher theorem for isolated singularity, were used crucially. Now we don't have these properties for fourth order elliptic equations.
This leads to an obstruction to using fourth order equations to develop local analysis.
We observe that along scalings the bounds of Green's function are preserved. In view of Green's representation, we develop a blow up analysis procedure for integral equations and answer the above two questions completely in dimensions less than $10$. This is inspired by our recent joint work with Jin \cite{JLX3} for a unified treatment of the Nirenberg problem and its generalizations, which in turn was stimulated by our previous work on a fractional Nirenberg problem \cite{JLX, JLX2}. The approach of the latter two papers were based on the Caffarelli-Silvestre extension developed in \cite{CaffS}. Our analysis is very flexible and can easily be adapted to deal with higher order and fractional order conformally invariant elliptic equations. The organization of the paper is shown in the table of Contents.
\medskip
\textbf{Notations.} Letters $x,y,z$ denote points in $\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$, and capital letters $X,Y,Z$ denote points on Riemannian manifolds. Denote by $B_r(x)\subset \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$ the ball centered at $x$ with radius $r>0$. We may write $B_r$ in replace of $B_r(0)$ for brevity. For $X\in M$, $\B_{\delta}(X)$ denotes the geodesic ball centered at $X$ with radius $\delta$. Throughout the paper, constants $C>0$ in inequalities may vary from line to line and are universal, which means they depend on given quantities but not on solutions. $f=O^{(k)}(r^m)$ denotes any quantity satisfying $|\nabla^j f(r)|\le C r^{m-j}$ for all integers $1\le j\le k$, where $k$ is a positive integer and $m$ is a real number. $|\mathbb{S}^{n-1}|$ denotes the area of the standard $n-1$-sphere. Here are specified constants used throughout the paper:
\begin{itemize}
\item $c(n)=n(n+2)(n-2)(n-4)$ appears in constant $Q$-curvature equation,
\item $\alpha} \newcommand{\lda}{\lambda_n=\frac{1}{2(n-2)(n-4)|\mathbb{S}^{n-1}|}$ appears in the expansion of Green's functions,
\item $c_n=c(n)\cdot \alpha} \newcommand{\lda}{\lambda_n =\frac{n(n+2)}{2|\mathbb{S}^{n-1}|}$.
\end{itemize}
\noindent\textbf{Added note on June 1, 2015:} Theorem \ref{thm:main theorem}
was announced by the first named author in his talk at the
International Conference on Local and Nonlocal Partial Differential
Equations, NYU Shanghai, China, April 24-26, 2015; while
the part of the theorem
for general manifolds of dimension $n=5,6,7$ and
for locally conformally flat manifolds of dimension $n\ge 5$
was announced in his talk at the
Conference on Partial Differential Equations,
University of Sussex, UK, September 15-17, 2014.
We noticed that two days ago an article
was posted on the arXiv, [Gang Li, A compactness theorem on Branson$'$s
$Q$-curvature equation,
arXiv:1505.07692v1 [math.DG] 28 May 2015], where
a compactness result in dimension $n=5$, under the assumption that $R_g>0$ and $Q_g\ge 0$ but not identically equal to zero, was proved independently.
\bigskip
\noindent\textbf{Acknowledgments:}
J. Xiong is grateful to Professor Jiguang Bao and Professor Gang Tian for their supports.
\medskip
\section{Preliminaries}
\label{section:pre}
\subsection{Paneitz operator in conformal normal coordinates}
Let $(M,g)$ be a smooth Riemannian manifold (with or without boundary) of dimension $n\ge 5$, and $P_g$ be the Paneitz operator on $M$.
For any point $X\in M$, it was proved in \cite{LP}, together with some improvement in \cite{Cao} and \cite{Guther}, that there exists a positive smooth function $\kappa $ (with control) on $M$ such that the conformal metric $\tilde g=\kappa^{\frac{-4}{n-4}}g$ satisfies, in $\tilde g$-normal coordinates $\{x_1,\dots,x_n\}$ centered at $X$,
\[
\det \tilde g=1 \quad \mbox{in }B_{\delta}
\] for some $\delta>0$. We refer such coordinates as conformal normal coordinates. Notice that $\det \tilde g=1+O(|x|^N)$ will be enough for our use if $N$ is sufficiently large. Since one can view $x$ as a tangent vector of $M$ at $X$, thus $
\det g(x)=1+O(|x|^2)$. It follows that $\kappa(x)=1+O(|x|^2)$. In particular,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:cfn-1}
\kappa(0)=1, \quad \nabla \kappa(0)=0.
\ee
In the $\tilde g$-normal coordinates,
\begin{eqnarray*}
R_{ij}(0)=0, & & Sym_{ijk} R_{ij,k}(0)=0, \\
R_{,i}(0)=0, & & \Delta_{ \tilde g} R(0)=-\frac{1}{6}|W_{\tilde g}(0)|^2,
\end{eqnarray*}
where the Ricci tensor $R_{ij}$, scalar curvature $R$, and Weyl tensor $W$ are with respect to $\tilde g$.
We also have \[
\Delta_{\tilde g} =\Delta+\pa_l \tilde g^{kl}\pa_k+(\tilde g^{kl}-\delta^{kl})\pa_{kl}=:\Delta+d^{(1)}_k\pa_k+d^{(2)}_{kl}\pa_{kl},
\]
and
\[
\Delta_{\tilde g}^2=\Delta^2+f^{(1)}_{k}\pa_k +f^{(2)}_{kl}\pa_{kl}+f^{(3)}_{kls}\pa_{kls}+f^{(4)}_{klst}\pa_{klst},
\]
where
\begin{align*}
f^{(1)}_{k}:&=\Delta d^{(1)}_k+d^{(1)}_s\pa_s d^{(1)}_k+d^{(2)}_{st}\pa_{st}d^{(1)}_k=O(1), \\
f^{(2)}_{kl}:&=\pa_k d^{(1)}_l+\Delta d^{(2)}_{kl}+d^{(1)}_kd^{(1)}_l+d^{(1)}_s\pa_sd^{(2)}_{kl}+d^{(2)}_{sl}\pa_s d^{(1)}_k+d^{(2)}_{st}\pa_{st}d^{(2)}_{kl}=O(1), \\
f^{(3)}_{kls}:&=2d^{(1)}_s\delta^{kl}+\pa_s d^{(2)}_{kl}+2d^{(1)}_s d^{(2)}_{kl}+d^{(2)}_{st}\pa_t d^{(2)}_{kl}=O(|x|),\\
f^{(4)}_{klst}:&=2d^{(2)}_{kl}\delta^{st}+d^{(2)}_{kl}d^{(2)}_{st}=O(|x|^2).
\end{align*}
Now the second term of the Paneitz operator $P_{\tilde g}$ can be expressed as
\[
-\mathrm{div}_{\tilde g}(a_n R_{\tilde g} \tilde g+b_nRic_{\tilde g})d=-\pa_{l}((a_nR{\tilde g}_{st}+b_nR_{st}){\tilde g}^{sk}{\tilde g}^{tl}\pa_k)=:f^{(5)}_k\pa_k+f^{(6)}_{kl}\pa_{kl},
\]
where
\begin{align*}
f^{(5)}_{k}:&=-\pa_{l}\big((a_nR{\tilde g}_{st}+b_nR_{st}){\tilde g}^{sk}{\tilde g}^{tl}\big)=O(1),\\
f^{(6)}_{kl}:&=-(a_nR{\tilde g}_{st}+b_nR_{st}){\tilde g}^{sk}{\tilde g}^{tl}=O(|x|).
\end{align*}
By abusing notations, we relabel $f^{(1)}_k$ as $f^{(1)}_{k}+f^{(5)}_k$, and $f^{(2)}_{kl}$ as $f^{(1)}_{kl}+f^{(6)}_{kl}$. Hence,
\begin{align}
E(u):&=P_{\tilde g}u-\Delta^2 u\nonumber\\&
=\frac{n-4}{2}Q_{\tilde g}u+ f^{(1)}_{k}\pa_ku +f^{(2)}_{kl}\pa_{kl}u+f^{(3)}_{kls}\pa_{kls}u+f^{(4)}_{klst}\pa_{klst}u,
\label{eq:Eu}
\end{align}
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:co-Eu}
f^{(1)}_{k}(x)=O(1), \quad f^{(2)}_{kl}(x)=O(1),\quad f^{(3)}_{kls}(x)=O(|x|), \quad f^{(4)}_{klst}(x)=O(|x|^2).
\ee
We point out that each term of $f^{(1)}_k$ takes up to three times derivatives of $\tilde g$ totally, each term of $f^{(2)}_{kl}(x)$ takes twice, each term of $f^{(3)}_{kls}(x)$ takes once, and no derivative of $\tilde g$ is taken in any term of $f^{(4)}_{klst}$. Hence, we see that
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:r1}
\begin{split}
\|f^{(1)}_{k}\|_{L^\infty(B_\delta)}+ \|f^{(2)}_{kl}\|_{L^\infty(B_\delta)}& +\|\nabla f^{(3)}_{kls} \|_{L^\infty(B_\delta)}+\|\nabla^2 f^{(4)}_{klst}\|_{L^\infty(B_\delta)} \\&
\le C\sum_{k\ge 1, 2\le k+1\le 4} \|\nabla^k g\|_{L^\infty(B_\delta)}^l
\end{split}
\ee
\begin{lem}\label{lem:GM2.8} In the $\tilde g$-normal coordinates, we have, for any smooth radial function $u$,
\begin{align*}
P_{\tilde g}u=&\Delta^2 u+\frac{1}{2(n-1)}R_{,kl}(0)x^kx^l (c_1^*\frac{u'}{r} +c_2^*u'')-\frac{4}{9(n-2)r^2}\sum_{kl}(W_{ikjl}(0)x^i x^j)^2(u''-\frac{u'}{r})\\&+\frac{n-4}{24(n-1)}|W_g(0)|^2 u+(\frac{\psi_5(x)}{r^2}+\psi_3(x))u'' -(\frac{\psi_5(x)}{r^3} +\frac{\psi_3'(x)}{r}) u'+\psi_1(x)u \\&+O(r^4)u''+O(r^3) u'+O(r^2) u,
\end{align*}
where $r=|x|$, $\psi_k(x),\psi_k'(x)$ are homogeneous polynomials of degree $k$, and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:c-star}
c_1^*=\frac{2(n-1)}{(n-2)}-\frac{(n-1)(n-2)}{2}+6-n,\quad c_2^*=-\frac{n-2}{2}-\frac{2}{n-2}.
\ee
\end{lem}
\begin{proof} Since $\det \tilde g=1$ and $u$ is radial, we have $\Delta_{\tilde g}^2 u =\Delta^2 u$. The rest of the proof is same as that of Lemma 2.8 of \cite{GM}. It suffices to expand the coefficients of lower order terms of $P_{\tilde g}$ in Taylor series to a higher order so that $(\frac{\psi_5(x)}{r^2}+\psi_3(x))u'' -(\frac{\psi_5(x)}{r^3} +\frac{\psi_3'(x)}{r}) u'+\psi_1(x)u$ appears.
\end{proof}
If $\mathrm{Ker} P_{g}=\{0\}$, then $P_g$ has unique Green function $G_g$, i.e., $P_gG_g(X,\cdot)=\delta_{X}(\cdot)$ for every $X\in M$, where $\delta_X(\cdot)$ is the Dirac measure at $X$ on manifolds $(M,g)$. It is easy to check that $\mathrm{Ker} P_{g}=\{0\}$ is conformally invariant.
\begin{prop}[\cite{GM}, \cite{HY14b}]\label{prop:GM} Let $(M,\tilde g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$, on which $\mathrm{Ker} P_{\tilde g}=\{0\}$. Then there exists a small constant $\delta>0$, depending only on $(M,\tilde g)$, such that if $\det \tilde g=1$ in the normal coordinate $\{x_1,\dots, x_n\}$ centered at $\bar X$, the Green's function $G(\bar X, \exp_{\bar X}x)$ of $P_{\tilde g}$ has the expansion, for $x\in B_\delta(0)$,
\begin{itemize}
\item If $n=5,6,7$, or $M$ is flat in a neighborhood of $\bar X$,
\[
G(\bar X, \exp_{\bar X}x)=\frac{\alpha} \newcommand{\lda}{\lambda_n}{|x|^{n-4}}+A+O^{(4)}(|x|),
\]
\item If $n=8$,
\[
G(\bar X, \exp_{\bar X}x)=\frac{\alpha} \newcommand{\lda}{\lambda_n}{|x|^{n-4}}-\frac{\alpha} \newcommand{\lda}{\lambda_n}{1440}|W(\bar X)|^2 \log |x|+O^{(4)}(1),
\]
\item If $n\ge 9$, \begin{align*}
G(\bar X, \exp_{\bar X}x)=\frac{\alpha} \newcommand{\lda}{\lambda_n}{|x|^{n-4}}\Big(1+\psi_4(x)\Big)+O^{(4)}(|x|^{9-n}),
\end{align*}
\end{itemize}
where $\alpha} \newcommand{\lda}{\lambda_n=\frac{1}{2(n-2)(n-4)|\mathbb{S}^{n-1}|}$, $A$ is a constant, $W(\bar X)$ is the Weyl tensor at $\bar X$, and $\psi_4(x)$ a homogeneous polynomial of degree $4$.
\end{prop}
\begin{cor}\label{cor:Q-gf-expansion} Suppose the assumptions in Proposition \ref{prop:GM}. Then in the normal coordinate centered at $\bar X$ we have
\[
G(\exp_{\bar X}x, \exp_{\bar X}y)=\frac{\alpha} \newcommand{\lda}{\lambda_n(1+O^{(4)}(|x|^2)+O^{(4)}(|y|^2))}{|x-y|^{n-4}}+\bar a+O^{(4)}(|x-y|^{6-n}),
\]
where $x,y\in B_\delta$, $x-y=(x_1-y_1, \dots, x_n-y_n)$, $|x-y|=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}$, $\bar a$ is a constant and $\bar a=0$ if $n\ge 6$.
\end{cor}
\begin{proof} We only prove the case that $M$ is non-locally formally flat. Denote $X=\exp_{\bar X}x$ and $Y= \exp_{\bar X}y$ for $x,y\in B_\delta$, where $\delta>0$ depends only on $(M,\tilde g)$. For $X\neq \bar X$, we can find $g_{X}=v^{\frac{4}{n-4}} \tilde g$ such that in the $g_{X}$-normal coordinate centered at $X$ there hold $\det g_{X}=1$ and $v(Y)=1+O^{(4)}(dist_{g_{X}}(X,Y)^2)$. Let $G_{g_{X}}$ be the Green's function of $P_{g_{X}}$. By Proposition \ref{prop:GM},
\[
G_{g_{X}}(X,Y)=\alpha} \newcommand{\lda}{\lambda_{n} dist_{g_{X}}(X,Y)^{4-n}+A+O^{(4)}(dist_{g_{X}}(X,Y)^{6-n}),
\] where $A$ is a constant and $A=0$ if $n\ge 6$. By the conformal invariance of the Paneitz operator, we have the transformation law
\[
G(X,Y) =G_{g_{X} }(X,Y) v(X)v(Y)=G_{g_{X} }(X,Y) v(Y).
\]
Since $g_{X}=v^{\frac{4}{n-4}} \tilde g$ and $v(Y)=1+O^{(4)}(dist_{g_{X}}(X,Y)^2)$, we obtain
\[
\begin{split}
dist_{g_{X}}(X,Y)&=dist_{g_{X}}(\exp_{\bar X}x,\exp_{\bar X}y)\\&
=(1+O^{(4)}(|x-y|^2))dist_{\tilde g}(\exp_{\bar X}x,\exp_{\bar X}y)\\&
=(1+O^{(4)}(|x-y|^2))(1+O^{(4)}(|x|^2)+O^{(4)}(|y|^2))|x-y|,
\end{split}
\]
where $\tilde g$ is viewed as a Riemannian metric on $B_\delta$ because of the exponential map $\exp_{\bar X}$.
Therefore, we get
\[
G(\exp_{\bar X}x, \exp_{\bar X}y)=\alpha} \newcommand{\lda}{\lambda_{n}\frac{1+O^{(4)}(|\bar x|^2)+O^{(4)}(|y|^2)}{|\bar x-y|^{n-4}}+O^{(4)}(|\bar x-y|^{6-n}).
\]
If $X=\bar X$, it follows Proposition \ref{prop:GM}.
We complete the proof.
\end{proof}
The following positive mass type theorem for Paneitz operator was proved through \cite{HuR}, \cite{GM} and \cite{HY14b}.
\begin{thm}\label{thm:positive mass} Let $(M,g)$ be a compact manifold of dimension $n\ge 5$, and $\bar X\in M$ be a point. Let $g$ be a conformal metric of $g$ such that $\det \tilde g=1$ in the $\tilde g$-normal coordinate $\{x_1,\dots, x_n\}$ centered at $\bar X$. Suppose also that $\lda_1(-L_{g})>0$ and \eqref{condition:main2} holds. If $n=5,6,7$, or $(M,g)$ is locally conformally flat, then the constant
$A$ in Proposition \ref{prop:GM} is nonnegative, and $A=0$ if and only if $(M, g)$ is conformal to the standard $n$-sphere.
\end{thm}
\begin{rem}\label{rem:positive mass} Suppose the assumptions in Theorem \ref{thm:positive mass}. If $W(\bar X)=0$, it follows from Proposition 2.1 of \cite{HY14b} that, in the $\tilde g$-normal coordinates centered at $\bar X$, the Green's function $G$ of $P_{\tilde g}$ has the expansion
\[
G(\bar X, \exp_{\bar X}x)= \begin{cases} \alpha} \newcommand{\lda}{\lambda_8|x|^{-4}+\psi(\theta) +\log |x| O^{(4)}(|x|),& \quad n=8,\\[2mm]
\alpha} \newcommand{\lda}{\lambda_9|x|^{-5}(1+\frac{R_{,ij}(\bar X) x^ix^j |x|^2}{384}) +A+ O^{(4)}(|x|), &\quad n=9,
\end{cases}
\]
where $x=|x|\theta$, $\psi$ is a smooth function of $\theta$, and $A$ is constant. In dimension $n=8,9$, we say the positive mass type theorem holds for Paneitz operator if $\int_{\mathbb{S}^{n-1}} \psi(\theta)\,\ud \theta>0 $ and $A>0$ respectively.
\end{rem}
Let
\[
U_\lda(x):=\left(\frac{\lda}{1+\lda^2|x|^2}\right)^{\frac{n-4}{2}}, \quad \lda>0,
\]
which is the unique positive solution of $\Delta^2 u=c(n)u^{\frac{n+4}{n-4}}$ in $\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$, $n\ge 5$, up to translations by Lin \cite{Lin}. By Lemma \ref{lem:GM2.8}, in the $\tilde g$-normal coordinates we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:cor-GM}
P_{\tilde g} U_{\lda}= c(n)U_{\lda}^{\frac{n+4}{n-4}}+ f_{\lda}U_{\lda},
\ee
where $f_{\lda}(x)$ is a smooth function satisfying that $\lda^{-k}|\nabla_x^kf_\lda(x)| $, $k=0,1,\dots, 5$, is uniformly bounded in $B_\delta$ independent of $\lda \ge 1$.
Indeed,
by direct computations
\begin{align*}
\pa_rU_\lda&=(4-n)\lda^{\frac{n}{2}}(1+\lda^2r^2)^{\frac{2-n}{2}}r\\
\pa_{rr}^2 U_\lda &=(4-n)(2-n)\lda^{\frac{n+4}{2}}(1+\lda^2r^2)^{\frac{-n}{2}}r^2+(4-n)\lda^{\frac{n}{2}}(1+\lda^2r^2)^{\frac{2-n}{2}}.
\end{align*}
Inserting them to the expression in Lemma \ref{lem:GM2.8}, \eqref{eq:cor-GM} follows.
\begin{cor}\label{cor:GM2.8} Let $(M,\tilde g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$, on which $\mathrm{Ker} P_{\tilde g}=\{0\}$. Then there exists a small constant $\delta>0$, depending only on $(M,\tilde g)$, such that if $\det \tilde g=1$ in the normal coordinate $\{x_1,\dots, x_n\}$ centered at $\bar X$, then
\[
U_{\lda}(x)=c(n)\int_{B_{\delta}}G(\exp_{\bar X} x, \exp_{\bar X} y) \{U_{\lda}(y)^{\frac{n+4}{n-4}}+c_\lda'(x) U_{\lda}(y)\}\,\ud y+c_\lda''(x),
\]
where $\delta>0$ depends only on $M,\tilde g$, and $c_\lda', c_\lda''$ are smooth functions satisfying
\[
\lda^{-k}|\nabla^kc_\lda'(x)| \le C, \quad |\nabla ^kc_\lda''(x)|\le C\lda^{\frac{4-n}{2}},
\]
for $k=0,1,\dots, 5$ and some $C>0$ independent of $\lda\ge 1$.
\end{cor}
\begin{proof} Let $\eta(x)=\eta(|x|)$ be a smooth cutoff function satisfying
\[
\eta(t)=1 ~ \mbox{for }t<\delta/2, \quad \eta(t)=0 ~ \mbox{for }t>\delta.
\] By the Green's representation formula, we have
\[
(U_\lda\eta)(x)=\int_{B_\delta} G(\exp_{\bar X} x, \exp_{\bar X} y) P_{\tilde g}(U_\lda\eta)(y)\,\ud y.
\]
Making use of \eqref{eq:cor-GM} and Lemma \ref{lem:GM2.8}, we see that $c_\lda'=\frac{f_\lda}{c(n)}$ and proof is finished.
\end{proof}
\subsection{Two Pohozaev type identities}
For $r>0$, define in Euclidean space
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Poho-a}\begin{split}
\mathcal{P}(r,u):=\int_{\pa B_r}&\frac{n-4}{2}\Big(\Delta u \frac{\pa u}{\pa \nu}-u\frac{\pa }{\pa \nu}(\Delta u)\Big) -\frac{r}{2}|\Delta u|^2
\\&-x^k \pa_k u \frac{\pa }{\pa \nu}(\Delta u)+\Delta u \frac{\pa }{\pa \nu}(x^k\pa_ku)\,\ud S,
\end{split}
\ee
where $\nu=\frac{x}{r}$ is the outward normal to $\pa B_r$.
\begin{prop}\label{prop:4-pohozaev} Let $0<u\in C^4(\bar B_r)$ satisfy
\[
\Delta^2 u+E(u)=Ku^{p}\quad \mbox{in }B_r,
\]
where $E:C^4(\bar B_r)\to C^0(\bar B_{r})$ is an operator, $p>0, r>0$ and $K\in C^1(\bar B_{r})$. Then
\begin{align*}
\mathcal{P}(r,u)=& \int_{B_r} (x^k\pa_k u +\frac{n-4}{2} u) E(u)\,\ud x+\mathcal{N}(r,u),
\end{align*}
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:right-pohozaev}
\begin{split}
\mathcal{N}(r,u):=& (\frac{n}{p+1}-\frac{n-4}{2}) \int_{B_r} Ku^{p+1}\,\ud x +\frac{1}{p+1}\int_{B_r} x^k\pa_k K u^{p+1}\,\ud x \\& -\frac{r}{p+1}\int_{\pa B_{r}} K u^{p+1}\,\ud S.
\end{split}
\ee
\end{prop}
\begin{proof} A similar Pohozaev identity without $E(u)$ was derived in \cite{DMO}.
We present the proof for completeness.
For any $u\in C^4(\bar B_r)$, by Green's second identity we have
\[
\int_{B_r} u \Delta^2 u\,\ud x= \int_{B_r} (\Delta u)^2\,\ud x+ \int_{\pa B_r} u\frac{\pa }{\pa \nu}( \Delta u)- \frac{\pa u}{\pa \nu}\Delta u\,\ud S
\]
and
\[
\int_{B_r} x^k\pa_k u \Delta^2 u\,\ud x =\int_{B_r} \Delta(x^k\pa_k u) \Delta u\,\ud x + \int_{\pa B_r} x^k\pa_k u \frac{\pa }{\pa \nu}( \Delta u)- \frac{\pa }{\pa \nu}(x^k\pa_ku)\Delta u\,\ud S.
\]
Using Green's first identity, we have
\begin{align*}
\int_{B_r} \Delta(x^k\pa_k u) \Delta u\,\ud x &=2\int_{B_r} (\Delta u)^2\,\ud x+\frac12\int_{B_r} x^k\pa_k (\Delta u)^2\,\ud x \\&
=\frac{4-n}{2}\int_{B_r} (\Delta u)^2\,\ud x+ \int_{\pa B_r} \frac{r}{2}(\Delta u)^2\,\ud S.
\end{align*}
Therefore, we obtain
\begin{align*}
&\frac{n-4}{2} \int_{B_r} u \Delta^2 u\,\ud x+\int_{B_r} x^k\pa_k u \Delta^2 u\,\ud x\\
&=\frac{n-4}{2} \int_{\pa B_r} u\frac{\pa }{\pa \nu}( \Delta u)- \frac{\pa u}{\pa \nu}\Delta u\,\ud S
\\&\quad +
\int_{\pa B_r} \frac{r}{2}|\Delta u|^2 +x^k \pa_k u \frac{\pa }{\pa \nu}(\Delta u)- \frac{\pa }{\pa \nu}(x^k\pa_ku)\Delta u\,\ud S.
\end{align*}
By the equation of $u$ we get
\[
\mathcal{P}(r,u)=\int_{B_r} (x^k\pa_k u +\frac{n-4}{2} u) E(u)\,\ud x-\int_{B_r} (x^k\pa_k u +\frac{n-4}{2} u) Ku^p\,\ud x.
\]
Since
\begin{align*}
\int_{B_r} x^k\pa_k u Ku^p\,\ud x&=\frac{1}{p+1}\int_{B_r} K x^k\pa_k u^{p+1}\,\ud x\\
&=-\frac{n}{p+1}\int_{B_r} Ku^{p+1}\, \ud x-\frac{1}{p+1}\int_{B_r}x^k \pa_k K u^{p+1}\\& \quad +\frac{r}{p+1}\int_{B_r} K u^{p+1}\,\ud S,
\end{align*}
we complete the proof.
\end{proof}
\begin{lem}\label{lem:test-poho} For $G(x)=|x|^{4-n}+A+O^{(4)}(|x|)$, where $A$ is constant. Then
\[
\lim_{r\to 0}\mathcal{P}(r,G)=-(n-4)^2(n-2)A|\mathbb{S}^{n-1}|.
\]
\end{lem}
The following proposition is a special case of Proposition 2.15 of \cite{JLX3}.
\begin{prop} \label{prop:pohozaev} For $R>0$, let $0\le u\in C^1(\bar B_R)$ be a solution of
\[
u(x)= \int_{B_R} \frac{K(y)u(y)^{p}}{|x-y|^{n-4}}\,\ud y+ h_R(x),
\]
where $p>0$, and $h_R(x)\in C^1(B_R)$, $\nabla h_R\in L^1(B_R)$.
Then
\begin{align*}
&\left(\frac{n-4}{2}-\frac{n}{p+1}\right) \int_{B_R} K(x)u(x)^{p+1}\,\ud x-\frac{1}{p+1} \int_{B_R} x\nabla K(x) u(x)^{p+1}\,\ud x \\ &
=\frac{n-4}{2} \int_{B_R} K(x) u(x)^p h_R(x)\,\ud x+ \int_{B_R} x\nabla h_R(x) K(x)u(x)^p \,\ud x
\\& \quad - \frac{R}{p+1} \int_{\pa B_R} K(x) u(x)^{p+1}\,\ud S.
\end{align*}
\end{prop}
\section{Blow up analysis for integral equations}
\label{s:blowup}
In the section, the idea of dealing with integral equation is inspired by \cite{JLX3}, but we have to consider general integral kernels and remainder terms. We will use $A_1, A_2,A_3 $ to denote positive constants, and $\{\tau_i\}_{i=1}^\infty$ to denote a sequence of nonnegative constants satisfying $\lim_{i\to \infty}\tau_i=0$. Set
\begin{equation}} \newcommand{\ee}{\end{equation} \label{p}
p_i=\frac{n+4}{n-4}-\tau_i.
\ee
Let $\{G_i(x,y)\}_{i=1}^\infty$ be a sequence of functions on $B_3\times B_3$ satisfying
\begin{equation}} \newcommand{\ee}{\end{equation} \label{G}
\begin{aligned}
&G_i(x,y)=G_i(y,x),\qquad G_i(x,y)\ge A_1^{-1}|x-y|^{4-n},\\[2mm]
& |\nabla^l_x G_i(x,y)|\le A_1|x-y|^{4-n-l}, \quad l=0,1,\dots, 5 \\[2mm]
&G_i(x,y)=c_{n}\frac{1+O^{(4)}(|x|^2)+O^{(4)}(|y|^2)}{|x-y|^{n-4}}+\bar a_i+O^{(4)}(\frac{1}{|x-y|^{n-6}})
\end{aligned}
\ee
for all $x,y\in \bar B_3$, where $
c_n=\frac{n(n+2)}{2|\mathbb{S}^{n-1}|}
$ is the constant given towards the end of the introduction, $f=O^{(4)}(r^m)$ denotes any quantity satisfying $|\nabla^j f(r)|\le A_1 r^{m-j}$ for all integers $1\le j\le 4$, and $\bar a_i$ is a constant and $\bar a_i=0$ if $n\ge 6$.
Let $\{K_i\}_{i=1}^\infty \in C^\infty(\bar B_3)$ satisfy
\begin{equation}} \newcommand{\ee}{\end{equation} \label{K}
\lim_{i\to \infty}K_i(0)=1, \quad K_i\ge A_2^{-1} , \quad
\| K_i\|_{C^5(B_3)} \le A_2.
\ee
Let $\{ h_i\}_{i=1}^\infty$ be a sequence of nonnegative functions in $C^\infty( B_3)$ satisfying
\begin{equation}} \newcommand{\ee}{\end{equation} \label{H}
\begin{aligned}
\max_{\bar B_{r}(x)} h_i &\le A_2 \min_{\bar B_r(x)} h_i \\
\sum_{j=1}^5r^j|\nabla^j h_i(x)| &\le A_2 \|h_i\|_{L^\infty(B_r(x))}
\end{aligned}
\ee
for all $x\in B_{2}$ and $0<r<1/2$.
Given $p_i, G_i, K_i, $ and $ h_i$ satisfying \eqref{p}-\eqref{H}, let $0\le u_i\in L^{\frac{2n}{n-4}}(B_3)$
be a solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:s1}
u_i(x)=\int_{B_3 }G_i(x,y) K_i(y) u_i^{p_i}(y)\,\ud y +h_i(x) \quad \mbox{in } B_3.
\ee
It follows from \cite{Li04} and Proposition \ref{prop:local estimates} that $u_i\in C^{4}(B_3)$. In the following we will always assume $u_i\in C^{4}(B_3)$.
We say that $\{u_i\}$ blows up if $\|u_i\|_{L^\infty(B_3)}\to \infty$ as $i\to \infty$.
\begin{defn}\label{def4.1}
We say a point $\bar x\in B_3$ is an isolated blow up point of $\{u_i\}$ if there exist
$0<\overline r<dist(\overline x,\pa B_3)$, $\overline C>0$, and a sequence $x_i$ tending to $\overline x$, such that,
$x_i$ is a local maximum of $u_i$, $u_i(x_i)\to \infty$ and
\[
u_i(x)\leq \overline C |x-x_i|^{-4/(p_i-1)} \quad \mbox{for all } x\in B_{\overline r}(x_i).
\]
\end{defn}
Let $x_i\to \overline x$ be an isolated blow up of $u_i$. Define
\begin{equation}} \newcommand{\ee}{\end{equation}\label{def:average}
\overline u_i(r)=\frac{1}{|\pa B_r|} \int_{\pa B_r(x_i)}u_i\,\ud S,\quad r>0,
\ee
and
\[
\overline w_i(r)=r^{4/(p_i-1)}\overline u_i(r), \quad r>0.
\]
\begin{defn}\label{def4.2}
We say $x_i \to \overline x\in B_3$ is an isolated simple blow up point, if $x_i \to \overline x$ is an isolated blow up point, such that, for some
$\rho>0$ (independent of $i$) $\overline w_i$ has precisely one critical point in $(0,\rho)$ for large $i$.
\end{defn}
\begin{lem}\label{lem:harnack} Given $p_i, G_i, K_i$ and $h_i$ satisfying \eqref{p}-\eqref{H}, let $0\le u_i\in C^{4}(B_3)$ be a solution of \eqref{eq:s1}. Suppose that $0$ is an isolated blow up point of $\{u_i\}$ with $\bar r=2$, i.e., for some positive constant $A_3$ independent of $i$,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{4.7}
u_i(x)\leq A_3|x|^{-4/(p_i-1)}\quad \mbox{for all } x\in B_{2}.
\ee
Then for any $0<r<1/3 $ we have
\[
\sup_{B_{2r}\setminus B_{r/2}} u_i\leq C \inf_{B_{2r}\setminus B_{r/2}} u_i,
\]
where $C$ is a positive constant depending only on $n, A_1, A_2, A_3$.
\end{lem}
\begin{proof} For every $0<r<1/3$, set
\[
w_i(x)=r^{4/(p_i-1)}u_i(rx).
\]
By the equation of $u_i$, we have
\[
w_i(x)= \int_{B_{3/r}} G_{i,r}(x,y)K_i(ry)w_i(y)^{p_i}\,\ud y + \tilde h_i(x)\quad x\in B_{3/r},
\]
where
\[
G_{i,r}(x,y)=r^{n-4}G_i(rx,ry) \quad \mbox{for }r>0
\]
and $\tilde h_i(x):=r^{4/(p_i-1)} h_i(rx)$.
Since $0$ is an isolated blow up point of $u_i$,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:out}
w_i(x)\leq A_3 |x|^{-4/(p_i-1)}\quad \mbox{for all } x\in B_3.
\ee
Set $\Omega} \newcommand{\pa}{\partial_1=B_{5/2}\setminus B_{1/4}$, $\Omega} \newcommand{\pa}{\partial_2=B_{2}\setminus B_{1/2}$ and $
V_i(y)=K_i(ry)w_i(y)^{p_i-1}$. Thus $w_i$ satisfies the linear equation
\[
w_i(x)= \int_{\Omega} \newcommand{\pa}{\partial_1} G_{i,r}(x,y)V_i(y)w_i(y)\,\ud y + \bar h_i(x)\quad \mbox{for } x\in B_{5/2}\setminus B_{1/4},
\]
where
\[
\bar h_i(x)=\tilde h_i(x)+\int_{B_{3/r}\setminus \Omega} \newcommand{\pa}{\partial_1} G_{i,r}(x,y)K_i(ry)w_i(y)^{p_i}\,\ud y.
\]
By \eqref{eq:out} and \eqref{K}, $\|V_i\|_{L^\infty(\Omega} \newcommand{\pa}{\partial_1)} \le C(n,A_1,A_2,A_3)<\infty$.
Since $K_i$ and $w_i$ are nonnegative, by \eqref{G} on $G_i$ and \eqref{H} on $h_i$ we have $\max_{\bar \Omega} \newcommand{\pa}{\partial_2}\bar h_i \le C(n,A_1,A_2) \min_{\bar \Omega} \newcommand{\pa}{\partial_2} \bar h$. Applying Proposition \ref{prop:har} to $w_i$ gives
\[
\max_{\bar \Omega} \newcommand{\pa}{\partial_2} w_i \le C \min_{\bar \Omega} \newcommand{\pa}{\partial_2} w_i,
\]
where $C>0$ depends only on $n, A_1,A_2 $ and $A_3$. Rescaling back to $u_i$, the lemma follows.
\end{proof}
\begin{prop}\label{prop:blow up a bubble}
Suppose that $0\le u_i\in C^{4}(B_3)$ is a solution of \eqref{eq:s1} and all assumptions in Lemma \ref{lem:harnack} hold.
Let $R_i\rightarrow \infty$ with $R_i^{\tau_i}=1+o(1)$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}_i\rightarrow 0^+$, where $o(1)$ denotes some quantity tending to $0$ as $i\to \infty$. Then we have,
after passing to a subsequence (still denoted as $\{u_i\}$, $\tau_i$ and etc . . .),
\[
\|m_i^{-1}u_i(m_i^{-(p_i-1)/4} \cdot)-(1+ |\cdot|^2)^{(4-n)/2}\|_{C^3(B_{2R_i}(0))}\leq \varepsilon} \newcommand{\ud}{\mathrm{d}_i,
\]
\[
r_i:=R_im_i^{-(p_i-1)/4}\rightarrow 0\quad \mbox{as}\quad i\rightarrow \infty,
\]
where $m_i=u_i(0)$.
\end{prop}
\begin{proof} Let
\[
\varphi_i(x)=m_i^{-1} u_i(m_i^{-(p_i-1)/4} x) \quad \mbox{for }|x|<3 m_i^{(p_i-1)/4}.
\]
By the equation of $u_i$, we have,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:scal}
\varphi_i(x)= \int_{B_{3 m_i^{(p_i-1)/4}}} \tilde G_i (x,y) \tilde K_i(y)\varphi_i(y)^{p_i}\,\ud y+\tilde h_i(x),
\ee
where $\tilde G_i (x,y)=G_{i,m_i^{-\frac{p_i-1}{4}} }(x,y)$, $\tilde K_i(y)=K_i(m_i^{-\frac{p_i-1}{4}} y )$ and $\tilde h_i(x)= m_i^{-1} h_i(m_i^{-\frac{p_i-1}{4}} x)$.
First of all, $\max_{\pa B_1 } h_i\le \max_{\pa B_1 } u_i\le A_3$, by \eqref{H} we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq: h goes}
\tilde h_i \to 0\quad \mbox{in } C^5_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n) \quad \mbox{as } i\to \infty.
\ee
Secondly, since $0$ is an isolated blow up point of $u_i$,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:scalbound}
\varphi_i(0)=1, \quad \nabla \varphi_i(0)=0, \quad 0<\varphi_i(x)\leq A_3 |x|^{-4/(p_i-1)}.
\ee
For any $R>0$, we claim that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:scalbound2}
\|\varphi_i\|_{C^{4}(B_R)}\leq C(R)
\ee
for sufficiently large $i$.
Indeed, by Proposition \ref{prop:local estimates} and \eqref{eq:scalbound}, it suffices to prove that $\varphi_i \le C$ in $B_1$.
If $\varphi_i(\bar x_i)=\sup_{B_{1}} \varphi_i \to \infty $, set
\[
\tilde \varphi_i(z)=\varphi_i(\bar x_i)^{-1}\varphi_i(\varphi_i(\bar x_i)^{-(p_i-1)/4}z+\bar x_i)\leq 1\quad \mbox{for }|z| \le \frac12 \varphi_i(\bar x_i)^{(p_i-1)/4}.
\]
By \eqref{eq:scalbound},
\[
\tilde \varphi_i(z_i)= \varphi_i(\bar x_i)^{-1} \varphi_i(0)\to 0
\] for $z_i=-\varphi_i(\bar x_i)^{(p_i-1)/4}\bar x_i$. Since $\varphi_i(\bar x_i) \leq A_3 |\bar x_i|^{-4/(p_i-1)}$, we have $|z_i|\leq A_3^{4/(p_1-1)}$. Hence, we can find $t>0$ independent of $i$ such that such that $z_i\in B_t$.
Applying Proposition \ref{prop:har} to $\tilde \varphi_i$ in $B_{2t}$ (since $\tilde\varphi_i$ satisfies a similar equation to \eqref{eq:scal}), we have
\[
1=\tilde \varphi_i(0)\le C \tilde \varphi_i(z_i)\to 0,
\]
which is impossible. Hence, $\varphi_i \le C$ in $B_1$.
It follows from \eqref{eq:scalbound2} that there exists a function $\varphi\in C^4(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ such that, after passing subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:phigeos}
\varphi_i(x)\to \varphi \quad \mbox{in }C^3_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n) \quad \mbox{as }i\to \infty.
\ee
Thirdly, for every $R>0$, let
\[
g_i(R,x):= \int_{B_{3 m_i^{(p_i-1)/4}}\setminus B_{R}} \tilde G_i (x,y) \tilde K_i(y)\varphi_i(y)^{p_i}\,\ud y.
\]
Since $K_i$ and $\varphi_i$ are nonnegative, a simple computation using \eqref{G} gives that, for any $x\in B_{R-1}$,
\[
|\nabla^k g_i(R,x)| \le C g_i(R,x), \quad k=1,\dots,5.
\]
Note that $g_{i}(R,x)\le \varphi_i(x) \le C(R)$. It follows that, after passing to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:ggoes}
g_i(R,x) \to g(R,x)\ge 0 \quad \mbox{in } C^{4}(B_{R-1}) \quad \mbox{as }i\to \infty.
\ee
By \eqref{G} and \eqref{K}, we have
\[
\tilde G_i(x,y)\to c_n \frac{1}{|x-y|^{n-4}}\quad \forall~ x\neq y
\]
and $\tilde K_i(y)\to K_i(0)=1.$
Combining \eqref{eq: h goes}, \eqref{eq:phigeos} and \eqref{eq:ggoes} together, by \eqref{eq:scal} we have that for any fixed $R>0$ and $x\in B_{R-1}$
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:limit1}
g(R,x)=\varphi(x)-c_{n}\int_{B_R}\frac{ \varphi(y)^{\frac{n+4}{n-4}}}{|x-y|^{n-4}}\,\ud y.
\ee
By \eqref{eq:limit1}$, g(R,x)$ is non-increasing in $R$. For any fixed $x$ and $|y|\ge R>>|x|$, by \eqref{G} we have
\begin{align*}
\frac{G_{i,m_i^{-(p_i-1)/4} } (x,y)}{G_{i,m_i^{-(p_i-1)/4} } (0,y)}=\frac{G_{i,|y|m_i^{-(p_i-1)/4} } (\frac{x}{|y|},\frac{y}{|y|})}{G_{i,|y| m_i^{-(p_i-1)/4} } (0,\frac{y}{|y|})}=1+O(\frac{|x|}{|y|}).
\end{align*}
Hence, $g_i(R,x)=(1+O(\frac{|x|}{R}))g_i(R,0)$, which implies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:remainder1}
\lim_{R\to \infty}g(R,x)= \lim_{R\to \infty} g(R,0):=c_0\ge 0.
\ee
Sending $R$ to $\infty$ in \eqref{eq:limit1}, it follows from Lebesgue's monotone convergence theorem that
\[
\varphi(x)=c_{n} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{\varphi(y)^{\frac{n+4}{n-4}}}{|x-y|^{n-4}} \ud y+c_0 \quad x\in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n.
\]
We claim that $c_0=0$. If not,
\[
\varphi(x)-c_0= c_{n}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{\varphi(y)^{\frac{n+4}{n-4}}}{|y|^{n-4}}\,\ud y >0,
\]
which implies that
\[
1=\varphi(0)\geq c_{n} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{c_0^{\frac{n+4}{n-4}}}{|x-y|^{n-4}}=\infty.
\]
This is impossible.
The use of monotonicity in the above argument is taken from \cite{JLX3}.
It follows from the classification theorem in \cite{CLO} or \cite{Li04} that
\[
\varphi(x)=\left(1+|x|^2\right)^{-\frac{n-4}{2}},
\]
where we have used that $\varphi(0)=1$ and $\nabla \varphi(0)=0$.
The proposition follows immediately.
\end{proof}
Since passing to subsequences does not affect our proofs, in the rest of the paper we will always choose $R_i\to\infty$ with $R_i^{\tau_i}=1+o(1) $ first, and then $\varepsilon} \newcommand{\ud}{\mathrm{d}_i\to 0^+$ as small as we wish (depending on $R_i$) and then choose our subsequence $\{u_{j_i}\}$ to work with. Since $i\le j_i$ and $\lim_{i\to \infty}\tau_i=0$, one can ensure that $R_i^{\tau_{j_i}}=1+o(1) $ as $i\to \infty$. In the sequel, we will still denote the subsequences as $u_i, \tau_i$ and etc.
\begin{rem}\label{rem:blow} By checking the proof of Proposition \ref{prop:blow up a bubble}, together with the fact $\nabla^2 (1+|x|^2)^{-\frac{n-4}{2}}$ is negatively definite near zero and the $C^2$ convergence in a fixed neighborhood of zero, the following statement holds. Let $0\le u_i\in C^{4}(B_3)$ be a solution of \eqref{eq:s1} and satisfy \eqref{4.7}. Suppose that $u_i(0)\to \infty$ as $i\to \infty$, $\nabla u_i(0)=0$ and $\max_{B_3}u_i\le b u_i(0)$ for some constant $b\ge 1$ independent of $i$. Then, after passing to a subsequence, $0$ must be a local maximum point of $u_i$ for $i$ large. Namely, $0$ is an isolated blow up point of $u_i$ after passing to a subsequence.
\end{rem}
\begin{prop}\label{prop:lower bounded by bubble} Under the hypotheses of Proposition \ref{prop:blow up a bubble}, there exists a constant $C>0$, depending only on $n, A_1,A_2$ and $A_3$, such that,
\[
u_i(x)\geq C^{-1}m_i(1+m_i^{(p_i-1)/2}|x|^2)^{(4-n)/2}, \quad |x|\leq 1.
\]
In particular, for any $e\in \mathbb{R}^n$, $|e|=1$, we have
\[
u_i(e)\geq C^{-1}m_i^{-1+((n-4)/4)\tau_i}.
\]
\end{prop}
\begin{proof}
By change of variables and using Proposition \ref{prop:blow up a bubble}, we have for $r_i\le |x|\le 1$,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:lower bound bubble}
\begin{split}
u_i(x)&\ge C^{-1}\int_{|y|\le r_i} \frac{u_{i}(y)^{p_i}}{|x-y|^{n-4}} \,\ud y\\&
\ge C^{-1} m_i\int_{|z|\le R_i} \frac{\big(m_i^{-1}u_{i}(m_i^{-(p_i-1)/4}z)\big)^{p_i}}{|m_i^{(p_i-1)/4}x-z|^{n-4}} \,\ud z
\\&
\ge C^{-1} m_i \int_{|z|\le R_i} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4}x-z|^{n-4}} \,\ud z\\&
\ge \frac 12 C^{-1} m_i \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(z)^{\frac{n+4}{n-4}}}{|m_i^{(p_i-1)/4} x-z|^{n-4}} \,\ud z\\&
=\frac 12 C^{-1}m_i U_1(m_i^{(p_i-1)/4}x).
\end{split}
\ee
Recall that
\[
U_\lda(z)=\left(\frac{\lda}{1+\lda^2|z|^2}\right)^{(n-4)/2}, \quad \lda>0.
\]
The proposition follows immediately.
\end{proof}
\begin{lem} \label{lem:upbound1} Suppose the hypotheses of Proposition \ref{prop:blow up a bubble} and in addition that $ 0$ is also an isolated simple blow up point with the constant $\rho>0$. Then there exist $\delta_i>0$, $\delta_i=O(R_i^{-4})$,
such that
\[
u_i(x)\leq C u_i(0)^{-\lda_i}|x|^{4-n+\delta_i},\quad \mbox{for all }r_i\leq |x|\leq 1,
\]
where $\lda_i=(n-4-\delta_i)(p_i-1)/4-1$ and $C>0$ depends only on $n, A_1,A_3$ and $\rho$.
\end{lem}
\begin{proof} We divide the proof into several steps.
Step 1. From Proposition \ref{prop:blow up a bubble}, we see that
\begin{align}
u_i(x)&\le C m_i \left(\frac{1}{1+|m_i^{(p_i-1)/4}x|^2}\right)^{\frac{n-4}{2}} \nonumber \\ &
\le C m_iR_i^{4-n} \quad \mbox{for all } |x|=r_i=R_i m_i^{-(p_i-1)/4}.
\label{4.8}
\end{align}
Let $\overline u_i(r)$ be the average of $u_i$ over the sphere of radius $r$ centered at $0$.
It follows from the assumption of isolated simple
blow up points and Proposition \ref{prop:blow up a bubble} that
\begin{equation}} \newcommand{\ee}{\end{equation}\label{4.9}
r^{4/(p_i-1)}\overline u_i(r) \quad \mbox{is strictly decreasing for $r_i<r<\rho$}.
\ee
By Lemma \ref{lem:harnack}, \eqref{4.9} and \eqref{4.8}, we have, for all $r_i<|x|<\rho$,
\[
\begin{split}
|x|^{4/(p_i-1)}u_i(x)&\leq C|x|^{4/(p_i-1)}\overline u_i(|x|)\\&
\leq C r_i^{4/(p_i-1)}\overline u_i(r_i)\\&
\leq CR_i^{\frac{4-n}{2}},
\end{split}
\]
where we used $R_i^{\tau_i}=1+o(1)$. Thus,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:coeff}
u_i(x)^{p_i-1}\leq C R_i^{-4}|x|^{-4} \quad \mbox{for all } r_i\leq |x|\le \rho.
\ee
\medskip
Step 2. Let
\[
\mathcal{L}_i\phi(y):= \int_{B_3} G_i(x,z)K_i(z) u_i(z)^{p_i-1}\phi(z)\,\ud z.
\]
Thus
\[
u_i=\mathcal{L}_i u_i+h_i.
\]
Note that for $4<\mu<n$ and $0<|x|<2$,
\begin{align*}
\int_{B_3} G_i(x,y)|y|^{-\mu}\,\ud y&\le A_1\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{1}{|x-y|^{n-4}|y|^{\mu}}\,\ud y
\\&=A_1 |x|^{4-n} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}
\frac{1}{||x|^{-1}x-|x|^{-1}y|^{n-4}|y|^{\mu}}\,\ud y \\&
= A_1|x|^{-\mu+4} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{1}{||x|^{-1}x-z|^{n-4}|z|^{\mu}}\,\ud z \\&
\le C\Big( \frac{1}{n-\mu}+\frac{1}{\mu- 4} \Big)|x|^{-\mu+4},
\end{align*}
where we did the change of variables $y=|x|z$.
By \eqref{eq:coeff}, one can properly choose $0<\delta_i=O(R_i^{-4})$ such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a1}
\int_{r_i<|y|<\rho} G_i(x,y)K_i(y)u_i(y)^{p_i-1} |y|^{-\delta_i} \,\ud y\leq \frac{1}{4} |x|^{-\delta_i},
\ee
and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a2}
\int_{r_i<|y|<\rho} G_i(x,y)K_i(y)u_i(y)^{p_i-1} |y|^{4-n+\delta_i} \,\ud y\leq \frac{1}{4} |x|^{4-n+\delta_i},
\ee
for all $r_i< |x|<\rho$.
Set $M_i:=4^n A_1^2\max_{\pa B_\rho} u_i+2\max_{\bar B_\rho}h_i$,
\[
f_i(x):=M_i \rho^{\delta_i} |x|^{-\delta_i}+A m_i^{-\lda_i} |x|^{4-n+\delta_i},
\] and
\[
\phi_i(x)=\begin{cases}
f_i(x), & \quad r_i< |x|< \rho,\\
u_i(x),&\quad \mbox{otherwise} ,
\end{cases}
\]
where $A>1$ will be chosen later.
By \eqref{eq:a1} and \eqref{eq:a2}, we have for $r_i<|x|< \rho$.
\begin{align*}
\mathcal{L}_i \phi_i (x)&= \int_{B_3} G_i(x,y)K_i(y)u_i(y)^{p_i-1} \phi_i(y)\,\ud y\\&
=\left(\int_{|y|\le r_i} + \int_{r_i<|y|<\rho} + \int_{\rho\le |y|<3} \right)G_i(x,y)K_i(y)u_i(y)^{p_i-1} \phi_i(y)\,\ud y
\\&
\leq A_1 \int_{|y|\le r_i} \frac{u_{i}(y)^{p_i}}{|x-y|^{n-4}} \,\ud y+ \frac{f_i}{4}+ \frac{M_i}{2^{n-4}},
\end{align*}
where we used, in view of \eqref{G},
\[
\begin{split}
\int_{\rho\le |y|<3}& G_i(x,y)K_i(y)u_i(y)^{p_i-1} \phi_i(y)\,\ud y\\&
=\int_{\rho\le |y|<3} G_i(x,y)K_i(y)u_i(y)^{p_i} \,\ud y\\ &\le A_1^2 2^{n+4}\int_{\rho\le |y|<3} G_i(\frac{\rho x}{|x|},y)K_i(y)u_i(y)^{p_i}\,\ud y \\ &\le A_1^2 2^{n+4} \max_{\pa B_{\rho}} u_i \le 2^{4-n} M_i.
\end{split}
\]
By change of variables and using Proposition \ref{prop:blow up a bubble}, we have, similar to \eqref{eq:lower bound bubble},
\begin{align*}
\int_{|y|\le r_i} \frac{u_{i}(y)^{p_i}}{|x-y|^{n-4}} \,\ud y&
=m_i\int_{|z|\le R_i} \frac{\big(m_i^{-1}u_{i}(m_i^{-(p_i-1)/4}z)\big)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-4}} \,\ud z
\\&
\le 2m_i \int_{|z|\le R_i} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-4}} \,\ud z\\&
\le C m_i \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(z)^{\frac{n+4}{n-4}}}{|m_i^{(p_i-1)/4}x-z|^{n-4}} \,\ud z\\&
=Cm_i U_1(m_i^{(p_i-1)/4}x),
\end{align*}
where we used $R^{(n-4)\tau_i}=1+o(1)$.
Since $|x|>r_i$, we see
\begin{align*}
m_i U_1(m_i^{(p_i-1)/4} x) &\le Cm_i^{1-(p_i-1)(n-4)/4}|x|^{4-n} \\&
\le Cm_i^{-\lda_i}|x|^{4-n+\delta_i}.
\end{align*}
Therefore, we conclude that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:intineq}
\mathcal{L}_i \phi_i(x)+h_i(x) \leq \phi_i(x)\quad \mbox{for all } r_i\le |x| \leq \rho,
\ee
provided $A$ is large independent of $i$.
\medskip
Step 3. Note that
\[
\liminf_{|x|\to r_i ^+}f_i(x) >A m_i^{-\lda_i}R_i^{4-n+\delta_i} m_i^{(p_i-1)(n-4-\delta_i)/4}=A R_i^{4-n+\delta_i} m_i.
\] In view of \eqref{4.8}, we may choose $A$ large such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:strict}
\liminf_{|x|\to r_i ^+}(f_i(x)-u_i(x))>0 .
\ee We claim that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:upper bounded}
u_i(x)\le \phi_i(x).
\ee Indeed, if not, let
\[
1<t_i:=\inf\{t>1, t\phi_i(x)\ge u_i(x) \quad \mbox{for all } r_i\le |x|\le \rho \}<\infty.
\] By \eqref{eq:strict}, $t_i>1$, together with $f_i>u_i$ on $\pa B_{\rho}$, we can find a sufficient small open neighborhood of $\pa B_{r_i}\cup \pa B_\rho$ in which $t_i \phi_i> u_i $. By the continuity there exists $y_i\in B_\rho \setminus \bar B_{r_i}$ such that
\[
0=t_i\phi_i(y_i)-u_i(y_i)\ge \mathcal{L}_i(t_i\phi_i-u_i)(y_i)+(t_i-1)h_i(y_i)>0.
\]
We derived a contradiction and thus \eqref{eq:upper bounded} is valid.
\medskip
Step 4. By \eqref{H}, we have $\max_{\bar B_{\rho}} h_i \le A_2\max_{\pa \bar B_{\rho}}h_i \le A_2 \max_{\pa \bar B_{\rho}}u_i$.
Hence,
\[
M_i\le C\max_{\pa B_{\rho}} u_i .
\]
For $r_i<\theta<\rho$,
\[
\begin{split}
\rho^{4/(p_i-1)}M_i&\leq C \rho^{4/(p_i-1)}\overline u_i(\rho)\\
&\leq C\theta^{4/(p_i-1)}\overline u_i(\theta)\\
&\leq C\theta^{4/(p_i-1)}\{M_i\rho^{\delta_i}\theta^{-\delta_i}+Am_i^{-\lda_i}\theta^{4-n+\delta_i}\}.
\end{split}
\]
Choose $\theta=\theta(n,\rho,A_1,A_2, A_3)$ sufficiently small so that
\[
C\theta^{4/(p_i-1)}\rho^{\delta_i}\theta^{-\delta_i}\leq \frac12 \rho^{4/(p_i-1)}.
\]
Hence, we have
\[
M_i\le C m_i^{-\lda_i}.
\]
It follows from \eqref{eq:upper bounded} that
\[
u_i(x)\le \phi_i(x) \le Cm_i^{-\lda_i}|x|^{-\delta_i}+A m_i^{-\lda_i} |x|^{4-n+\delta_i} \le Cm_i^{-\lda_i} |x|^{4-n+\delta_i}.
\]
We complete the proof of the lemma.
\end{proof}
\begin{lem} \label{lem:aux1} Under the assumptions in Lemma \ref{lem:upbound1}, for $k<n$ we have
\[
I_k[u_i](x) \le C\begin{cases}
m_i^{\frac{n-2k+4}{n-4}+o(1)}, &\quad \mbox{if } |x|<r_i, \\
m_i^{-1+o(1)}|x|^{k-n}, &\quad \mbox{if }r_i \le |x|<1,
\end{cases}
\]
where
\[
I_k[u_i](x) =\int_{B_1}|x-y|^{k-n} u_i(y)^{p_i}\,\ud y.
\]
\end{lem}
\begin{proof} Making use of Proposition \ref{prop:blow up a bubble} and Lemma \ref{lem:upbound1}, we have
\begin{align*}
I_k[u_i](x) & = \int_{B_{r_i}}\frac{u_i(y)^{p_i}}{ |x-y|^{n-k}} \,\ud y+ \int_{B_1\setminus B_{ r_i}} \frac{u_i(y)^{p_i}}{ |x-y|^{n-k}}\,\ud y\\&
\le Cm_i^{\frac{n-2k+4}{n-4}+o(1)} \int_{B_{R_i}} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-k}}\,\ud z\\& \quad +Cm_i^{-\frac{n+4}{n-4}+o(1)}\int_{B_1\setminus B_{ r_i}}\frac{1}{|x-y|^{n-k}|y|^{n+4}}\,\ud y.
\end{align*}
If $|x|<r_i$, we see that
\[
\int_{B_{R_i}} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-k}}\,\ud z \le C
\]
by Lemma \ref{lem:aux2},
and
\begin{align*}
\int_{B_1\setminus B_{ r_i}}\frac{1}{|x-y|^{n-k}|y|^{n+4}}\,\ud y \le \int_{B_1\setminus B_{ r_i}}\frac{1}{|y|^{2n-k+4}}\,\ud y
\le C(n) R_i^{-(n-k+4)}m_i^{\frac{2(n-k+4)}{n-4}+o(1)}.
\end{align*}
Hence, $I_k[u_i](x) \le Cm_i^{\frac{n-2k+4}{n-4}+o(1)}$.
If $r_i<|x|<1 $, then $|m_i^{(p_i-1)/4} x|\ge 1$. It follows from Lemma \ref{lem:aux2} that
\begin{align*}
\int_{B_{R_i}} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-k}}\,\ud z & \le \int_{B_{R_i}} \frac{1}{|m_i^{(p_i-1)/4} x-z|^{n-k}(1+|z|)^{n+4+o(1)}}\,\ud z \\&
\le C|m_i^{(p_i-1)/4} x|^{k-n}.
\end{align*}
By change of variables $z=m_i^{(p_i-1)/4} y$,
\begin{align*}
\int_{B_1\setminus B_{ r_i}}\frac{1}{|x-y|^{n-k}|y|^{n+4}}\,\ud y&=m_i^{\frac{2(n-k+4)}{n-4}+o(1)} \int_{B_{m_i^{(p_i-1)/4}}\setminus B_{R_i}} \frac{1}{|m_i^{(p_i-1)/4}x-z|^{n-k}|z|^{n+4}}\,\ud z\\&
\le Cm_i^{\frac{2(n-k+4)}{n-4}+o(1)} |m_i^{(p_i-1)/4} x|^{k-n}.
\end{align*}
Thus
\[
I_k[u_i](x) \le Cm_i^{\frac{n-2k+4}{n-4}+o(1)} |m_i^{(p_i-1)/4} x|^{k-n} =m_i^{-1+o(1)} |x|^{k-n}.
\]
Therefore, the proof of the lemma is completed.
\end{proof}
\begin{lem}\label{lem:error}
Under the assumptions in Lemma \ref{lem:upbound1}, we have
\[
\tau_i=O(u_{i}(0)^{-2/(n-4)+o(1)}).
\]
Consequently, $m_i^{\tau_i}=1+o(1)$.
\end{lem}
\begin{proof} For $x\in B_1$, we write equation \eqref{eq:s1} as
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:flat}
u_i(x)= c_{n}\int_{B_1} \frac{K_i(y)u_i(y)^{p_i}}{|x-y|^{n-4}}\,\ud y +b_i(x),
\ee
where $b_i(x):=Q'_i(x)+Q''_i(x) +h_i(x)$,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{F}
Q_i' (x):=\int_{B_1}(G_i(x,y)-c_{n}|x-y|^{4-n}) K_i(y) u_i(y)^{p_i}\,\ud y
\ee
and
\[
Q_i''(x):= \int_{B_3\setminus B_1} G_i(x,y)K_i(y) u_i(y)^{p_i}\,\ud y.
\]
Notice that
\[
|G_i(x,y)-c_{n}|x-y|^{4-n}| \le \frac{C|x|^2 }{|x-y|^{n-4}}+|\bar a_i|+C|x-y|^{6-n}.
\]
\[
|\nabla_x(
G_i(x,y)-c_{n}|x-y|^{4-n})| \le \frac{C|x|^2 }{|x-y|^{n-3}}+\frac{C|x| }{|x-y|^{n-4}}+C|x-y|^{5-n}.
\]
Hence,
\begin{align*}
|Q_i'(x)| & \le C(|x|^2 u_i(x)+|a_i| \|u_i^{p_i}\|_{L^1(B_1)}+I_{6} [u_i^{p_i}](x)), \\
|\nabla Q_i'(x)| &\le C ( |x|^2 I_{3}+|x|I_{4} +I_{5}) [u_i^{p_i}](x),
\end{align*}
where $I_k [u_i^{p_i}](x)=\int_{B_1}|x-y|^{k-n} u_i(y)^{p_i}\,\ud y $.
By Lemma \ref{lem:upbound1}, we have $u_i(x)\le C m_i^{-\lda_i}$ for all $x\in B_{3/2}\setminus B_{1/2}$. Hence, $Q_i''(x) +h_i(x)\le u_i(x)\le C m_i^{-1+o(1)}$ for any $x\in \pa B_1$ . It follows from \eqref{H} that
\[
\max_{\bar B_2}h_i(x) \leq C \min_{\pa B_1} h_i(x)\le C m_i^{-1+o(1)}.
\]
and
\[
|\nabla h_i(x)|\le C\max_{\bar B_2}h_i(x) \le C m_i^{-1+o(1)} \quad \mbox{for all }x\in B_1.
\]
Since $u_i$ is nonnegative, by \eqref{G} it is easy to check that
\[
|Q_i''(x)|+|\nabla Q_i''(x)|\le Cm_i^{-1+o(1)} \quad \mbox{for all }x\in B_1.
\]
Applying Proposition \ref{prop:pohozaev} to \eqref{eq:flat}, we have
\begin{align}
\tau_i & \int_{B_1} u_i(x)^{p_i+1}-A_2\int_{B_1} |x| u_i(x)^{p_i+1} \,\ud x \nonumber\\& \leq C\Big(\int_{B_1}(|Q'_i(x)|+|x||\nabla Q_i'(x)) u_i(x)^{p_i}+ m_i^{-1+o(1)}\int_{ B_1} u_i^{p_i}+ \int_{\pa B_1} u_i^{p_i+1}\, \ud s\Big).
\label{eq:a6}
\end{align}
By Proposition \ref{prop:blow up a bubble} and change of variables,
\begin{align*}
\int_{B_1} u_i(x)^{p_i+1} \,\ud x &\ge C^{-1}\int_{B_{r_i}} \frac{m_i^{p_i+1}}{(1+|m_i^{(p_i-1)/4}y|^2)^{(n-4)(p_i+1)/2}}\,\ud y \\&
\ge C^{-1} m_i^{\tau_i(1-n/4)} \int_{R_i} \frac{1}{(1+|z|^2)^{(n-4)(p_i+1)/2}}\,\ud z \\&
\ge C^{-1} m_i^{\tau_i(1-n/4)} ,
\end{align*}
By Proposition \ref{prop:blow up a bubble}, Lemma \ref{lem:upbound1}, we have
\[
\int_{B_{1}} u_i^{p_i} \le C m_i^{-1+o(1)},
\]
\[
\int_{B_{1}} |x|^su_i^{p_i+1} \le C m_i^{-2s/(n-4)+o(1)}, \quad \mbox{for }-n<s<n,
\]
and
\[
\int_{\pa B_1} u_i^{p_i+1}\, \ud s \le Cm_i^{-2n/(n-4) +o(1)}.
\]
It follows from Lemma \ref{lem:aux1} that
\[
\int_{B_{1}} (|Q'_i(x)|+|x||\nabla Q_i'(x)|) u_i(x)^{p_i+1}\,\ud x\le Cm_i^{-2/(n-4)+o(1)}.
\]
Therefore, we complete the proof.
\end{proof}
\begin{lem}\label{lem:aux3} For $-4<s<4$, we have, as $i\to \infty$,
\[
m_i^{1+\frac{2s}{n-4}} \int_{B_{r_i}}|y|^s u_i(y)^{p_i}\,\ud y \to \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}|z|^s (1+|z|^2)^{-\frac{n+4}{2}}\,\ud z\]
and
\[
m_i^{1+\frac{2s}{n-4}} \int_{B_1\setminus B_{r_i}}|y|^s u_i(y)^{p_i}\,\ud y \to 0.
\]
\end{lem}
\begin{proof} By a change of variables $y=m_i^{-(p_i-1)/4} z$, we have
\[
\int_{B_{r_i}}|y|^s u_i(y)^{p_i}\,\ud y =m_i^{-\frac{(p_i-1)(s+n)}{4}+p_i} \int_{B_{R_i}}|z|^s (m_i^{-1}u_i(m_i^{-(p_i-1)/4} z))^{p_i}\,\ud z
\]
By Lemma \ref{lem:error}, $m_i^{-\frac{(p_i-1)(s+n)}{4}+p_i}=(1+o(1)) m_i^{-1-\frac{2s}{n-4}}$. In view of Proposition \ref{prop:blow up a bubble} and $-4<s<4$, it follows from Lebesgue's dominated convergence theorem that
\[ \int_{B_{R_i}}|z|^s (m_i^{-1}u_i(m_i^{-(p_i-1)/4} z))^{p_i}\,\ud z \to \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}|z|^s (1+|z|^2)^{-\frac{n+4}{2}}\,\ud z.
\]
Hence, the first convergence result in the lemma follows.
By Lemma \ref{lem:upbound1},
\begin{align*}
\int_{r_i\le |y|<1}|y|^s u_i(y)^{p_i}\,\ud y&\le Cm_i^{-\lda_i p_i} \int_{r_i\le |y|<1}|y|^s |y|^{(4-n+\delta_i)p_i}\,\ud y\\&
\le C m_i^{-\lda_i p_i}m_i^{\frac{(p_i-1)((n-4-\delta_i)p_i-s-n)}{4}} R_i^{n+s-(n-4-\delta_i)p_i}\\&
= Cm_i^{-\frac{(p_i-1)(s+n)}{4}+p_i} R_i^{n+s-(n-4-\delta_i)p_i},
\end{align*}
where $0<\delta_i=O(R_i^{-4})$ and $\lda_i=(n-4-\delta_i)(p_i-1)/4-1$. Since $m_i^{-\frac{(p_i-1)(s+n)}{4}+p_i}=(1+o(1)) m_i^{-1-\frac{2s}{n-4}}$ and $n+s-(n-4-\delta_i)p_i \to s-4<0$ as $i\to \infty$, we have
the second convergence result in the lemma.
In conclusion, the lemma is proved.
\end{proof}
\begin{prop}\label{prop:upbound2} Under the assumptions in Lemma \ref{lem:upbound1}, we have
\[
u_i(x)\leq Cu_i^{-1}(0)|x|^{4-n},\quad \mbox{for all } |x|\leq 1.
\]
\end{prop}
\begin{proof} For $|x|\le r_i$, the proposition follows immediately from Proposition \ref{prop:blow up a bubble} and Lemma \ref{lem:error}.
We shall show first that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a7'}
\sup_{|e|=1}u_i( \rho e) u_i(0)\le C.
\ee If not, then along a subsequence we have, for some unit vectors $\{e_i\}$,
\[
\lim_{i\to\infty}u_i(\rho e_i) u_i(0)=+\infty.
\]
Since $u_i(x)\le A_3 |x|^{-4/(p_i-1)}$ in $B_2$, it follows from Proposition \ref{prop:har} that for any $0<\varepsilon} \newcommand{\ud}{\mathrm{d}<1$ there exists a positive constant
$C(\varepsilon} \newcommand{\ud}{\mathrm{d})$, depending only on $n, A_1, A_2, A_3$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$, such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:extra1}
\sup_{B_{3/2}\setminus B_\varepsilon} \newcommand{\ud}{\mathrm{d}} u_i \le C(\varepsilon} \newcommand{\ud}{\mathrm{d})\inf_{B_{3/2}\setminus B_\varepsilon} \newcommand{\ud}{\mathrm{d}} u_i.
\ee
Let $\varphi_i (x)=u_i(\rho e_i) ^{-1} u_i(x)$. Then for $|x|\le 1$,
\[
\varphi_i(x)= \int_{B_3} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y+\tilde h_i(x),
\]
where $\tilde h_i(x)=u_i(\rho e_i) ^{-1} h_i(x)$.
Since $\varphi_i(\rho e_i)=1$, by \eqref{eq:extra1}
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a8}
\|\varphi_i\|_{L^\infty(B_{3/2}\setminus B_\varepsilon} \newcommand{\ud}{\mathrm{d})} \le C(\varepsilon} \newcommand{\ud}{\mathrm{d}) \quad \mbox{for }0<\varepsilon} \newcommand{\ud}{\mathrm{d}<1.
\ee
By \eqref{H}, we have that for any $x\in B_1$
\[
\tilde h_i(x) \le A_2 \tilde h_i(\rho e_i) \le A_2.
\]
Besides,
by Lemma \ref{lem:upbound1},
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:a8'}
u_i(\rho e_i) ^{p_i-1} \to 0
\ee
as $i\to \infty$.
Because of \eqref{G}-\eqref{H}, by applying Proposition \ref{prop:local estimates} to $\varphi_i$ we conclude that there exists $\varphi\in C^3(B_1\setminus \{0\})$ such that $\varphi_i\to \varphi$ in $C^3_{loc}(B_1\setminus \{0\})$ after passing to a subsequence.
Let us write the equation of $\varphi_i$ as
\begin{equation}} \newcommand{\ee}{\end{equation}
\varphi_i(x)= \int_{B_1} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y+b_i(x),
\ee
where $b_i(x):= \int_{B_3\setminus B_1} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y +\tilde h_i(x)$.
By \eqref{eq:a8}, there exists $b\in C^3(B_1)$ such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a9}
b_i(x) \to b(x)\ge 0\quad \mbox{in }C_{loc}^3(B_1)
\ee
after passing to a subsequence. Therefore,
\[
\int_{B_1} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y= \varphi_i(x)-b_i(x) \to \varphi(x) -b(x)
\]
in $C^3_{loc}(B_1\setminus \{0\})$. Denote $\Gamma(x):=\varphi(x)-b(x)$.
For any $|x|>0$ and $0< \varepsilon} \newcommand{\ud}{\mathrm{d}<\frac{1}{2}|x|$, in view of \eqref{eq:a8} and \eqref{eq:a8'} we have
\begin{align}
\Gamma(x)&= \lim_{i\to \infty} \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y \nonumber\\ &
=\lim_{i\to \infty} \left(G_i(x,0) \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y +O(m_i)\int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}}|y| u_i(y)^{p_i}\,\ud y\right)\nonumber \\&
= \lim_{i\to \infty} (G_i(x,0) \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y + O(m_i^{-\frac{2}{n-4}}))\nonumber\\&
=:G_\infty(x,0) a(\varepsilon} \newcommand{\ud}{\mathrm{d}),
\label{eq:the constant a}
\end{align}
where we used Lemma \ref{lem:aux3} in the third identity, $a(\varepsilon} \newcommand{\ud}{\mathrm{d})$ is a bounded nonnegative function of $\varepsilon} \newcommand{\ud}{\mathrm{d}$,
\[
G_\infty(x,0)=c_{n} |x|^{4-n}+\bar a+O'(|x|^{6-n})
\] by \eqref{G}, $\bar a\ge 0$ and $\bar a=0$ if $n\ge 6$.
Clearly, $a(\varepsilon} \newcommand{\ud}{\mathrm{d})$ is nondecreasing in $\varepsilon} \newcommand{\ud}{\mathrm{d}$, so $\lim_{\varepsilon} \newcommand{\ud}{\mathrm{d}\to 0} a(\varepsilon} \newcommand{\ud}{\mathrm{d})$ exists which we denote as $a$.
Sending $\varepsilon} \newcommand{\ud}{\mathrm{d}\to 0$, we obtain
\[
\Gamma(x)= a G_\infty(x,0).
\]
Since $ 0$ is an isolated simple blow point of $\{u_i\}_{i=1}^\infty$, we have $r^{\frac{n-4}{2}}\bar \varphi(r) \ge \rho^{\frac{n-4}{2}}\bar \varphi(\rho)$ for $0<r<\rho$. It follows that $\varphi$ is singular at $0$, and thus, $a>0$. Hence,
\[
\lim_{i\to \infty} \int_{B_{1/8}}K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y\ge a(\varepsilon} \newcommand{\ud}{\mathrm{d})\ge a>0.
\]
However,
\[
\begin{split}
&\int_{B_{1/8}}K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y\\
&\quad\le C u_i(\rho e_i)^{-1} \int_{B_{1/8}} u_i(y)^{p_i}\,\ud y\\
& \quad\le \frac{C}{u_i(\rho e_i) u_i(0)} \to 0 \quad\mbox{as } i \to \infty,
\end{split}
\]
where we used Lemma \ref{lem:aux3} in the last inequality. This is a contradiction.
Without loss of generality, we may assume that $\rho\le 1/2$.
It follows from Proposition \ref{prop:har} and \eqref{eq:a7'} that Proposition \ref{prop:upbound2} holds for $\rho\le |x| \le 1$.
To establish the inequality in the Proposition for $r_i\le |x|\le \rho$, we only need to rescale and reduce it to the case of $|x|=1$. Suppose the contrary that there exists a subsequence $ x_i$ satisfying $|x_i|\le \rho$
and $\lim_{i\to \infty}u_i( x_i) u_i(0)|x_i|^{n-4}= +\infty$.
Set $\tilde r_i:=| x_i|$, $\tilde u_i(x)= \tilde r_i^{4/(p_i-1)}u_i(\tilde r_i x)$. Then $\tilde u_i$ satisfies
\[
\tilde u_i(x)= \int_{B_3} G_{i,\tilde r_i}(x,y)K_i(\tilde r_i y)\tilde u_i(y)^{p_i}\,\ud y +\tilde h_i(x) \quad \mbox{for } x\in B_2,
\]
where $\tilde h_i(x)=\int_{B_{3/\tilde r_i}\setminus B_3} G_{i,\tilde r_i}(x,y)K_i(\tilde r_i y)\tilde u_i(y)^{p_i}\,\ud y + \tilde r_i^{4/(p_i-1)}h_i(\tilde r_i x)$. One can easily check that $\tilde u_i$ and the above equation
satisfy all hypotheses of Proposition \ref{prop:upbound2} for $u_i$ and its equation. It follows from
\eqref{eq:a7'} that
\[
\tilde u_i(0) \tilde u_i(\frac{x_i}{\tilde r_i}) \le C.
\]
It follows (using Lemma \ref{lem:error}) that
\[
\lim_{i\to \infty} u_i(x_i) u_i(0) |x_i|^{n-4} <\infty.
\]
This is again a contradiction.
Therefore, the proposition is proved.
\end{proof}
\begin{prop}\label{prop:upbound3} Under the assumptions in Lemma \ref{lem:upbound1}, we have
\[
|\nabla^k u_i(x)|\leq Cu_i^{-1}(0)|x|^{4-n-k},\quad \mbox{for all } r_i\le |x|\leq 1,
\]
where $k=1,\dots, 4$.
\end{prop}
\begin{proof} Since $0$ is an isolated blow up point in $B_2$, by Proposition \ref{prop:har} we see that Proposition \ref{prop:upbound2} holds for all $|x|\le \frac32$. For any $r_i\le |x|<1$, let
\[
\varphi_i(z)=\Big( \frac{|x|}{4}\Big)^{\frac{4}{p_i-1}} u_i(x+\frac{|x|}{4} z).
\]
By the equation of $u_i$, we have
\[
\varphi_i(z)=\int_{\{y:|x+\frac{|x|}{4} y|\le 3\}} \tilde G_{i}(z,y) \tilde K_i(y) \varphi_i(y)^{p_i-1} \varphi_i(y)\,\ud y+\tilde h_i(z),
\]
where $\tilde G_{i}(z,y)=(\frac{|x|}{4})^{n-4} G_i(x+\frac{|x|}{4} z, x+\frac{|x|}{4} y)$, $\tilde K_i(y)=K_i(x+\frac{|x|}{4} y)$, and $\tilde h_i(z) =( \frac{|x|}{4})^{\frac{4}{p_i-1}} h_i(x+\frac{|x|}{4} z)$. Since $0$ is an isolated blow up point of $u_i$, we have $\varphi_i(z)^{p_i-1}\le A_2^{p_i-1}$ for all $|z|\le 1$. Since $\varphi_i, \tilde G_i, \tilde K_i$ and $\tilde h_i$ are nonnegative, by Proposition \ref{prop:local estimates} we have
\[
|\nabla^k \varphi_i(0)|\le C(\|\varphi_i\|_{L^\infty(B_1)} +\| \tilde h_i\|_{C^4(B_1)}).
\]
This gives
\begin{align*}
(\frac{|x|}{4})^k |\nabla^k u_i(x)| &\le C\|u_i\|_{L^\infty(B_{\frac{|x|}{4}}(x))}+Cm_i^{-1}\\
& \le C u_i(0)^{-1}|x|^{4-n}.
\end{align*}
Therefore, the proposition follows.
\end{proof}
\begin{cor}\label{cor:energy} Under the hypotheses of Lemma \ref{lem:upbound1}, we have
\[
\int_{B_1}|x|^{s}u_i(x)^{p_i+1}\le C u_i(0)^{-2s/(n-4)}, \quad\mbox{for } -n< s<n,
\]
\end{cor}
\begin{proof} Making use of Proposition \ref{prop:blow up a bubble}, Lemma \ref{lem:error} and Proposition
\ref{prop:upbound2}, the corollary follows immediately.
\end{proof}
By Proposition \ref{prop:upbound2} and its proof, we have the following corollary.
\begin{cor} \label{cor:convergence} Under the assumptions in Lemma \ref{lem:upbound1}, if we let $T_i(x)=T'_i(x)+T''_i(x)$, where
\[
T'_i(x):=u_i(0)\int_{B_1} G_{i}(x,y)K_i(y) u_i(y)^{p_i}\,\ud y
\]
and
\[
T''_i(x):= u_i(0) \int_{B_3\setminus B_1} G_{i}(x,y) K_i(y) u_i(y)^{p_i}\,\ud y+u_i(0) h_i(x).
\] then, after passing a subsequence,
\[
T'_i(x) \to aG_\infty(x,0) \quad \mbox{in } C^3_{loc}(B_1\setminus \{0\})
\]
and
\[
T''_i(x) \to h(x) \quad \mbox{in } C_{loc}^3(B_1)
\] for some $h(x)\in C^3(B_2)$,
where $G_\infty$ is the limit of a subsequence of $G_i$ in $C^3_{loc}(B_1\setminus \{0\})$,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:number a}
a=\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\left(\frac{1}{1+|y|^2}\right)^\frac{n+4}{2}\ud y.
\ee
Consequently, we have
\[u_i(0) u_i(x)\to aG_\infty(x,0) +h(x) \quad \mbox{in }C^3_{loc}(B_1\setminus \{0\}).
\]
\end{cor}
\begin{proof} Similar to that in the proof of Proposition \ref{prop:upbound2}, we set
$\varphi_i(x)=u_i(0) u_i(x)$, which satisfies
\begin{align*}
\varphi_i(x)&=\int_{B_3} G_{i}(x,y)K_i(y) u_i(0)^{1-p_i} \varphi_i(y)^{p_i}\,\ud y+u_i(0)h_i(x)\\&
= : \int_{B_1} G_{i}(x,y)K_i(y) u_i(0)^{1-p_i} \varphi_i(y)^{p_i}\,\ud y +T''_i(x) =T'_i(x) +T''_i(x) .
\end{align*}
We have all the ingredients as in the proof of Proposition \ref{prop:upbound2}. Hence, we only need to evaluate the positive constant $a$. By \eqref{eq:the constant a} and Lemma \ref{lem:aux3}, we have
\[
a= \lim_{\varepsilon} \newcommand{\ud}{\mathrm{d}\to 0}\lim_{i\to \infty} u_i(0) \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} K_i (y)u_i(y)^{p_i}\,\ud y=\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\left(\frac{1}{1+|y|^2}\right)^\frac{n+4}{2}\ud y.
\]
\end{proof}
\section{Expansions of blow up solutions of integral equations}
\label{section:bubble-expansion}
In this section, we are interested in stronger estimates than that in Proposition \ref{prop:upbound2}. To make statements closer to the main goal of the paper, we restrict our attention to a special $K_i$. Namely, given $p_i, G_i$, and $ h_i$ satisfying \eqref{p}, \eqref{G} and \eqref{H} respectively, $\kappa_i$ satisfying \eqref{K} with $K_i$ replaced by $\kappa_i$, let $0\le u_i\in C^4(B_3)$
be a solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:s1'}
u_i(x)=\int_{B_3 }G_i(x,y) \kappa_i(y)^{\tau_i} u_i^{p_i}(y)\,\ud y +h_i(x) \quad \mbox{in } B_3.
\ee
We also assume
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:56}
\nabla \kappa_i(0)=0.
\ee Suppose that $0$ is an isolated simple blow up point of $\{u_i\}$ with $\rho=1$, i.e.,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:A_3}
u_i(x)\leq A_3 |x|^{-4/(p_i-1)}\quad \mbox{for all } x\in B_2.
\ee
and $
r^{4/(p_i-1)} \bar u_i(r)$ has precisely one critical point in $(0,1)$.
Let us first introduce a non-degeneracy result.
\begin{lem}\label{lem:non-degeneracy} Let $v\in L^\infty(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ be a solution of
\[
v(x)=c_{n}\frac{n+4}{n-4}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(y)^{\frac{8}{n-4}}v(y)}{|x-y|^{n-4}}\,\ud y.
\]
Then
\[
v(z)=a_0 \left(\frac{n-4}{2}U_1(z)+z\cdot \nabla U_1(z)\right)+\sum_{j=1}^n a_j \pa_j U_1(z),
\]
where $a_0,\dots,a_n$ are constants.
\end{lem}
\begin{proof} Since $v\in L^\infty(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$, by using Lemma \ref{lem:aux2} iteratively a finite number of times we obtain
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:d1}|v(x)|\le C (1+|x|)^{4-n}.
\ee Let $F:\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n\to \mathbb{S}^n\setminus\{N\}$,
\[
F(x)=\left(\frac{2x}{1+|x|^2}, \frac{1-|x|^2}{1+|x|^2}\right)
\]
denote the inverse of the inverse of the stereographic projection and $ h(F(x)):=v(x) J_{F}(x)^{-\frac{n-4}{2n}} $, where $J_F=(\frac{2}{1+|x|^2})^{n}$ is the Jacobian determinant of $F$ and $N$ is the north pole. It follows from \eqref{eq:d1} that $h\in L^\infty(\Sn)$. Let $\xi=F(x)$ and $\eta=F(y)$.
Then
\[
|\xi-\eta| = \frac{2|x-y|}{\sqrt{(1+|x|^2)(1+|y|^2)}} \quad\mbox{and} \quad
\ud \eta =\left(\frac{2}{1+|y|^2}\right)^{n}\ud y
\]
are respectively the distance between $\xi$ and $\eta$ in $\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^{n+1}$ and the surface measure of $\Sn$.
It follows that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{inte eq}
h(\xi)=2^{-4}n(n-2)(n+2)(n+4) \alpha} \newcommand{\lda}{\lambda_n\int_{\Sn} |\xi-\eta|^{4-n} h(\eta)\,\ud \eta.
\ee
By the regularity theory for Riesz potentials, $h\in C^\infty(\Sn)$. Note that the Paneitz operator
\[
P_{g_{\Sn}}=\Delta^2_{g_{\Sn}}-\frac{n^2-2n-4}{2}\Delta_{g_{\Sn}} +\frac{n(n-2)(n+2)(n-4)}{16}
\]
with respect to the standard metric $g_{\Sn}$ on $\Sn$ satisfies $P_{g_{\Sn}} \phi=|J_{F}|^{-\frac{n+4}{2n}} \Delta^2 (|J_{F}|^{\frac{n-4}{2n}} \phi \circ F)$ for any $\phi\in C^\infty(\Sn)$. By the integral equation of $v$ we have
\begin{align*}
P_{g_{\Sn}} h= 2^{-4}n(n-2)(n+2)(n+4)h.
\end{align*}
Let $Y^{(k)}$ be a spherical harmonics of degree $k\ge 0$. We have
\[
P_{g_{\Sn}}Y^{(k)}=2^{-4}(2k+n+2)(2k+n)(2k+n-2)(2k+n-4) Y^{(k)}.
\]
Hence, $h$ must be a spherical harmonics of degree one. Transforming $h$ back, we complete the proof.
\end{proof}
In view of Corollary \ref{cor:GM2.8}, we assume in this and next section that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:IE-cond}
U_{\lda}(x)=\int_{B_3} G_i(x,y)\{U_\lda(y)^{\frac{n+4}{n-4}}+c_{\lda,i}'(y) U_{\lda}(y)\}\,\ud y+c_{\lda,i}''(x) \quad \forall ~\lda\ge 1 ,~x\in B_3
\ee
where $c_{\lda,i}',c_{\lda,i}''\in C^5(B_3)$ satisfy
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:non-flat}
\Theta_i:=\sum_{k=0}^5\|\lda^{-k}\nabla^kc_{\lda,i}'\|_{L^\infty(B_2)}\le A_2
,\ee and $\|c_{\lda,i}''\|_{C^5(B_2)}\le A_2 \lda^{\frac{4-n}{2}}$, respectively.
\begin{lem}
\label{lem:expansion-a}
Let $0\le u_i\in C^4(B_3)$ be a solution of \eqref{eq:s1'} and $0$ is an isolated simple blow up point of $\{u_i\}$ with some constant
$\rho$, say $\rho=1$. Suppose \eqref{eq:IE-cond} holds and let $\Theta_i$ be defined in \eqref{eq:non-flat}. Then we have
\[
|\varphi_i(z)-U_1(z)| \le C \begin{cases}
\max\{\tau_i, m_i^{-2}\},& \quad \mbox{if } 5\le n\le 7, \\
\max\{ \tau_i, \Theta_i m_i^{-2}\log m_i, m_i^{-2}\},& \quad \mbox{if }n=8,\\
\max\{ \tau_i, \Theta_i m_i^{-\frac{8}{n-4}}, m_i^{-2}\},& \quad\mbox{if } n\ge 9,
\end{cases}
\quad \forall~ |z|\le m_i^{\frac{p_i-1}{4}},
\]where $\varphi_i(z)=\frac{1}{m_i}u_i(m_i^{-\frac{p_i-1}{4}}z)$, $m_i=u_i(0)$, and $C>0$ depends only on $n,A_1,A_2$ and $A_3$.
\end{lem}
\begin{proof}
For brevity, set $\ell_i= m_i^{\frac{p_i-1}{4}}$. By the equation of $u_i$, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:varphi}
\varphi_i(z)=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y)\tilde \kappa_i(y)^{\tau_i}\varphi_i(y)^{p_i}\,\ud y +\bar h_i(z),
\ee
where $G_{i,\ell_i^{-1}}(z,y)=\ell_i^{4-n}G_i(\ell_i^{-1}x,\ell_i^{-1} y)$, $\tilde \kappa_i(z)=\kappa_i(\ell_i^{-1} z)$, and
$
\bar h_i(z)=m_i^{-1}\tilde h_i(\ell_i^{-1}z)
$ with
\[
\tilde h_i (x)= \int_{B_3\setminus B_1} G_i(x,y) \kappa_i(y)^{\tau_i} u_i(y)^{p_i} \,\ud y+h_i(x).
\]
Since $0$ is an isolated simple blow up point of $u_i$, by Proposition \ref{prop:upbound2} we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:exp-1}
u_i(x) \le C m_i^{-1}|x|^{4-n} \quad \mbox{for }|x|<1.
\ee It follows that $\tilde h_i(x)\le C m_i^{-1}$ for $x\in B_1$ and $\bar h_i(z) \le Cm_i^{-2}$ for $z\in B_{\ell_i}$.
Notice that $U_{\ell_i}(x)\le Cm_i^{-1}$ for $1\le |x|\le 3$. Let $z=\ell_i x$. By \eqref{eq:IE-cond} with $\lda=\ell_i$ we have for $|z|\le \ell_i$
\begin{align}
U_1(z)&=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y)(U_1(y)^{\frac{n+4}{n-4}}+m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1}y)U_1(y))\,\ud y+\mathcal{O}(m_{i}^{-2})\nonumber \\&
=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i}+T_i(y))\,\ud y+\mathcal{O}(m_i^{-2}),
\label{eq:U1}
\end{align}
where we used $m_i^{\tau_i}=1+o(1)$, and
\[
T_i(y):=U_1(y)^{\frac{n+4}{n-4}}-\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i} +m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1} y)U_1(y).
\]
\emph{Here and throughout this section, $\mathcal{O}(m_i^{-2})$ denotes some function $f_i$ satisfying $\|\nabla ^k f_i\|_{B_{(1-\varepsilon} \newcommand{\ud}{\mathrm{d})\ell_i}} \le C(\varepsilon} \newcommand{\ud}{\mathrm{d}) m_i^{-2-\frac{2k}{n-4}}$ for small $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ and $k=0,\dots, 5$.}
In the following, we adapt some arguments from Marques \cite{Marques} for the Yamabe equation; see also the proof of Proposition 2.2 of Li-Zhang \cite{Li-Zhang05}.
Let
\[
\Lambda} \newcommand{\B}{\mathcal{B}_i=\max_{|z|\le \ell_i} |\varphi_i-U_1|.
\]
By \eqref{eq:exp-1}, for any $0<\varepsilon} \newcommand{\ud}{\mathrm{d}<1$ and $\varepsilon} \newcommand{\ud}{\mathrm{d} \ell_i\le |z|\le \ell_i$, we have $|\varphi_i (z)-U_1(z)|\le C(\varepsilon} \newcommand{\ud}{\mathrm{d}) m_i^{-2}$, where we used $m_i^{\tau_i}=1+o(1)$. Hence, we may assume that $\Lambda} \newcommand{\B}{\mathcal{B}_i$ is achieved at some point $|z_i|\le \frac12 \ell_i$, otherwise the proof is finished.
Set
\[
v_i(z)= \frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}(\varphi_i(z)-U_1(z)).
\]
It follows from \eqref{eq:varphi} and \eqref{eq:U1} that $v_i$ satisfies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:newscale1}
v_i(z)= \int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (b_i(y) v_i(y)+\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}T_i(y))\,\ud y+\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}\mathcal{O}(m_i^{-2}),
\ee
where
\[
b_i=\tilde \kappa_i^{\tau_i}\frac{\varphi_i^{p_i}-U_1^{p_i}}{\varphi_i-U_1}.
\]
Since
\[
G_{i,\ell_i^{-1}}(z,y)\le A_1|z-y|^{4-n}
\] and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Ti-estimate}
|T_i(y)|\le C\tau_i (|\log U_i|+|\log \tilde \kappa_i|)(1+|y|)^{-p_i(n-4)}+\Theta_i m_i^{-\frac{8}{n-4}} (1+|y|)^{4-n},
\ee we obtain
\[
\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |T_i(y)|\,\ud y\le C (\tau_i+\Theta_i \alpha} \newcommand{\lda}{\lambda_i)\quad \mbox{for }|z|\le \frac{\ell_i}{2},
\]
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:ali}
\alpha} \newcommand{\lda}{\lambda_i= \begin{cases}
m_i^{-2},& \quad \mbox{if }5\le n\le 7, \\
m_i^{-2}\log m_i,& \quad \mbox{if }n=8,\\
m_i^{-\frac{8}{n-4}},& \quad \mbox{if }n\ge 9.
\end{cases}
\ee
Since $\kappa_i(x)$ is bounded and $\varphi_i\le C U_1$, we see
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:bi-estimate}
|b_i(y)|\le CU_1(y)^{p_i-1}\le C (1+|y|)^{-7.5}, \quad y\in B_{\ell_i}.
\ee
Noticing that $\|v_i\|_{L^\infty(B_{\ell_i})}\le 1$, by Lemma \ref{lem:aux2} we have
\[
\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |b_i(y) v_i(y)|\,\ud y\le C(1+|z|)^{-\min\{n-4,3.5\}}.
\] Hence, we get
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:b-expansion-1}
v_i (z) \le C((1+|z|)^{-\min\{n-4,3.5\}}+\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}(\tau_i+\Theta_i\alpha} \newcommand{\lda}{\lambda_i+m_i^{-2})) \quad \mbox{for }|z|\le \frac{\ell_i}{2}.
\ee
Suppose the contrary that $\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}\max\{\tau_i,\Theta_i\alpha} \newcommand{\lda}{\lambda_i,m_i^{-2} \}\to 0$ as $i \to \infty$. Since $v(z_i)=1$, by \eqref{eq:b-expansion-1} we see that
\[
|z_i|\le C.
\]
Differentiating the integral equation \eqref{eq:newscale1} up to three times, together with \eqref{eq:Ti-estimate} and \eqref{eq:bi-estimate}, we see that the $C^3$ norm of $v_i$ on any compact set is uniformly bounded. By Arzel\`a-Ascoli theorem let $v:=\lim_{i\to \infty}v_i$ after passing to a subsequence. Using Lebesgue's dominated convergence theorem, we obtain
\[
v(z)=c_{n}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{U_1(y)^{\frac{8}{n-4}} v(y)}{|z-y|^{n-4}}\,\ud y.
\]
It follows from Lemma \ref{lem:non-degeneracy} that
\[
v(z)=a_0 (\frac{n-4}{2}U_1(z)+z\cdot \nabla U_1(z))+\sum_{j=1}^n a_j \pa_j U_1(z),
\]
where $a_0,\dots,a_n$ are constants.
Since $v(0)=0$ and $\nabla v(0)=0$, $v$ has to be zero. However, $v(z_i)=1$. We obtain a contradiction.
Therefore, $\Lambda} \newcommand{\B}{\mathcal{B}_i\le C (\tau_i+\alpha} \newcommand{\lda}{\lambda_i)$ and the proof is completed.
\end{proof}
\begin{lem}\label{lem:expansion-b} Under the same assumptions in Lemma \ref{lem:expansion-a}, we have
\[
\tau_i \le C \begin{cases}
m_i^{-2},& \quad \mbox{if }5\le n\le 7, \\
\max\{ \Theta_i m_i^{-2}\log m_i,m_i^{-2}\},& \quad \mbox{if }n=8,\\
\max\{ \Theta_i m_i^{-\frac{8}{n-4}},m_i^{-2}\},& \quad \mbox{if }n\ge 9.
\end{cases}
\]
\end{lem}
\begin{proof} The proof is also by contradiction. Recall the definition of $\alpha} \newcommand{\lda}{\lambda_i$ in \eqref{eq:ali}. Suppose the contrary that $\frac{1}{\tau_i}\max\{\Theta_i\alpha} \newcommand{\lda}{\lambda_i,m_i^{-2} \}\to 0$ as $i\to \infty$.
Set
\[
v_i(z)=\frac{\varphi_i(z)-U_1(z)}{\tau_i}.
\]
It follows from Lemma \ref{lem:expansion-a} that $|v_i(z)|\le C$ in $B_{\ell_i}$, where $\ell_i= m_i^{\frac{p_i-1}{4}}$. As \eqref{eq:newscale1}, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:newscale2}
v_i(z)= \int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (b_i(y) v_i(y)+\frac{1}{\tau_i}T_i(y))\,\ud y+\frac{1}{\tau_i}\mathcal{O}(m_i^{-2}),
\ee
where
\[
b_i=\tilde \kappa_i^{\tau_i}\frac{\varphi_i^{p_i}-U_1^{p_i}}{\varphi_i-U_1},
\]
and
\[
T_i(y):=U_1(y)^{\frac{n+4}{n-4}}-\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i} +m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1} y)U_1(y).
\]
By the estimates \eqref{eq:bi-estimate} and \eqref{eq:Ti-estimate} for $b_i$ and $T_i$ respectively, we conclude from the integral equation that $\|v_i\|_{C^3}$ is uniformly bounded over any compact set. It follows that $v_i\to v$ in $C^2_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ for some $v\in C^3(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$.
Multiplying both sides of \eqref{eq:newscale2} by $b_i(z)\phi(z)$, where $\phi(z)=\frac{n-4}{2} U_1(z)+z\cdot \nabla U_1(z)$, and integrating over $B_{\ell_i}$, we have, using the symmetry of $G_{i,\ell_i^{-1}}$ in $y$ and $z$,
\begin{align*}
&\int_{B_{\ell_i}}b_i(z)v_i(z)\left(\phi(z)-\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y\right)\,\ud z\\&
=\frac{1}{\tau_i}\int_{B_{\ell_i}}T_i(z)\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y\,\ud z+\frac{1}{\tau_i}\mathcal{O}(m_i^{-2})\int_{B_{\ell_i}} b_i(z)\phi(z)\,\ud z.
\end{align*}
As $i\to \infty$, we have
\[
\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y \to c_{n}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(y)^{\frac{8}{n-4}}\phi(y)}{|z-y|^{n-4}}\,\ud z=\phi(z),
\]
\[
\frac{1}{\tau_i}\mathcal{O}(m_i^{-2})\int_{B_{\ell_i}} b_i(z)\phi(z)\,\ud z \to 0\mbox{ by the contradiction hypothesis},
\]
and
\[
\frac{T_i(z)}{\tau_i} \to (\log U_1(z))U_1(z)^{\frac{n+4}{n-4}}.
\]
Hence, by Lebesgue's dominated convergence theorem we obtain
\[
\begin{split}
\lim_{i\to \infty}&\frac{1}{\tau_i}\int_{B_{\ell_i}}T_i(z)\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y\,\ud z\\&=\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \phi(z) (\log U_1(z))U_1(z)^{\frac{n+4}{n-4}}\ud z=0.
\end{split}
\]
This is impossible, because
\begin{align*}
&\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \phi(z) (\log U_1(z))U_1(z)^{\frac{n+4}{n-4}}\ud z\\&=\frac{(n-4)^2|\mathbb{S}^{n-1}|}{4} \int_{0}^\infty \frac{(r^2-1)r^{n-1}}{(1+r^2)^{n+1}} \log (1+r^2)\,\ud r\\&
=\frac{(n-4)^2|\mathbb{S}^{n-1}|}{2} \int_{1}^\infty \frac{(r^2-1)r^{n-1}}{(1+r^2)^{n+1}} \log r\,\ud r>0,
\end{align*} where we used
\[
\int_{0}^1\frac{(r^2-1)r^{n-1}}{(1+r^2)^{n+1}} \log (1+r^2)\,\ud r= -\int_{1}^\infty \frac{(s^2-1)s^{n-1}}{(1+s^2)^{n+1}} (\log (1+s^2)-\log s^2)\,\ud s
\]
by the change of variable $r=\frac{1}{s}$.
We obtain a contradiction and thus $\tau_i\le \alpha} \newcommand{\lda}{\lambda_i$. Therefore, the lemma is proved.
\end{proof}
\begin{prop}\label{prop:expansion} Under the hypotheses in Lemma \ref{lem:expansion-a}, we have
\[
|\varphi_i (z)-U_1(z)| \le C \begin{cases}
m_i^{-2},& \quad \mbox{if } 5\le n\le 7, \\
\max\{ \Theta_i m_i^{-2}\log m_i,m_i^{-2}\},& \quad \mbox{if }n=8,\\
\max\{ \Theta_i m_i^{-\frac{8}{n-4}},m_i^{-2}\},& \quad\mbox{if } n\ge 9,
\end{cases}
\quad \forall~ |z|\le m_i^{\frac{p_i-1}{4}}.
\]
\end{prop}
\begin{proof} It follows immediately from Lemma \ref{lem:expansion-a} and Lemma \ref{lem:expansion-b}.
\end{proof}
\begin{prop}\label{prop:expansion8+} Under the hypotheses in Lemma \ref{lem:expansion-a}, we have, for every $|z|\le m_i^{\frac{p_i-1}{4}}$,
\[
|\varphi_i (z)-U_1(z)| \le C \begin{cases}
\max\{ \Theta_im_i^{-2}m_i^{\frac{2}{n-4}}(1+|z|)^{-1}, m_i^{-2}\},& \quad \mbox{if } n=8, \\
\max\{ \Theta_i m_i^{-2}m_i^{\frac{2(n-8)}{n-4}}(1+|z|)^{8-n}, m_i^{-2}\} ,& \quad\mbox{if } n\ge 9.
\end{cases}
\]
\end{prop}
\begin{proof} Let $\alpha} \newcommand{\lda}{\lambda_i$ be defined in \eqref{eq:ali}. We may assume that $\frac{m_i^{-2}}{\Theta_i\alpha} \newcommand{\lda}{\lambda_i}\to 0$ as $i\to \infty$ for $n\ge 8$; otherwise the proposition follows immediately from Proposition \ref{prop:expansion}. Set
\[
\alpha} \newcommand{\lda}{\lambda'_i=\begin{cases}
m_i^{-2}m_i^{\frac{2}{n-4}},& \quad \mbox{if } n=8, \\
m_i^{-2}m_i^{\frac{2(n-8)}{n-4}},& \quad\mbox{if } n\ge 9,
\end{cases}
\]
and
\[
v_i(z)=\frac{\varphi_i(z)-U_1(z)}{\Theta_i \alpha} \newcommand{\lda}{\lambda'_i}, \quad |z|\le m_i^{\frac{p_i-1}{4}}.
\]
Since $\frac{m_i^{-2}}{\Theta_i\alpha} \newcommand{\lda}{\lambda_i}\to 0$, it follows from Proposition \ref{prop:expansion} that $|v_i|\le C$. Since $0<\varphi_i\le CU_1$, we only need to prove the proposition when $|z|\le \frac{1}{2}\ell_i$, where $\ell_i=m_i^{\frac{p_i-1}{4}}$. Similar to \eqref{eq:newscale1}, $v_i$ now satisfies
\[
v_i(z)=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (b_i(y) v_i(y)+\frac{1}{\Theta_i \alpha} \newcommand{\lda}{\lambda'_i}T_i(y))\,\ud y+\frac{1}{\Theta_i \alpha} \newcommand{\lda}{\lambda_i'}\mathcal{O}(m_i^{-2}),
\]
where
\[
b_i=\tilde \kappa_i^{\tau_i}\frac{\varphi_i^{p_i}-U_1^{p_i}}{\varphi_i-U_1}
\]
and
\[
T_i(y):=U_1(y)^{\frac{n+4}{n-4}}-\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i} +m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1} y)U_1(y).
\]
Noticing that
\[
|T_i(y)|\le C\tau_i (|\log U_1|+|\log \tilde \kappa_i|)(1+|y|)^{-4-n}+m_i^{-\frac{8}{n-4}} \Theta_i (1+|y|)^{4-n},
\]
we have
\begin{align*}
\frac{1}{\Theta_i \alpha} \newcommand{\lda}{\lambda_i'}\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |T_i(y)|\,\ud y&\le C\int_{B_{\ell_i}} \frac{1}{|z-y|^{n-4}(1+|y|)^4 m_i^{\frac{2}{n-4}}} \,\ud y\\&
\le C \int_{B_{\ell_i}} \frac{1}{|z-y|^{n-4}(1+|y|)^5} \,\ud y\\&
\le C(1+|z|)^{-1},
\end{align*}
if $n=8$, and
\[
\frac{1}{\Theta_i\alpha} \newcommand{\lda}{\lambda_i'}\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |T_i(y)|\,\ud y \le C(1+|z|)^{8-n}
\]
if $n\ge 9$, where we used Lemma \ref{lem:aux2}.
Thus
\[
|v_i(z)|\le C((1+|z|)^{-3.5}+(1+|z|)^{-1})
\]
for $n=8$, and
\[
|v_i(z)|\le C((1+|z|)^{-3.5}+(1+|z|)^{8-n})
\] for $n\ge 9$. If $n=8,9,10,11$, the conclusion follows immediately from multiplying both sides of the above inequalities by $\alpha} \newcommand{\lda}{\lambda'_i$. If $n\ge 12$, the above estimate gives $|v_i(z)|\le C(1+|z|)^{-3.5}$. Plugging this estimate to the term $\int G_{i,\ell_i^{-1}}(z,y)b_i(y)v_i(y)\,\ud y$ yields $|v_i(z)|\le C(1+|z|)^{8-n}$ as long as $n\le 14$. Repeating this process, we complete the proof.
\end{proof}
\begin{cor}\label{cor:hot-expansion} Under the hypotheses in Lemma \ref{lem:expansion-a}, we have, for very $|z|\le m_i^{\frac{p_i-1}{4}}$,
\begin{align*}
&|\nabla^k(\varphi_i-U_1)(z)|\\& \le C(1+|z|)^{-k} \begin{cases}
m_i^{-2},& \quad \mbox{if } 5\le n\le 7,\\
\max\{ \Theta_im_i^{-2}m_i^{\frac{2}{n-4}}(1+|z|)^{-1}, m_i^{-2}\},& \quad \mbox{if } n=8, \\
\max\{ \Theta_i m_i^{-2}m_i^{\frac{2(n-8)}{n-4}}(1+|z|)^{8-n}, m_i^{-2}\} ,& \quad\mbox{if } n\ge 9.
\end{cases}
\end{align*}
where $k=1,2,3,4$.
\end{cor}
\begin{proof} Considering the integral equation of $v_i=\varphi_i-U_1$, the conclusion follows from Lemma \ref{lem:aux2}.
Indeed, if $k<4$, we can differentiate the integral equation of $v_i$ directly and then use Lemma \ref{lem:aux2}. If $k=4$, we can use a standard technique (see the proof of Proposition \ref{prop:local estimates}) for proving the higher order regularity of Riesz potential since $v_i$ and the coefficients are of $C^1$.
\end{proof}
\section{Blow up local solutions of fourth order equations}
\label{section:Q equation blow}
In the previous two sections, we have analyzed the blow up profiles of the blow up local solutions of integral equations. In this section, we will assume that those blow up solutions also satisfy differential equations, which is only used to check the Pohozaev identity in Proposition \ref{prop:4-pohozaev}. It should be possible to completely avoid using differential equations after improving Corollary \ref{cor:Q-gf-expansion}. This is the case on the sphere; see our joint work with Jin \cite{JLX3}. On the other hand, as mentioned in the Introduction, without extra information fourth order differential equations themselves are not enough to do blow up analysis for positive local solutions.
\begin{prop}\label{prop:one-side} In addition to the hypotheses in Lemma \ref{lem:expansion-a}, assume that $u_i$ also satisfies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Q-sub}
P_{ g_i} u_i=c(n) \kappa_i^{\tau_i} u_i^{p_i}\quad \mbox{in }B_3,
\ee
where $\det g_i=1$, $B_3$ is a normal coordinates chart of $g_i$ at $0$ and $\|g_i\|_{C^{10}(B_3)}\le A_1$.
\begin{itemize}
\item[(i)] If either $n\le 9$ or $g_i$ is flat, then
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sign restrict}
\liminf_{r\to 0}\mathcal{P}(r,\Gamma)\ge 0,
\ee
where $\Gamma$ is a limit of $u_i(0)u_i(x)$ along a subsequence.
\item[(ii)] If $n\ge 8$, then $|W_{g_i}(0)|^2 \le C^* \mathcal{G}_i \beta_i$ with $C^*>0$ depending only on $n,A_1,A_2, A_3$,
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:gi}
\mathcal{G}_i:= \sum_{k\ge 1,~ 2\le k+l\le 4}\Theta_i \|\nabla^k g_i\|_{L^\infty(B_3)}^l+\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l ,
\ee
and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:beta_i}
\beta_i:=\begin{cases} (\log m_i)^{-1},& \quad \mbox{if }n=8,\\
m_i^{-\frac{2}{n-4}}, &\quad \mbox{if }n=9,\\
m_i^{-\frac{4}{n-4}}\log m_i, &\quad \mbox{if }n=10,\\
m_i^{-\frac{4}{n-4}}, &\quad \mbox{if }n\ge 11.\\
\end{cases}
\ee
\item[(iii)] \eqref{eq:sign restrict} holds if $n\ge 10$ and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:r2}
|W_{g_i}(0)|^2 > C^* \mathcal{G}_i \beta_i.
\ee
\end{itemize}
\end{prop}
\begin{proof} It follows from Corollary \ref{cor:convergence} that after passing a subsequence
\[
\lim u_i(0)u_i(x)=:\Gamma(x),
\] where $\Gamma(x)$ is in $C^3(B_{1}\setminus \{0\})$. We will still denote the subsequence as $u_i$.
Notice that for every $0<r<1$
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-0}
m_i^{2}\mathcal{P}(r,u_i)\to \mathcal{P}(r,\Gamma)\quad \mbox{as }i\to \infty.
\ee
By Proposition \ref{prop:4-pohozaev},
\begin{align*}
\mathcal{P}(r,u_i)= \int_{B_r} (x^k \pa_k u_i +\frac{n-4}{2} u_i) E(u_i)+\mathcal{N}(r,u_i),
\end{align*}
where $E(u_i)$ is as in \eqref{eq:Eu} with $\tilde g$ and $u$ replaced by $g_i$ and $u_i$ respectively, i.e.,
\begin{align*}
E(u_i):&=P_{g_i}u_i-\Delta^2 u_i\\&
=\frac{n-4}{2}Q_{g_i}u_i+ f^{(1)}_{i,k}\pa_ku_i +f^{(2)}_{i,kl}\pa_{kl}u_i+f^{(3)}_{i,kls}\pa_{kls}u_i+f^{(4)}_{i,klst}\pa_{klst}u_i,
\end{align*}
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:fi}
f^{(1)}_{i,k}(x)=O(1), \quad f^{(2)}_{i,kl}(x)=O(1),\quad f^{(3)}_{i,kls}(x)=O(|x|), \quad f^{(4)}_{i,klst}(x)=O(|x|^2),
\ee
and
\begin{align*}
\mathcal{N}(r,u_i)=\frac{c(n)\tau_i}{p_i+1} \int_{B_r} (\frac{n-4}{2}\kappa_i^{\tau_i}+ x^k\pa_k \kappa_i \kappa_i^{\tau_i-1}) u_i^{p_i+1}
-\frac{r}{p_i+1}\int_{\pa B_{r}} c(n)\kappa_i^{\tau_i} u_i^{p_i+1}.
\end{align*}
By Proposition \ref{prop:upbound2}, for $0<r<1$ we have, for some $C>0$ independent of $i$ and $r$,
\begin{align*}
m_i^{2}\mathcal{N}(r,u_i) \ge
-\frac{m_i^{2}r}{p_i+1}\int_{\pa B_{r}} c(n)\kappa_i^{\tau_i} u_i^{p_i+1}\ge -Cr^{-n}m_i^{1-p_i},
\end{align*}
where we used the facts that $\kappa_i(x)=1+O(|x|^2)$ and $|\nabla \kappa_i(x)|=O(|x|)$ with $O(\cdot)$ independent of $i$. Hence, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-1}
\liminf_{i\to \infty}m_i^{2}\mathcal{N}(r,u_i)\ge 0.
\ee
Throughout this section, without otherwise stated, we use $C$ to denote some constants independent of $i$ and $r$.
If $g_i$ is flat, then we complete the proof because $E(u_i)=0$.
Now we assume $g_i$ is not flat. By a change of variables $z=\ell_i x$ with $\ell_i=m_i^{\frac{p_i-1}{4}}$, we have
\begin{align*}
\mathcal{ E}_i(r):&= m_i^{2}\int_{B_r} (x^k \pa_k u_i +\frac{n-4}{2} u_i) E(u_i) \,\ud x\\&
=m_i^{2} m_i^{2+(4-n)\frac{p_i-1}{4}} \int_{B_{\ell_i r}}\Big( z^k\pa_k\varphi_i+\frac{n-4}{2}\varphi_i\Big)\cdot\\& \quad
\Big(\frac{n-4}{2}\ell_i^{-4}Q_{g_i}(\ell_i^{-1}z)\varphi_i+ \sum_{j=1}^4\ell_i^{-4+j}f_i^{(j)}(\ell_i^{-1}z)\nabla^j \varphi_i\Big)\,\ud z,
\end{align*}
where $\varphi_i(z)=m_i^{-1}u_i(m_i^{-\frac{p_i-1}{4}}z)$, $
f_i^{(1)}(\ell_i^{-1}z)\nabla^1 \varphi_i=f^{(1)}_{i,k}(\ell_i^{-1}z)\pa_k\varphi_i$
and $f_i^{(j)}(\ell_i^{-1}z)\nabla^j \varphi_i$ is defined in the same fashion for $j\neq 1$. Define
\begin{align*}
\mathcal{\hat E}_i(r):&=m_i^{2} m_i^{2+(4-n)\frac{p_i-1}{4}} \int_{B_{\ell_i r}}\Big( z^k\pa_kU_1+\frac{n-4}{2}U_1\Big)\cdot\\& \quad
\Big(\frac{n-4}{2}\ell_i^{-4}Q_{g_i}(\ell_i^{-1}z)U_1+ \sum_{j=1}^4\ell_i^{-4+j}f_i^{(j)}(\ell_i^{-1}z)\nabla^j U_1\Big)\,\ud z.
\end{align*}
Notice that $m_i^{2+(4-n)\frac{p_i-1}{4}}=1+o(1)$, and $Q_{\tilde g}=O(1)$. By Proposition \ref{prop:expansion}, Proposition \ref{prop:expansion8+}, Corollary \ref{cor:hot-expansion}, \eqref{eq:fi} and \eqref{eq:r1}, we have
\begin{align}
&|\mathcal{ E}_i(r)-\mathcal{\hat E}_i(r)|\\&\le Cm_i^{2}m_i^{-\frac{4}{n-4}}\int_{B_{\ell_i r}} \sum_{j=0}^4|\nabla^j(\varphi_i-U_1)|(z)(1+|z|)^{2-n+j}\,\ud z\nonumber \\&
\le C \sum_{k\ge 1,~ 2\le k+l\le 4}\|\nabla^k g_i\|_{L^\infty(B_3)}^l
\begin{cases}
r^2, & \quad \mbox{if }n=5,6,7,\\
\max\{\Theta_i r, r^2\}, & \quad \mbox{if }n=8,9,\\
\max\{\Theta_i \log (rm_i), r^2\},& \quad \mbox{if }n=10,\\
\max\{\Theta_i m_i^{\frac{2(n-10)}{n-4}}, r^2\},& \quad \mbox{if }n\ge 11.
\end{cases}
\label{eq:tos-2}
\end{align}
Now we estimate $\mathcal{\hat E}_i(r)$.
If $n=5,6,7$, we have
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:why 8d}
\begin{split}
\mathcal{\hat E}_i(r)&= m_i^{2+(n-4)\tau_i}\int_{B_r} (x^k \pa_k U_{\ell_i} +\frac{n-4}{2} U_{\ell_i}) E(U_{\ell_i}) \,\ud x\\
&= m_i^{2+(n-4)\tau_i}\int_{B_r} (x^k \pa_k U_{\ell_i} +\frac{n-4}{2} U_{\ell_i}) (P_{ g_i}-\Delta^2)U_{\ell_i}\,\ud x\\&
=O(1)m_i^{2}\int_{B_r} |x^k \pa_k U_{\ell_i}+\frac{n-4}{2} U_{\ell_i}| U_{\ell_i},
\end{split}
\ee
where we have used $(P_{ g_i}-\Delta^2)U_{\ell_i}=O(1) U_{\ell_i}$ because of \eqref{eq:cor-GM}, and
\[
||x|^k\nabla_x^k U_{\ell_i}(x)|\le C(n,k) U_{\ell_i}(x)\quad \mbox{for } k\in \mathbb{N}.
\]
Hence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-3}
|\mathcal{\hat E}_i(r)|\le Cr^{8-n}\le Cr.
\ee
Therefore, \eqref{eq:sign restrict} follows from \eqref{eq:tos-1}, \eqref{eq:tos-2} and \eqref{eq:tos-3} when $n=5,6,7$.
If $n\ge 8$, by Lemma \ref{lem:GM2.8} we have
\begin{align*}
\mathcal{\hat E}_i(r) =& -\frac{2}{n}\gamma_i\int_{B_{\ell_i r}}(s\pa_s U_1+\frac{n-4}{2}U_1)(c_1^*s\pa_s U_1+c_2^*s^2 \pa_{ss}U_1)\,\ud z\\&
-\frac{32(n-1)\gamma_i}{3(n-2)n^2} \int_{B_{\ell_i r}}(s\pa_s U_1+\frac{n-4}{2}U_1)s^2(\pa_{ss}U_1-\frac{\pa_s U_1}{s})\,\ud z \\&
+(n-4)\gamma_i\int_{B_{\ell_i r}}(s\pa_s U_1+\frac{n-4}{2}U_1)U_1\,\ud z+O(\alpha} \newcommand{\lda}{\lambda_i'')\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l,
\end{align*}
where we used the symmetry so that those terms involving homogeneous polynomials of odd degrees are gone, $s=|z|$, $\gamma_i=\frac{m_i^{\frac{2(n-8)}{n-4}+(n-4)\tau_i}|W_{g_i}(0)|^2}{24(n-1)}\ge 0$,
\[
\alpha} \newcommand{\lda}{\lambda_i''=\int_{B_r} |x|^2 U_{\ell_i}(x)^2\,\ud x=O(1)\begin{cases}
r^{10-n}, & \quad \mbox{if }n=8,9,\\
\log rm_i,& \quad \mbox{if }n=10,\\
m_i^{\frac{2(n-10)}{n-4}},& \quad \mbox{if }n\ge 11.
\end{cases}
\]
and $c_1^*, c_2^*$ are given in \eqref{eq:c-star}.
By direct computations,
\[
r\pa_r U_1+\frac{n-4}{2}U_1=\frac{n-4}{2}\frac{1-r^2}{(1+r^2)^{\frac{n-2}{2}}},
\]
\begin{align*}
c_1^*r\pa_r U_1+c_2^*r^2 \pa_{rr}U_1&=(4-n)\frac{(c_1^*+c_2^*)r^2}{(1+r^2)^{\frac{n-2}{2}}}+(4-n)(2-n)\frac{c_2^*r^4}{(1+r^2)^{\frac{n}{2}}}\\
&=(4-n)\frac{(c_1^*+c_2^*)r^2+(c_1^*+(3-n)c_2^*)r^4}{(1+r^2)^{\frac{n}{2}}},
\end{align*}
\[
\pa_{rr} U_1-\frac{\pa_r U_1}{r}=(n-4)(n-2)(1+r^2)^\frac{-n}{2} r^2.
\]
Thus
\begin{align*}
\mathcal{\hat E}_i(r) = \frac{(n-4)^2}{n}\gamma_i |\mathbb{S}^{n-1}| J_i+O(\alpha} \newcommand{\lda}{\lambda_i'')\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l,
\end{align*}
where
\[
J_i:=\int^{\ell_i r}_0 \frac{(1-s^2)[\frac{n}{2}+(c_1^*+c_2^*+n)s^2+(c_1^*+(3-n)c_2^*-\frac{16(n-1)}{3n}+\frac{n}{2})s^4]s^{n-1}}{(1+s^2)^{n-1}}\,\ud s.
\]
If $n=8$, we have $-(c_1^*+(3-n)c_2^*+\frac{n}{2})= (2n-12)+\frac{14}{3}-4=\frac{14}{3}$. Since $\int^{\ell_i r}_0 \frac{s^{13}}{(1+s^2)^7}\,\ud s\to \infty$ as $\ell_i\to \infty$, Hence, $J_i\to \infty$ as $i\to \infty$.
For $n\ge 9$, we notice that for positive integers $2< m+1<2k$,
\[
\int_0^\infty \frac{t^m}{(1+t^2)^k}\,\ud t=\frac{m-1}{2k-m-1}\int_{0}^\infty \frac{t^{m-2}}{(1+t^2)^k}\,\ud t.
\]
If $\ell_i r =\infty$, we have
\begin{align*}
J_i=&\Big\{-\frac{2n}{n-4}-(c_1^*+c_2^*+n)\frac{8n}{(n-6)(n-4)}\\&-(c_1^*+(3-n)c_2^*-\frac{16(n-1)}{3n}+\frac{n}{2}) \frac{12n(n+2)}{(n-8)(n-6)(n-4)}\Big\} \int_0^\infty \frac{s^{n-1}}{(1+s^2)^{n-1}}\,\ud s.
\end{align*}
We compute the coefficients of the integral, \begin{align*}
&-\frac{2n}{n-4}-(c_1^*+c_2^*+n)\frac{8n}{(n-6)(n-4)}\\&\quad -(c_1^*+(3-n)c_2^*-\frac{16(n-1)}{3n}+\frac{n}{2}) \frac{12n(n+2)}{(n-8)(n-6)(n-4)}\\&
=\frac{2n}{n-4}\Big\{-1+(\frac{n(n-2)}{2}-8)\frac{4}{(n-6)}+(\frac{3n}{2}+\frac{16(n-1)}{3n}-12) \frac{6(n+2)}{(n-8)(n-6)}\Big\}\\&
= \frac{2n}{n-4}\Big\{\frac{2n^2 -5n-26}{n-6}+\frac{9(n+2)}{(n-6)}+\frac{32(n-1)(n+2)}{n(n-8)(n-6)}\Big\}\\&
\ge \frac{4n(n^2+2n-4)}{(n-4)(n-6)}>0.
\end{align*}
Therefore, for any $0<r<1$ and sufficiently large $i$ (the largeness of $i$ may depend on $r$), we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-5}
J_i\ge 1/C(n)>0.
\ee In conclusion, we obtain
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-6}
\mathcal{\hat E}_i(r) \ge \begin{cases} \frac{1}{C}|W_{g_i}(0)|^2 \log m_i+O(\alpha} \newcommand{\lda}{\lambda''_i)\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l, &\quad \mbox{if }n=8,\\
\frac{1}{C} |W_{g_i}(0)|^2 m_i^{\frac{2(n-8)}{n-4}}+O(\alpha} \newcommand{\lda}{\lambda''_i)\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l, &\quad \mbox{if }n\ge 9. \end{cases}
\ee
Combing \eqref{eq:tos-1}, \eqref{eq:tos-2} and \eqref{eq:tos-6} together, we see that, for $n\ge 8$,
\begin{align}
m_i^2 \mathcal{P}(r,u_i)&=m_i^2 \mathcal{N}(r, u_i)+(\mathcal{ E}_i(r)-\mathcal{\hat E}_i(r))+\mathcal{\hat E}_i(r)
\nonumber\\& \ge m_i^2 \mathcal{N}(r, u_i) +\frac{1}{2} \mathcal{\hat E}_i(r) -Cr
\label{eq:tos-7}
\end{align}
where $Cr$ can be set to zero when $n\ge 9$. If $n=8,9$, by sending $i\to \infty$ in \eqref{eq:tos-7} we have $ \mathcal{P}(r,\Gamma) \ge -Cr$. Thus \eqref{eq:sign restrict} follows and the conclusion (i) is proved.
If $n\ge 10$ and $|W_{g_i}(0)|^2 $ satisfies \eqref{eq:r2} for large $C^*>0$, by \eqref{eq:tos-6} we see that $(\mathcal{ E}_i(r)-\mathcal{\hat E}_i(r))+\mathcal{\hat E}_i(r) \ge 0$. Hence, the conclusion (iii) follows.
Since $|\mathcal{P}(r,\Gamma)|\le C$, it follows from \eqref{eq:tos-7} that for large $i$, $
\mathcal{\hat E}_i(r) \le C$. In view of \eqref{eq:tos-6} and the definition of $\alpha} \newcommand{\lda}{\lambda''$, the conclusion (ii) follows.
\end{proof}
\begin{prop} \label{prop:isolated to isolated simple} Given $p_i, G_i$, and $ h_i$ satisfying \eqref{p}, \eqref{G} and \eqref{H} respectively, $\kappa_i$ satisfying \eqref{K} with $K_i$ replaced by $\kappa_i$, let $0\le u_i\in C^4(B_3)$ solve both \eqref{eq:s1'} and \eqref{eq:Q-sub}, and assume \eqref{eq:IE-cond} holds. Suppose that $0$ is an isolated blow up point of $\{u_i\}$ with \eqref{eq:A_3} holds. Then $0$ is an isolated simple blow up point, if one of the three cases happens:
\begin{itemize}
\item $g_i$ is flat;
\item $n\le 9$;
\item $n\ge 10$ and \eqref{eq:r2} holds.
\end{itemize}
\end{prop}
\begin{proof} By Proposition \ref{prop:blow up a bubble}, $r^{4/(p_i-1)}\overline u_i(r)$ has precisely
one critical point in the interval $0<r<r_i$,
where $R_i\to \infty$ $r_i=R_iu_i(0)^{-\frac{p_i-1}{4}}$ as in Proposition \ref{prop:blow up a bubble}.
Suppose the contrary that $0$ is not an isolated simple blow up point and let $\mu_i$ be the second critical point of $r^{4/(p_i-1)}\overline u_i(r)$. Then we must have
\begin{equation}} \newcommand{\ee}{\end{equation}\label{5.2}
\mu_i\geq r_i,\quad \displaystyle\lim_{i\to \infty}\mu_i=0.
\ee
Set
\[
v_i(x)=\mu_i^{4/(p_i-1)}u_i(\mu_i x),\quad x\in B_{3/\mu_i}.
\]
By the assumptions of Proposition \ref{prop:blow up a bubble}, $v_i$ satisfies
\begin{align*}
v_i(x)&=\int_{B_{3/\mu_i}}\tilde G_{i}(x,y)\tilde \kappa_i(y)^{\tau_i} v_i(y)^{p_i}\,\ud y+\tilde h_i(x)
\\[2mm]
|x|^{4/(p_i-1)}v_i(x)&\leq A_3,\quad |x|<2/\mu_i \to \infty,
\\[2mm]
\lim_{i\to \infty}v_i(0)&=\infty,
\end{align*}
\[
r^{4/(p_i-1)}\overline v_i(r)\mbox{ has precisely one critical point in } 0<r<1,
\]
and
\[
\frac{\mathrm{d}}{\mathrm{d}r}\left\{ r^{4/(p_i-1)}\overline v_i(r)\right\}\Big|_{r=1}=0,
\]
where $\tilde G_i=G_{i,\mu_i}$, $\tilde \kappa_i(y)=\kappa_i(\mu_i y)$,
$\tilde h_i(x)=\mu_i^{4/(p_i-1)}h_i(\mu_i x)
$ and $\overline v_i(r)=|\pa B_r|^{-1}\int_{\pa B_r}v_i$.
Therefore, $0$ is an isolated simple blow up point of $\{v_i\}$.
\textbf{Claim.} We have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:converg2}
v_i(0) v_i (x) \to \frac{ac_{n}}{|x|^{n-4}} + ac_{n} \quad \mbox{in }C^3_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n \setminus \{0\}).
\ee
where $a>0
$ is given in \eqref{eq:number a}.
First of all, by Proposition \ref{prop:upbound2} we have $\tilde h_i(e)\le v_i(e)\le C v_i(0)^{-1}$ for any $e\in \mathbb{S}^{n-1}$, where $C>0$ is independent of $i$. It follows from the assumption \eqref{H} on $h_i$ that
\[
v_i(0)\tilde h_i(x)\le C \quad \mbox{for all } |x|\le 2/\mu_i
\] and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:iso to isos0a}
\|\nabla (v_i(0)\tilde h_i)\|_{L^\infty(B_{\frac{1}{9\mu_i}})} \le \mu_i \|v_i(0)\tilde h_i\|_{L^\infty(B_{\frac{1}{4\mu_i}})} \le C\mu_i.
\ee Hence, for some constant $c_0\ge 0$, we have, along a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:iso to isos0b}
\lim_{i\to \infty}\|v_i(0)\tilde h_i(x)-c_0\|_{L^\infty(B_t)} =0, \quad \forall~t>0.
\ee
Secondly, by Corollary \ref{cor:convergence} and Proposition \ref{prop:upbound2} we have, up to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:iso to isos1}
v_i(0)\int_{B_t}\tilde G_{i}(x,y)\tilde \kappa_i(y)^{\tau_i} v_i(y)^{p_i}\,\ud y \to \frac{a c_n}{|x|^{n-4}} \quad \mbox{in }C^3_{loc}(B_t \setminus \{0\}) \mbox{ for any } t>0,
\ee
where we used that $\tilde G_i(x,0)\to c_{n}|x|^{4-n}$.
Notice that for any $x\in B_{t/2}$
\[
Q''_i(x):= \int_{B_{3/\mu_i}\setminus B_t}
\tilde G_{i}(x,y)\tilde \kappa_i(y) ^{\tau_i}v_i(y)^{p_i}\,\ud y \le C(n, A_1)\max_{\pa B_t} v_i.
\]
Since $\max_{\pa B_t} v_i\le Ct^{4-n} v_i(0)^{-1}$, we have as in the proof of \eqref{eq:phigeos}, after passing to a subsequence,
\[
v_i(0) Q''_i(x)\to q(x) \quad \mbox{in }C^3_{loc}(B_t) \quad \mbox{as }i\to \infty
\] for some $q\in C^3(B_t)$. For any fixed large $R>t+1$, it follows from \eqref{eq:iso to isos1} that \[
v_i(0)\int_{t \le |y|\le R}
\tilde G_{i}(x,y)\tilde \kappa_i(y)^{\tau_i}v_i(y)^{p_i}\,\ud y \to 0
\]
as $i\to \infty$, since the constant $a$ is independent of $t$. By the assumption \eqref{G} on $G_i$, for any $x\in B_t$ and $|y|>R$, we have
\[
|\nabla_x \tilde G_{i}(x,y)| \le A_1 |x-y|^{3-n} \le \frac{A_1}{R-t} |x-y|^{4-n} \le \frac{A_1^2}{R-t} \tilde G_{i}(x,y).
\]
Therefore, we have $
|\nabla q(x)| \le \frac{A_1^2}{R-t} q(x).$
By sending $R\to \infty$, we have $|\nabla q(x)| \equiv 0$ for any $x\in B_t$. Thus,
\[
q(x)\equiv q(0)\quad \mbox{for all } x\in B_t.
\]
Since
\[
\frac{\mathrm{d}}{\mathrm{d}r}\left\{ r^{4/(p_i-1)}v_i(0)\overline v_i(r)\right\}\Big|_{r=1}=
v_i(0)\frac{\mathrm{d}}{\mathrm{d}r}\left\{ r^{4/(p_i-1)}\overline v_i(r)\right\} \Big|_{r=1}=0,
\]
we have, by choosing, for example, $t=2$ and sending $i$ to $\infty$, that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:iso to isos3}
q(0)+c_0=ac_{n}>0.
\ee
Therefore, \eqref{eq:converg2} is proved.
It follows from \eqref{eq:converg2} and Lemma \ref{lem:test-poho} that
\[
\liminf_{i\to \infty} v_i(0)^2\mathcal{P}(r,v_i)=-(n-4)^2(n-2)a^2c_n^2 |\mathbb{S}^{n-1}|<0\quad \mbox{for all }0<r<1.
\]
On the other hand, by \eqref{eq:Q-sub} $v_i$ satisfies
\[
P_{\tilde g_i} v_i=c(n)\tilde \kappa_i^{\tau_i}v_i^{p_i} \quad \mbox{in }B_{3/\mu_i},
\]where $\tilde g_i(z)=g_i(\mu_i z)$. It is easy to see that \eqref{eq:IE-cond} is still correct with $G_i$ replaced by $\tilde G_i$. If $n\le 9$ or $g_i$ is flat, it follows from Proposition \ref{prop:one-side} that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:another-side}
\liminf_{r\to 0}\liminf_{i\to \infty} v_i(0)^2\mathcal{P}(r,v_i)\ge 0.
\ee
If $n \ge 10$, by \eqref{eq:IE-cond}, we have
\begin{align*}
U_{\lda}(x)&=\int_{B_{3/\mu_i}} G_{i,\mu_i}(x,y)\{U_\lda(y)^{\frac{n+4}{n-4}}+\mu_i^4 c_{\lda/\mu_i,i}'(\mu_i y) U_{\lda}(y)\}\,\ud y+\mu_i^{\frac{n-4}{2}}c_{\lda/\mu_i,i}''(\mu_ix) \\& = \int_{B_3} \tilde G_i(x,y) \{U_\lda(y)^{\frac{n+4}{n-4}}+ \tilde c_{\lda,i}'(y) U_{\lda}(y)\}\,\ud y+\tilde c_{\lda,i}''(x) \quad \forall ~\lda\ge 1 ,~x\in B_3,
\end{align*}
where $c_{\lda,i}'(y):=\mu_i^4 c_{\lda/\mu_i,i}'(\mu_i y) $ and
\[
\tilde c_{\lda,i}''(x) = \int_{B_{3/\mu_i}\setminus B_3} G_{i,\mu_i}(x,y)\{U_\lda(y)^{\frac{n+4}{n-4}}+\mu_i^4 c_{\lda/\mu_i,i}'(\mu_i y) U_{\lda}(y)\}\,\ud y+\mu_i^{\frac{n-4}{2}}c_{\lda/\mu_i,i}''(\mu_ix).
\]
By the assumptions for $c'_{\lda,i}$ and $c''_{\lda,i}$, we have
\[
\tilde \Theta_i:=\sum_{i=0}^5 \|\lda^{-k} \nabla^k \tilde c'_{\lda, i}\|_{L^\infty(B_3)} \le \mu_i^4\Theta_i,
\]
and $\|\tilde c_{\lda,i}''\|_{C^5(B_2)}\le CA_2 \lda^{\frac{4-n}{2}}$, where $C>0$ depends only on $n,A_1,A_2$. Clearly, we have $|W_{\tilde g_i}(0)|^2=\mu_i^4 |W_{ g_i}(0)|^2$. Hence \eqref{eq:r2} is satisfied. By Proposition \ref{prop:one-side}, we also have \eqref{eq:another-side}. We obtain a contradiction.
Therefore, $0$ must be an isolated simple blow up point of $u_i$
and the proof is completed.
\end{proof}
\begin{lem}\label{lem:isolated to isolated simple} Let $0\le u_i\in C^4(B_3)$ solve both \eqref{eq:s1'} and \eqref{eq:Q-sub} with $n\ge 10$, and assume \eqref{eq:IE-cond} holds. For $\mu_i\to 0$, let
\[
v_i(x)=\mu_i^{\frac{4}{p_i-1}} u_i(\mu_i x).
\]
Suppose that $0$ is an isolated blow up point of $\{v_i\}$ and \eqref{eq:r2} holds. Then $0$ is also an isolated simple blow up point.
\end{lem}
\begin{proof} From the end of proof of Proposition \ref{prop:isolated to isolated simple}, we see that the condition \eqref{eq:r2} is preserved under the scaling $v_i(x)=\mu_i^{\frac{4}{p_i-1}} u_i(\mu_i x)$. Hence, the lemma follows from Proposition \ref{prop:isolated to isolated simple}.
\end{proof}
\section{Global analysis, and proof of Theorem \ref{thm:energy}}
\label{section:thm1.1}
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. Suppose that $\mathrm{Ker} P_g=\{0\}$ and the Green's function $G_g$ of $P_g$ is positive. Consider the equation
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sub DE}
P_gu= c(n)u^p, \quad u \ge 0 \quad \mbox{on } M,
\ee
where $1<p\le \frac{n+4}{n-4}$.
\begin{prop} \label{prop:reduction} Assume the above. For any given $R>0$ and $0\le \varepsilon} \newcommand{\ud}{\mathrm{d}<\frac{1}{n-4}$, there exist positive constants $C_0=C_0(M,g,R,\varepsilon} \newcommand{\ud}{\mathrm{d})$, $C_1=C_1(M, g,R,\varepsilon} \newcommand{\ud}{\mathrm{d})$ such that, for any smooth positive solution of \eqref{eq:sub DE} with
\[
\max_{M} u(X)\ge C_0,
\]
then $\frac{n+4}{n-4}-p<\varepsilon} \newcommand{\ud}{\mathrm{d}$ and there exists a set of finite distinct points
\[
\mathscr{S}(u):=\{Z_1,\dots,Z_N\}\subset M
\] such that the following statements are true.
(i) Each $Z_i$ is a local maximum point of $u$ and
\[
\overline{\B_{\bar r_i}(Z_i)} \cap \overline{\B_{\bar r_j}(Z_j)}=\emptyset \quad \mbox{for }i\neq j,
\]
where $\bar r_i=R u(Z_i)^{(1-p)/4}$, and $\B_{r_i}(Z_i)$ denotes the geodesic ball in $B_2$ centered at $Z_i$ with radius $\bar r_i$
(ii) For each $Z_i$,
\[
\left\| \frac{1}{u(Z_i)}u\left(\exp_{Z_i}\left(\frac{y}{u(Z_i)^{(p-1)/4}}\right)\right)-\left(\frac{1}{1+ |y|^2}\right)^{\frac{n-4}{2}}\right\|_{C^4(B_{2R})} <\varepsilon} \newcommand{\ud}{\mathrm{d}.
\]
(iii) $u(X)\le C_1 dist_g(x,\{Z_1,\dots,Z_N\})^{-4/(p-1)}$ for all $X\in M$.
\end{prop}
\begin{proof}
The proof is standard by now.
\end{proof}
\begin{prop}\label{prop:ruling out accumulation} If either $n\le 9$ or $(M,g)$ is locally conformally flat, then, for $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$, $R>1$ and any solution \eqref{eq:sub DE} with $\max_M u>C_0$, we have
\[
|Z_1-Z_2|\ge \delta^*>0 \quad \mbox{for any }Z_1,Z_2\in \mathscr{S}(u), ~Z_1\neq Z_2,
\]
where $\delta^*$ depends only on $M,g$.
\end{prop}
\begin{proof} Suppose the contrary, then there exist a sequence $0\le \frac{n+4}{n-4}-p_i<\varepsilon} \newcommand{\ud}{\mathrm{d}$ and $u_i$ satisfying \eqref{eq:sub DE} with $p=p_i$,
\[
\max_{M}u_i(X)> C_0,
\] and
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:e1}
dist_g(Z_{1i},Z_{2i})=\min_{1\le k,l\le N_i, ~k\neq l} dist_g(Z_{ki},Z_{li})\to 0
\ee
as $i\to \infty$, where $\mathscr{S}(u_i)=(Z_{1i},\dots, Z_{N_ii} )$ be the local maximum points of $u_i$ as selected by Proposition \ref{prop:reduction}. Without loss of generality, we may assume
\[
u_i(Z_{1i})\ge u_i(Z_{2i}).
\]
Since $\B_{Ru_i(Z_{1i})^{-(p_i-1)/4}}(Z_{1i})$ and $\B_{Ru_i(Z_{2i})^{-(p_i-1)/4}}(Z_{2i})$ have to be disjoint,
we have, because of \eqref{eq:e1}, that
$ u_i(Z_{1i})\to \infty$ and $ u_i(Z_{2i})\to \infty$.
Let $\{x_1,\dots, x_n\}$ be the conformal normal coordinates centered at $Z_{1i}$. We write \eqref{eq:sub DE} as
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sub DE-a}
P_{ g_i}\tilde u_i=c(n) \kappa_i^{\tau_i}\tilde u_i^{p_i} \quad \mbox{on }M,
\ee
where $g_i=\kappa_i^{\frac{-4}{n-4}} g$, $\tilde u_i=\kappa_i u_i$, $\kappa_i>0$, $\kappa_i(Z_{1i})=1$, $\nabla_g \kappa_i(Z_{1i})=0$, and $\tau_i=\frac{n+4}{n-4}-p_i$.
Since $dist_g(Z_{1i},Z_{2i})\to 0$, for large $i$ we let $z_{2i} \in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$ such that $\exp_{Z_{1i}} z_{2i}=Z_{2i}$, and let
\[
\vartheta_i:=|z_{2i}|\to 0.
\]
We will sit in the conformal normal coordinates chart $B_t$ at $Z_{1i}$, where $t>0$ is independent of $i$, and write $f(\exp_{Z_{1,i}}x)$ simply as $f(x)$.
Set
\[
\varphi_i(x)=\vartheta_i^{4/(p_i-1)}\tilde u_i(\vartheta_i x) \quad \mbox{for } |x|\le t/\vartheta_i.
\]
By the equation \eqref{eq:sub DE-a}, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sub DE-b}
P_{\tilde g_i} \varphi_i(x)=c(n)\tilde \kappa_i(x)^{\tau_i} \varphi_i(x)^{p_i} \quad \mbox{in }B_{t/\vartheta_i},
\ee
where $\tilde \kappa_i(x)=\kappa_i(\vartheta_i x)$, $\tilde g_i(x)=g_i(\vartheta_i x)$. Using the Green representation for \eqref{eq:sub DE-a},
\[
\tilde u_i(x)=c(n)\int_{B_t}G_i(x,y)\kappa_i(y)^{\tau_i} \tilde u_i(y)^{p_i}\,\ud y+h_i(x),
\]
where $G_i(x,y)=G_{g_i}(\exp_{Z_{1,i}}x,\exp_{Z_{1,i}}y)$ and
\[
h_i(x)=\int_{M\setminus \exp_{Z_{1,i}} B_t} G_{g_i}(\exp_{Z_{1,i}} x, Y)\kappa_i(Y)^{\tau_i}u_i(Y)\,\ud vol_{g_i}(Y).
\]
Hence, $\varphi_i$ also satisfies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:e5}
\varphi_i(x)=\int_{B_{t/\vartheta_i}}G_{i,\vartheta_i}(x,y)K_i(\vartheta_iy)\varphi_i(y)^{p_i} \,\ud y+\tilde h_i(x)
\quad \mbox{for all } x\in B_{t/\vartheta_i},
\ee
where $G_{i,\vartheta_i}(x,y) =\vartheta_i^{n-4}G(\vartheta_ix,\vartheta_iy)$ and $\tilde h_i=
\vartheta_i^{4/(p_i-1)}h_i(\vartheta_i y)$.
By proposition \ref{prop:reduction},
we have
\begin{align*}
\tilde u_i(x)&\leq C_1^*|x|^{-4/(p_i-1)}\quad \mbox{for all }|x|\leq 3\vartheta_i/4,\\
\tilde u_i(x) &\le C_1^* |x-z_{2i}|^{-4/(p_i-1)} \quad \mbox{for all }|x-z_{2i}|\leq 3\vartheta_i/4,
\end{align*}
where $C_1^*$ depending only on $C_1$, $M$ and $g$.
Hence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{9-4}
\begin{split}
\varphi_i(x)&\leq C_1^*|x|^{-4/(p_i-1)}\quad \mbox{for all }|x|\leq 3/4,\\
\varphi_i(x) &\le C_1^* |x-\vartheta_i^{-1}z_{2i}|^{-4/(p_i-1)} \quad \mbox{for all }|x-\vartheta_i^{-1}z_{2i}|\leq 3/4.
\end{split}
\ee
Set $\xi_i=\vartheta_i^{-1}z_{2i}$. We claim that, after passing to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:9-5}
\varphi_i(0), \varphi_i(\xi_i)\to \infty \quad \mbox{as }i\to \infty.
\ee
It is clear that $\varphi_i(0)$ and $\varphi_i(\xi_i)$ are bounded from below by
some positive constant independent of $i$.
If there exists a subsequence (still denoted as $\varphi_i$) such that $\displaystyle\lim_{i\to \infty}\varphi_i(0)=\infty $ but $\varphi_i(\xi_i)$ stays bounded, we have that $0$ is an isolated blow up point for $\varphi_i$ in $B_{3/4}$ when $i$ is large; see Remark \ref{rem:blow}. Using equation \eqref{eq:e5} and \eqref{9-4}, by the same proof of \eqref{eq:scalbound2} we have $\sup_{B_{1/2}(\xi_i)} \varphi_i<\infty$. It follows from Proposition \ref{prop:upbound2} and Proposition \ref{prop:har} that $\displaystyle\lim_{i\to \infty}\varphi_i(\xi_i) = 0$, but this is impossible since $\vartheta_i>Ru_i(Z_{2i})^{-(p_i-1)/4} $ and thus $
\varphi_i(\xi_i)\ge \frac{1}{C}R$ for some $C>0$ depending only on $M $, $g$, $R$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$. On the other hand, if there exists a subsequence (still denoted as $\varphi_i$) such that $\varphi_i(0)$ and $\varphi_i(\xi_i)$ remain bounded, we know from a similar argument as above that $\varphi_i$ is locally bounded. The same proof of Proposition \ref{prop:blow up a bubble} yields that after passing to a subsequence $\varphi_i\to \varphi$ in $C^3_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ for some $\varphi$ satisfying
\[
\varphi(x)=c_n \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{\varphi(y)^{\frac{n+4}{n-4}}}{|x-y|^{n-4}}\,\ud y \quad \mbox{for all }x\in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n,
\]
$\nabla \varphi(0)=0$, $\nabla \varphi(\bar z)=\lim_{i\to \infty} \nabla \varphi_i(\xi_i)= \lim_{i\to \infty} \frac{\vartheta_i \varphi_{i}(\xi_i) \nabla \kappa_i(z_{2i})}{\kappa_i(z_{2i})} =0$, where $|\bar z|=1$ is the limit of $\xi_i$ up to passing a subsequence. This contradicts to the Liouville theorem in \cite{CLO} or Li \cite{Li04}. Hence, \eqref{eq:9-5} is proved.
Since $\nabla \varphi_i(0)=0$, it follows from the first inequality of \eqref{9-4} and \eqref{eq:9-5} that $0$ is an isolated blow up point of $\{\varphi_i\}$. Since $n\le 9$ or $(M,g)$ is locally conformally flat, by Proposition \ref{prop:isolated to isolated simple} we conclude that $0$ is an isolated simple blow up point of $\{\varphi_i\}$. It follows from Corollary \ref{cor:convergence} that for all $x\in B_{1/2}$
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:e6}
\varphi_i(0)\int_{B_{1/2}}G_{i,\vartheta_i}(x,y)\varphi_i(y)^{p_i} K_i(\vartheta_i y)\,\ud y \to a c_{n}|x|^{4-n}
\ee
and
\[
\varphi_i(0) (Q_i''(x)+\tilde h_i(x))\to h(x)\ge 0 \quad \mbox{in } C^3_{loc}(B_{1/2}),
\] where $a>0$ is given in \eqref{eq:number a}, $h(x)\in C^5(B_{1/2})$ and
\[
Q''_i(x)=\int_{B_{t/\vartheta}\setminus B_{1/2}}G_{i,\vartheta_i}(x,y)\varphi_i(y)^{p_i}K_i(\vartheta_i y)\,\ud y \quad x\in B_{1/2}.
\]
Note that \[
\varphi_i(0)Q_i''(x) \ge \frac{1}{C} \varphi_i(0) \int_{B_{1/2}(\xi_i)}\varphi_i(y)^{p_i} \,\ud y.
\]
It follows from \eqref{9-4}, \eqref{eq:9-5} and the proof of
\eqref{eq:scalbound2} that there exists a constant $C>0$, depending only on $M,g,R $ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$ such that $\varphi_i(x)\le C \varphi_i(\xi_i)$ for all $|x-\xi_i|\le \frac{1}{2}$. It follows from the proof of Proposition \ref{prop:blow up a bubble} that there exist a constant $\lda$ and an point $x_0\in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$ with $1\le \lda\le C$ and $|x_0|\le C$ such that for any fixed $\bar R>0$ we have
\[
\lim_{i\to \infty} \left \|\frac{1}{\varphi_i(\xi_i)} \varphi_i(\xi_i+\varphi_i(\xi_i)^{-(p_i-1)/4} x)-U_{\lda}(x-x_0) \right\|_{C^4(\bar B_{R})}=0
\]
By changing of variables $y=\xi_i+\varphi_i(\xi_i)^{-(p_i-1)/4} x$, we have
\begin{align*}
&\varphi_i(0) \int_{B_{1/2}(\xi_i)}\varphi_i(y)^{p_i} \,\ud y\\&=\varphi_i(0) \varphi_i(\xi_i)^{p_i-\frac{(p_i-1)n}{4}} \int_{B_{\varphi_i(\xi_i)/2}(0)}\left( \frac{1}{\varphi_i(\xi_i)} \varphi_i(\xi_i+\varphi_i(\xi_i)^{-(p_i-1)/4} x)\right)^{p_i} \,\ud x\\ &
\ge \frac{1}{C}\varphi_i(0) \varphi_i(\xi_i)^{p_i-\frac{(p_i-1)n}{4}} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} U_{\lda}(x-x_0)^{p_i}\,\ud x\\&
\ge \frac{1}{C}\varphi_i(0) \varphi_i(\xi_i)^{p_i-\frac{(p_i-1)n}{4}} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} (1+|x|^2)^{\frac{n+4}{2}}\,\ud x,
\end{align*}
where we used $1\le \lda\le C$ and $|x_0|\le C$. Since $u_i(Z_{1i})\ge u_i(Z_{2i})$, we have $ \varphi_i(0) \ge \frac{1}{C} \varphi_i(\xi_i)$ for some $C$ depending only on $M,g$. By Lemma \ref{lem:error}, we have $\varphi_i(\xi_i)^{1+p_i-\frac{(p_i-1)n}{4}}=1+o(1)$. Therefore, we obtain
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:e7}
\lim_{i\to \infty} \varphi_i(0)Q_i''(x) \ge \frac{1}{C} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} (1+|\xi|^2)^{\frac{n+4}{2}}\,\ud \xi=:a_0>0.
\ee
In conclusion,
\[
\varphi_i(0)\varphi_i(x)\to ac_n|x|^{4-n}+h(x) \quad \mbox{in }C^3_{loc} (B_{1/2} \setminus \{0\})
\]
for some nonnegative bounded function in $C^3(B_{1/2})$ with $h(0)\ge a_0$.
It follows from Lemma \ref{lem:test-poho} that
\[
\liminf_{r\to 0}\liminf_{i\to \infty} \varphi_i^2\mathcal{P}(r,\varphi_i)<-(n-4)^2(n-2)aa_0c_n|\mathbb{S}^{n-1}|.
\]
On the other hand, notice that $\varphi_i$ satisfies \eqref{eq:sub DE-a}. We also have Corollary \ref{cor:GM2.8}. It follows from Proposition \ref{prop:one-side} that
\[
\liminf_{r\to 0}\liminf_{i\to \infty} \varphi_i(0)^2\mathcal{P}(r,\varphi_i)\ge 0.
\]
We arrive at a contradiction. Therefore, \eqref{eq:e1} is not valid and the proposition follows.
\end{proof}
Theorem \ref{thm:energy} is a part of the following theorem.
\begin{thm}\label{thm:final-a} Let $u_i\in C^4(M)$ be a sequences of positive solutions of $P_g u_i=c(n)u_i^{p_i}$ on $M$, where $0\le (n+4)/(n-4)-p_i \to 0$ as $i\to \infty$. Assume \eqref{condition:main2}. If either $n\le 9$ or $(M,g)$ is locally conformally flat, then
\[
\|u_i\|_{H^2(M)} \le C,
\]
where $C>0$ depending only on $M,g$. Furthermore, after passing to a subsequence, $\{u_i\}$ is uniformly bounded or has only isolated simple blow up points and the distance between any two blow up points is bounded below by some positive constant depending only on $M,g$.
\end{thm}
\begin{rem}\label{rem:isolated on manifolds} On $(M,g)$, we say a point $\bar X\in M$ is an isolated blow up point for $\{u_i\}$ if there exists a sequence $X_i\in M$, where each $X_i$ is a local maximum point for $u_i$ and $X_i\to \bar X$, such that $u_i(X_i)\to \infty$ as $i\to \infty$ and $u_i(X) \le C dist_g(X,X_i)^{-\frac{4}{p_i-1}} $ in $\B_{\delta}(X_i)$ for some constants $C,\delta>0$ independent of $i$. Under the assumptions that $u_i$ is a positive solution of $P_{g}u_i=c(n)u_i^{p_i}$ with $0\le \frac{n+4}{n-4}-p_i\to 0$, $\mathrm{Ker} P_g=\{0\}$ and that the Green's function $G_g$ of $P_g$ is positive, it is easy to see that if $X_i\to \bar X\in M$ is an isolated blow up point of $\{u_i\}$, then in the conformal normal coordinates centered at $X_i$, $0$ is an isolated blow up point of $\{\tilde u_i(\exp_{X_i} x)\}$, where the exponential map is with respect to conformal metric $g_i=\kappa_i^{\frac{-4}{n-4}}g$, $\kappa_i>0$ is under control on $M$, and $\tilde u_i=\kappa_i u_i$; see Remark \ref{rem:blow}.
Since in Theorem \ref{thm:final-a} and the sequel those assumptions will always be assumed, the notation of isolated simple blow up points on manifolds is understood in conformal normal coordinates.
\end{rem}
\begin{proof}[Proof of Theorem \ref{thm:final-a}] The last statement follows immediately from Proposition \ref{prop:reduction}, Proposition \ref{prop:ruling out accumulation} and Proposition \ref{prop:isolated to isolated simple}. Consequently, it follows from Corollary \ref{cor:energy} and Proposition \ref{prop:upbound2} and Proposition \ref{prop:har} that $\int_{M} u_i^{\frac{2n}{n-4}}\,\ud vol_g\le C$. By the Green's representation and standard estimates for Riesz potential, we have the $H^2$ estimates.
\end{proof}
Now we consider that $n\ge 10$.
\begin{prop}\label{prop:r} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 10$.
Assume (\ref{condition:main2}).
Let $u_i$ be a sequence of positive solutions of $P_g u=c(n)u^{p_i}$,
where $p_i\le \frac{n+4}{n-4}$,
$p_i\to \frac{n+4}{n-4}$ as $i\to \infty$. Suppose that there is a sequence $X_i\to \bar X\in M$ such that $u_i(X_i)\to \infty$. For any small $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ and $R>1$, let $\mathscr{S}(u_i)$ denote the set selected as in Proposition \ref{prop:reduction} for $u_i$. If $|W_g(\bar X)|^2\ge \varepsilon} \newcommand{\ud}{\mathrm{d}_0>0$ on $M$ for some constant $\varepsilon} \newcommand{\ud}{\mathrm{d}_0$,
then there exists $\delta^*>0$ depending only on $M,g$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}_0$ such that $\B_{\delta^*}(\bar X)\cap \mathscr{S}(u_i)$ contains precisely one point.
\end{prop}
\begin{proof}
Let $\bar\delta>0$ such that $|W_g(X)|^2\ge \varepsilon} \newcommand{\ud}{\mathrm{d}_0/2$
for $X\in \B_{\bar \delta}(\bar X)$.
Assume the contrary of the proposition, then for a
subsequence
of $\{u_i\}$ (still denoted as $\{u_i\}$)
there exist distinct points $X_{1i},\hat X_{1i}\in \mathscr{S}(u_i)$ such that $X_{1i}, \hat X_{1i}\to \bar X$.
Define $f_i:\mathscr{S}(u_i)\to (0,\infty)$ by
\[
f_i(X):=\min_{X'\in \mathscr{S}(u_i)\setminus \{X\}} dist_g(X',X).
\]Let $R_i\to \infty$ satisfying $R_i f_i(X_{1i})\to 0$.
\textbf{Claim.} There exists a subsequence of $i\to \infty$ such that one
can find $X_{i}'\in \mathscr{S}(u_i)\cap \B_{\bar \delta/9}(\bar X)$ satisfying
\[
f_i(X_i') \le (2R_i+1)f_i(X_{1i})
\]
and
\[
\min_{X\in \mathscr{S}(u_i)\cap \B_{R_i f_i(X_i')} (X_i')} f_i(X) \ge \frac12 f_i(X_i').
\]
Indeed, suppose the contrary, then there exists $I\in \mathbb{N}$ such that for any $i\ge I$, $X_i'$ in the claim can not been selected. Since $f_i(X_{1i})\le (2R_i+1) f_i(X_{1i})$, by the contradiction hypothesis, there must exist $X_{2i}\in \mathscr{S}(u_i)\cap \B_{R_i f_i(X_{1i})} (X_{1i})$ such that $f_{i}(X_{2i})<\frac12 f_{i}(X_{1i})$. We can define $X_{li}\in \mathscr{S}(u_i)$, $l=3\dots$, satisfying
$f_i(X_{li})<\frac{1}{2} f_i(X_{(l-1)i})$
and $0<dist_g(X_{li},X_{(l-1)i})< R_i f_i(X_{(l-1)i})$
inductively as follows. Once $X_{li}$, $l\ge 2
$, is defined, we have, for $2\le m\le l$, that
\[
dist_g(X_{mi},X_{(m-1)i})<R_i f_i(X_{(m-1)i})<R_i 2^{-1} f_i(X_{(m-2)i})<\cdots <R_i 2^{2-m} f_i(X_{1i}),
\]
which implies
$$
dist_g(X_{li},X_{1i})\le \sum_{m=2}^l dist_g(X_{mi},X_{(m-1)i})
< R_i f_i(X_{1i}) \sum_{m=2}^l 2^{2-m}
< 2R_i f_i(X_{1i}),
$$
and
$$
f_i(X_{li})\le dist_g(X_{li}, X_{1i})+f_i(X_{1i})\le (2R_i+1) f_i(X_{1i}).
$$
so $X_i':=X_{li}$ satisfies
$X_{i}'\in \mathscr{S}(u_i)\cap \B_{\bar \delta/9}(\bar X)$
and the first inequality
of the claim.
By the contradiction hypothesis, there must exist $X_{(l+1)i}\in \mathscr{S}(u_i)\cap \B_{R_i f_i(X_{li})} (X_{li})$ such that $f_{i}(X_{(l+1)i})<\frac12 f_{i}(X_{li})$.
But $\mathscr{S}(u_i)$ is a finite set and we can not work for all $l\ge 2$. Therefore, the claim follows.
By the claim, we can follow the proof of Proposition \ref{prop:ruling out accumulation} with $Z_{1i}$ replaced by $X_i'$. We then derive a contradiction to Proposition \ref{prop:one-side}. Therefore, we complete the proof.
\end{proof}
\section{Proof of Theorems \ref{thm:main theorem}, Theorem \ref{thm:main-b}, Theorem \ref{thm:main-c}}
\label{section:thm1.2}
\begin{proof}[Proof of Theorem \ref{thm:main-b}] For $n=8,9$, since $u_i(X_i)\to \infty$, it follows from Proposition \ref{prop:ruling out accumulation} and Theorem \ref{thm:final-a} that
$\bar X$ is an
isolated simple blow up points of $\{u_i\}$. Then Theorem \ref{thm:main-b} follows from item (ii) of Proposition \ref{prop:one-side}. When $n\ge 10$, we may argue by contradiction. Suppose that $|W_{g}(\bar X)|^2>0$.
By Proposition \ref{prop:r},
Proposition \ref{prop:isolated to isolated simple},
and in view of the proof of
(\ref{eq:scalbound2}) under (\ref{eq:scalbound}),
that there exists a sequence of $X_i'\to \bar X$ which is an isolated simple blow up point of $\{u_i\}$; see Remark \ref{rem:isolated on manifolds}. By item (ii) of Proposition \ref{prop:one-side}, we have $|W_g(X_i')|^2 \to 0$. It gives
$|W_{g}(\bar X)|^2=0$. We obtain a contradiction. Hence, we complete the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main-c}] Suppose the contrary,
then, after passing to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{c-hy}
|W_g(X_i)|^2> \frac 1{ |o(1)| } \begin{cases} u_i(X_i)^{-\frac{4}{n-4}}\log u_i(X_i), &\quad \mbox{if }n=10,\\
u_i(X_i)^{-\frac{4}{n-4}}, &\quad \mbox{if }n\ge 11.
\end{cases}
\ee
For any $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ and $R>1$, let $\mathscr{S}(u_i)=\{Z_{1i},\dots, Z_{N_i i}\}$ be the set selected as in Proposition \ref{prop:reduction} for $u_i$, where $N_i\in \mathbb{N}^+$. Let, without loss of generality,
\[
dist_g(X_i, Z_{2i}) =\inf_{Z_{ji}\in \mathscr{S}(u_i), Z_{ji}\neq X_i}dist_g(X_i, Z_{ji}).
\] If there exists a constant $\delta^*>0$
independent of $i$
such that $dist_g(X_i, Z_{2i})\ge \delta^*$, then $X_i\in \mathscr{S}(u_i)$
for large $i$. It follows from item (iii) of Proposition \ref{prop:reduction} and Proposition \ref{prop:isolated to isolated simple} that $X_i\to \bar X$ has to be an isolated simple blow up point of $\{u_i\}$,
using the fact that
\eqref{c-hy} guarantees
\eqref{eq:r2}. By item (ii) of Proposition \ref{prop:one-side}, we obtain an opposite side inequality of
\eqref{c-hy}. Contradiction.
If $dist_g(X_i, Z_{2i}) \to 0$ as $i\to \infty$. Let $\{x_1,\dots, x_n\}$ be the conformal normal coordinates centered at $X_i$. Define $\varphi_i$ as that in the proof of Proposition \ref{prop:ruling out accumulation} with $Z_{1i}$ replaced by $X_i$. Since $\sup_{\Omega} \newcommand{\pa}{\partial}u_i\le \bar bu_i(X_i)$, we must have $\varphi_i(0) \to \infty$ by the Liouville theorem in \cite{CLO} or \cite{Li04}; see the proof of \eqref{eq:9-5}.
Because of \eqref{eq:technical condition 12} and
\eqref{eq:technical condition},
$0$ has to be an isolated blow up point of $\{\varphi_i\}$; see Remark \ref{rem:blow}. It follows from the contradiction hypothesis
\eqref{c-hy}, which guarantees \eqref{eq:r2},
as in the proof of
Lemma \ref{lem:isolated to isolated simple} that $0$ has to be an isolated simple blow up point of $\{\varphi_i\}$ for $i$ large. Then we we arrive at a contradiction by item (ii) of Proposition \ref{prop:one-side} again.
Therefore, we complete the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main theorem}]
If $n\ge 8$ and $|W_g|^2>0$ on $M$, Theorem \ref{thm:main theorem} is a direct corollary of Theorem \ref{thm:main-b}. Hence, we only need to consider $n\le 9$ or $(M,g)$ is locally conformally flat.
By Proposition \ref{prop:reduction}, it suffices to consider that $p$ is close to $\frac{n+4}{n-4}$. Suppose the contrary that there exists a sequences of positive solutions $u_i\in C^4(M)$ of $P_g u_i=c(n)u_i^{p_i}$ on $M$, where $p_i\to (n+4)/(n-4)$ as $i\to \infty$, such that $\max_{M}u_i\to \infty$. By Theorem \ref{thm:final-a}, let $X_i\to \bar X\in M$ be an isolated simple blow up point of $\{u_i\}$; see Remark \ref{rem:isolated on manifolds}. It follows from Proposition \ref{prop:one-side} that, in the $g_{\bar X}$-normal coordinates centered at $\bar X$,
\[
\liminf_{r\to 0} \mathcal{P}(r, c(n)G)\ge 0,
\]
where $g_{\bar X}$ a conformal metric of $g$ with $\det g_{\bar X}=1$ in an open ball $B_{\delta}$ of the $g_{\bar X}$-normal coordinates, $G(x)=G_{g_{\bar X}}(\bar X,\exp_{\bar X} x)$ and $G_{g_{\bar X}}$ is the Green's function of $P_{g_{\bar X}}$.
On the other hand, if $n=5,6,7$ or $(M,g)$ is locally conformally flat, by Theorem \ref{thm:positive mass} and Lemma \ref{lem:test-poho} we have
\[
\mathcal{P}(r, c(n)G) <-A \quad \mbox{for small }r,
\]
where $A>0$ depends only on $M,g$. We obtain a contradiction.
If $n=8,9$, by Theorem \ref{thm:main-b} we have $W_g(\bar X)=0$. In view of Remark \ref{rem:positive mass}, we have
\[
\lim_{r\to 0}\mathcal{P}(r,G)=\begin{cases} -2 \Xint-_{\mathbb{S}^{n-1}} \psi(\theta), &\quad n=8, \\
-\frac{5}{2}A, &\quad n=9,
\end{cases}
\]
where $\psi(\theta)$ and $A$ are as in Remark \ref{rem:positive mass}.
If the positive mass type theorem holds for Paneitz operator in dimension $n=8,9$, we obtain $\lim_{r\to 0}\mathcal{P}(r,G)<0$. Again, we derived a contradiction.
Therefore, $u_i$ must be uniformly bounded and the proof is completed.
\end{proof}
\section{Proof of Theorem \ref{thm:compact1}}
This section will not use previous analysis and thus is independent. The proof of Theorem \ref{thm:compact1} is divided into two steps.
\emph{Step 1}. $L^p$ estimate. Let $u\ge 0$ be a solution of \eqref{eq:-Q}. Integrating both sides of \eqref{eq:-Q} and using H\"older inequality, we have
\[
\int_{M}u^p \ud vol_g =\left| -\int_{M} uP_g(1) \ud vol_g\right| \le \frac{n-4}{2} \|u\|_{L^p(M)} \|Q_g\|_{L^{p'}(M)},
\]
where $\frac{1}{p'}+\frac1p=1$. It follows that $\|u\|_{L^p(M)} ^{p-1} \le \frac{n-4}{2} \|Q_g\|_{L^{p'}(M)}$.
\emph{Step 2.} If $\mathrm{Ker} P_{g}=\{0\}$, there exist a unique Green function of $P_g$. If the kernel of $P_{g}$ is non-trivial, since the spectrum of Paneitz operator is discrete, there exists a small constant $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ such that the kernel of $P_g-\varepsilon} \newcommand{\ud}{\mathrm{d}$ is trivial. Let $G_g$ be the Green function of the operator $P_g-\varepsilon} \newcommand{\ud}{\mathrm{d}$, where $\varepsilon} \newcommand{\ud}{\mathrm{d}\ge 0$. Then there exists a constant $\delta>0$, depending only $M,g$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$, such that, for every $X\in M$, we have $G(X,Y)>0$ for $Y\in \mathcal{B}_{\delta}(X)$ and $|G_g(X,Y)|\le C(\delta,\varepsilon} \newcommand{\ud}{\mathrm{d})$ for $Y\in M\setminus \mathcal{B}_{\delta}(X)$. Rewrite the equation of $u$ as
\[
P_g u-\varepsilon} \newcommand{\ud}{\mathrm{d} u=-(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u).
\]
It follows from the Green representation theorem that
\begin{align*}
u(X)& =-\int_{M}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y)\\&
=-\int_{\mathcal{B}_{\delta}(X)}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y)\\& \quad -\int_{M\setminus \mathcal{B}_{\delta}(X)}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y)\\&
\le -\int_{M\setminus \mathcal{B}_{\delta}(X)}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y) \\&
\le C\max\{\|u\|_{L^p(M)}^p, \|u\|_{L^p(M)}\} \le C.
\end{align*}
By the arbitrary choice of $X$, we have $\|u\|_{L^\infty}\le C$. The higher order estimate follows from the standard linear elliptic partial differential equation theory; see Agmon-Douglis-Nirenberg \cite{ADN}.
Therefore, we complete the proof.
|
2106.05253
|
\section{Introduction}
It has long been recognised that matter in the universe is arranged along an intricate pattern known as the cosmic web \citep{bond1996}. This highly structured web results from the anisotropic gravitational collapse and hosts four main classes of substructures: massive halos, long filaments, sheets, and vast low-density regions known as cosmic voids \citep{zeldovich1982}. Filaments are highly defining features in this network, as they delimit voids and provide bridges along which matter is expected to be channeled towards halos and galaxy clusters.
Filamentary structures in the large-scale distribution of galaxies were originally observed with the Center for Astrophysics (CfA) galaxy survey \citep{delapparent1986}. Since then, ample evidence for such structures has been gathered by galaxy redshifts surveys, such as the Sloan Digital Sky Survey (SDSS, \citealp{york2000}), the Two-degree-Field Galaxy Redshift Survey (2dFGRS, \citealp{cole2005}), the Six-degree-Field Galaxy Survey (6dFGS, \citealp{jones2004}), the Cosmic Evolution Survey (COSMOS, \citealp{scoville2007}), the Galaxy and Mass Assembly (GAMA, \citealp{driver2011}), the Two Micron All Sky Survey (2MASS, \citealp{huchra2012}), and the VIMOS Public Extragalactic Redshift Survey (VIPERS, \citealp{guzzo2014}). Concurrently, further corroboration for the filamentary arrangement of matter also came from cosmological N-body simulations ({\it e.g.}, \citealp{springel2005}, \citealp{vogelsberger2014}).
Detecting and reconstructing cosmic filaments, however, still poses some challenges due to the lack of a standard and unique definition. As a consequence, a variety of approaches and algorithms has been explored in the literature. Among these there are: Multiscale Morphology Filters, such as MMF (\citealp{aragoncalvo2007,aragoncalvo2010}) or NEXUS (\citealp{cautun2013}); segmentation-based approaches, such as the Candy model (\citealp{stoica2005}) or the watershed method implemented in SpineWeb (\citealt{aragoncalvo2010a}); skeleton analyses (\citealp{novikov2006,sousbie2008}); tessellations, like in DisPerSE (\citealp{sousbie2011}); the Bisous model (\citealp{tempel2016}); methods based on Graph Theory such as T-Rex (\citealp{bonnaire2019}) or Minimal Spanning Trees (\citealp{pereyra2019}); or methods based on machine learning classification algorithms, such as Random Forests (\citealp{buncher2020}) or Convolutional Neural Networks (\citealp{aragoncalvo2019}). Comparisons between different methods can be found, e.g., in \citealp{libeskind2017,rost2020}. All these methods rely on different assumptions and they can be more or less advantageous depending on the specific application.
In this paper we adopt and extend the technique proposed in \citet{chen2015} (hereafter \YCC{}), which implements the Subspace-Constrained Mean-Shift (SMCS) algorithm to identify the filamentary structure in the distribution of galaxies. This technique is based on a modified gradient ascent method that models filaments as ridges in the galaxy density distribution. It has already been successfully applied to SDSS data \citep{chen2016}, providing a filament catalogue which has been used to study the effect of filaments on galaxy properties, such as color and mass \citep{chen2017}, or orientation \citep{chen2019,krolewski2019}. The catalogue has also allowed the first detection of Cosmic Microwave Background (CMB) lensing by cosmic filaments \citep{he2018} and it has been applied to study the possible effect of cosmic strings on cosmic filaments \citep{fernandez2020}. Our implementation of the algorithm presents two main differences with respect to \YCC{}. In their work, the filament finder is applied on 2D galaxy density maps which are built from the original catalogue assuming a flat-sky approximation. Here, we extend the formalism to work on spherical coordinates, as we expect this approach to be less sensitive to projection effects, especially for wide surveys. Secondly, we complement the method with a machine learning technique that combines information from different smoothing scales in the density field. This procedure has been designed to be robust against the choice of a particular smoothing scale, and allows the algorithm to exploit the information present at lower redshift (with more available galaxies) in order to robustly predict the filaments at higher redshift (with fewer available galaxies).
We apply this new implementation of the algorithm to the DR16 SDSS data \citep{ahumada2020} from BOSS (Baryon Oscillation Spectroscopic Survey) and eBOSS (extended BOSS) clustering surveys. The latter, in particular, has not been used in previous filament extraction studies. It creates the largest volume survey to date and, thanks to its sample of quasars, we have been able to build a filament catalogue that extends to very high-redshifts, up to $z=2.2$. In our approach, we divide the data in $269$ redshift shells of width $20$ Mpc, and provide the filament reconstruction in 2D spherical slices. We also deliver the uncertainties associated to the filaments positions, which have been estimated with a bootstrap method.
This tomographic approach to filament reconstruction has several advantages. For example, the 2D filament maps can be directly employed in cross-correlation studies with maps of the CMB lensing convergence or thermal Sunyaev-Zel'dovich signal (see {\it e.g.}, \citealt{tanimura2020}). We also expect this method to be particularly suitable for the extraction of filament maps from future wide photometric surveys, such as Euclid \citep{laureijs2011} and the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST, \citealp{ivezic2019}), where uncertainties on redshift estimates might limit the applicability of $3D$ methods. The tomographic approach has also been shown to provide more robust detection near galaxy clusters and to be less sensitive to Finger of God effect, even when it is statistically corrected \citep{kuchner2021}. Moreover, the tomographic catalogue will naturally trace the evolution of the universe filamentary structure as a function of redshift and, therefore, can be used to test structure formation history and cosmological models.
The paper is organised as follows. We start by providing a detailed description of the methodology and the algorithm in \Cref{s:methodology}. In \Cref{s:data} we introduce the data sets used in our analysis, while in \Cref{s:results} we present the results of the filament extraction. In \Cref{s:comparison} we compare the results with the catalogue from \YCC{}. In \Cref{s:validation}
we discuss the validation of the catalogue through several metrics. Finally, in \Cref{s:Conclusions} we draw our conclusions.
\section{Methodology}
\label{s:methodology}
In this section, we will explain the theoretical framework that we use and the complete methodology that we implement. The scheme of this section is the following. In \Cref{ss:definition} we explain the definition of cosmic filament that we are going to exploit. In \Cref{ss:tomo} we introduce the tomographic strategy. In \Cref{ss:scms} we explain the basis of the Subspace-Constrained Mean Shift (SCMS) algorithm to detect these filaments from galaxy distribution maps. In \Cref{ss:err} we study how this algorithm can be extended to yield an estimation of the error of the detection. In \Cref{ss:size} we introduce the two methods (based on SCMS) that we implement to detect filaments, including a version of SCMS boosted with Machine Learning techniques. Finally, in \Cref{ss:pros} we discuss the strengths and limitations of our methodology. All the details explaining the training of the Machine Learning step, and the procedure to clean the final filament catalogue and remove outliers can be found in \Cref{ap:train} and \Cref{ap:clean}, respectively.
\subsection{Definition of cosmic filaments}
\label{ss:definition}
Cosmic filaments can be defined in multiple ways. In this work, we adopt the mathematical framework of the \textit{Ridge formalism} \citep{eberly1996}. This formalism has been widely studied in the mathematical literature, e.g. \citet{ozertem2011,genovese2014}. We refer the reader to these references for a more complete overview of the properties of ridges. Here, we summarise the definition and some key properties.
A ridge of a density field $d(x)$ in $D$ dimensions is defined as follows. Let $g(x)$ and $H(x)$ be the gradient and the hessian of $d(x)$. Let $\lambda_1\geq...\geq\lambda_D$ be the eigenvalues of $H(x)$, with associated eigenvectors $v_1,...,v_D$. We define $V(x)$ to be the subspace spanned by all eigenvectors except $v_1$ (therefore perpendicular to it). We define the gradient projected into this subspace as $G(x)$. Finally, the ridges of the density field $d(x)$ are defined as
\begin{equation}
\text{Ridge}(d) \equiv \{x\ |\ G(x)=0,\ \lambda_2<0\}
\end{equation}
This definition is conceptually analogous to the criteria for maxima of a one-dimensional function: ``first derivative equals $0$'' and ``second derivative is negative''. In the case of higher dimension, one has to restrict those criteria to the subspace perpendicular to the filament. The largest eigenvalue $\lambda_1$ correspond to an eigenvector $v_1$ parallel to the filament, given that the density decreases with the distance to a filament, but not in the direction of the filament (or not as fast). Note that the gradient on a point of the ridge is always parallel to the direction of the ridge. This observation is not only important for the algorithm we use to find the filaments, but it also provides a method for calculating the direction of a filament in a point.
By definition, ridges are one-dimensional, overdense structures, the two properties that define cosmic filaments. We note that other cosmological structures are given by slightly modified definitions. If the subspace $V(x)$ is the whole space and $\lambda_1<0$, we obtain overdense 0-dimensional structures, similar to nodes in the cosmic web, or halos. If the subspace $V(x)$ is the whole space and $\lambda_3>0$, we obtain underdense 0-dimensional structures, similar to voids centers. Finally, if the subspace $V(x)$ is spanned only by $v_3$ and $\lambda_3<0$, we obtain overdense 2-dimensional structures, similar to cosmic walls. It is important to notice that these definitions always include the lower-dimension objects: this means that filaments include nodes of the cosmic web (therefore, galaxy clusters) by definition.
Alternative definitions of cosmic filaments can be found in the literature. In particular, it is common to classify the Cosmic Web according only to the eigenvalues of the hessian \citep{cautun2013, cautun2014}: all positive for halos, two positive for cosmic walls, one positive for filaments, and all negative for voids. Therefore, the space is completely divided into these four classes. This definition implies that regions classified as filaments are more clumpy, as opposed to one-dimensional. In other words, it does not provide information about the center of the filaments, which may be relevant for some applications. We note that ridges satisfy this definition, while also being local maxima in the direction perpendicular to the filament. Another common approach is to detect filaments with graph-based tools, such in \citet{bonnaire2019, pereyra2019}. These methods do not define filament using an underlying physical quantity like the density. Instead, they try to connect most galaxies with links subject to some regularizing criteria. They result in a one-dimensional network, but they are usually sensitive to a range of parameters which must be tuned.
\subsection{Tomographic analysis}
\label{ss:tomo}
The previous definition of filaments can be applied to the Cosmic Web with $D=3$ dimensions, by converting redshift into distance; or in $D=2$ dimensions, by taking redshift slices and applying it independently to each slice. In this work, we use the second approach with thin slices in order to obtain tomographic information of the filaments network. Given that we work on spherical slices, all computations are done by taking into account the spherical geometry of the space, i.e. the computation of the gradient, the hessian, and the definition of the density estimate $d$.
There are different techniques to slice the redshift information. An important aspect to take into account is that the width of the shells needs to be thin enough to avoid different filaments to overlap in a single slice; while it needs to be thick enough to include a large number of galaxies so we can detect the filaments in each slice. Additionally, we ensure that this width is larger than the redshift uncertainty of the galaxies within it, by at least a factor of $5$.
\subsection{Subspace-Constrained Mean Shift}
\label{ss:scms}
The Subspace-Constrained Mean Shift (SCMS) algorithm exploits the previous definition of ridge in order to find one-dimensional maxima. It has been used in Cosmology to obtain Cosmic Filaments in galaxy catalogues \citep{chen2015a,chen2014} and simulations \citep{chen2015b}. This algorithm has also been applied to weak lensing maps by the Dark Energy Survey \citep{moews2020}; to galactic simulations, in order to study filaments formed in interactions with satellite galaxies \citep{hendel2018}; and even in the analysis of the location of crimes in cities \citep{moews2019}.
SCMS is an iterative algorithm. We start with a set of test points that uniformly fill the analysed area and at each step they move towards the filaments. In order to do that, we first compute an estimate of the galaxy density, and obtain its gradient and hessian. These points move in the direction given by the eigenvector of the hessian with the smallest eigenvalue. The magnitude of the movement is given by the projection of the gradient onto this eigenvector. In a ridge, the gradient is parallel to the direction of the ridge by definition, while this eigenvector is always perpendicular. This means that points stay still once they reach a ridge.
A detailed description of the algorithm can be found in \citet{chen2015}. Here, we reproduce the steps as we have implemented them:
\begin{alg}[Filament detection]
\label[alg]{alg:1}
\begin{enumerate
\item[\emph{In}:] A collection of points on the sphere, $\{x_i\} \equiv \{(\theta_i,\phi_i)\}$, representing the galaxies observed within a given redshift slice.
\item Obtain a density function $d(\theta,\phi)$ (defined over the sphere) by smoothing the point distribution with a certain kernel.
\item Calculate the gradient of the density: $g(\theta,\phi)\equiv\nabla d(\theta,\phi)$
\item Calculate the hessian of the density: $\mathcal{H} (\theta,\phi)$
\item Diagonalize the hessian ($2\times2$ matrix) at every point and find the eigenvectors $v_1(\theta,\phi)$, $v_2(\theta,\phi)$, with the smallest eigenvalue corresponding to $v_2$
\item Project the gradient $g$ onto the eigenvector $v_2$. Let $p(\theta,\phi)$ be this projection.
\item Select a set of points $\{y_j\}$ that will be used to search for the filaments. In our case, it will be a uniform grid over the sphere. Iterate until convergence:
\begin{enumerate}
\item At step $s$, move every point $y_j^{(s)}$ in the direction of the projection at that point $y_j^{(s+1)} = y_j^{(s)} + c \cdot p(y_j^{(s)})$. \footnote{The value of $c$ for optimal convergence varies with the pixel, and it can be calculated with the derivative of the smoothing kernel. See \citet{comaniciu2002}.}
\end{enumerate}
\item When the points do not move between steps, they have reached a filament.
\item[\emph{Out}:] A collection of points on the sphere $y_j^{(S)}$ placed on the filaments.
\end{enumerate}
\end{alg}
\Cref{f:example} illustrates several steps of this algorithm.
\begin{figure*}
\centering
\includegraphics[width=0.32\textwidth]{example1}
\includegraphics[width=0.32\textwidth]{example2}
\includegraphics[width=0.32\textwidth]{example3}
\includegraphics[width=0.32\textwidth]{example4}
\includegraphics[width=0.32\textwidth]{example5}
\includegraphics[width=0.32\textwidth]{example6}
\caption{Different steps of the algorithms. \textit{Top left}: Initial distribution of galaxies in a redshift slice. \textit{Top center}: a density distribution is estimated from the galaxies. \textit{Top right}: a grid of points $\{y_j\}$ is selected as the initial positions of the iterative steps; they correspond to the centers of HEALPix pixels. \textit{Bottom left}: the positions of the points after 1 step of the iteration. \textit{Bottom center}: the positions of the points after they converge, with the initial galaxy distribution overlapped. \textit{Bottom right}: same as bottom center, but the uncertainty of each point in the filaments is calculated with Algorithm 2; a redder color representing a lower certainty. All images are gnomic projections centered at $RA=\ang{180}$, $dec=\ang{40}$ with a side size of \ang{60}. The data used in this example corresponds to BOSS data at $z=0.570$ (see \Cref{s:data}).
\label{f:example}}
\end{figure*}
All quantities defined over the sky ($d$, $g$, $\mathcal{H}$, $v_1$, $v_2$ and $p$) are computed using the HEALPix pixelisation scheme at resolution $N_{side}=1024$ \citep{gorski2005}. This corresponds to pixels with an approximate size of \SI{3.44}{\arcminute}. This is chosen to be much lower than the expected scales of any the of fields of interest (typically in the order of degrees), to avoid pixelisation effects.
In order to implement the algorithm, we must choose the kernel to be used in the density estimation: in this work, we use a spherical Gaussian kernel (i.e., a Gaussian kernel evaluated on the geodesic distance). This requires the choice of a free parameter: the full width half maximum of the kernel, \textit{fwhm}. The relation between the size of the kernel and the number of points can significantly affect the quality of the reconstruction in a redshift slice. To mitigate this effect, we implement two independent solutions: \begin{enumerate*}\item[a)] adapt the \textit{fwhm} of the kernel to the number of points of each redshift slice; \item[b)] combine the results of the algorithm at different scales using a Machine Learning approach. Both approaches are explained in \Cref{ss:size}. \end{enumerate*}
\subsection{Estimate of the uncertainty}
\label{ss:err}
We measure the robustness of each detection as explained in \citet{chen2015}. The idea of this method is to simulate different realizations of the galaxy distribution using bootstrapping: for each simulation, we take a random sample of the real galaxies in order to obtain a modified realization of galaxies.
We then compare the filaments obtained in the real data with the filaments obtained in this new realization of galaxies. In practice, we do it in the following way:
\begin{alg}[Uncertainty of the detection]
\label[alg]{alg:2}
\begin{enumerate
\item The \textit{true} filaments are computed on all the real data, $y_j$.
\item A new set of simulated galaxies is generated by bootstrapping the original galaxy catalogue. The new set has the same number of galaxies, but with possible repetitions.
\item \Cref{alg:1} is run on these galaxies to obtain a \textit{new} set of simulated filaments ${y^n_j}$.
\item For every point on the true filament, we compute the minimum distance to the closest new filament $\rho_n(y_j) = \min (d(y_j, y_j^n))$. Small distances correspond to more consistent detections.
\item We repeat steps 2 to 4 a total of $N=100$ times. For every point of the true filament, we have a minimum distance for each simulation; we find the (quadratic) mean of all simulations $\rho(y_j) = \sqrt{\frac{1}{N}\sum_{n=1}^{N} \rho_n(y_j)^2}$
\item[\textit{Out}:] A single error estimate (average minimum distance) for every point in the true filaments, $\rho(y_j)$.
\end{enumerate}
\end{alg}
The value of $\rho(y_j)$ has a natural interpretation: it is the typical error in the location of a particular point in a filament, when the galaxy distribution changes slightly. Since it is given in degrees, is can be readily represented as a confidence region on the sky. This is illustrated on the left image of \Cref{f:uncs}. We note that most filaments are determined with an accuracy well below \ang{1}, while the ends of filaments and smaller filaments present higher uncertainties. This uncertainty estimate may be very useful in applications where the user needs a certain position accuracy, as they can keep only the filaments that satisfy their criteria.
We note that \Cref{alg:2} can be run on any set of points $y_j$, not necessarily the filament points. In this case, one obtains the typical distance to a filament for any point on the sky, averaged over the $100$ bootstrapping realizations. We use this approach to construct full maps of the regions of interest, setting $y_j$ to be the set of all pixels in the region of interest. This will be the basis of the second method to obtain filaments, as explained in the next section. An example of such a map can be seen in the right image of \Cref{f:uncs}. It can be seen that filaments are mostly detected at regions with minimum uncertainty. This figure also presents some spurious detections, which are characterised by a high uncertainty and can be removed a posteriori.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{conffils}
\includegraphics[width=\columnwidth]{confmap}
\caption{Representations of the uncertainty of the detection. On the left, we plot a circle around each point of the filament with a radius equal to the uncertainty; this is a way of visualizing the regions of the sky where a certain filament is expected to be. On the right, we apply the uncertainty estimate to each pixel of the map, so we obtain an average distance to bootstrapped filaments for every pixel; actual filaments are found mostly where the mean distances are low. In both cases, color represents uncertainty in degrees, and filaments found with real data are represented in white. Both figures are gnomic projections centered at $RA=\ang{180}$, $dec=\ang{40}$ with a side size of \ang{60}. The data used in this example corresponds to BOSS data $z=0.570$ (see \Cref{s:data}).
\label{f:uncs}}
\end{figure*}
\subsection{Choice and combination of scales}
\label{ss:size}
As mentioned before, the shape and size of the filter kernel is a free parameter of \Cref{alg:1}. We use a Gaussian kernel to perform this step. We use two independent methods in order to select the best $fwhm$ for each redshift slice, and mitigate the importance of this parameter. They will be explained in the next section.
Note that we also tested the Mexican needlets of different orders \citep[see][]{narcowich2006,geller2008,marinucci2008}, as they have yielded better results than Gaussian profiles in several applications in Cosmology \citep[e.g.,][]{oppizzi2019, planckcollaboration2016b}. However, in this particular case, they did not improve the results. This is due to the oscillating shape of these filters in pixel space, which may introduce artificial ridges in the maps.
\subsubsection[Method A]{Method A: optimize the \textit{fwhm}}
\label{ss:size_A}
The first method consists of finding an optimal value for the $fwhm$ at each redshift slice. In order to avoid dependence on any theoretical model, we do not use the possible evolution of filaments as information to find them. The only parameter we consider is the number of galaxies per unit area within each redshift slice. Slices with a low density number do not have enough reliable information to reconstruct smaller filaments, so the kernel has to be wider in order to compensate for this. On the other hand, highly populated slices have enough information to reconstruct both large and small filaments, so a narrower kernel can be used.
The goal of changing the $fwhm$ with redshift is to obtain the most complete reconstruction possible at each redshift, while maintaining a similar quality of the reconstruction, in terms of accuracy and error.
In practice, we scale the $fwhm$ with the number of galaxies using the following expression:
\begin{equation}
fwhm = A \cdot \left(\frac{n}{f}\right)^{-\frac{1}{6}}
\end{equation}
where $n$ is the number of galaxies within a redshift slice, $f$ is the observed sky fraction, and $A$ is a constant parameter. The exponent $-1/6$ is a common scaling factor for kernel density estimators in two dimensions, used for example in the \textit{Silverman's rule of thumb} \citep{silverman1998}. This is proven to be optimal in the case of reconstruction of a Gaussian profile, in which case $A$ can be optimally calculated. However, since we do not expect a Gaussian profile, we tune the prefactor $A$ of this rule of thumb to
\begin{equation}
A = 21\, \deg .
\end{equation}
We have tested different values in simulations with filaments and noise. This value yields better results than higher ($28, 35$) or lower ($14$, $7$) values; with a false detection rate below $15\%$ and discovery power above $90\%$. This value seems to perform better in real data as well.
The filaments catalogue obtained with method A will be called Catalogue A.
\subsubsection[Method B]{Method B: combine the information at different scales}
\label{ss:combine_B}
The alternative method aims to eliminate the choice of a $fwhm$ by combining different scales. It combines the results of filtering at different scales in order to obtain a single predictor of the filament locations. The key idea is that the quantities computed in \Cref{alg:1} at different scales (density, gradient, hessian, ...), contain enough information to distinguish a point in a filament from a point far from it. By combining these quantities, we may be able to predict the location of filaments more accurately than applying the algorithm on a single scale.
In order to combine the information at different scales, we deploy a Machine Learning algorithm: Gradient Boosting. This algorithm generates a series of \textit{decision trees} based on the input values for each quantity. The first tree will be trained to predict the desired output, the second is trained to learn the error caused by the first one, the $n$-th one is trained to predict the error of the sum of the first $n-1$. By generating a large number of them and combining their predictions in this way, the ensemble prediction is greatly improved. For more information about the algorithm, see \citet{tkh1995,friedman2001,geron2019}. The choices for some of the most important hyperparameters are the following: number of trees ($100$), learning rate ($0.1$), loss function (least squares), and maximum depth of each tree ($3$ layers). These are the baseline settings for this algorithm and we have verified results are stable to small changes in these parameters. This would resolve the problem of fixing the $fwhm$ and eliminate the dependence with redshift. Additionally, the algorithm should be able to learn the typical characteristics of filaments in the most populated redshift slices at low redshift, where filaments are easier to detect. Then, it can use this learnt information in order to identify filaments at higher redshift or less populated slices.
The idea of this algorithm is similar to the well-established technique of convolutional neural networks (see \citet{krachmalnicoff2019} for an implementation of convolutional neural networks compatible with the HEALPix pixelization). These networks typically convolve the map with a large number of small filters which are trained to maximize the information extracted from the map. However, in our case, we can extract more useful information from the density map by obtaining relevant quantities such as the modulus of the gradient or the eigenvalues of the hessian at different scales, as these are very good indicators of the location of ridges. These quantities can not be easily reproduced by training linear filters. Computing these quantities also implies isotropic filters, as opposed to trained filters, where the orientation is important. Lastly, extracting these features explicitly allows us to use different fixed scales; these scales are much larger than the pixel size ($\sim50$ to $100$ times larger), which can usually be obtained only by stacking a large number of convolutional layers.
The input of the Gradient Boosting is some of the quantities computed in \Cref{alg:1} at each pixel. We select five quantities that could be physically correlated (positively or negatively) with the presence of a filament: the density estimate $d$, the magnitude of the gradient $\left\| g \right\| $, the magnitude of the projection $\left\| p \right\| $, the value of the second eigenvalue of the Hessian of the density $\lambda_2$, and its \textit{eigengap} $\lambda_1-\lambda_2$. We compute these quantities for each pixel at several smoothing scales. For example, for the data we are going to analyse, we have verified that four scales provide a stable reconstruction, while adding more scales does not carry any significant improvement. In particular, we will use $fwhm = $\;\SIlist{2.7; 3.4; 4.1; 4.8}{\deg}. Therefore, there are $20$ input features ($5$ quantities times $4$ scales) and as many points as pixels in the region of interest.
As output of the Gradient Boosting, we want a measure of the ``filament-ness'' for each pixel. In order to do this, we exploit the method to estimate the error described in the last paragraph of \Cref{ss:err}. There we derived a map in which the value of each pixel is the average distance to a filament in the bootstrapping simulations, as in the right image of \Cref{f:uncs}. The machine learning algorithm will use the value of the quantities at each pixel to predict the typical distance of a filament from a pixel with certain characteristics. In particular, the algorithm will learn the characteristics of the pixels with filaments. The exact procedure to produce the training maps is explained in \Cref{ap:train}.
After the algorithm is trained, we obtain a prediction of the estimated error map for each redshift slice. In order to compare with the previous method, we obtain a filament distribution by applying \Cref{alg:1} again on these maps. We report the filament catalogue, which will be called Catalogue B.
Doing the prediction with the Gradient Boosting algorithm has three key advantages:
\begin{itemize}
\item It generalizes better. The algorithm is able to learn what range of values are expected at a filament, close to a filament, and far from a filament. This means that results from redshift slices with a greater number of observed galaxies will improve the algorithm at all redshift slices. In particular, the algorithm becomes more robust at higher redshifts where there are fewer observations.
\item It reduces the impact of anomalies. For the same reason, spurious detections become less likely, as they are sometimes caused by local ridges that do not correspond to values expected from real filaments (e.g., produced by a small number of galaxies).
\item It is much faster than computing the real scale-combined estimated uncertainty maps, as explained in \Cref{ap:train}.
\end{itemize}
\subsection{Strengths and limitations}
\label{ss:pros}
Given the choices explained in this section and the algorithms used, we identify the following strengths ($+$) and limitations ($-$) of our methodology:
\begin{itemize
\item[$+$] The implementation works natively in the surface of the sphere, taking into account its geometry, and it is compatible with the widely used HEALPix scheme. This also makes the implementation rotationally invariant and independent of the chosen reference system.
\item[$+$] We reduce the sensitivity to the density estimate, thanks to the development of a boosted version of the standard SCMS algorithm with Machine Learning (Method B).
\item[$+$] The approach provides a tomographic reconstruction. This eases the study of the evolution of different parameters or characteristics of filaments as function of redshift. Additionally, it avoids or reduces artefacts due to the conversion from redshift to distance, such as the \textit{Finger of God} effect or the asymmetrical error in the 3-dimensional location.
\item[$+$] The method provides uncertainty estimates for each detection using bootstrapping simulations (see \Cref{ss:err}).
\item[$+$] The algorithm is fast, it takes only few seconds per redshift slice on a laptop.
\item[$\pm$] Physical center and width: the ridge definition applies to the center of the filaments, so the algorithm is able to locate the one-dimensional center; however, it gives no direct information about its physical width.
\item[$-$] This methodology is less sensitive to filaments in the line of sight direction.
\item[$-$] The ridge definition does not determine the individual filaments, it only determines which points are in one; filaments separation needs to be done afterwards through another method.
\end{itemize}
\section{Data}
\label{s:data}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{data_lowz}
\includegraphics[width=0.9\columnwidth]{data_cmass}
\includegraphics[width=0.9\columnwidth]{data_lrg}
\includegraphics[width=0.9\columnwidth]{data_elg}
\caption{Sky footprint of the data reported by BOSS LOWZ (top), BOSS CMASS (second) and the three eBOSS samples: LRG and QSO (third), and ELG (bottom). The representation uses a mollweide projection in equatorial coordinates (J2000), with a rotation of \ang{90} along the $z$ axis. The footprints on the left and right sides correspond to the North and South galactic caps, respectively.
\label{f:skies}}
\end{figure}
The Sloan Digital Sky Survey (SDSS) \citep{blanton2017} currently provides the most large and complete spectroscopic galaxy catalogue. In particular, we use two of its surveys: BOSS (Baryon Oscillation Spectroscopic Survey) and eBOSS (extended BOSS). Both of these surveys report a specific catalogue tailored to the study of Large Scale Structure. We use these catalogues in our work. They are divided into different samples, each optimized to a specific redshift range. In this section we briefly describe the different samples and how we use and combine them.
\subsection{BOSS}
BOSS mapped the distribution of luminous red galaxies (LRG) and quasars (QSO) on the sky and in redshift. The final data release is part of the SDSS DR12 \citep[see][]{dawson2013,alam2015,anderson2014}. They report two samples:
\begin{itemize}
\item LOWZ: galaxies in this sample are mostly located between $0.05<z<0.5$ in both the North and South galactic cap. The sky footprint can be seen in \Cref{f:skies} (top row). The sky fraction is approximately $17.0\%$ and $7.5\%$ for the North and South galactic caps, respectively. \footnote{The North galactic cap presents an unobserved region; note that we remove all filaments detected close to it to avoid border effects, see \Cref{ap:clean}.}
\item CMASS: galaxies in this sample are mostly located between $0.4<z<0.8$ in both the North and South galactic cap. The sky footprint can be seen in \Cref{f:skies} (second row). The sky fraction is approximately $19.0\%$ and $7.5\%$ for the North and South galactic caps, respectively.
\end{itemize}
\subsection{eBOSS}
The eBOSS survey uses three different tracers: LRG, ELG (emission line galaxies), and QSO. The most recent data is reported in SDSS DR16 \citep[see][]{dawson2016,ahumada2020,ross2020}. Catalogues for each type are reported separately:
\begin{itemize}
\item LRG: galaxies in this sample are located between $0.6<z<1.0$ in both the North and South galactic cap. The sky footprint can be seen in \Cref{f:skies} (third row). The sky fraction is approximately $8.4\%$ and $6.0\%$ for the North and South galactic caps, respectively. The North galactic cap is a subregion of the BOSS footprint, around half of the size and very regular. The South galactic cap is irregular and presents a gap. Given that the area close to borders may introduce spurious detections (\Cref{ap:clean}), the region available for the analysis is too small to be used for our filament finding algorithm: we do not include this cap.
\item ELG: galaxies in this sample are located between $0.6<z<1.1$ in both the North and South galactic cap. The sky footprint can be seen in \Cref{f:skies} (bottom row). The sky fraction is $1.6\%$ and $2.0\%$ for the North and South galactic caps, respectively. Both of these caps are too small and irregular for the algorithm, so we do not use this sample.
\item QSO: quasars in this sample are located between $0.8<z<2.2$ in the North and South galactic cap. The sky footprint is identical to the one for LRG, which can be seen in \Cref{f:skies} (third row). The sky fraction is approximately $8.4\%$ and $6.0\%$ for the North and South galactic caps, respectively. As before, we are unable to use the data in the South galactic cap, we use only the North galactic cap.
\end{itemize}
\subsection{Combining data}
\label{ss:combine}
In order to produce the best possible filament catalogue we carefully combine the previous data maximizing the density of galaxies and the covered sky fraction. Combining different samples by just merging the catalogues may generate an inhomogeneous sample on the sky. This would introduce artefacts in the regions when the number of galaxies varies abruptly due to sampling inhomogeneities. Therefore, when combining samples with different footprints, we need to select the most restrictive footprint. This means that there is a natural trade-off between the observed number of galaxies and the sky footprint to be considered.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Block1}
\caption{Galaxy distribution in Block 1. On top, stacked number of galaxies per redshift bin from both the used surveys. On the bottom, the fraction of galaxies corresponding to each survey as a function of redshift.
\label{f:block1}}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Block2}
\caption{Galaxy distribution in Block 2. Number of galaxies per redshift bin. All galaxies correspond to the CMASS sample.
\label{f:block2}}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Block3}
\caption{Galaxy distribution in Block 3. On top, stacked number of galaxies per redshift bin from all of the used surveys. On the bottom, the fraction of galaxies corresponding to each sample as a function of redshift.
\label{f:block3}}
\end{figure*}
In practice, we combine the data into the following blocks (i.e. data samples), each with a different mask:
\begin{itemize}
\item Block 1: $0.05<z<0.45$. Data from LOWZ and CMASS; in the sky footprint from LOWZ, which is the most restrictive of the two. The number of galaxies per redshift bin and the ratio of both samples can be seen in \Cref{f:block1}. Note that this block is dominated by LOWZ, with CMASS data providing a relevant contribution (more than $10\%$ of the galaxies) only at $z>0.4$. At the higher redshift end, the majority of galaxies come from the CMASS sample. However, there is a large number of LOWZ galaxies which we decide not to neglect as they improve the filament reconstruction, at the expense of reducing the sky region slightly.
\item Block 2: $0.45<z<0.7$. Data from CMASS only. The number of galaxies per redshift bin can be seen in \Cref{f:block2}.
\item Block 3: $0.6<z<2.2$. Data from CMASS, LRG and QSO; in the footprint from LRG (equal to the QSO footprint), the most restrictive of the two regions. There are no QSO data below $z=0.8$, while all filaments above $z=1.0$ are detected using QSO data exclusively. The number of galaxies per redshift bin and the fraction of each sample can be seen in \Cref{f:block3}.
\end{itemize}
We note that the redshift range $0.6<z<0.7$ is covered by two blocks: Block 2 which is made up only by CMASS data in a larger sky region; and Block 3 which is made up by CMASS+LRG data, but in a smaller region given by LRG. We perform the filament reconstruction in this redshift range separately for the two blocks. This is done in order to extract the most information out of the samples, while ensuring that the entire region is homogeneously sampled in order not to introduce artefacts. As an interesting by product, we will be able to compare filaments extracted from BOSS alone and from BOSS+eBOSS.
\section{The Filament Catalogue}
\label{s:results}
\subsection{Filament reconstruction}
We apply Method A and Method B, as explained in \Cref{ss:size}, to extract the cosmic filaments catalogues from the three blocks of data introduced in \Cref{ss:combine}. These three blocks correspond to three different redshift ranges and sky fractions $f$, inherited from the used SDSS data: a) Block 1 ($0.05<z<0.45$), with $78$ bins and $f_1=0.186$; b) Block 2 ($0.45<z<0.7$), with $40$ bins and $f_2=0.219$; and c) Block 3 ($0.6<z<2.2$), with $166$ bins and $f_3=0.062$.
The width of the slices in the line-of-sight direction is taken as $20$ Mpc, computed with the $\Lambda$CDM parameters obtained by Planck \citep{planck2019vi}; in particular, $H_0=67 \frac{km}{s \, Mpc}$. We note that the exact parameters to be used have a limited impact, as small variations of this width do not significantly alter our results. This width is chosen to be small enough to avoid several overlapping filaments in a single slice, but several times larger than the uncertainty on the galaxy measured redshift.
Method A requires the choice of the size of the smoothing kernel. As explained in \Cref{ss:size_A}, this is taken to be a function of the number of galaxies within a redshift slice. The value of $fwhm$ of the Gaussian kernel, in degrees, can be seen in \Cref{f:fwhm}. Method B uses four fixed scales which can also be seen in the figure ($2.7\, \deg, 3.4\, \deg, 4.1\, \deg, 4.8 \, \deg$); they contain the range of sizes used for Method A.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{fwhm}
\caption{Value of the $fwhm$ of the Gaussian kernel chosen for Method A, as a function of redshift, for each block. The scales that are combined in Method B are fixed and can be seen with markers on both redshift ends.
\label{f:fwhm}}
\end{figure*}
We will compare the results obtained with both methods in \Cref{ss:comp}. We will see that Method B generally outperforms Method A. Therefore, we recommend the use of the catalogue produced with Method B, which is publicly available at \href{https://www.javiercarron.com/catalogue}{javiercarron.com/catalogue}. The catalogue produced with Method A is also available upon request.
The columns present at the catalogues are explained in \Cref{t:cata}.
\begin{table}
\caption{Columns in the catalogues. \textit{RA} and \textit{dec} refers to the equatorial system J2000. \textit{Angle} refers to the angle of the filament with the parallels of the sphere in these coordinates, in a counter-clockwise direction. Column \textit{dens} refers to the density estimate of the filament. Column \textit{ini\_dens} refers to the density estimate of the starting points at the beginning of \Cref{alg:1}, before the points move towards the ridges; this can be used to set a threshold, see \Cref{ap:clean}. Units of density estimates are arbitrary.
\label{t:cata}}
\centering
\begin{tabular}{ll}
\textbf{Column} & \textbf{Name} \\
RA & Right ascension ($\deg$) \\
dec & Declination ($\deg$) \\
dens & Density estimate \\
unc & Estimated uncertainty ($\deg$) \\
grad\_RA & Gradient (RA component) \\
grad\_dec & Gradient (dec component) \\
angle & Angle of the filament ($\deg$) \\
z\_low & Start of the redshift bin \\
z\_high & End of the redshift bin \\
ini\_dens & Initial density estimate \\
\end{tabular}
\end{table}
\subsection{Comparison between methods}
\label{ss:comp}
In this section we present and compare the catalogues obtained with Method A and Method B explained in \Cref{ss:size}. We use the final results in the complete redshift range $0.05<z<2.2$ and all available sky.
The first difference between the two catalogues is that Method B is able to detect more filaments. An example at $z=0.20$ can be seen in the top row of \Cref{f:comp}, this is representative of the results at lower redshift ($z<0.5$). It can be seen that most filaments in Catalogue A are also detected in Catalogue B. The latter have a slightly larger associated uncertainty. However, more importantly, there is an additional filament population in Catalogue B, which is detected at lower significance. These additional filaments are not detected with Method A because they are erased when smoothing the galaxy density map at a single scale. On the other hand, as expected, Method B is able to correctly recover the same filaments detected with method A, plus additional filaments from other (usually smaller) scales.
At higher redshift ($z>0.7$), we continue to observe an additional filament population due to the same reason. However, the filaments which are observed in both catalogues are detected with a significantly lower uncertainty estimate in catalogue B. A representative example at $z=1.00$ can be seen in the bottom row of \Cref{f:comp}. At this redshift, the number of galaxies is lower; Method A only uses the information at the given slice, so it is more sensitive to noise and incomplete information. On the other hand, the Machine Learning algorithm in Method B is able to ``learn'' the typical characteristics of filaments in slices where there are more galaxies and less noise. It is then able to use some of this information in all redshift slices to obtain a more confident and uniform reconstruction of the filaments.
\begin{figure*}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{compA1}
\end{subfigure}
\begin{subfigure}{0.08\textwidth}
\centering
\includegraphics[width=\textwidth]{color}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{compB1}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{compA3}
\end{subfigure}
\begin{subfigure}{0.08\textwidth}
\centering
\includegraphics[width=\textwidth]{color}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{compB3}
\end{subfigure}
\caption{Example of reconstructed filaments at $z=0.20$ (top) and $z=1.00$ (bottom); with Method A (left) and Method B (right).
The color of the points represents the error estimate, in degrees. The visualization is a Gnomonic projection around the point ($RA=\ang{140}$, $\delta=\ang{40}$) with a side size of \ang{60}. Meridians and parallels are both represented at a step of \SI{20}{\deg}.
\label{f:comp}}
\end{figure*}
\subsubsection{Distance from galaxies to filaments}
\label{ss:distance}
The filament catalogues are built to trace the regions with a galaxy number overdensity. Consequently, there is a correlation between filaments and galaxies by construction. We use this correlation as a proxy to study the quality of the reconstruction and to quantify the differences between the two methods. We consider the distance from every galaxy to its closest filament. Galaxies are typically not located at the exact centre of the filaments, so this distance will have a dispersion around $0$. In general, the main causes for this dispersion are the following:
\begin{enumerate}[nolistsep,label=\alph*)]
\item physical width of the filament,
\item error in the location of the centre of the filament,
\item non-detection of actual filaments, so the galaxies in that filament will appear to be far from detected filaments.
\end{enumerate}
The second point can be approximated with the estimated uncertainty of the detections. The third point is not significant in the full catalogue, but could be relevant if one takes a small subset, such as a strong threshold in the uncertainty of the detection. In our analysis we consider the full catalogue.
We compute the distance between galaxies and their closest filament for the two methods. A comparison of the median distance over redshift can be seen in \Cref{f:dists} both in degrees and proper distance. We choose to use the median as it is more robust in slices with lower amount of galaxies (especially at high and very low redshift). This distance is lower overall in Method B, partly due to the fact that it reports a higher number of filaments.
Another way to analyse the distance between galaxies and filaments is to look at their distribution compared to that obtained from a random distribution of points. In this case, we partially mitigate the effect induced by having a larger number of filaments in method B, as it also decreases the overall distance from random points to filaments. The number of galaxies at a distance $r$ from a filament, with respect to the number of random points at the same distance, $\delta(r)=\frac{n_{gal}(r)}{n_{ran}(r)}$, can be seen in \Cref{f:ratio} for both catalogues. Both catalogues present a similar ratio, although Catalogue A produces a slightly higher peak at a very close distance to the filaments.
Both methods yield significant detection levels throughout the entire redshift range. This includes $z>1$, where data come from QSO detections only. The quality of the reconstruction in this range remains roughly constant. Distances from galaxies to filaments are significantly lower than for random points.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{compdist}
\includegraphics[width=\columnwidth]{compdistmpc}
\caption{Mean distance between galaxies and their closest filament, as a function of redshift. Blue corresponds to Method A. Orange corresponds to Method B. Different shades of color correspond to the three different Blocks with different masks. Top: angular distance in degrees; Bottom: proper distance in Mpc.
\label{f:dists}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{ratio}
\caption{Average distribution of distances between galaxies and their closest filament, divided by the same distribution for random points, as described in the text. Left: Method A. Right: Method B. Different colors correspond to the three different Blocks.
\label{f:ratio}}
\end{figure}
This distance between galaxies and filaments can be understood as a typical width of the filament and it is compatible with values in the literature. This quantity has been studied in hydrodynamical simulations in \citet{galarragaespinosa2020} using Illustris-TNG, Illustris, and Magneticum simulations. They observe that the radii of filaments are typically $\sim3$ to $\sim5$ Mpc at $z=0$. Although the radius definition is different and we do not reach $z=0$, we have verified that our measurement in the lowest redshift bins is compatible with their measurement in simulations.
\subsubsection{Uncertainty of the detections}
The second metric we use in the comparison is the estimated uncertainty of the detections. This is calculated by using bootstrapping of the original galaxies, as explained in \Cref{alg:2}. This procedure estimates the variability in the location of the detected filaments, given in degrees. It is therefore a measure of the robustness of the results.
We compute the median of the estimated uncertainty of detection. A comparison can be seen in \Cref{f:errs} in degrees and proper distance. The median uncertainty for Method B is approximately constant over $z$. As mentioned before, at low redshift, the higher median uncertainty with respect to method A is mainly driven by the additional population of detected filaments with higher uncertainty. At higher redshift this effect also happens, but it is compensated by a much lower estimated uncertainty for the filaments found in both catalogues.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{comperr}
\includegraphics[width=\columnwidth]{comperrmpc}
\caption{Mean uncertainty estimate for the detected filaments, as a function of redshift. Blue corresponds to Method A. Orange corresponds to Method B. Different shades of color correspond to the three different Blocks with different masks. Top: angular distance in degrees; Bottom: proper distance in Mpc.
\label{f:errs}}
\end{figure}
It is important to note that, at low redshift, Method B yields a result of similar quality to Method A (as measured with galaxy-filament distances), even if Method A reports a lower estimated uncertainty. At higher redshift ($z>1$), Method B yields both lower distances and lower uncertainty estimates.
\paragraph*{}
We note that we expect most applications of these catalogues to require an additional threshold, for example, in the limit of the estimated uncertainty required\footnote{This can be done manually on the catalogue or with the Python functions that we provide at \href{https://www.javiercarron.com/catalogue}{javiercarron.com/catalogue}}. After a cut of this kind (for example, using the Catalogue B while removing points with an estimated uncertainty over \SI{1}{\deg}), the described results would be slightly modified in the following way: a) The median distance would increase overall by around \SI{0.05}{\deg}, b) The peak in $\delta$ would increase to be very similar to the one in Catalogue A, and c) The estimated uncertainty would be very uniform in $z$ at a value of $(0.55\pm0.02)$\si{\deg}. This is a consequence of the natural trade-off between confidence of the detections and amount of filaments detected that arises when imposing a threshold of this kind. Different applications will require different choices.
\section{Comparison with previous work}
\label{s:comparison}
As stated before, this work follows a similar framework as the one described in \citet{chen2015, chen2016} (\YCC{}). In this section, we explore the improvements in the catalogues obtained by our implementation.
A first improvement concerns the data itself. In this work we perform the reconstruction on a wider sky region encompassing both the North and South Galactic Caps, while the \YCC{} catalogue is produced only in the North Galactic Cap. We also include the eBOSS sample, which allows a deeper reconstruction in redshift. However, to make a fairer comparison of the methodologies in the rest of the section we restrict to the redshift range given by \YCC{}: $0.05<z<0.7$, equivalent to Block 1 and Block 2 of our catalogues. These two blocks are produced only with BOSS data. Additionally, in \YCC{} they use the NYU Value Added Catalogue \citep{blanton2005} to have more data in the lowest redshift bins. We decided against using this data in order to have a consistent dataset over the whole redshift range, produced only by SDSS. This means that comparisons at the lowest redshift bins ($z\lesssim0.2$) will be affected by the different used data, so we do not consider them in the comparison.
We use our Catalogue A in the comparisons, as it is the most similar to the methodology followed to produce \YCC{}. The main difference is the spherical treatment of the sky; other minor differences are the binning strategy ($20$ Mpc instead of $\Delta z = 0.005$) and slightly different criterion for the smoothing kernel. In \Cref{ss:comp} we compared the results obtained by our two methods: Method A, our implementation of the SCMS algorithm, and Method B, a version boosted with a Machine Learning approach.
\subsection{Uncertainty of the detections}
The first metric we use to compare the catalogues is the estimated uncertainty for each point in a filament. This is calculated by using bootstrapping of the original galaxies, as explained in \Cref{alg:2}. This estimates the variability in the location of the detected filaments, given in degrees. It is therefore a measure of the robustness of the results.
For each redshift, we compute the average uncertainty estimate. This can be seen in \Cref{f:yccerr}. We observe that the estimated uncertainty is lower in our catalogue over the entire redshift range, approximately by a factor $2$. It is also more uniform with redshift. We stress that the uncertainty has been estimated using the same method adopted in \YCC{} (\Cref{alg:2}), thus the difference we find comes from the reconstruction of filaments.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{yccerr}
\caption{Average estimated uncertainty of the detected filaments, as a function of redshift. Orange corresponds to our Method A, blue corresponds to \YCC{}. For $z<0.2$ (gray band) filaments are reconstructed from different data in the two catalogues. Lines are non-linear fits shown for visualization only.
\label{f:yccerr}}
\end{figure}
\subsection{Distance from galaxies to filaments}
The second metric we use to compare the catalogues is the distance between galaxies and their closest filament. As explained in \Cref{ss:distance}, this quantity is mainly determined by the physical width of the filaments and the error in their detected positions.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{yccdist}
\caption{Average distance between galaxies and their closest filament, as a function of redshift. Orange corresponds to our Method A, blue corresponds to \YCC{}. For $z<0.2$ (gray band) filaments are reconstructed from different data in the two catalogues. Lines are non-linear fits shown for visualization only.
\label{f:yccdist}}
\end{figure}
We compute the distribution of these distances and obtain the average for each redshift slice. The comparison of average distances can be seen in \Cref{f:yccdist}. Our catalogue produces lower distances between galaxies and filaments for $z<0.45$. \YCC{} presents a sharp decrease at redshift $z\approx0.45$, corresponding to the transition from LOWZ data to CMASS data. As we will explore in \Cref{s:iso}, this is mainly driven by artefacts at high latitudes, where the flat sky approach is less accurate. We observe a smoother curve with less dispersion in our catalogue, corresponding to more consistent detections across redshifts.
Additionally, we can compare the average distances between filaments and galaxies with the estimated uncertainties for the centre of the filaments. These two quantities are comparable for \YCC{}, whereas in our Catalogue A the uncertainty is lower by approximately a factor $2$. This means that, in our case, the dispersion in distances is not dominated by the uncertainty in the location of filament centres, although this effect can not be neglected.
\subsection{Isotropy}
\label{s:iso}
One of the main differences between our work and \YCC{} is the full treatment of the spherical geometry of the sky. The catalogue in \YCC{} is obtained by approximating the observed area to a flat sky, using spherical coordinates as euclidean. As a consequence, there is some risk that results could show anisotropic features at high latitudes. On the other hand, since we do not apply such approximation, the algorithm is expected to behave isotropically.
We test this expectation by considering two stripes at different latitudes\footnote{We are always working on equatorial reference system, so the latitude of the sphere corresponds to the declination $\delta$.}:
\begin{equation}
\begin{split}
45<& \: lat < 65 \, \deg\\
0<& \: lat < 20 \, \deg
\end{split}
\end{equation}
We look at the same metrics discussed above, restricting the analysis to these stripes. The estimated uncertainty and the galaxy-filament distance can be seen in \Cref{f:isotropy}. The first thing to notice is that our catalogue reports highly compatible results at high and low latitudes, both for the estimated uncertainty and the galaxy-filament distance. Our algorithm is performing isotropically, as expected.
\begin{figure}
\centering
\centering
\includegraphics[width=\columnwidth]{isodist}
\centering
\includegraphics[width=\columnwidth]{isoerr}
\caption{Top: Average distance from galaxies to their closest filaments, in degrees. Bottom: average estimated uncertainty in the location of the centre of the filaments, in degrees. Both metrics are calculated separately for high and low latitude filaments. For $z<0.2$ (gray band) filaments are reconstructed from different data in the two catalogues. Lines are non-linear fits shown for visualization only.
\label{f:isotropy}}
\end{figure}
The \YCC{} catalogue, on the other hand, shows a slightly different behaviour for results at higher and lower latitudes. An example of this difference can be seen in \Cref{f:iso_highz}, where we compare the filaments obtained at $z=0.510$ for both catalogues. It can be seen that the filaments at low latitudes are similar, while there is a larger discrepancy at high latitudes. In that region, \YCC{} produces a larger number of filaments, with a preferred vertical orientation. This is an artefact due to the flat sky approximation and, indeed, it is hidden when looking at the filaments in a flat sky, since this region gets dilated (e.g., see Figure 4 in \citealp{chen2016}). This anisotropic feature artificially reduces the distances from galaxies to filaments in this region, compared to low-latitude results with no anisotropic effects. This reduction reaches up to a $25\%$ value in the average distance. Heuristically, it seems that the intrinsic physical meaning of the derivative in the $\phi$ direction is altered by flat sky projection effects at high latitudes and this has an effect on the features of the algorithm.
\begin{figure*}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{isoz_ycc}
\end{subfigure}
\begin{subfigure}{0.08\textwidth}
\centering
\includegraphics[width=\textwidth]{color_iso}
\vspace{0.5cm}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{isoz_our}
\end{subfigure}
\caption{Example of obtained filaments at $z=0.510$ for \YCC{} (\textit{left}) and our result (\textit{right}).
The color of the points represents the uncertainty estimate, in degrees. The visualization is a Gnomonic projection around the point ($RA=140$, $\delta=40$) with a side size of \ang{60}. Meridians and parallels are both represented at a step of \SI{20}{\deg}.
\label{f:iso_highz}}
\end{figure*}
Similarly, the estimated uncertainty reported by \YCC{} is also affected by this anisotropy. Estimated uncertainties are calculated as an average of distances between real filaments and simulated filaments (as explained in \Cref{alg:2}). However, distances calculated in flat space are artificially increased at high latitudes. This implies an increased value of the estimated uncertainty for this region. Indeed, this is the observed effect, as presented in \Cref{f:isotropy}.
This effect can be quantified by defining $\Delta dist$ and $\Delta unc$ as the average difference between values at high and low latitudes of the distance and the estimated uncertainty. For Method A we have the following results:
\begin{equation}
\begin{split}
\Delta dist &= 0.005\,\deg\\
\Delta unc &= -0.01\,\deg,
\end{split}
\end{equation}
while for \YCC{} we obtain the following:
\begin{equation}
\begin{split}
\Delta dist &= -0.075\,\deg\\
\Delta unc &= 0.23\,\deg
\end{split}
\end{equation}
We note that this anisotropy is only relevant at high latitudes. At low and intermediate latitudes, our catalogue reports similar results to \YCC. Therefore, this effect does not invalidate the multiple results in the literature based on this catalogue. If anything, we expect these results to be more significant when this anisotropy is corrected.
\section{Validation of the Catalogue}
\label{s:validation}
In this section, we evaluate the quality of the reconstruction in several regards: we check that the results are isotropic, that the catalogue is strongly correlated to independent galaxy cluster catalogues, that the results are stable even if new data is included, and that the length distribution of filaments is compatible with other works in the literature.
\subsection{Isotropy}
An interesting aspect related to the isotropy of our catalogue is the result at both hemispheres. For $0.05<z<0.7$ (i.e. Block 1 and 2), we have data at both the Southern and Northern hemispheres. The sky fractions\footnote{The sky fraction is defined as the observed area over the total sky area; the maximum value of $f$ for a hemisphere is therefore $0.5$.} are $f_S=0.054$, $f_N=0.132$ for Block 1, and $f_S=0.054$, $f_N=0.165$ for Block 2. As a sanity check, we test that the results at both hemispheres are similar.
On each redshift slice and each hemisphere, we compute the mean distance from galaxies to filaments, and the mean estimated uncertainty. Then, we compute the difference between the metrics at both hemispheres (North $-$ South). The average differences for Method A are:
\begin{equation}
\begin{split}
\Delta dist &= 4.4 \cdot 10^{-5}\,\deg\\
\Delta unc &= 7.0 \cdot 10^{-3}\,\deg
\end{split}
\end{equation}
And the average differences for Method B are:
\begin{equation}
\begin{split}
\Delta dist &= 5.7 \cdot 10^{-3}\,\deg\\
\Delta unc &= 5.4 \cdot 10^{-2}\,\deg
\end{split}
\end{equation}
We see that none of these differences is significant: all of them are below $1$ arcminute, except the difference in estimated error with Method B, which is \ang{;3.2;}. Additionally, we do not observe trends with redshift.
The algorithms is performing isotropically, yielding compatible results at both hemispheres.
\subsection{Galaxy clusters}
According to the evolution of the Cosmic Web, the largest overdensities are located in the cosmic halos, which are usually located at the intersection of several filaments. These overdensitites can seed the formation of galaxy clusters. Therefore, known galaxy clusters are expected to be located at a low distance from cosmic filaments \citep{hahn2007}.
We test this expectation with two catalogues of galaxy clusters available in the literature, listed below. Both catalogues have been built using the Sunyaev-Zel'dovich effect as a tracer. Therefore, the detection of these galaxy clusters is completely independent of the SDSS data we use to detect the filaments. The catalogues are the following:
\begin{itemize}
\item Planck galaxy cluster catalogue: \citet{planckcollaboration2016a}. This catalogue presents a total of $1653$ clusters, mainly at $z<0.6$, corresponding to our Blocks 1 and 2. They are reported to have a $SNR\geq4.5$. They are distributed over the full sky; a total of $301$ clusters are located within the region where we extract the filaments ($254$ for Block 1 and $47$ in Block 2).
\item Atacama Cosmology Telescope (ACT) galaxy cluster catalogue: \citet{hilton2020}. This catalogue presents a total of $4195$ clusters, with $SNR\geq4.0$. This catalogue covers a restricted sky area, as it presents clusters with latitudes up to \ang{20}. Therefore, there is some overlap with the footprints for our Blocks 1 and 2. Unfortunately, there is no overlap with Block 3. In order to minimize possible systematics, we select the subsample of clusters with spectroscopic determination of redshift. The number of selected clusters in the footprint of the filaments is $951$ ($479$ in Block 1 and $472$ in Block 2).
\end{itemize}
Two examples of the location of clusters and filaments can be found in \Cref{f:clusmap} for $z=0.249$ (Method A) and $z=0.552$ (Method B). All clusters are found to be close to filaments and, in particular, they tend to be close to intersections of filaments. We observe this trend for both catalogues and throughout the entire redshift range.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{clusmap1}
\vspace{1cm}
\includegraphics[width=0.8\textwidth]{clusmap2b}
\caption{Example of filament maps at different redshift slices with the two methods. The color of the filament represent the estimated uncertainty of the detection, in degrees. Independently-measured galaxy clusters from Planck and ACT are also represented. The grey area is excluded from the catalogue due to lack of data and border effects.
\label{f:clusmap}}
\end{figure*}
In order to test the statistical significance of the correlation between these clusters and our filaments, we produce $500$ random mocks for each cluster catalogue. For each simulation, we take all the clusters located in the footprint of our filaments and randomly place them in a different location, keeping their original redshift. The location is sampled uniformly from the intersection between the cluster catalogue footprint and the filaments footprint. We compute the average distance from these new points to the filaments. We repeat this calculation for the actual data and see how different it is from the random realizations. We can see the result in \Cref{f:clus}. It can be seen that the actual clusters are much closer to the filaments than the random catalogue. In particular, the average distance with actual data is between $14\sigma$ and $23\sigma$ lower than random realizations.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{cluster}
\caption{Distribution of the average distance between clusters and filaments for random realizations and real data. In the y-axis, the average distance, in degrees; different histograms for the two cluster catalogues and methods are represented. The average distance for real data is represented with squares.
\label{f:clus}}
\end{figure}
The average distance between clusters and the closest filaments is \ang{0.54} for ACT and \ang{0.62} for Planck. Therefore, there is a strong correlation between the filament catalogues presented in this work and cluster catalogues in the literature. We emphasize that the detection of these clusters is obtained with a different physical observable and they are independent from our data.
\subsection{Stability to new data}
One question that arises when analysing the results is how robust these filaments are with respect to new data. If a future survey is able to detect more galaxies at a certain redshift, we would like to know whether we would obtain a similar set of filaments. Equivalently, this can be seen as the predictive power of the filaments: whether new observations of galaxies will be located in the regions predicted by the detected filaments.
We can test this exact situation in the redshift range of $0.6<z<0.7$. In this range, we have data coming from BOSS that we use in our Block 2. But we also have more recent data coming from eBOSS; we use an aggregation of both in the Block 3. The two blocks overlap in this redshift range, so we can test how much the filaments differ when adding the more recent eBOSS data. We note that eBOSS sky coverage is smaller, so we limit this comparison to the common sky region.
In order to compute the difference, we take the filaments obtained with BOSS data only and consider an area around them with a radius of \ang{1}. We then take the filaments obtained with BOSS+eBOSS data and compute the fraction of filament points which lay inside this region. This procedure is done for all redshift slices in $0.6<z<0.7$. Slices at lower redshift have a higher percentage of common galaxies; i.e., new data coming from eBOSS is less important here, and we expect to find similar filaments. At higher redshift, the majority of galaxies come from new data, and we expect the filaments to differ more.
We can see the fraction of common filaments as a function of the fraction of common galaxies in \Cref{f:stability}. Each point corresponds to a single redshift slice, and the two curves correspond to Method A and B. The grey region corresponds to the expected results when doing this procedure with uncorrelated slices. The fraction of common filaments (alternatively, the fraction of filaments that are correctly predicted by older data) is between $72\%$ and $96\%$, even when the fraction of shared galaxies goes below $40\%$.
This means that new observations in this range are unlikely to change the results significantly: duplicating the number of galaxies only produced $25\%$ more filaments. New data may have a larger effect if a different tracer is used, or with further improvements to the algorithms.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{compat}
\caption{Stability of the results to new data. On the y-axis, the fraction of filaments found using all data (older and newer) which were also found using only older data. On the x-axis, the fraction of galaxies corresponding to older data. Each point corresponds to the result for a single redshift slice. See text.
\label{f:stability}}
\end{figure}
\subsection{Length distribution of the Cosmic Filaments}
The study of the topology of the Cosmic Web is highly non-trivial and it is an active topic in the literature; see, for example \citet{neyrinck2018,wilding2020}. This is in part due to the difficulty of defining \textit{a single filament}. In this work, we have used a definition of filaments based on the ridge formalism, which is very useful to find the network of filaments, but does not provide information about individual filaments (see \Cref{ss:definition}).
For this section, as a matter of example, we consider that every intersection of the web is the starting point of a filament, regardless of the angle or density of such intersections. The filament ends at another intersection or an end point. We note that this kind of definition is sensitive to the depth of the detection: detection of fainter and shorter filaments will of course increase the number of shorter filaments, but it will also increase the number intersections and split longer filaments into shorter ones.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{lengths}
\caption{Distribution of the lengths of the filaments in the catalogues, for the three different blocks (Block 1: $0.05<z<0.45$, Block 2: $0.45<z<0.7$, Block 3: $0.6<z<2.2$). The dotted line represents the results when we impose an additional threshold on the catalogue, considering only points with uncertainty of the detection under \ang{0.8}.
\label{f:length}}
\end{figure}
The distribution of filament lengths of the catalogue, using the previous definition on Catalogue B, can be seen in \Cref{f:length}. It is characterised by a redshift-dependent peak followed by a power-law decrease. This distribution is also observed in \citet{malavasi2020} (see their Figs. 18-22); their length distribution is almost exactly the same that our result for Block 1, even when the method they use to find filaments is completely different (i.e., $3D$ reconstruction with the DisPerSE scheme). For Blocks 2 and 3, at higher redshifts, we notice that there are fewer shorter filaments; this could be an observational effect (shorter filaments are more difficult to detect at higher redshifts) or a Physical effect (if, for example, filaments tend to fragment with time). We note that a similar effect has been observed in simulations: less densely populated simulations report longer filaments \citep{galarragaespinosa2020}; albeit these simulations report shorter filaments overall. More investigation is needed to further investigate these trends.
Finally, we also report the length distribution when imposing an additional threshold to the catalogue. The fainter lines in \Cref{f:length} represent the result when considering only points with associated uncertainty lower than \ang{0.8}. As expected, the number of shorter filaments decreases, which decreases the number of intersections, increasing the number of longer filaments. We note that this threshold provides a similar result than the produced from Method A, since this method is less sensitive to short filaments, as explained in \Cref{ss:comp}.
\section{Conclusions}
\label{s:Conclusions}
In this work we have presented a new catalogue of Cosmic Filaments found with the SDSS Data Release 16. We have implemented a version of the Subspace Constrained Mean Shift algorithm on spherical coordinates and applied it to thin redshift slices in order to obtain a tomographic reconstruction of the Cosmic Web. We have boosted the algorithm with Machine Learning techniques, which improves the results at higher redshift. We summarize the main results in the following points:
\begin{itemize}
\item We report a new public catalogue of Cosmic Filaments, which inherits the sky footprint of SDSS, and its redshift coverage from $z=0.05$ up to $z=2.2$. All detected points of the Cosmic Web come with an associated uncertainty.
\item We study the distances from galaxies to Cosmic Filaments, and the uncertainty associated to these filaments, both as a function of redshift. We use these as metrics to assess the goodness of the reconstruction. They remain within reasonable bounds even at high redshift.
\item We build upon the SCMS algorithm, which has already proven useful for filament detection and cosmological applications \citep{chen2015}. We show improvements in the isotropy of our results and we extend the coverage to higher redshift and the Southern Galactic Hemisphere, thanks to more recent data.
\item We additionally boost the algorithm with a Machine Learning algorithm that ``learns'' the characteristics of filaments from the most populated redshift slices and use that knowledge to better detect filaments at less populated redshift slices. This proves especially useful to extend the reconstruction to higher redshift slices. We show that, according to the metrics above, this approach also provides more uniform results across redshifts.
\item We study the correlation between our Cosmic Filament catalogue and external Galaxy Cluster catalogues from Planck and ACT as part of the validation of our catalogue. These Galaxy Cluster catalogues are built using the Sunyaev-Zel'dovich effect, a completely independent physical probe. We show a very strong correlation with both catalogues, up to more than $20\sigma$ compared to random points.
\item We study the filament persistence with a realistic example by comparing the filaments found with older data (BOSS) and with all data (BOSS+eBOSS) at redshift slices between $z=0.6$ and $z=0.7$. We show that excluding the new data (up to $60\%$ of total points) only results in a loss of less than $25\%$ of the reconstructed Cosmic Web.
\item Although the algorithm is not designed to isolate single filaments, we performed a first analysis of the filament length distribution assuming that a filament endpoints are in the intersections. We find similar trends to those obtained in other works \citep{malavasi2020, galarragaespinosa2020}
\end{itemize}
The catalogue is publicly available in \href{https://www.javiercarron.com/catalogue}{javiercarron.com/catalogue}, along with python codes to perform standard operations.
We hope that this new catalogue opens the door to more complete studies of the Cosmic Web. In particular, we are currently working on the extension and validation of previously found correlations between filaments and weak lensing of the Cosmic Microwave Background and the Sunyaev-Zeldovich effect in order to constrain mass profiles and their hot gas content \citep{he2018, tanimura2020}. The methodology in this work can be also applied to N-body simulations in order to assess the feasibility of using cosmic filaments as cosmological probes. Finally, we believe that the tomographic reconstruction of the Cosmic Web will be extremely valuable for upcoming wide photometric galaxy surveys, like those expected from Euclid and LSST, for which tridimensional reconstruction is more difficult.
\section*{Acknowledgements}
We thank Yen-Chi Chen for the valuable discussions regarding the SCMS algorithm and the comparison of the catalogues.
We acknowledge support from INFN through the InDark initiative. This work was also supported by ASI/COSMOS grant n. 2016-24-H.0 and ASI/LiteBIRD grant n. 2020-9-HH.0. M.M. is supported by the program for young researchers ``Rita Levi Montalcini" year 2015. D.M. acknowledges support from the MIUR Excellence Project awarded to the Department of Mathematics, Universit\`{a} di Roma Tor Vergata, CUP E83C18000100006.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is \url{www.sdss.org}.
SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard \& Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
Based on observations obtained with Planck (\url{http://www.esa.int/Planck}), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.
This work used the public data produced by the Atacama Cosmology Telescope (\url{https://act.princeton.edu}).
\paragraph*{}
\textit{Software:} We have developed the code used in this work using \textit{Python 3.8} (available at \href{https://www.python.org/}{python.org}) with the JupyterLab environment \citep{jupyter}, and the following packages. For numerical computations, we use \textit{NumPy} \citep{numpy}, \textit{SciPy} \citep{scipy}, and \textit{Astropy} \citep{astropy}. For treatment of the spherical maps, we use \textit{healpy} \citep{gorski2005,healpy} and \textit{MTNeedlet} \citep{carronduque2019}. For treatment of tables and databases, we use \textit{pandas} \citep{pandas}. For parallelization of the code, we use \textit{Ray} \citep{ray}. For visualization, we use \textit{Matplotlib} \citep{matplotlib} and \textit{seaborn} \citep{seaborn}.
\bibliographystyle{aa}
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\section{#1} \setcounter{equation}{0}}
\newtheorem{theo}{Theorem}[section]
\begin{document}
\newtheorem{lem}[theo]{Lemma}
\newtheorem{prop}[theo]{Proposition}
\newtheorem{coro}[theo]{Corollary}
\newtheorem{rk}[theo]{Remark}
\title{Eta Invariant and Holonomy: the Even Dimensional Case}
\author{Xianzhe Dai\thanks{Math Dept, UCSB, Santa Barbara, CA 93106, USA \tt{Email:
dai@math.ucsb.edu}. Partially supported by NSF and NSFC.} \and
Weiping Zhang\thanks{Chern Institute of Mathematics and LPMC,
Nankai University, Tianjin 300071, P. R. China. \tt{Email:
weiping@nankai.edu.cn}. Partially supported by MOE and NNSFC} }
\maketitle
\begin{abstract}
In previous work, we introduced eta invariants for even
dimensional manifolds. It plays the same role as the eta invariant
of Atiyah-Patodi-Singer, which is for odd dimensional manifolds.
It is associated to $K^1$ representatives on even dimensional
manifolds and is closely related
to the so called WZW theory in physics. In fact, it is an intrinsic interpretation
of the Wess-Zumino term without passing to the bounding
$3$-manifold. Spectrally the eta invariant is defined on a finite cylinder, rather than on the
manifold itself. Thus it is an interesting question to find an
intrinsic spectral interpretation of this new invariant. We
address this issue here using adiabatic limit technique. The
general formulation relates the (mod $\mathbb Z$ reduction of) eta
invariant for even dimensional manifolds with the holonomy of the determinant
line bundle of a natural family of Dirac type operators. In this
sense our result might be thought of as an even dimensional
analogue of Witten's holonomy theorem proved by Bismut-Freed
and Cheeger independently.
\end{abstract}
\section{Introduction}
The $\eta$-invariant is introduced by Atiyah-Patodi-Singer in their
seminal series of papers \cite{aps1,aps2,aps3} as the correction
term from the boundary for the index formula on a manifold with
boundary. It is a spectral invariant associated to the natural
geometric operator on the (boundary) manifold and it vanishes for even
dimensional manifolds (in this case the corresponding manifold with
boundary will have odd dimension). In our previous work \cite{dz},
we introduced an invariant of eta type for even dimensional
manifolds. It plays the same role as the eta invariant of
Atiyah-Patodi-Singer.
Any elliptic differential operator on an odd dimensional closed manifold will
have index zero. In this case, the appropriate index to consider is
that of Toeplitz operators. This also fits perfectly with the
interpretation of the index of Dirac operator on even dimensional
manifolds as a pairing between the even $K$-group and $K$-homology.
Thus in the odd dimensional case one considers the odd $K$-group and
odd $K$-homology. For a closed manifold $M$, an element of
$K^{-1}(M)$ can be represented by a differentiable map from $M$ into
the unitary group
\begin{equation} \label{kor} g: \ M \longrightarrow { U}(N), \end{equation} where $N$ is a positive integer. As we mentioned the
appropriate index pairing between the odd $K$-group and $K$-homology
is given by that of the Toeplitz operator, defined as follows.
Consider $L^2(S(TM)\otimes E)$, the space of $L^2$ spinor
fields\footnote{In this paper, for simplicity, we will generally
assume that our manifolds are spin, although our discussion extends
trivially to the case of Dirac type opreatros.} twisted by an
auxilliary vector bundle $E$. It decomposes into an orthogonal
direct sum
$$L^2(S(TM)\otimes E)=\bigoplus_{\lambda\in {\rm Spec}(D^E)} E_\lambda,$$
according to the eigenvalues $\lambda$ of the Dirac operator $D^E$. The ``Hardy space" will be
$$L^2_{\geq 0}(S(TM)\otimes E) =\bigoplus_{\lambda\geq 0}E_\lambda. $$
The corresponding orthogonal projection from $L^2(S(TM)\otimes E)$
to $L^2_{\geq 0}(S(TM)\otimes E)$ will be denoted by $P^E_{\geq 0}$.
The Toeplitz operator $T^E_g$ is then defined as \begin{equation} \label{toe}
T^E_g=P^E_{\geq 0}gP^E_{\geq 0}:L^2_{\geq 0}\left(S(TM)\otimes E
\otimes {\bf C}^N\right) \longrightarrow L^2_{\geq
0}\left(S(TM)\otimes E \otimes {\bf C}^N\right).\end{equation} This is a
Fredholm operator whose index is given by \begin{equation} \label{tifcm} {\rm
ind}\, T^E_g=-\left\langle \widehat{A}(TM){\rm ch}(E){\rm
ch}(g),[M]\right\rangle ,\end{equation} where ${\rm ch}(g)$ is the odd Chern
character associated to $g$ \cite{bd}. It is represented by the
differential form (cf. [Z1, Chap. 1])
$$
{\rm ch}(g)= \sum_{n=0}^{\dim M-1\over 2} {n!\over (2n+1)!}{\rm
Tr}\left[\left(g^{-1}dg\right)^{2n+1}\right].
$$
In \cite{dz} we established an index theorem which generalizes
(\ref{tifcm}) to the case where $M$ is an odd dimensional spin
manifold with boundary $\partial M$. The definition of the
Toeplitz operator now uses Atiyah-Patodi-Singer boundary
conditions on $\partial M$. The self adjoint Atiyah-Patodi-Singer
boundary conditions depend on choices of Lagrangian subspaces $L
\subset \ker D^E_{\partial M}$. We will denote the corresponding
boundary condition by $P_{\partial M}(L)$. The resulting Toeplitz
operator will then be denoted by $T^E_g(L)$.
We recall the main result in \cite{dz} as follows.
\begin{theo} \label{dz} The Toeplitz operator $T^E_g(L)$ is Fredholm with index given
by \begin{eqnarray} \label{titfmwb}
{\rm ind}\, T^E_g(L)& = & -\left({1\over 2\pi\sqrt{-1}}\right)^{(\dim M+1)/2}\int_M
\widehat{A}\left(R^{TM}\right){\rm
Tr}\left[\exp\left(-R^E\right)\right]{\rm ch}(g) \nonumber \\
& & - \ \overline{\eta}(\partial M, E, g) + \tau_\mu
\left(gP_{\partial M}(L) g^{-1},P_{\partial M}(L) , {\mathcal
P}_M\right) .
\end{eqnarray}
\end{theo}
Here $\overline{\eta}(\partial M, E, g)$ denotes the
invariant of $\eta$-type for even dimensional manifold $\partial M$ and
the $K^1$ representative $g$. The third term is an interesting new
{\em integer} term, a triple Maslov index introduced in [KL]. See
\cite{dz} for details.
\noindent{\bf Remark}. Our index formula is closely related
to the so called WZW theory in physics \cite{w0}. When $\partial M=S^2$ or
a compact Riemann surface and $E$ is trivial, the local term in
(\ref{dz}) is precisely the Wess-Zumino term, which allows an integer
ambiguity, in the WZW theory. Thus, our eta invariant
$\overline{\eta}(\partial M, g)$ gives an intrinsic interpretation
of the Wess-Zumino term without passing to the bounding
$3$-manifold. In fact, for $\partial M=S^2$, it can be further reduced to
a local term on $S^2$ by using Bott's periodicity, see \cite[Remark
5.9]{dz}.
The eta invariant $\overline{\eta}(\partial M, E, g)$ is defined
on a finite cylinder $[0, 1]\times \partial M$, rather than on $\partial M$
itself. Thus it is an interesting question to find an intrinsic
spectral interpretation of this new invariant. In this paper we
answer this question by using the adiabatic limit technique.
First, under invertibility assumptions, we give an explicit
formula for our eta invariant in terms of a natural family of
Dirac type operators on the manifold. This family arises from the
original Dirac operator by a perturbation involving the $K^1$
representative. The general formulation relates (the mod $\mathbb
Z$ reduction of) the eta invariant for even dimensional manifolds
with the holonomy of the determinant line bundle of this natural
family of Dirac type operators. The work of \cite{d} on the
adiabatic limit of eta invariants for manifolds with boundary and
that of \cite{df} on Witten's Holonomy Theorem play an important
role here.
This paper is organized as follows. In Section 2, we review the
definition of the eta invariant for an even dimensional closed
manifold introduced in \cite{dz}. In Section 3, we give an intrinsic
spectral interpretation of the eta invariant under certain
invertibility assumption. Section 4 deals with the general case. And
we end with a conjecture and a few remarks in the last section.
Some of the results in this paper have been described in \cite{d3}.
\section{ An invariant of $\eta$ type for even dimensional
manifolds}
For an even dimensional closed manifold $X$ (which may or may not be
the boundary of an odd dimensional manifold) and a $K^1$
representative $g: X \rightarrow U(N)$, the eta invariant will be defined in
terms of an eta invariant on the cylinder $[0, 1]\times X$ with
appropriate APS boundary conditions.
In general, for a compact manifold $M$ with boundary $\partial M$ with
the product structure near the boundary, the Dirac operator $D^E$
twisted by an hermitian vector bundle $E\otimes{\bf C}^N$
decomposes near the boundary as \begin{equation} \label{ps} D^E =
c\left({\partial \over
\partial x}\right) \left( {\partial \over
\partial x} + D^E_{\partial M} \right). \end{equation}
The APS projection $P_{\partial M}$ is an elliptic global boundary
condition for $D^E$. However, for self adjoint boundary
conditions, we need to modify it by a Lagrangian subspace of $\ker
D^E_{\partial M}$, namely, a subspace $L$ of $\ker D^E_{\partial M}$
such that $c({\partial \over
\partial x})L=L^\perp \cap (\ker D^E_{\partial M})$. Since $\partial
M$ bounds $M$, by the cobordism invariance of the index, such
Lagrangian subspaces always exist.
The modified APS projection is then obtained by reducing the kernel
part of the projection to the projection onto the Lagrangian
subspace. More precisely, denote
$$L^2_{+}((S(TM) \otimes E\otimes{\bf
C}^N)|_{\partial M}) =\bigoplus_{\lambda>
0}E_\lambda(D^{E\otimes{\bf C}^N}_{\partial M}), $$ where $\lambda$
runs over the positive eigenvalues of $D^{E\otimes{\bf
C}^N}_{\partial M}$. Denote by $P_{\partial M}$ the orthogonal
projection from $L^2((S(TM) \otimes E\otimes{\bf C}^N)|_{\partial
M})$ to $L^2_{+}((S(TM)\otimes E\otimes{\bf C}^N)|_{\partial M})$.
Let $P_{\partial M}(L)$ denote the orthogonal
projection operator from $L^2((S(TM) \otimes E\otimes{\bf
C}^N)|_{\partial M})$ to $L^2_{+}((S(TM)\otimes E\otimes{\bf
C}^N)|_{\partial M})\oplus L$: \begin{equation} \label{mapsp} P_{\partial M}(L)
=P_{\partial M}+P_L, \end{equation} where $P_L$ denotes the orthogonal
projection from $L^2((S(TM)\otimes E\otimes{\bf C}^N)|_{\partial
M})$ to $L$.
The pair $(D^E, P^E_{\partial M}(L) )$ forms a self-adjoint elliptic
boundary problem, and $P_{\partial M}(L) $ is called an
Atiyah-Patodi-Singer boundary condition associated to $L$. We will
denote the corresponding elliptic self-adjoint operator by
$D^E_{P_{\partial M}(L) }$.
In \cite{dz}, we originally intend to consider the conjugated
elliptic boundary value problem $D^{E}_{gP_{\partial M}(L)
g^{-1}}$ (cf. \cite{z2}). However, the analysis turns out to be
surprisingly subtle and difficult. To circumvent this difficulty,
a perturbation of the original problem was constructed.
Let $\psi=\psi(x)$ be a cut off function which is identically $1$ in
the $\epsilon$-tubular neighborhood of $\partial M$ ($\epsilon
>0$ sufficiently small) and vanishes outside the $2\epsilon$-tubular
neighborhood of $\partial M$. Consider the Dirac type operator
$$
D^{\psi}=(1-\psi)D^E + \psi gD^E g^{-1}.
$$
The motivation for considering this perturbation is that, near the boundary, the
operator $D^{\psi}$ is actually given by the conjugation of $D^E$,
and therefore, the elliptic boundary problem $(D^{\psi},
gP_{\partial M}(L) g^{-1})$ is now the conjugation of the APS
boundary problem $(D^E, P_{\partial M}(L))$, i.e., this is now
effectively standard APS situation and we have a self adjoint
boundary value problem $(D^{\psi}, gP_{\partial M}(L) g^{-1})$
together with its associated self adjoint elliptic operator
$D^{\psi}_{gP_{\partial M}(L) g^{-1}}$.
The same thing can be said about the conjugation of $D^{\psi}$: \begin{equation}
\label{pdo} D^{\psi, g}=g^{-1} D^{\psi} g =D^E + (1-\psi)
g^{-1}[D^E, g] . \end{equation}
We will in fact use $D^{\psi, g}$.
We are now ready to construct the eta invariant for even
dimensional manifolds. Given an even dimensional closed spin
manifold $X$, we consider the cylinder $[0, 1]\times X$ with the
product metric. Let $g: \ X \rightarrow U(N)$ be a map from $X$ into the
unitary group which extends trivially to the cylinder. Similarly,
$E \rightarrow X$ is an Hermitian vector bundle which is also extended
trivially to the cylinder. We assume that ${\rm ind}\, D^E_+=0$ on
$X$ which guarantees the existence of the Lagrangian subspaces
$L$.
Consider the analog of $D^{\psi,g}$ as defined in (\ref{pdo}), but
now on the cylinder $[0, 1]\times X$ and denote it by
$D^{\psi,g}_{[0, 1]}$. Here $\psi=\psi(x)$ is a cut off function on $[0, 1]$ which is identically $1$ for $0\leq x \leq \epsilon$ ($\epsilon
>0$ sufficiently small) and vanishes when $1-2\epsilon\leq x \leq 1$. We equip it with the boundary condition
$P_{X}(L) $ on one of the boundary components $\{0\}\times X$ and
the boundary condition ${\rm Id}-g^{-1}P_{X}(L) g $ on the other
boundary component $\{1\}\times X$ (Note that the Lagrangian
subspace $L$ exists by our assumption of vanishing index). Then
$(D^{\psi,g}_{[0, 1]}, P_X(L) , {\rm Id}-g^{-1}P_{X}(L) g)$ forms a
self-adjoint elliptic boundary problem. For simplicity, we will
still denote the corresponding elliptic self-adjoint operator by
$D^{\psi, g}_{[0, 1]}$.
Let $\eta(D^{\psi, g}_{[0, 1]},s)$ be the $\eta$-function of
$D^{\psi, g}_{[0, 1]}$ which, when ${\rm Re}(s)>>0$, is defined by
\begin{equation} \label{ef} \eta(D^{\psi, g}_{[0, 1]},s) =\sum_{\lambda \neq
0}{{\rm sgn}(\lambda)\over |\lambda|^s}, \end{equation}
where $\lambda$ runs
through the nonzero eigenvalues of $D^{\psi, g}_{[0, 1]}$.
By \cite{mu,df}, one knows that the $\eta$-function $\eta(D^{\psi,
g}_{[0, 1]},s)$ admits a meromorphic extension to ${\bf C}$ with
$s=0$ a regular point (and it has only simple poles). One then
defines, as in \cite{aps1}, the $\eta$-invariant of
$D^{\psi,g}_{[0,1]}$ by $\eta(D^{\psi,g}_{[0, 1]})= \eta
(D^{\psi,g}_{[0, 1]},0 )$, and the reduced $\eta$-invariant by
\begin{equation} \label{rei} \overline{\eta}\left(D^{\psi,g}_{[0,
1]}\right)={\dim \ker D^{\psi,g}_{[0, 1]} +
\eta\left(D^{\psi,g}_{[0, 1]}\right) \over 2}. \end{equation}
\noindent {\bf Definition 2.1.} {We define an invariant of $\eta$
type for the Hermitian vector bundle $E$ on the even dimensional
manifold $X$ (with vanishing index) and the $K^1$ representative
$g$ by \begin{equation} \label{eifedm} \overline{\eta}(X,E, g)=
\overline{\eta} \left(D^{\psi, g}_{[0, 1]}\right) - {\rm sf} \left\{D^{\psi,g}_{[0, 1]}(s);
0 \leq s \leq 1 \right\},
\end{equation} where $D^{\psi,g}_{[0, 1]}(s)$ is a path connecting $g^{-1} D^E
g$ with $D^{\psi,g}_{[0, 1]}$ defined by
$$
D^{\psi,g}(s)= D^E + (1-s\psi) g^{-1}[D^E, g]
$$
on $[0, 1]\times X$, with the boundary condition $P_{X}(L) $ on
$\{0\}\times X$ and the boundary condition ${\rm
Id}-g^{-1}P_{X}(L) g$ at $\{1\}\times X$.}
\newline
It was shown in \cite{dz} that $\overline{\eta}(X, E,
g)$ does not depend on the cut off function $\psi$.
\section{An intrinsic spectral interpretation, the invertible case}
The usefulness of the eta invariant of Atiyah-Patodi-Singer comes,
at least partially, from the spectral nature of the invariant,
i.e. that it is defined via the spectral data of the Dirac
operator on the (odd dimensional) manifold. Our eta invariant for
even dimensional manifold is defined via the eta invariant on the
corresponding odd dimensional cylinder by imposing APS boundary
conditions. Thus, it will be desirable to have a direct spectral
interpretation in terms of the spectral data of the original
manifold (and the $K^1$ representative). In this section we give
such an interpretation under certain invertibility assumption.
This invertibility condition will be removed in the next section.
The crucial point here is the following observation. As in the
previous section, we can also consider the invariant
$\overline{\eta}(D^{\psi,g}_{[0, a]})$, similarly constructed on a
cylinder $[0, a] \times X$ of radial size $a>0$.
\begin{lem} Assuming that $ \ker [D_X + s\ c(g^{-1} dg)] =0,\ \ \forall\ 0
\leq s \leq 1$, then $\overline{\eta}(D^{\psi,g}_{[0, a]})$, and
hence $\overline{\eta}(X,E, g)$, is independent of $a$. Without
the invertibility assumption, the mod $\mathbb Z$ reduction of
$\overline{\eta}(X,E, g)$ is independent of $a$.
\end{lem}
This can be seen by a rescaling argument (cf. [M\"u, Proposition
2.16], see also \cite[Theorem 3.2]{d2}).
On the other hand, as we mentioned before,
\begin{lem}
$\overline{\eta}(X,E, g)$, is independent of the choice of the cut
off function $\psi$.
\end{lem}
This is Proposition 5.1 of \cite{dz}.
These two lemmas together show that \begin{equation} \label{ko}
\overline{\eta}(X,E, g)= \lim_{a \rightarrow \infty}
\overline{\eta}(D^{\psi,g}_{[0, a]}) \end{equation} for any cut off function
which may depend on $a$ ((\ref{ko}) is to be interpreted as an
equation mod $\mathbb Z$ without the invertibility assumption).
This is exactly the adiabatic limit.
\newline
We now recall the setup and result from \cite{d} on the adiabatic limit of eta invariant, which is an
extension of \cite{bc} to manifolds with boundary. More precisely,
let \begin{equation} Y\rightarrow M\stackrel{\pi}{\rightarrow} B \end{equation} be a fibration where the
fiber $Y$ is closed but the base $B$ may have nonempty boundary.
Let $g_B$ be a metric on $B$ which is of the
product type near the boundary $\partial B$. Now equip $M$ with a
submersion metric $g$,
\[ g = \pi^{*} g_{B} + g_Y \]
so that $g$ is also product near $\partial M$. This is equivalent to
requiring $g_Y$ to be independent of the normal variable near $\partial
B$, given by the distance to $\partial B$.
The adiabatic metric $g_x$ on $M$ is given by \begin{equation} g_x =x^{-2}
\pi^{*} g_{B} + g_Y, \end{equation} where $x$ is a positive parameter.
For simplicity we assume that $M$ as well as the vertical tangent
bundle $T^VM$ are spin. Associated to these data we have in
particular the total Dirac operator $D_x$ on $M$, the boundary Dirac
operator $D_x^{\partial M}$ on $\partial M$, and the family of Dirac operators
$D_Y$ along the fibers. If the family $D_Y$ is invertible, then,
according to \cite{bc}, the boundary Dirac operator $D_x^{\partial M}$ is
also invertible for all small $x$, therefore the eta invariant of
$D_x$ with the APS boundary condition, $\eta(D_x)$, is
well-defined. We have the following result from \cite{d}.
\begin{theo} \label{al} Consider the fibration $Y\rightarrow M\rightarrow B$ as above. Assume that the
Dirac family along the fiber, $D_Y$, is invertible. Consider the
total Dirac operator $D_x$ on $X$ with respect to the adiabatic
metric $g_x$ and let $\eta(D_x)$ denote the eta invariant of $D_x$
with the APS boundary condition. Then the limit $\lim_{x
\rightarrow 0} \bar{\eta}(D_x) = \lim_{x \rightarrow 0}
\frac{1}{2} \eta(D_x)$ exists in ${\Bbb R}$ and \begin{equation} \label{alf}
\lim_{x \rightarrow 0} \bar{\eta}(D_x) =
\int_B\hat{A}\left(\frac{R^B}{2\pi}\right) \wedge \tilde{\eta},
\end{equation} where $R^B$ is the curvature of $g_B$, $\hat{A}$ denotes the
the $\hat{A}$-polynomial and $\tilde{\eta}$ is the $\eta$-form of
Bismut-Cheeger \cite{bc}.
\end{theo}
Recall that the (unnormalized) $\eta$-form of Bismut-Cheeger, the
$\hat{\eta}$ form, is defined as \begin{equation} \label{ehf} \hat{\eta} =\left\{
\begin{array}{ll} {\displaystyle
\int}_{0}^{\infty}{\rm tr}_{s}\left[\left(D_{Y} +
\frac{c(T)}{4t}\right) e^{-B_{t}^{2}}\right]
\frac{dt}{2t^{1/2}} & \mbox{if $\dim Y=2l$} \\
& \\
{\displaystyle \int}_{0}^{\infty} {\rm
tr}^{\rm even}\left[\left(D_Y + \frac{c(T)}{4t}\right)
e^{-B_{t}^{2}}\right] \frac{dt}{2t^{1/2}} & \mbox{if $\dim
Y=2l-1$}
\end{array} \right. , \end{equation}
assuming that $\ker D_Y$ does define a vector bundle on $B$. Here
$B_t$ denotes the rescaled Bismut superconnection: \begin{equation} \label{rbs}
B_t =\tilde{\nabla}^u + t^{1/2}D_Y - \frac{c(T)}{4t^{1/2}}. \end{equation}
We normalize $\hat{\eta}$ by defining \begin{equation} \label{etf} \tilde{\eta} =
\left\{
\begin{array}{ll} {\displaystyle \sum \frac{1}{(2\pi
i)^j} [\hat{\eta}]_{2j-1}} & \mbox{if $\dim Y=2l$} \\
{\displaystyle \sum \frac{1}{(2\pi i)^j} [\hat{\eta}]_{2j}} &
\mbox{if $\dim Y=2l-1$}
\end{array} \right.. \end{equation}
We now turn to the intrinsic spectral interpretation of our eta
invariant.
\begin{theo} \label{mtic} Under the assumption that $\label{ta} \ker [D_X + s\ c(g^{-1} dg)] =0,\ \ \forall\ 0
\leq s \leq 1$,
$$
\overline{\eta} \left(X, E,g\right)= \frac{i}{4\pi} \int_0^1
\int_{0}^{\infty}
{\rm tr}_{s}\left[c\left(g^{-1} dg\right)\left(D_{X} + s\,
c\left(g^{-1} dg\right)\right) e^{-t\left(D_{X} + s\,
c\left(g^{-1} dg\right)\right)^2}
\right] dt \ ds .
$$
\end{theo}
\noindent {\em Proof.} We apply Theorem \ref{al} to our current situation where $M=[0,
1] \times X$ fibers over $[0, 1]$ with the fibre $X$. The operator
\[ D_{[0,1]}^{\psi, g} =D^E + (1-\psi)
g^{-1}[D^E, g]=D^E + (1-\psi) c(g^{-1} dg) \] is of Dirac type,
and of product type near the boundaries. Hence the result still
applies.
By the invertibility assumption there is no spectral flow
contribution and hence, by (\ref{ko}), $ \overline{\eta} \left(X,
E,g\right)$ is given by the adiabatic limit formula.
The Dirac family along the fiber is $D_X + (1-\psi(x))c(g^{-1} dg)$.
The curvature of the Bismut superconnection is given by
\[ B_t^2= t \left[D_X + (1-\psi(x))c\left(g^{-1} dg\right)\right]^2 - t^{1/2} \psi'(x) dx\, c\left(g^{-1}
dg\right). \]
Thus,
\[ \hat{\eta} = \frac{\psi'(x)dx}{2}
\int_{0}^{\infty} {\rm tr}_{s}\left[c\left(g^{-1} dg\right)\left(D_X
+ (1-\psi(x))c\left(g^{-1} dg\right)\right) e^{-t \left(D_X +
(1-\psi(x))c\left(g^{-1} dg\right)\right)^2}\right] dt .\]
Since $\tilde{\eta}= \frac{1}{2\pi i} \hat{\eta}$ here, the
adiabatic limit formula in Theorem \ref{al} gives
\[ \begin{array}{l}
\overline{\eta}(X, E, g) \\
= \displaystyle\int_0^1
\frac{\psi'(x)}{4\pi i}
\int_{0}^{\infty} {\rm tr}_{s}\left[c\left(g^{-1} dg\right)\left(D_X + (1-\psi(x))c\left(g^{-1}
dg\right)\right) e^{-t \left(D_X + (1-\psi(x))c\left(g^{-1} dg\right)\right)^2}\right] dt dx \\
= \displaystyle \frac{i}{4\pi } \int_0^1
\int_{0}^{\infty} {\rm tr}_{s}\left[c\left(g^{-1} dg\right)\left
(D_{X} + s\, c\left(g^{-1} dg\right)\right) e^{-t\left(D_{X} + s\,
c\left(g^{-1} dg\right)\right)^2}
\right] dt \ ds \end{array} \]
as claimed.
\hspace*{\fill}\Box
\section{The noninvertible case}
For a fibration over the circle, Witten's Holonomy Theorem
\cite{w,bf,c} says that the adiabatic limit of the eta invariant
of the total space is related to the holonomy of the determinant
line bundle of the family operators along the fibers. Indeed, in
the invertible case, namely the family operators along the fibers are invertible,
there is an explicit formula for the adiabatic limit of the eta
invariant in terms of the family operators, \cite[(3.166)]{bf},
\cite[(1.56)]{c}, which states \begin{equation} \label{altic} \lim_{x
\rightarrow 0} \bar{\eta}(D_x) = \frac{i}{4\pi} \int_{S^1}
\int_0^{\infty} {\rm tr}_{s}\left[\left(\tilde{\nabla}^u
D_Y\right)D_Y e^{-tD_Y^2}\right] dt. \end{equation} Of course, the integrand
in the formula (\ref{altic}) is just the degree one term of the
Bismut-Cheeger $\eta$-form.
If one applies (\ref{altic}) to the family $s \in [0,1]
\longrightarrow D_X + s\ c(g^{-1} dg)$, we would obtain Theorem
\ref{mtic}. However, the family here is not periodic. Nevertheless,
it is almost periodic in the sense that the operators at the
endpoints differ by a conjugation. This leads us to the
generalization to the general noninvertible case.
To deal with the noninvertible case, we make use of the framework
and result of \cite{df}. We first recall the setup of \cite{df}.
Suppose $M$ is a compact odd dimensional Riemannian manifold with nonempty
boundary. For simplicity, we assume $M$ is spin so that one can
consider the Dirac operator $D_M$ (the same consideration can be
adapted to Dirac type operators). Further, assume that the metric is of product type near the boundary.
In order to consider eta invariant, one needs to impose boundary conditions and the self adjoint APS boundary condition
amounts to a ``trivialization" of the graded kernel of the boundary Dirac operator $D_{\partial M}$. Taking this into consideration, the result of \cite{df} says that the exponentiated eta invariant of $D_M$ actually defines an element of the inverse determinant line of the boundary Dirac operator $D_{\partial M}$.
More precisely, let $K_{\partial M}^+\oplus K_{\partial M}^-$ be the (graded) kernel of $D_{\partial M}$ and $\mbox{\rm Det}^{-1}_{\partial M}$ the inverse determinant line of $D_{\partial M}$:
\begin{equation} \label{idl} \mbox{\rm Det}^{-1}_{\partial M}=\Lambda^{{\rm max}}K_{\partial M}^+\hat{\otimes} [\Lambda^{{\rm max}} K_{\partial M}^-]^{-1} .\end{equation}
Here inverse denotes the dual.
A self adjoint APS boundary condition is determined by a choice of isometry
\begin{equation} \label{sdbc} T\ :K_{\partial M}^+ \longrightarrow K_{\partial M}^-. \end{equation}
Let $\overline{\eta}(D_M(T))$ denote the reduced eta invariant of $D_M$ with
the self adjoint APS boundary condition determined by $T$ (cf.
\cite{df}). A basic result of \cite{df} says that
\begin{equation} \label{brodf} \tau_M = e^{2\pi i \overline{\eta}(D_M(T))}( \det T)^{-1} \in \mbox{\rm Det}^{-1}_{\partial M} \end{equation}
is independent of $T$ (and satisfies the laws of TQFT as well as a variation formula).
Relevant to our discussion here is Witten's Holonomy Theorem as
formulated in this framework. Let $\pi :Y\to Z$ be a fibration whose
typical fiber is a closed even dimensional manifold, and as before
we assume that both $Y$ and $T^VY$ are spin for simplicity. Let
$L\to Z$ denote the corresponding inverse determinant line bundle.
It comes equipped with a (Quillen) metric and a natural unitary
(Bismut-Freed) connection $\nabla $. The curvature of $\nabla $ is
\cite[Theorem 1.21]{bf}
$$ \Omega ^L = -2 \pi i \left[\int_{Y/Z}\hat{A}(\Omega ^{Y/Z})\right]_{(2)}. $$
Given $\gamma: [0, 1]\to Z$ a smooth path, let $Y_\gamma =\gamma
^*Y$ denote the pullback of $\pi :Y\to Z$ via $\gamma $; then $\pi
_\gamma :Y_\gamma \to [0, 1]$ is a fibration, the induced fibration.
Let $g_{[0,1]}$ denote an arbitrary metric on the unit interval and
$g_{Y/Z}$ the metric on the vertical tangent bundle $T^VY$. Define a
family of metrics on $Y_\gamma $ by the formula
$$ g_\epsilon = \frac{g_{[0,1]}}{\epsilon ^2}\oplus g_{Y/Z},\qquad \epsilon \not= 0. $$
(We assume that $\gamma$ is constant near the two endpoints so that $g_\epsilon$ is of the product type near the boundary.)
The construction above gives rise to a linear map
\begin{equation} \label{bcodf} \tau _{Y_\gamma }(\epsilon ) :L_{\gamma (0)}\longrightarrow L_{\gamma (1)}. \end{equation}
\begin{theo}[Dai-Freed] The adiabatic limit $\tau _\gamma =\lim\limits_{\epsilon \to0}\tau _{Y_\gamma
}(\epsilon )$ exists and gives the holonomy along $\gamma$ of the Bismut-Freed connection.
\end{theo}
Consider now the fibration $\pi: \mathbb R \times X \longrightarrow \mathbb R$ given by the projection, with the family of Dirac type operators
\begin{equation} \label{fdo} s\in \mathbb R \rightarrow D_s = D_X + s c(g^{-1}dg).\end{equation}
Let $L\rightarrow \mathbb R$ be the inverse determinant line bundle with the
Quillen metric and the Bismut-Freed connection. Denote by $L_s$ the fiber of $L$
at $s\in \mathbb R$. Since $D_1=D_X +
c(g^{-1}dg)=g^{-1}D_X g=g^{-1}D_0 g$, there is an isomorphism
\begin{equation} \label{isom} g^{-1}: \, L_0 \simeq L_1\end{equation}
determined by the isomorphism $g^{-1}$ between the graded kernels
$\ker D_0$ and $\ker D_1$. On the other hand, since $\mathbb R$ is one dimensional, any
two monotonic paths from $0$ to $1$ are reparametrizations of each
other. Hence there is a unique holonomy map
\begin{equation} \label{hm} \tau_{0,1}: L_0 \rightarrow L_1. \end{equation}
Composing with the isomorphism (\ref{isom}) gives rise to a map
\begin{equation} \label{im} L_0 \rightarrow L_1 \simeq L_0 \end{equation}
which can then be identified with a complex number $\tau\in \mathbb
C$. In fact, since both the holonomy map (\ref{hm}) and the isomorphism (\ref{isom}) are
unitary maps, $\tau$ has modulus one.
We can now state the main result of this paper as follows.
\begin{theo} \label{mt} We have
\[ \tau = e^{2\pi i \overline{\eta} \left(X, E,g\right)}. \]
\end{theo}
\noindent {\em Proof.} By taking the exponential we discount the contribution from the
spectral flow to our eta invariant. Thus we are only concerned with
$\overline{\eta}(D^{\psi, g}_{[0, 1]})$. By definition, $\overline{\eta}(D^{\psi, g}_{[0, 1]})$
is the reduced eta invariant of
\[ D^{\psi, g}=D^E + (1-\psi)
g^{-1}[D^E, g] \] on the cylinder $[0, 1]\times X$ with the
boundary condition $P_{X}(L) $ on one of the boundary components
$\{0\}\times X$ and the boundary condition ${\rm Id}-g^{-1}P_{X}(L)
g $ on the other boundary component $\{1\}\times X$, where $L$ is a
Lagrangian subspace of $\ker D_X$. Let $\ker D_X=K^+_X \oplus K^-_X$
be its $\mathbb Z_2$ grading. Then an isometry
\[ T:\ K^+_X \rightarrow K^-_X \]
gives rise to a Lagrangian subspace, namely the graph of $T$. A
little linear algebra shows that the boundary condition ${\rm
Id}-g^{-1}P_{X}(L) g $ corresponds to the isometry
\[ g^{-1}T^{-1}g:\ g^{-1}K^-_X \rightarrow g^{-1} K^+_X. \]
Hence, we have by the previous theorem and the definition (using the
notation $a$-$\lim$ to denote the adiabatic limit)
\begin{eqnarray*} \tau_{0, 1} & = & a\mbox{-} \lim e^{2\pi i
\overline{\eta}(D^{\psi, g}_{[0, 1]})} (\det T)^{-1} \det g^{-1}T g
\\
&=& \lim_{a\rightarrow \infty} e^{2\pi i \overline{\eta}(D^{\psi, g}_{[0,
a]})} (\det T)^{-1} \det g^{-1}T g \\
&=& e^{2\pi i \overline{\eta}(D^{\psi, g}_{[0, 1]})} (\det T)^{-1}
\det g^{-1}T g. \end{eqnarray*} Therefore
\[ \tau = e^{2\pi i \overline{\eta} \left(X, E,g\right)} \]
using the identification. \hspace*{\fill}\Box
$\ $
\begin{rk} Recall that in [DZ, Remarks 2.5 and 5.9], the
$\eta$-invariant is used to give an intrinsic analytic
interpretation of the Wess-Zumino term in the WZW theory. Now by
Theorem 4.2, this term is further interpreted by using holonomy.
\end{rk}
$\ $
\begin{rk} Theorem 4.1 gives an adiabatic limit formula for
(reduced) eta invariants without invertibility assumption for one
dimensional manifolds with boundary, namely the interval. Theorem
3.3, on the other hand, is such a formula with invertibility
assumption, but for any compact manifold with boundary as the
base. It will be interesting to have a general result combining
these two. This will be addressed elsewhere.
\end{rk}
\section{Final remarks}
We end this paper by recalling a conjecture from \cite{dz}, and
also by some remarks.
As we mentioned before, the eta type invariant $\overline{\eta}(X,
E, g)$, which we introduced using a cut off function, is in fact
independent of the cut off function. This leads naturally to the
question of whether $\overline{\eta}(X, E, g)$ can actually be
defined directly. The following conjecture is stated in \cite{dz} and \cite{z2}.
Let $D^{[0, 1]}$ be the Dirac operator on $[0, 1]\times X$. We equip
the boundary condition $gP_{X}(L) g^{-1}$ at $\{0\}\times X$ and the
boundary condition ${\rm Id}-P_{X}(L)$ at $\{ 1 \}\times X$.
Then $(D^{[0, 1]}, gP_{X}(L) g^{-1} , {\rm Id}-P_{X}(L)
)$ forms a self-adjoint elliptic boundary problem. We denote
the corresponding elliptic self-adjoint operator by
$D^{[0,1]}_{gP_{X}(L) g^{-1} , P_{X}(L) }$.
Let $\eta(D^{[0,1]}_{gP_{X}(L) g^{-1}, P_{X}(L) },s)$ be the
$\eta$-function of $D^{[0,1]}_{gP_{X}(L) g^{-1} , P_{X}(L)
}$. By [KL, Theorem 3.1], one knows
that the $\eta$-function $\eta(D^{[0,1]}_{gP_{X}(L) g^{-1} ,
P_{X}(L) },s)$ admits a meromorphic extension to ${\bf C}$ with
poles of order at most 2. One then defines, as in [KL, Definition
3.2], the $\eta$-invariant of $D^{[0,1]}_{gP_{X}(L) g^{-1}, P_{X}(L)
}$, denoted by $\eta(D^{[0,1]}_{gP_{X}(L) g^{-1} , P_{X}(L) })$, to
be the constant term in the Laurent expansion of
$\eta(D^{[0,1]}_{gP_{X}(L) g^{-1}, P_{X}(L) },s)$ at $s=0$.
Let $\overline{\eta}(D^{[0,1]}_{gP_{X}(L) g^{-1}, P_{X}(L)
})$ be the associated reduced $\eta$-invariant.
$\ $
\noindent {\bf Conjecture}:
$$
\overline{\eta}(X, E, g)=
\overline{\eta}\left(D^{[0,1]}_{gP_{X}(L) g^{-1} , P_{X}(L)
}\right).
$$
\newline
We would also like to say a few words about the technical assumption that
${\rm ind}\, D^E_+=0$ imposed in order to define the eta invariant $\overline{\eta}(X, E, g)$.
The assumption guarantees the existence of the Lagrangian subspaces
$L$ which are used in the boundary conditions. In the Toeplitz index theorem, this
assumption is automatically satisfied since $X=\partial M$ is a boundary. In general, of course,
it may not. However, if one is willing to overlook the integer contribution (as one often does
in applications), this technical issue can be overcome by using another eta invariant, this time
on $S^1 \times X$, as follows. Note that we now have no boundary, hence no need for boundary conditions!
Consider $S^1 \times X= [0, 1] \times X / \sim$ where $\sim$ is the equivalence relation that identifies $0\times X$ with $1\times X$. Let $E_g \rightarrow S^1 \times X$ be the vector bundle which is $E\otimes {\mathbb C}^N$ over $(0, 1) \times X$
and the transition from $0\times X$ to $1\times X$ is given by $g:\ X \rightarrow U(N)$. Denote by $D_{E_g}$ the Dirac operator on $S^1 \times X$ twisted by $E_g$.
\begin{prop}\footnote{We thank Jean-Michel Bismut for pointing this out to us several years ago.} One has
\[ \overline{\eta}(X, E, g) \equiv \overline{\eta}(D_{E_g}) \ \ \ {\rm mod} \ \ \mathbb Z . \]
\end{prop}
This is an easy consequence of the so called gluing law for the
eta invariant, see \cite{bu,bl,df}. An analog of this result in
the noncommutative setting plays an important role in \cite{x},
which also contains an odd dimensional analog of \cite{lmp}.
$\ $
\begin{rk} An application of the Witten holonomy theorem (\cite{w,
bf, c}) to the right hand side of the above formula leads to an analogous result as
Theorem \ref{mt}. However the family of operators here is not as explicit as in Theorem \ref{mt}.
\newline
\end{rk}
\begin{rk} It might be interesting to note the duality that
$\overline{\eta} (X, E,g)$
is a spectral invariant associated to a $K^1$-representative
on an {\it even} dimensional manifold, while the usual
Atiyah-Patodi-Singer $\eta$-invariant ([APS1]) is a spectral
invariant associated to a $K^0$-representative on an {\it odd}
dimensional manifold.
\end{rk}
$\ $
|
1205.0034
|
\section{Introduction}
Cluster algebra was invented by Fomin-Zelevinsky in 2000,
attempting to understand total positivity in algebraic groups and canonical bases in quantum groups.
It has been heavily studied during the last decade due to
its wide connection to many areas in mathematics,
(for more details, see the introduction survey \cite{Kel12}).
The combinatorial ingredient in the cluster theory is quiver mutation,
which leads to the categorification of cluster algebra via quiver representation theory
due to Buan-Marsh-Reineke-Reiten-Todorov in 2005.
Recently, Keller spotted a remarkable special case of quiver mutation by adding certain restrictions,
known as the green quiver mutation (Definition~\ref{def:green}); using which,
he obtained results concerning Kontsevich-Soibelman's noncommutative Donaldson-Thomas invariant
via quantum cluster algebras.
Inspired by Keller \cite{Kel11} and Nagao \cite{Nag10},
King-Qiu \cite{KQ11} studied the exchange graphs of hearts and clusters
in various categories associated to cluster categories,
with applications to stability conditions and quantum dilogarithm identities in \cite{Qiu11}.
Another motivation studying green mutation sequences is coming from
theoretical physics where they yield the complete spectrum of BPS states, cf. \cite{CCV}.
Our first aim in this paper is to interpret
Keller's green mutation in terms of tilting.
More precisely, a green sequence $\mathbf{s}$ induces
a path $\mathrm{P}(\mathbf{s})$ in the exchange graph $\EG_Q$ (cf. Definition~\ref{def:eg}),
that is, a sequence of simple (backward) tilting.
Thus $\mathbf{s}$ corresponds to a heart $\h_\mathbf{s}$.
Then we can obtain Keller's results about green mutations via studying this heart.
Here is a summarization of the results in Section~\ref{sec:keller}.
\begin{theorem}\label{thm:0.1}
Let $Q$ be an acyclic quiver.
\begin{itemize}
\item A sequence $\mathbf{s}$ is a green mutation sequence if and only if
$\h\geq\h_Q[-1]$ for any $\h$ in the path $\mathrm{P}(\mathbf{s})$.
\item A vertex $j\in Q_0$ for some green mutation sequence $\mathbf{s}$
is either green or red. Moreover, it is green if and only if
the corresponding simple $S_j^{\mathbf{s}}$ in $\h_\mathbf{s}$ is in $\h_Q$ and it is red
if and only if $S_j^{\mathbf{s}}$ is in $\h_Q[-1]$.
\item A green sequence $\mathbf{s}$ is maximal if and only if $\h_\mathbf{s}=\h_Q[-1]$.
Hence the mutated quivers associated to two maximal green mutation sequences are isomorphic.
\item The simples of the wide subcategory $\mathcal{W}{\mathbf{s}}$
associated to the torsion class $\mathcal{T}_\mathbf{s}$ are precisely
the red simples in $\h_\mathbf{s}$ shifting by one.
\end{itemize}
\end{theorem}
Our second focus is on c-sortable words (c for Coxeter element),
defined by Reading \cite{R07},
who showed bijections between c-sortable words, c-clusters and noncrossing partitions
in finite case (Dynkin case).
Ingalls-Thomas extended Reading's result in the direction of representation theory
and gave bijections between many sets (see \cite[p.~1534]{IT09}).
The bijection between c-sortable words and finite torsion classes
was first generalized by Thomas \cite{Tho}
and also obtained by Amiot-Iyama-Reiten-Todorov \cite{AIRT10} via layers for preprojective algebras.
We will interpret a c-sortable word as a green mutation sequence (Theorem~\ref{thm:main})
and obtain many consequences,
summarized by the following theorem.
\begin{theorem}\label{thm:0.2}
For an acyclic quiver $Q$ with an admissible Coxeter element $c$.
Then any $c$-sortable word $\mathbf{w}$ induces a green mutation sequence $\widetilde{\mathbf{w}}$
and we have the following bijections.
\begin{itemize}
\item $\{$the $c$-sortable word $\mathbf{w}\} \overset{_{1-1}}{\longleftrightarrow}
\{$the finite torsion class $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$ in $\h_Q=\mod \k Q\}$.
\item $\{$the inversion $t_{T}$ for $\mathbf{w}\} \overset{_{1-1}}{\longleftrightarrow}
\{$the indecomposable $T$ in $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}\}$.
\item $\{$the descent $s_j$ for $\mathbf{w}\} \overset{_{1-1}}{\longleftrightarrow}
\{$the red vertex $j$ for $\mathbf{w}\}$.
\item $\{$the cover reflection $t_{T}$ for $\mathbf{w}\} \overset{_{1-1}}{\longleftrightarrow}
\{$the red simple $T$ in $\h_{\w}%{\widetilde{\mathbf{w}}}\}$.
\end{itemize}
Further, if $Q$ is of Dynkin type,
the noncrossing partition $\mathrm{nc}_c(\mathbf{w})$ associated to $\mathbf{w}$ can be calculated as
\[
\mathrm{nc}_c(\mathbf{w})=
\prod_{j\in \VR{\w}%{\widetilde{\mathbf{w}}}} s_j^{\w}%{\widetilde{\mathbf{w}}},
\]
with $\mathrm{rank}\mathrm{nc}_c(\mathbf{w})=\#\VR{\w}%{\widetilde{\mathbf{w}}}$,
where $\VR{\w}%{\widetilde{\mathbf{w}}}$ is the set of the red vertices and $s_j^{\w}%{\widetilde{\mathbf{w}}}$
is the reflection corresponding the $j$-th simple in $\h_\mathbf{w}$.
Also,
the tree of $c$-sortable words (with respect to the week order)
is isomorphic to a supporting tree of the exchange graph $\EG_Q$.
\end{theorem}
These results give a deeper understanding of the results of Ingalls-Thomas \cite{IT09}.
Note that all our bijections are consistent with theirs,
cf. Table~\ref{table} and \cite[Table~1]{IT09}.
Also, the construction from c-sortable words to the green mutation sequences
should be the `dual' construction of Amiot-Iyama-Reiten-Todorov \cite{AIRT10} (cf. \cite{BIRS09})
and provides a combinatorial perspective to attack their problems at end of their paper.
Finally, a more systematic study of maximal green sequences can be found in \cite{BDP}.
\subsection*{Acknowledgements}
Some of the ideas in the work was developed when I was a Ph.D student of Alastair King
and when I visited Bernhard Keller in March 2011.
I would like to thank them, as well as Thomas Brustle for helpful conversations.
\section{Preliminaries}
Fixed an algebraically-closed field $\k$.
Throughout this paper, $Q$ will be a finite acyclic quiver
with vertex set $Q_0=\{1,\ldots,n\}$ (unless otherwise stated).
The path algebra is denoted by $\k Q$.
Let $\h_Q:=\mod\k Q$ be the category of finite dimensional $\k Q$-modules,
which is an abelian category,
and $\mathrm{\mathcal{D}}(Q):=\mathcal{D}^b(\h_Q)$ be its bounded derived category,
which is a triangulated category.
We denote by $\mathrm{Sim}\mathcal{A}$ a complete set of non-isomorphic simples in an abelian category $\mathcal{A}$
and let
\[
\mathrm{Sim}\h_Q=\{S_1,\ldots,S_n\},
\]
where $S_i$ is the simple $\k Q$-module corresponding to vertex $i\in Q_0$.
\subsection{Coxeter group and words}
Recall that the \emph{Euler form}
\[
\<-,-\>:\mathbb{Z}^{Q_0}\times\mathbb{Z}^{Q_0}\to\mathbb{Z}
\]
associated to the quiver $Q$ is defined by
\[
\<\mathbf{a}, \mathbf{b}\>
=\sum_{i\in Q_0}a_i b_i-\sum_{(i\to j)\in Q_1}a_i b_j.
\]
Denote by $(-,-)$ the symmetrized Euler form,
i.e. $(\mathbf{a},\mathbf{b})=\<\mathbf{a},\mathbf{b}\>+\<\mathbf{b},\mathbf{a}\>$.
Moreover for $M,L\in\mod \k Q$, we have
\begin{gather}\label{eq:euler form}
\<\dim M,\dim L\>=\dim\mathrm{Hom}(M,L)-\dim\mathrm{Ext}^1(M,L),
\end{gather}
where $\dim E\in\mathbb{N}^{Q_0}$ is the \emph{dimension vector} of any $E\in\mod \k Q$.
Let $V=K(\k Q)\otimes\mathbb{R}$,
where $K(\k Q)$ is the Grothendieck group of $\k Q$.
For any non-zero $v\in V$, define a \emph{reflection}
\[
s_v(u)=u-\frac{2(v,u)}{(u,u)}v.
\]
We will write $s_M=s_{\dim M}$ for $M\in \h_Q\sqcup\h_Q[-1]$.
The \emph{Coxeter group} $W=W_Q$ is the group of transformations generated by
the \emph{simple reflections} $s_i=s_{\dim S_i}, i\in Q_0$.
The (real) \emph{roots} in $W$ are $\{w(e_i)\mid w\in W, i\in Q_0\}$,
where $e_i$ are the idempotents;
the \emph{positive roots} are those root which are a non-negative (integral) combination of the $e_i$.
Note that, the reflection of a positive root is in $W$.
Denote by $\mathrm{T}$
the set of all the reflections of $W$,
that is, the set of all conjugates of the simple reflections of $W$.
A \emph{Coxeter element }for $W$ is the product of the simple reflections in some order.
For a Coxeter element $c=s_{\sigma_1}\ldots s_{\sigma_n}$,
we say it is \emph{admissible} with respect to the orientation of $Q$,
if there is no arrow from $\sigma_i$ to $\sigma_j$ in $Q$ for any $i>j$.
A word $\mathbf{w}$ in $W$ is an expression in the free monoid generated by $s_i, i\in Q_0$.
For $w\in W$, denote by $l(w)$ its \emph{length},
that is, the length of the shortest word for $w$ as a product of simple reflections.
A \emph{reduced} word $\mathbf{w}$ for an element $w\in W$ is a word such that $\mathbf{w}=w$ with minimal length.
The notion of reduced word leads to the \emph{weak order} $\leq$ on $W$,
i.e. $x\leq y$ if and only if $x$ has a reduced expression which is a prefix of some reduced word for $y$.
For a word $\mathbf{w}$ in $W_Q$, we have the following notions.
\begin{itemize}
\item An \emph{inversion} of $\mathbf{w}$ is a reflection $t$ such that $l(t\mathbf{w})\leq l(\mathbf{w})$.
The set of inversions of $\mathbf{w}$ is denoted by $\mathrm{Inv}(w)$.
\item A \emph{descent} of $\mathbf{w}$ is a simple reflection $s$ such that $l(\mathbf{w} s)\leq l(\mathbf{w})$.
The set of descent of $\mathbf{w}$ is denoted by $\mathrm{Des}(w)$.
\item A \emph{cover reflection} of $w$ is a reflection $t$ such that $t\mathbf{w}=\mathbf{w} s$ for some descent $s$ of $\mathbf{w}$.
The set of cover reflections of $\mathbf{w}$ is denoted by $\mathrm{Cov}(w)$.
\end{itemize}
\begin{definition}[Reading, \cite{R07}]
For a word $a=a_1\ldots a_k$, define the support $\mathrm{supp}(a)$ to be $\{a_1,\ldots,a_k\}$.
Fix a Coxeter element $c=s_{\sigma_1}\ldots s_{\sigma_n}$.
A word $\mathbf{w}$ is called \emph{$c$-sortable} if it has the form
$\mathbf{w}=c^{(0)}c^{(1)}\ldots c^{(m)}$, where $c^{(i)}$ are subwords of $c$ satisfying
\[
\mathrm{supp}(c^{(0)})\subseteq\mathrm{supp}(c^{(1)})\subseteq\cdots\subseteq\mathrm{supp}(c^{(m)})\subseteq Q_0.
\]
\end{definition}
Similarly to normal words,
a $\mathrm{T}$-word is an expression in the free monoid generated by elements in
the set $\mathrm{T}$ of all reflections.
Denote by by $l_T(w)$ its \emph{absolute length},
that is, the length of the shortest word for $w$ as a product of arbitrary reflections.
So we have the notion of reduced $\mathrm{T}$-words,
which induces the \emph{absolute order} $\leq_{\mathrm{T}}$ on $W$.
The \emph{noncrossing partitions}, with respect to a Coxeter element $c$,
for $W$ are elements
between the identity and $c$, with respect to the absolute order.
The \emph{rank} of a noncrossing partition is its absolute length.
\subsection{Hearts and t-structures}
We collect some facts from \cite{KQ11} about tilting theory.
A \emph{(bounded) t-structure} on a triangulated category $\mathcal{D}$ is
a full subcategory $\mathcal{P} \subset \mathcal{D}$
with $\mathcal{P}[1] \subset \mathcal{P}$ satisfies the following
\begin{itemize}
\item if one defines
$\mathcal{P}^{\perp}=\{ G\in\mathcal{D}: \mathrm{Hom}_{\mathcal{D}}(F,G)=0,
\forall F\in\mathcal{P} \}$,
then, for every object $E\in\mathcal{D}$, there is
a unique triangle $F \to E \to G\to F[1]$ in $\mathcal{D}$
with $F\in\mathcal{P}$ and $G\in\mathcal{P}^{\perp}$;
\item for every object $M$,
the shifts $M[k]$ are in $\mathcal{P}$ for $k\gg0$ and in $\mathcal{P}^{\perp}$ for $k\ll0$,
or equivalently,
\[
\mathcal{D}= \displaystyle\bigcup_{i,j \in \mathbb{Z}} \mathcal{P}^\perp[i] \cap \mathcal{P}[j].
\]
\end{itemize}
It follows immediately that we also have
\[
\mathcal{P}=\{ F\in\mathcal{D}: \mathrm{Hom}_{\mathcal{D}}(F,G)=0, \forall G\in\mathcal{P}^\perp \}.
\]
Note that $\mathcal{P}^{\perp}[-1]\subset \mathcal{P}^{\perp}$.
The \emph{heart} of a t-structure $\mathcal{P}$ is the full subcategory
\[
\h= \mathcal{P}^\perp[1]\cap\mathcal{P}
\]
and a t-structure is uniquely determined by its heart.
More precisely, any bounded t-structure $\mathcal{P}$
with heart $\h$ determines, for each $M$ in $\mathcal{D}$,
a canonical filtration
\begin{equation}
\label{eq:canonfilt}
\xymatrix@C=0,5pc{
0=M_0 \ar[rr] && M_1 \ar[dl] \ar[rr] && \cdots\ar[rr] && M_{m-1}
\ar[rr] && M_m=M \ar[dl] \\
& H_1[k_1] \ar@{-->}[ul] && && && H_m[k_m] \ar@{-->}[ul]
}
\end{equation}
where $H_i \in \h$ and $k_1 > \ldots > k_m$ are integers.
Moreover, the $k$-th homology of $M$, with respect to $\h$ is
\begin{gather}\label{eq:homology}
\Ho{k}(M)=
\begin{cases}
H_i & \text{if $k=k_i$} \\
0 & \text{otherwise.}
\end{cases}
\end{gather}
Then $\mathcal{P}$ consists of those objects
with no (nonzero) negative homology,
$\mathcal{P}^\perp$ those with only negative homology
and $\h$ those with homology only in degree 0.
There is a natural partial order on hearts given by inclusion of their corresponding t-structures.
More precisely, for two hearts $\h_1$ and $\h_2$ in $\mathcal{D}$,
with t-structures $\mathcal{P}_1$ and $\mathcal{P}_2$,
we say
\begin{equation}
\label{def:ineq}
\h_1 \leq \h_2
\end{equation}
if and only if $\mathcal{P}_2\subset\mathcal{P}_1$ ,
or equivalently $\h_2\subset \mathcal{P}_1$,
or equivalently $\mathcal{P}^\perp_1\subset\mathcal{P}^\perp_2$,
or equivalently $\h_1\subset \mathcal{P}^\perp_2[1]$.
\subsection{Torsion pair and tilting}
A similar notion to a t-structure on a triangulated category
is a torsion pair in an abelian category.
Tilting with respect to a torsion pair in the heart of a t-structure
provides a way to pass between different t-structures.
\begin{definition}
A \emph{torsion pair} in an abelian category $\mathcal{C}$ is a pair of
full subcategories $\<\mathcal{F},\mathcal{T}\>$ of $\mathcal{C}$,
such that $\mathrm{Hom}(\mathcal{T},\mathcal{F})=0$ and furthermore
every object $E \in \mathcal{C}$ fits into a short exact sequence
$ \xymatrix@C=0.5cm{0 \ar[r] & E^{\mathcal{T}} \ar[r] & E \ar[r] & E^{\mathcal{F}} \ar[r] & 0}$
for some objects $E^{\mathcal{T}} \in \mathcal{T}$ and $E^{\mathcal{F}} \in \mathcal{F}$.
\end{definition}
\begin{proposition} [(Happel, Reiten, Smal\o)]
Let $\h$ be a heart in a triangulated category $\mathcal{D}$.
Suppose further that $\<\mathcal{F},\mathcal{T}\>$ is a torsion pair in $\h$.
Then the full subcategory
\[
\h^\sharp
=\{ E \in \mathcal{D}:\Ho1(E) \in \mathcal{F}, \Ho0(E) \in \mathcal{T}
\mbox{ and } \Ho{i}(E)=0 \mbox{ otherwise} \}
\]
is also a heart in $\mathcal{D}$, as is
\[
\h^\flat
=\{ E \in \mathcal{D}:\Ho{0}(E) \in \mathcal{F}, \Ho{-1}(E) \in \mathcal{T}
\mbox{ and } \Ho{i}(E)=0 \mbox{ otherwise} \}.
\]
\end{proposition}
Recall that the homology $\Ho{\bullet}$ was defined in \eqref{eq:homology}.
We call $\h^\sharp$ the \emph{forward tilt} of $\h$,
with respect to the torsion pair $\<\mathcal{F},\mathcal{T}\>$,
and $\h^\flat$ the \emph{backward tilt} of $\h$.
Note that $\h^\flat=\h^\sharp[-1]$.
Furthermore, $\h^\sharp$ has a torsion pair $\<\mathcal{T},\mathcal{F}[1]\>$ and we have
\[ \mathcal{T}=\h\cap\h^\sharp, \quad \mathcal{F}=\h\cap\h^\sharp[-1]. \]
With respect to this torsion pair,
the forward and backward tilts are
$\bigl(\h^\sharp\bigr)^\sharp=\h[1]$
and $\bigl(\h^\sharp\bigr)^\flat=\h$.
Similarly $\h^\flat$ has a torsion pair $\<\mathcal{T}[-1],\mathcal{F}\>$ with
\begin{gather}\label{eq:torsion}
\mathcal{F}=\h\cap\h^\flat, \quad \mathcal{T}=\h\cap\h^\flat[1].
\end{gather}
And with respect to this torsion pair,
we have $\bigl(\h^\flat\bigr)^\sharp=\h$, $\bigl(\h^\flat\bigr)^\flat=\h[-1]$.
Recall the basic property of the partial order between a heart and its tilts as follows.
\begin{lemma}[(King-Qiu \cite{KQ11})]
\label{lem:tiltorder}
Let $\h$ be a heart in $\mathcal{D}(Q)$.
Then $\h<\h[m]$ for $m>0$.
For any forward tilt $\h^\sharp$
and backward tilt $\h^\flat$, we have
\[
\h[-1] \leq \h^\flat \leq \h \leq \h^\sharp \leq \h[1].
\]
Further, the forward tilts $\h^\sharp$ can be characterized as
precisely the hearts between $\h$ and $\h[1]$.
similarly the backward tilts $\h^\flat$ are those between $\h[-1]$ and $\h$.
\end{lemma}
Recall that an object in an abelian category is \emph{simple}
if it has no proper subobjects, or equivalently
it is not the middle term of any (non-trivial) short exact sequence.
An object $M$ is \emph{rigid} if $\mathrm{Ext}^1(M,M)=0$.
\begin{definition}\label{def:simpletilt}
We say a forward tilt is \emph{simple},
if the corresponding torsion free part is generated by a
single rigid simple object $S$.
We denote the new heart by $\tilt{\h}{\sharp}{S}$.
Similarly, a backward tilt is simple
if the corresponding torsion part is generated by such a simple
and the new heart is denoted by $\tilt{\h}{\flat}{S}$.
\end{definition}
For the standard heart $\hua{H}_Q$ in $\mathrm{\mathcal{D}}(Q)$, an APR tilt,
which reverses all arrows at a sink/source of $Q$,
is an example of a simple (forward/backward) tilt.
The simple tilting leads to the notation of exchange graphs.
\begin{definition}(\cite{KQ11})\label{def:eg}
The \emph{exchange graph} $\EG\mathrm{\mathcal{D}}(Q)$ of a triangulated category $\mathrm{\mathcal{D}}$
to be the oriented graph, whose vertices are all hearts in $\mathrm{\mathcal{D}}$
and whose edges correspond to the simple \emph{\textbf{backward}} tilting between them.
We denote by $\EGp\mathrm{\mathcal{D}}(Q)$ the `principal' component of $\EG\mathrm{\mathcal{D}}(Q)$,
that is, the connected component containing the heart $\h_Q$.
Furthermore, denote by $\mathrm{Eg}_Q$ full subgraph of $\EGp\mathrm{\mathcal{D}}(Q)$ consisting of
those hearts which are backward tilts of $\h_Q$.
\end{definition}
We have the following proposition which ensures us
to tilt at any simple of any heart in $\mathrm{Eg}_Q$.
\begin{proposition}[{\cite[Theorem~5.7]{KQ11}}]
Let $Q$ be an acyclic quiver.
Then every heart in $\EGp\mathrm{\mathcal{D}}(Q)$ is finite and rigid
(i.e. has finite many simples, each of which is rigid) and
\[
\mathrm{Eg}_Q=\{\h\in\EGp(Q)\mid \h_Q[-1]\leq\h\leq\h_Q \}.
\]
\end{proposition}
\begin{remark}
Unfortunately, we take a different convention to \cite{KQ11} (backward tilting instead of forward).
Thus a exchange graph in this paper has the opposite orientation
of the exchange graph there.
\end{remark}
\begin{example}\label{ex:pentagon}
Let $Q\colon=(2 \to 1)$ be a quiver of type $A_2$.
A piece of AR-quiver of $\mathrm{\mathcal{D}}(Q)$ is:
\[\xymatrix@C=1pc@R=1pc{
\cdots\qquad & P_2[-1] \ar[dr] && S_1 \ar[dr] && S_2\ar[dr] \\
S_1[-1] \ar[ur] && S_2[-1] \ar[ur] && P_2 \ar[ur]&&\cdots}
\]
Then $\EG_Q$ is as follows:
\[{\xymatrix@R=1.5pc@C=.5pc{
&&\{S_1[-1], S_2\} \ar[drr]\\
\{S_1, S_2\} \ar[urr] \ar[ddr] &&&& \{S_1[-1], S_2[-1]\}\\\\
&\{P_2,S_2[-1]\} \ar[rr] && \{P_2[-1], S_1\} \ar[uur],
}}\]
where we denote a heart by the set of its simples.
\end{example}
\subsection{Simple (backward) tilting sequence}\label{sec:sst}
Let $\mathbf{s}=i_1\ldots i_m$ be a sequence with $i_j\in Q_0$
and we have a sequence of hearts $\h_{\mathbf{s},j}$ with simples
\[\mathrm{Sim}\h_{\mathbf{s},j}=\{S_i^{\mathbf{s},j} \mid i\in Q_0\},\quad 0\leq j \leq m,\]
inductively defined as follows.
\begin{itemize}
\item $\h_{\mathbf{s},0}=\h_Q$ with $S_i^{\mathbf{s},0}=S_i$ for any $i\in Q_0$.
\item For $0\leq j\leq m-1$, we have
\[\h_{\mathbf{s},{j+1}}=\tilt{(\h_{\mathbf{s},j})}{ \flat }{ S^{\mathbf{s},j}_j }.\]
\end{itemize}
Note that $\mathrm{Sim}\h_{\mathbf{s},{j+1}}$ is given by the formula \cite[Proposition~5.2~(5.2)]{KQ11}
in terms of $\mathrm{Sim}\h_{\mathbf{s},{j}}$ so that each simple in $\mathrm{Sim}\h_{\mathbf{s},{j+1}}$
inherits a labeling (in $Q_0=\{1,\ldots,n\}$) from the corresponding simple in $\mathrm{Sim}\h_{\mathbf{s},{j}}$
inductively.
Define
\[
\h_\mathbf{s}=\h_\mathbf{s}(Q)\colon=\h_{\mathbf{s}, m}
\]
and $\mathrm{P}(\mathbf{s})$ to be the path $\mathrm{P}(\mathbf{s})=T_m^\mathbf{s} \cdots T_1^\mathbf{s}$ as follow
\[
\mathrm{P}(\mathbf{s})=\colon
\h_Q\h_{\mathbf{s}, 0} \xrightarrow{ T_1^\mathbf{s} } \h_{\mathbf{s},1} \xrightarrow{ T_2^\mathbf{s} }
\ldots \xrightarrow{ T_m^\mathbf{s} } \h_{\mathbf{s},m}=\h_\mathbf{s},
\]
in $\EGp\mathrm{\mathcal{D}}(Q)$, where $T_j^\mathbf{s}=S_{j}^{\mathbf{s},{j-1}}$ is the $j$-th simple in $\h_{\mathbf{s}_{j-1}}$.
As usual, the support $\mathrm{supp}\mathrm{P}(\mathbf{s})$ of $\mathrm{P}(\mathbf{s})$ is the set $\{ T_1^\mathbf{s},\ldots, T_m^\mathbf{s} \}$.
\section{Green mutation}\label{sec:keller}
In this section, the interpretation of the green mutation
via King-Qiu's Ext-quiver of heart is given,
which provides a proof of Keller's theorem.
\subsection{Green quiver mutation}
\begin{definition}([Fomin-Zelevinsky])\label{def:mutation}
Let $R$ be a finite quiver without loops or $2$-cycles.
The \emph{mutation} $\mu_k$ on $R$ at vertex $k$ is a quiver $R'=\mu_k(R)$
obtaining from $R$ as follows
\begin{itemize}
\item adding an arrow $i\to j$ for any pair of arrows $i\to k$ and $k \to j$ in $R$;
\item reversing all arrows incident with $k$;
\item deleting as many $2$-cycles as possible.
\end{itemize}
\end{definition}
It is straightforward to see that the mutation is an involution, i.e. $\mu_k^2=id$.
A \emph{mutation sequence} $\mathbf{s}=i_1 \ldots i_m$ on $R$ is a sequence with $i_j\in R_0$
and define
\[
R_\mathbf{s}\colon=\mu_\mathbf{s}(R)=\mu_{i_m}( \mu_{i_{m-1}}(\ldots \mu_{i_1}(R) \ldots)).
\]
As in Section~\ref{sec:sst}, a (green) mutation sequence $\mathbf{s}$ induces a sequence of
simple (backward) tilting and a heart $\h_\mathbf{s}$.
Let $\widetilde{Q}$ be the \emph{principal extension} of $Q$,
i.e. the quiver obtained from $Q$ by adding
a new frozen vertex $i'$ and a new arrow $i'\to i$ for each vertex $i\in Q_0$.
Note that we will never mutate a quiver at a frozen vertex
and so mutation sequences $\mathbf{s}$ for $\widetilde{Q}$ are precisely mutation sequences of $Q$.
\begin{definition}[Keller \cite{Kel11}]\label{def:green}
Let $\mathbf{s}$ be a mutation sequence of $\widetilde{Q}$.
\begin{itemize}
\item A vertex $j$ in the quiver $\widetilde{Q}_\mathbf{s}$ is called \emph{green} if
there is no arrows from $j$ to any frozen vertex $i'$;
\item A vertex $j$ is called \emph{red} if
there is no arrows to $j$ from any frozen vertex $i'$.
Let $\VR{\mathbf{s}}$ be the set of red vertices in $\widetilde{Q}_\mathbf{s}$ for $\mathbf{s}$.
\item A \emph{green mutation sequence} $\mathbf{s}$ on $Q$ (or $\widetilde{Q}$) is
a mutation sequence on $Q$ such that every mutation in the sequence is at
a green vertex in the corresponding quiver.
Such a green mutation sequence $\mathbf{s}$ is \emph{maximal} if $\VR{\mathbf{s}}=Q_0$.
\end{itemize}
\end{definition}
\subsection{Principal extension of Ext-quivers}
Following \cite{KQ11}, we will also use Ext-quivers of hearts interpret green mutation.
\begin{definition}[King-Qiu]
\label{def:extquiv}
Let $\h$ be a finite heart in a triangulated category $\mathrm{\mathcal{D}}$ with
$\mathbf{S}_{\h}=\bigoplus_{S\in\mathrm{Sim}\h} S$.
The Ext-quiver $\Q{\h}$ is the (positively) graded quiver
whose vertices are the simples of $\h$ and
whose graded edges correspond to a basis of
$\mathrm{End}^\bullet(\mathbf{S}_{\h},\mathbf{S}_{\h})$.
\end{definition}
Further, the \emph{CY-3 double} of a graded quiver $\mathcal{Q}$,
denoted by $\CY{\mathcal{Q}}{3}$,
to be the quiver obtained from $\mathcal{Q}$
by adding an arrow $T\to S$ of degree $3-k$ for each arrow $S\to T$ of degree $k$
and adding a loop of degree 3 at each vertex.
See Table~\ref{quivers} for an example of Ext-quivers and CY-3 doubling.
For the principal extension $\widetilde{Q}$ of a quiver $Q$,
Consider its module category $\h_{\widetilde{Q}}$ and derived category $\mathrm{\mathcal{D}}(\widetilde{Q})$.
Since $Q$ is a subquiver of its extension $\widetilde{Q}$,
$\h_Q$ and $\mathrm{\mathcal{D}}(Q)$ are subcategories of $\h_{\widetilde{Q}}$ and $\mathrm{\mathcal{D}}(\widetilde{Q})$ respectively.
For a sequence $\mathbf{s}$, it also induces a simple tilting sequence in $\mathrm{\mathcal{D}}(\widetilde{Q})$
(starting at $\h_{\widetilde{Q}}$) and corresponds to a heart, denoted by $\widetilde{\h_\mathbf{s}}$.
Let the set of simples in $\mathrm{Sim}\h_{\widetilde{Q}}-\mathrm{Sim}\h_Q$ be
\[\mathrm{Sim}\h_{Q'}\colon=\{S_i'\mid i\in Q_0\}.\]
A straightforward calculation gives
\[
\mathrm{Hom}^k(S_i',S_j)=\delta_{ij}\delta_{1k}, \quad \forall i,j\in Q_0, k\in\mathbb{Z}.
\]
Hence, for any $M\in\h_Q$, we have
\begin{gather}\label{eq:ijk1}
\mathrm{Hom}^k(\bigoplus_{i\in Q_0} S_i',M)\neq0\quad \iff\quad k=1.
\end{gather}
We have the following lemma.
\begin{lemma}\label{lem:bubian}
For any sequence $\mathbf{s}$, we have $\mathrm{Sim}\widetilde{\h_\mathbf{s}}=\mathrm{Sim}\h_{\mathbf{s}}\cup\mathrm{Sim}\h_{Q'}$.
\end{lemma}
\begin{proof}
Use induction on the length of $\mathbf{s}$ starting from the trivial case when $\mathbf{s}=\emptyset$.
Suppose that $\mathbf{s}=\mathbf{t} j$ with $\mathrm{Sim}\widetilde{\h_\mathbf{t}}=\mathrm{Sim}\h_{\mathbf{t}}\cup\mathrm{Sim}\h_{Q'}$.
By \cite[Lemma~3.4]{KQ11}, we have $\h_\mathbf{t}\leq\h_Q$ and hence
the homology of any object in $\h_t$, with respect to $\h_Q$, lives in non-positive degrees.
Thus, any $M\in\h_\mathbf{t}$ admits a filtration with factors $S_i[k], i\in Q_0, k\leq0$.
As $s'$ is a source in $\widetilde{Q}$ for any $s\in Q_0$,
$S_s'$ is an injective object in $\h_{\widetilde{Q}}$ which implies that
$\mathrm{Ext}^1(S_i[k],S_s')=0$ for any $i\in Q_0$ and $k\leq0$.
Therefore, we have $\mathrm{Ext}^1(M, S_s')=0$ for any $M\in\h_\mathbf{t}$,
in particular, for $M=S_j^\mathbf{t}$.
Then applying \cite[formula~(5.2)]{KQ11} to the backward tilt
$\tilt{\h_\mathbf{t}}{\flat}{S_j^\mathbf{t}}$ and $\tilt{(\widetilde{\h_\mathbf{t}})}{\flat}{S_j^\mathbf{t}}$,
gives $\mathrm{Sim}\widetilde{\h_\mathbf{s}}=\mathrm{Sim}\h_{\mathbf{s}}\cup\mathrm{Sim}\h_{Q'}$.
\end{proof}
By the lemma, we know that $\Q{\h_\mathbf{s}}$ is a subquiver of $\Q{\widetilde{\h_\mathbf{s}}}$.
\begin{definition}
\label{def:p.e.}
Given a sequence $\mathbf{s}$, define the \emph{principal extension} of the Ext-quiver $\Q{\h_\mathbf{s}}$
to be the Ext-quiver $\Q{\widetilde{\h_\mathbf{s}}}$ while the vertices in $\mathrm{Sim}\h_{Q'}$ are
the frozen vertices.
\end{definition}
From the proof of Lemma~\ref{lem:bubian}, it is straightforward to see the following.
\begin{lemma}\label{lem:source}
Every frozen vertices $S_i'$ is a source in $\Q{\widetilde{\h_\mathbf{s}}}$.
\end{lemma}
\subsection{Green mutation as simple (backward) tilting}
Before we proof Keller's observations for green mutation,
we need the following result concerning the relation between
quivers (for clusters) and Ext-quivers.
Because the proof is technical, we leave it to the appendix.
\begin{lemma}\label{lem:KQ}
If $\widetilde{\h_\mathbf{s}}\in\EG_{\widetilde{Q}}$ for some sequence $\mathbf{s}$,
then $\widetilde{Q}_\mathbf{s}$ is canonically isomorphic to the degree one part of
$\CY{ \Q{\widetilde{\h_\mathbf{s}}} }{3}$.
\end{lemma}
\begin{proof}
See Appendix~\ref{app}.
\end{proof}
Now we proceed to prove our first theorem
(which is an interpretation of Keller's green mutation
in the language of tilting in the derived categories).
\begin{theorem}\label{thm:keller}
Let $Q$ be an acyclic quiver and $\mathbf{s}$ be a green mutation sequence for $Q$.
Then we have the following.
\begin{enumerate}[label=\arabic*{$^\circ$}.]
\item $\h_Q[-1]\leq\h_\mathbf{s}\leq\h_Q$ and hence $\h_\mathbf{s}\in\EG_Q$.
\item $\widetilde{Q}_\mathbf{s}$ is canonically isomorphic to the degree one part of
$\CY{ \Q{\widetilde{\h_\mathbf{s}}} }{3}$.
\item A vertex $j$ in $\widetilde{Q}_\mathbf{s}$ is either green or red.
Moreover, it is green if and only if
the corresponding simple $S_j^{\mathbf{s}}$ in $\h_\mathbf{s}$ is in $\h_Q$ and it is red
if and only if $S_j^{\mathbf{s}}$ is in $\h_Q[-1]$.
\end{enumerate}
\end{theorem}
\begin{proof}
We use induction on the length of $\mathbf{s}$ starting with trivial case when $l(\mathbf{s})=0$.
Now suppose that the theorem holds for any green mutation sequence of length less than $m$
and consider the case when $l(\mathbf{s})=m$.
Let $\mathbf{s}=\mathbf{t} j$ where $l(\mathbf{t})=m-1$ and $j$ is a green vertex in $\widetilde{Q}_{\mathbf{t}}$.
First, the simple $S_j^\mathbf{t}$ corresponding to $j$ is in $\h_Q$,
by $3^\circ$ of the induction step,
which implies $1^\circ$ by \cite[Lemma~5.4, $1^\circ$]{KQ11}.
Second, as $\mathrm{Sim}\widetilde{\h_\mathbf{s}}=\mathrm{Sim}\h_{\mathbf{s}}\cup\mathrm{Sim}\h_{Q'}$ by Lemma~\ref{lem:bubian},
$\h_\mathbf{s}\in\EG_Q$ is equivalent to $\widetilde{\h_\mathbf{s}}\in\EG_{\widetilde{Q}}$.
Then $2^\circ$ follows from Lemma~\ref{lem:KQ}.
Third, since $\h_Q$ is hereditary, $1^\circ$ implies
that any simple $S_j^\mathbf{s}\in\mathrm{Sim}\h_\mathbf{s}$ is in either $\h_Q$ or $\h_Q[-1]$.
If $S^\mathbf{s}_j$ is in $\h_Q$, by \eqref{eq:ijk1},
there are arrows $S_i'\to S^\mathbf{s}_j$ in $\Q{\widetilde{\h_\mathbf{s}}}$
and each of which has degree one.
Then, by $2^\circ$ any such degree one arrow
corresponds to an arrow $i'\to j$ in $\widetilde{Q}_\mathbf{s}$.
Therefore $j$ is green.
Similarly, if $S^\mathbf{s}_j$ is in $\h_Q[-1]$,
there are arrows $S_i'\to S^\mathbf{s}_j$ in $\Q{\widetilde{\h_\mathbf{s}}}$,
each of which has degree two and corresponds to
an arrow $i'\leftarrow j$ in $\widetilde{Q}_\mathbf{s}$.
Then $j$ is red and thus we have $3^\circ$.
\end{proof}
For a green mutation sequence $\mathbf{s}$ of $Q$,
we will call a simple $S^{\mathbf{s}}_j\in\mathrm{Sim}\h_{\mathbf{s}}$ green/red
if the vertex $j$ is green/red in $\widetilde{Q}_\mathbf{s}$.
The consequences of theorem
include a criterion for a sequence being green mutation sequence
and one of Keller's original statement about maximal green mutation sequences.
\begin{corollary}\label{cor:keller}
A sequence $\mathbf{s}$ is a green mutation sequence if and only if
$\h\geq\h_Q[-1]$ for any $\h\in\mathrm{supp}\mathrm{P}(\mathbf{s})$.
Further, a green mutation sequence $\mathbf{s}$ is maximal if and only if $\h_\mathbf{s}=\h_Q[-1]$.
Thus, for a maximal green mutation sequence $\mathbf{s}$,
$\widetilde{Q}_\mathbf{s}$ can be obtained from $\widetilde{Q}$ by reversing all
arrows that are incident with frozen vertices.
\end{corollary}
\begin{proof}
The necessity of first statement follows from $1^\circ$ of Theorem~\ref{thm:keller}.
For the sufficiency, we only need to show that if $\mathbf{t}$ is a green mutation sequence
and $\mathbf{s}=\mathbf{t} j$ satisfies $\h_\mathbf{s}\geq\h_Q[-1]$, for some $j\in Q_0$,
then $\mathbf{s}$ is also a green mutation sequence.
Since $\h_\mathbf{t}\geq\h_Q[-1]$, by \cite[Lemma~5.4, $1^\circ$]{KQ11}
we know that $\h_\mathbf{s}\geq\h_Q[-1]$ implies $S_j^\mathbf{t}$ is in $\h_Q$.
But this means $j$ is a green vertex for $\mathbf{t}$, by $3^\circ$ of Theorem~\ref{thm:keller},
as required.
For the second statement,
$\mathbf{s}$ is a maximal, if and only if $S_i^\mathbf{s}\in\h_Q[-1]$ for any $i\in Q_0$,
or equivalently, $\h_\mathbf{s}=\h_Q[-1]$.
This implies the statement immediately.
\end{proof}
\begin{example}\label{ex:keller}
We borrow an example of $A_2$ type green mutations from Keller \cite{Kel11}
(but the orientation slightly differs).
Figure~\ref{fig:keller} gives two different maximal green mutation sequences
($121$ and $21$) which end up being isomorphic to each other.
If we identify the isomorphic ones, we recover the pentagon in Example~\ref{ex:pentagon}.
\begin{figure}[b]\centering
\[\xymatrix@C=1pc@R=1pc{
&&&& \green{1}\ar@{<-}[dd]\ar[rr]&& \green{2}\ar@{<-}[dd]&&&&&
\red{1}\ar[dd]&& \green{2}\ar@{<-}[ddll]\ar[ll]\\
&&&\ar@{~>}[lldd]_{\mu_2} &&&& \ar@{~>}[rrr]^{\mu_1}&&& &&&&\ar@{~>}[rrdd]^{\mu_2}\\
&&&& 1' && 2' &&&&& 1' && 2'\ar[uu]\\
&&&&& \ar@{~>}[dddd]_{\mu_{21}} & \ar@{~>}[ddddrrrrr]^{\mu_{121}} &&&&&&&&&&\\
\green{1}\ar@{<-}[dd]\ar@{<-}[rr]&& \red{2}\ar[dd]&&&&&&&&&&&&&
\green{1}\ar@{<-}[ddrr]&& \red{2}\ar[ddll]\ar@{<-}[ll]\\
&&& \\
1' && 2' &&&&&&&&&&&&& 1' && 2'\ar@{<-}[uu]\\
& \ar@{~>}[ddrr]_{\mu_1} &&&&&&&&&&&&&&& \ar@{~>}[ddll]^{\mu_1}\\
&&&& \red{1}\ar[dd]\ar[rr]&& \red{2}\ar[dd]&&&&&
\red{1}&& \red{2}\ar[ll]\\
&&&&&&& \ar@{<->}[rrr]^{\text{iso.}} &&&&&&&\\
&&&& 1' && 2' &&&&& 1'\ar@{<-}[uurr] && 2'\ar@{<-}[uull]
}\]
\caption{Two maximal green mutation sequences for an $A_2$ quiver}
\label{fig:keller}
\end{figure}
\end{example}
\subsection{Wide subcategory via red simples}
In this section, we aim to show the red simples are precisely the simples in
the wide subcategory $\mathcal{W}_{\mathbf{s}}$ corresponds to
the torsion class $\mathcal{T}_{\mathbf{s}}$ in the sense of Ingalls-Thomas.
Recall that a wide subcategory is an exact abelian category closed under extensions
of some abelian category.
Further, given a finite generated torsion class $\mathcal{T}$ in $\h_Q$,
define the corresponding wide subcategory $\mathcal{W}(\mathcal{T})$ to be
(cf. \cite[Section~2.3]{IT09})
\begin{gather}\label{eq:defwide}
\{ M\in\mathcal{T} \mid \forall (f;X\to M)\in\mathcal{T}, \ker(f)\in\mathcal{T} \}.
\end{gather}
First, we give another characterization for $\mathcal{W}(\mathcal{T})$.
\begin{proposition}\label{pp:wide}
Let $\<\mathcal{F},\mathcal{T}\>$ be a finite generated torsion pair in $\h_Q$
and $\h^{\sharp}$ be the corresponding backward tilt.
Then we have
\begin{gather}\label{eq:wide}
\mathrm{Sim}\mathcal{W}(\mathcal{T})=\mathcal{T}\cap\mathrm{Sim}\h^\sharp.
\end{gather}
\end{proposition}
\begin{proof}
By \cite{IT09} and \cite{KQ11},
such torsion pair corresponds to a cluster tilting object
(in the cluster category of $\mathrm{\mathcal{D}}(Q)$) and thus the heart $\h^\sharp$ is in $\EG_Q$
and hence finite.
Noticing that $\h^\sharp$ admits a torsion pair $\<\mathcal{T},\mathcal{F}[1]\>$,
any its simple is either in $\mathcal{T}$ or $\mathcal{F}[1]$.
Let $\mathcal{W}$ be the wide subcategory of $\h^\sharp$
generated by simples in $\mathcal{T}\cap\mathrm{Sim}\h^\sharp$.
First, for any $S\in\mathcal{T}\cap\mathrm{Sim}\h^\sharp$ and
\[
(f:X\to S)\in\mathcal{T}\subset\h^\sharp,
\]
$f$ is surjective (in $\h^\sharp$) since $S$ is a simple.
Thus $\ker(f)$ is in $\mathcal{T}$ since $\mathcal{T}$ is a torsion free class in $\h^\sharp$,
which implies $S\in\mathcal{W}(\mathcal{T})$.
Therefore $\mathcal{W}\subset\mathcal{W}(\mathcal{T})$
and we claim that they are equal.
If not, let $M$ in $\mathcal{W}(\mathcal{T})-\mathcal{W}$
whose simple filtration in $\h^\sharp$ (with factors in $\mathrm{Sim}\h^\sharp$)
has minimal number of factors.
Let $S$ be a simple top of $M$ and then $X=\ker(M\twoheadrightarrow S)$ is in $\mathcal{T}$.
If $S$ is in $\mathcal{T}\cap\mathrm{Sim}\h^\sharp$,
then $X$ is in $\mathcal{W}(\mathcal{T})-\mathcal{W}$ with
less simple factors, contradicting to the choice of $M$.
Hence $S\in\mathcal{F}[1]\cap\mathrm{Sim}\h^\sharp$.
Then we obtain a short exact sequence
\[
0 \to X \hookrightarrow M \twoheadrightarrow S \to 0
\]
in $\h^\sharp$ which became a short exact sequence
\[
0 \to S[-1] \hookrightarrow X \overset{f}{\twoheadrightarrow} M \to 0
\]
in $\h_Q$. But $\ker(f)=S[-1]\in\mathcal{F}$,
which contradicts to the fact that $M$ is in $\mathcal{W}(\mathcal{T})$ (cf. \eqref{eq:defwide}).
Therefore $\mathcal{W}(\mathcal{T})=\mathcal{W}$ or \eqref{eq:wide}.
\end{proof}
An immediate consequence of this corollary is as follows.
Recall Bridgeland's notion of stability condition first.
\begin{definition}[Bridgeland]\label{def:stab}
A \emph{stability condition} $\sigma = (Z,\mathcal{P})$ on $\mathcal{D}$ consists of
a group homomorphism $Z:K(\mathcal{D}) \to \mathbb{C}$ called the \emph{central charge} and
full additive subcategories $\mathcal{P}(\varphi) \subset \mathcal{D}$
for each $\varphi \in \mathbb{R}$, satisfying the following axioms:
\begin{enumerate}[label=\arabic*{$^\circ$}.]
\item if $0 \neq E \in \mathcal{P}(\varphi)$
then $Z(E) = m(E) \exp(\varphi \pi \mathbf{i} )$ for some $m(E) \in \mathbb{R}_{>0}$,
\item for all
$\varphi \in \mathbb{R}$, $\mathcal{P}(\varphi+1)=\mathcal{P}(\varphi)[1]$,
\item if $\varphi_1>\varphi_2$ and $A_i \in \mathcal{P}(\varphi_i)$
then $\mathrm{Hom}_{\mathcal{D}}(A_1,A_2)=0$,
\item for each nonzero object $E \in \mathcal{D}$ there is a finite sequence of real numbers
$$\varphi_1 > \varphi_2 > ... > \varphi_m$$
and a collection of triangles
$$\xymatrix@C=0.8pc@R=1.4pc{
0=E_0 \ar[rr] && E_1 \ar[dl] \ar[rr] && E_2 \ar[dl] \ar[rr] && ... \
\ar[rr] && E_{m-1} \ar[rr] && E_m=E \ar[dl] \\
& A_1 \ar@{-->}[ul] && A_2 \ar@{-->}[ul] && && && A_m \ar@{-->}[ul]
},$$
with $A_j \in \mathcal{P}(\varphi_j)$ for all j.
\end{enumerate}
\end{definition}
We call the collection of subcategories $\{\mathcal{P}(\varphi)\}$,
satisfying $2^\circ \sim 4^\circ$ in Definition~\ref{def:stab},
the \emph{slicing}.
Note that $\mathcal{P}(\varphi)$ is always abelian for any $\varphi\in\mathbb{R}$ (cf. \cite{B1})
and we call it a \emph{semistable subcategory} of $\sigma$.
\begin{corollary}\label{cor:stab}
A finite generated wide subcategory in $\h_Q$
is a semistable subcategory of some Bridgeland stability condition on $\mathrm{\mathcal{D}}(Q)$.
\end{corollary}
\begin{proof}
Let $\mathcal{W}(\mathcal{T})$ be a finite generated wide subcategory in $\h_Q$
which corresponds to the torsion pair $\<\mathcal{F},\mathcal{T}\>$.
Let $\h^{\sharp}$ be the corresponding backward tilt.
Recall that we have the following (\cite{B1}:
\begin{itemize}
\item To give a stability condition on a triangulated category $\mathcal{D}$
is equivalent to giving a bounded t-structure on $\mathcal{D}$ and
a stability function on its heart with the HN-property.
\end{itemize}
Thus, a function $Z$ from $\mathrm{Sim}\h^\sharp$ to the upper half plane $\mathbb{H}$
gives a stability condition $\sigma(Z, \h^\sharp)$ on the triangulated category $\mathrm{\mathcal{D}}(Q)$.
Then choosing $Z$ as follows
\[
Z(S)=
\begin{cases}
i & \text{if $S\in\mathrm{Sim}\h^\sharp\cap\mathcal{T}=\mathrm{Sim}\mathcal{W}(\mathcal{T})$} \\
0 & \text{if $S\in\mathrm{Sim}\h^\sharp\cap\mathcal{F}[1]$}
\end{cases}
\]
will make $\mathcal{W}(\mathcal{T})$ a semistable subcategory with respect to $\sigma(Z, \h^\sharp)$.
\end{proof}
\begin{remark}
As Bridgeland is kind of an improved version of King's $\theta$-stability condition,
And in fact, Corollary~\ref{cor:stab} implies immediately
Ingalls-Thomas' result, that every wide subcategory in $\h_Q$ is a semistable subcategory
for some $\theta$-stability condition on $\h_Q$.
\end{remark}
We end this section by showing that the simples of the wide subcategory
associated to a green mutation sequence.
Let $\mathbf{s}$ be a green mutation sequence and
\begin{gather}\label{def:torsion}
\mathcal{T}_\mathbf{s}=\h_Q\cap\h_\mathbf{s}[1],
\end{gather}
which is a torsion class in $\h_Q$ by \eqref{eq:torsion}.
We will write $\mathcal{W}_\mathbf{s}$ for the wide subcategory $\mathcal{W}(\mathcal{T}_\mathbf{s})$ of $\mathcal{T}_\mathbf{s}$.
Recall that $\VR{\mathbf{s}}$ is the set of red vertices of a green mutation sequence $\mathbf{s}$.
Denote by $\VR{\h_\mathbf{s}}$ the set of red simples in $\h_{\mathbf{s}}$.
\begin{corollary}
Let $s$ be a green mutation sequence.
Then $\mathrm{Sim}\mathcal{W}_\mathbf{s}=\VR{\h_\mathbf{s}}[1]$.
\end{corollary}
\begin{proof}
By $3^\circ$ of Theorem~\ref{thm:keller}, we have
\[
\VR{\h_\mathbf{s}}=\h_Q[-1]\cap\mathrm{Sim}\h_\mathbf{s}.
\]
But $\VR{\h_\mathbf{s}}\subset\h_\mathbf{s}$, we have
\[
\VR{\h_\mathbf{s}}=(\h_Q[-1]\cap\h_\mathbf{s})\cap\mathrm{Sim}\h_\mathbf{s}=\mathcal{T}_\mathbf{s}[-1]\cap\mathrm{Sim}\h_\mathbf{s},
\]
where the second equality uses \eqref{eq:torsion}.
Noticing that $\h_\mathbf{s}[1]$ is the forward tilt of $\h_Q$ with respect
to the torsion class $\mathcal{T}_\mathbf{s}$, we have
\[
\mathrm{Sim}\mathcal{W}_\mathbf{s}=\mathcal{T}_\mathbf{s}\cap\mathrm{Sim}\h_\mathbf{s}[1]
\]
by Proposition~\ref{pp:wide}.
Thus the claim follows.
\end{proof}
\section{C-sortable words}
In this section, we will show that it is natural to interpret a c-sortable word
as a green mutation sequence, which produces many consequences.
\subsection{Main results}
Denote by $\widetilde{\w}%{\widetilde{\mathbf{w}}}=i_1\ldots i_k$ the sequence induced from
a $c$-sortable word $\mathbf{w}=s_{i_1}\ldots s_{i_k}$.
Note that $\w}%{\widetilde{\mathbf{w}}$ induces a path $\mathrm{P}(\widetilde{\w}%{\widetilde{\mathbf{w}}})$ and
a heart $\h_{\widetilde{\w}%{\widetilde{\mathbf{w}}}}$ as in Section~\ref{sec:sst}.
We will drop the tilde of $\widetilde{\mathbf{w}}$ later
when it appears in the subscript or superscript.
\begin{theorem}\label{thm:main}
Let $Q$ be an acyclic quiver and $c$ be an admissible Coxeter element
with respect to the orientation of $Q$.
Let $\mathbf{w}$ be a $c$-sortable word and we have the following.
\begin{enumerate}[label=\arabic*{$^\circ$}.]
\item $\widetilde{\w}%{\widetilde{\mathbf{w}}}$ is a green mutation sequence.
\item For any $i\in Q_0$, let $s_i^{\w}%{\widetilde{\mathbf{w}}}$ be the reflection of $S^{\w}%{\widetilde{\mathbf{w}}}_i$,
the $i$-th simple of $\h_{\w}%{\widetilde{\mathbf{w}}}$. Then
\begin{gather}
\label{eq:main}
s_i^{\w}%{\widetilde{\mathbf{w}}} \cdot \mathbf{w}=\mathbf{w} \cdot s_i .
\end{gather}
\item Let the torsion class $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$ is defined as in \eqref{def:torsion} and we have
$\mathrm{Ind}\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}=\mathrm{supp}\mathrm{P}({\w}%{\widetilde{\mathbf{w}}})$.
\end{enumerate}
\end{theorem}
\begin{proof}
We use induction on $l(\mathbf{w})+\#Q_0$ staring with the trivial case $l(\mathbf{w})=0$.
Suppose that the theorem holds for any $(Q, c, \mathbf{w})$ with $l(\mathbf{w})+\#Q_0<m$.
Now we consider the case when $l(\mathbf{w})+\#Q_0=m$.
Let $c=s_1 c_-$ without lose of generality.
If $s_1$ is not the initial of $\mathbf{w}$,
then the theorem reduces to the case for $(Q_-, c_-, \mathbf{w})$,
where $Q_-$ is the full subquiver with vertex set $Q_0-\{1\}$,
which is true by the inductive assumption.
Next, suppose that $s_1$ is the initial of $\mathbf{w}$, so $\mathbf{w}=s_1 \mathbf{v}$ for some $\mathbf{v}$.
Denote by $\widetilde{\mathbf{v}}$ the sequence induced by $\mathbf{v}$.
Let $Q_+=\mu_1(Q)$, $c_+=s_1 c s_1$ and
we identify
\[
\h_{Q_+}=\mod \k Q_+\quad\text{with}\quad\h_{s_1}=\tilt{(\h_Q)}{\flat}{S_1}
\]
via a so-called APR-tilting (reflecting the source $1$ of $Q$).
By \cite[Lemma~2.5]{R07}, $\mathbf{v}$ is $c_+$-sortable and hence
the theorem holds for $(Q_+, c_+, \mathbf{v})$ by inductive assumption.
Let $\mathbf{v}=\mathbf{u} s_j$, then the theorem also holds for $(Q, c, s_1 \mathbf{u})$.
Let $T=S_j^\mathbf{w}$ the $j$-th simple of $\h_{\mathbf{w}}$.
Use the criterion in Corollary~\ref{cor:keller} for being a green mutation sequence,
we know that
\begin{equation}
\label{eq:geq}
\left\{
\begin{array}{l}
\tilt{(\h_\mathbf{w})}{\sharp}{T}=\h_{s_1 \mathbf{u}}\geq\h_Q[-1],\\
\h_{\mathbf{w}}=\h_{\mathbf{w}}(Q)=\h_{\mathbf{v}}(Q_+)\geq\h_{Q_+}[-1].
\end{array}
\right.
\end{equation}
If $\h_{\mathbf{w}}\geq\h_Q[-1]$ fails, comparing \eqref{eq:geq} with
\[
\mathrm{Ind}\h_{Q_+}[-1]=\mathrm{Ind}\h_Q[-1]-\{S_1[-1]\}\cup\{S_1[-2]\},
\]
we must have $T=S_1[-2]$.
However, by formula \eqref{eq:main} for $(Q,c,s_1 \mathbf{u})$ and $j\in Q_0$,
noticing that the $j$-th simple of $\h_{s_1 \mathbf{u}}$ is $T[1]$, we have
\[
s_{T[1]} \cdot (s_1 \mathbf{u})=(s_1 \mathbf{u}) \cdot s_j.
\]
The RHS is $\mathbf{w}$ while the LHS equals to $s_1^2 \mathbf{u}=\mathbf{u}$,
which is a contradiction to the fact that the $c$-sortable word $\mathbf{w}$ is reduced.
So $\h_{\mathbf{w}}\geq\h_Q[-1]$, and thus $\widetilde{\mathbf{w}}$ is a green mutation sequence,
by Corollary~\ref{cor:keller}, as $1^\circ$ required.
For $2^\circ$,
consider the influence of the APR-tilting on the dimension vectors and Coxeter group.
we know that for any $M\in\h_Q-\{S_1\}$,
the $\dim_+M$ with respect to $Q_+$ equals $s_1(\dim M)$.
Thus the reflection $t_M$ of $M$ for $Q_+$ equals $s_1 s_M s_1$.
In particular, the reflection $t_i^{\mathbf{v}}$ of $S_i^{\w}%{\widetilde{\mathbf{w}}}$ for $Q_+$
equals $s_1 s_i^{\mathbf{w}} s_1$.
Then formula \eqref{eq:main} gives
\[
t_i^{\mathbf{v}} \cdot \mathbf{v}=\mathbf{v} \cdot s_i\,\quad
\text{or}
\quad\,s_i^{\w}%{\widetilde{\mathbf{w}}} \cdot \mathbf{w}=\mathbf{w} \cdot s_i,
\]
as required.
Finally, we have $\mathrm{Ind}\mathcal{T}(Q)_{\w}%{\widetilde{\mathbf{w}}}=\mathrm{Ind}\mathcal{T}_{\mathbf{v}}(Q_+)\cup\{S_1\}$
which implies $3^\circ$.
\end{proof}
\subsection{Consequences}
In this subsection,
we discuss various corollaries of Theorem~\ref{thm:main}.
First, we prove the bijection between $c$-sortable words
and finite torsion classed in $\h_Q$,
which is essentially equivalent to the result in \cite{AIRT10},
that there is a bijection between $c$-sortable words
and finite torsion-free classed in $\h_Q$.
\begin{corollary}
There is a bijection between the set of $c$-sortable words
and the set of finite torsion classes in $\h_Q$,
sending such a word $\mathbf{w}$ to $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$.
\end{corollary}
\begin{proof}
Clearly, every torsion class $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$ induced by
a $c$-sortable word $\mathbf{w}$ is finite.
To see two different $c$-sortable words $\mathbf{w}_1$ and $\mathbf{w}_2$ induce different
finite torsion classes, we use the induction on $l(\mathbf{w})$.
Then it is reduced to the case when the initials of $\mathbf{w}_1$ and $\mathbf{w}_2$
are different.
Without lose of generality, let the initial $s_1$ of $\mathbf{w}_1$
is on the left of the initial $s_2$ of $\mathbf{w}_2$ in expression
\[
c=\cdots s_1 \cdots s_2 c'
\]
of the Coxeter element $c$.
Now, the sequence of simple tilting $\widetilde{\mathbf{w}}_2$ takes place in the full subcategory
\[\mathrm{\mathcal{D}}(Q_{\mathrm{res}})\subset\mathrm{\mathcal{D}}(Q),\]
where $Q_{\mathrm{res}}$ is the full subquiver of $Q$ restricted to $\mathrm{supp}(s_2 c')$.
Thus the simple $S_1$ will never appear in the path $\mathrm{P}(\mathbf{w}_2)$
which implies $\mathcal{T}_{\widetilde{\mathbf{w}}_1}\neq\mathcal{T}_{\widetilde{\mathbf{w}}_2}$
by $3^\circ$ of Theorem~\ref{thm:main}.
Therefore, we have an injection from the set of $c$-sortable words
to the set of finite torsion classes in $\h_Q$.
To finish, we need to show the surjectivity, i.e.
any finite torsion class $\mathcal{T}$ is equal to $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$ for some $c$-sortable words.
This is again by induction for $(Q, c, \mathcal{T})$
on $\#\mathrm{Ind}\mathcal{T}+\#Q_0$, starting with the trivial case when $\#\mathrm{Ind}\mathcal{T}=0$.
Suppose that the surjectivity hold for any $(Q, c, \mathcal{T})$ with
$\#\mathrm{Ind}\mathcal{T}+\#Q_0<m$ and consider the case when $\#\mathrm{Ind}\mathcal{T}+\#Q_0=m$.
Let $c=s_1 c_-$ without lose of generality.
If the simple injective $S_1$ of $\h_Q$ is not in $\mathcal{T}$,
we claim that $\mathcal{T}\subset\h_{Q_-}\subset\h_Q$,
where $Q_-$ is the full subquiver with vertex set $Q_0-\{1\}$.
If so, the theorem reduces to the case for $(Q_-, c_-, \mathbf{s})$, which holds by
the inductive assumption.
To see the claim, choose any $M\in\h_Q-\h_{Q_-}$.
Then $S_1$ is a simple factor of $M$ in its canonical filtration
and hence the top, since $S_1$ is injective.
Thus $\mathrm{Hom}(M,S)\neq0$.
But $S_1\notin\mathcal{T}$ implies $S_1$ is in the torsion free class corresponds to $\mathcal{T}$.
So $M\notin\mathcal{T}$, which implies $\mathcal{T}\subset\h_{Q_-}$ as required.
If the simple injective $S_1$ of $\h_Q$ is in $\mathcal{T}$,
then consider the quiver $Q_+=\mu_1(Q)$ and the torsion class
\[
\mathcal{T}_+=\mathrm{add}\left(\mathrm{Ind}\mathcal{T}-\{S_1\}\right).
\]
Similar to the proof of Theorem~\ref{thm:main},
we know that the claim holds for $(Q_+, c+, \mathcal{T}_+)$,
where $c_+=s_1 c s_1$.
i.e. $\mathcal{T}_+=\mathcal{T}_{\mathbf{v}}$ for some $c_+$-sortable word $\mathbf{v}$.
But $\mathbf{w}=s_1 \mathbf{v}$ is a $c$-sortable word by \cite[Lemma~2.5]{R07} and we have
\[
\mathrm{Ind}\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}=\{S_1\}\cup\mathrm{Ind}\mathcal{T}_{\mathbf{v}}=\mathrm{Ind}\mathcal{T},
\]
or $\mathcal{T}=\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$, as required.
\end{proof}
Second, we claim that the path $\mathrm{P}(\mathbf{w})$ has maximal length.
\begin{corollary}
Let $\mathbf{w}$ be a $c$-sortable word.
Then $\mathrm{P}(\w}%{\widetilde{\mathbf{w}})$ is a directed path in $\EG_Q$ connecting $\h$ and $\h_{\w}%{\widetilde{\mathbf{w}}}$
with maximal length.
\end{corollary}
\begin{proof}
By $4^\circ$ of Theorem~\ref{thm:main},
the number of indecomposables in $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$ is exactly the length of $\mathrm{P}(\w}%{\widetilde{\mathbf{w}})$.
Then the corollary follows from the fact that,
each time we do a backward tilt in the sequence $\widetilde{\w}%{\widetilde{\mathbf{w}}}$,
the torsion class adds at least a new indecomposable, i.e.
the simple where the tilting is at.
\end{proof}
Third, we describe the properties of a $c$-sortable word $\mathbf{w}$
in terms of red vertices of the corresponding green mutation sequence $\widetilde{\w}%{\widetilde{\mathbf{w}}}$.
Recall that $\VR{\w}%{\widetilde{\mathbf{w}}}$ is the set of red vertices of
a green mutation sequence $\widetilde{\w}%{\widetilde{\mathbf{w}}}$
and $\VR{\h_{\w}%{\widetilde{\mathbf{w}}}}$ the set of (red) simples in $\h_{\w}%{\widetilde{\mathbf{w}}}$.
\begin{corollary}
Let $Q$ be an acyclic quiver and $c$ be an admissible Coxeter element
with respect to the orientation of $Q$.
For a $c$-sortable word $\mathbf{w}$, the set of its
inversions, descents and cover reflections are given as follows
\begin{gather}
\label{eq:Inv}
\mathrm{Inv}({\mathbf{w}})=\{s_T \mid T\in\mathrm{supp}\mathrm{P}({\w}%{\widetilde{\mathbf{w}}})\},\\
\label{eq:Des}
\mathrm{Des}(\mathbf{w})=\{s_i\mid i\in \VR{\w}%{\widetilde{\mathbf{w}}}\}, \\
\label{eq:Cov}
\mathrm{Cov}(\mathbf{w})=\{s_T \mid T\in \VR{\h_{\w}%{\widetilde{\mathbf{w}}}}\},
\end{gather}
where $\mathbf{s}_T$ is the reflection of $T$.
\end{corollary}
\begin{proof}
First of all, as the proof of Theorem~\ref{thm:main} or \cite[Theorem~4.3]{IT09},
we have \eqref{eq:Inv} by inducting on $l(\mathbf{w})+\#Q_0$.
For any $j\in\VR{\mathbf{s}}$, by $4^\circ$ of Theorem~\ref{thm:keller},
we have the corresponding simple $S_j^{\w}%{\widetilde{\mathbf{w}}}$ is in $\h_Q[-1]$
and hence $S_j^{\w}%{\widetilde{\mathbf{w}}}[1]$ the torsion class $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$.
By formula \eqref{eq:Inv}, we know that $s_i^{\w}%{\widetilde{\mathbf{w}}}$ is in $\mathrm{Inv}(\mathbf{w})$
and hence $s_i$ is in $\mathrm{Des}(\mathbf{w})$ by \eqref{eq:main}.
For any $j\notin\VR{\mathbf{s}}$, by $3^\circ$ of Theorem~\ref{thm:keller},
and hence the corresponding simple $S_j^{\w}%{\widetilde{\mathbf{w}}}$ is in $\h_Q$
but not in the torsion class $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$.
Then $\dim S_j^{\w}%{\widetilde{\mathbf{w}}}$ is not equal to any $\dim T, T\in\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$
since $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$ is a simple in $\h_{\w}%{\widetilde{\mathbf{w}}}\supset\mathcal{T}$.
Again, by formula \eqref{eq:Inv}, we know that $s_i^{\w}%{\widetilde{\mathbf{w}}}$ is not in $\mathrm{Inv}(\mathbf{w})$
and hence $s_i$ is not in $\mathrm{Des}(\mathbf{w})$ by \eqref{eq:main}.
Therefore, \eqref{eq:Des} and \eqref{eq:Cov} both follow.
\end{proof}
In the finite case, there are two more consequences.
The first one is about the supporting trees of the (cluster) exchange graphs.
\begin{corollary}
Let $Q$ be a Dynkin quiver.
For any $\h\in\EG_Q$, there is a unique $c$-sortable word $\mathbf{w}$
such that $\h=\h_{\w}%{\widetilde{\mathbf{w}}}$.
Equivalently, the tree of $c$-sortable word $\mathbf{w}$ (with respect to the week order)
is isomorphic to a supporting tree of the exchange graph $\EG_Q$.
\end{corollary}
\begin{proof}
First, notice that
all $c$-sortable words forms a tree with respect to the week order.
Then the corollary follows from $3^\circ$ of Theorem~\ref{thm:main}
and the fact that any torsion class in $\h_Q$ is finite.
\end{proof}
We finish this section by showing a formula of a
$T$-reduced expression for noncrossing partitions via red vertices.
Let $\mathrm{nc}_c$ be Reading's map from $c$-sortable words to noncrossing partitions.
We have the following formula.
\begin{corollary}
Let $Q$ be a Dynkin quiver.
Keep the notation of Theorem~\ref{thm:main}, we have
the following formula
\begin{gather*}
\mathrm{nc}_c(\mathbf{w})=
\prod_{j\in \VR{\w}%{\widetilde{\mathbf{w}}}} s_j^{\w}%{\widetilde{\mathbf{w}}},
\end{gather*}
with $\mathrm{rank}\mathrm{nc}_c(\mathbf{w})=\#\VR{\w}%{\widetilde{\mathbf{w}}}$.
\end{corollary}
\begin{proof}
The corollary follows from \eqref{eq:Cov} and Reading's map (\cite[Section~6]{R07}).
\end{proof}
\section{Example: Associahedron}
\begin{figure}[b]\centering
\begin{tikzpicture}[scale=1.2, rotate=-90, xscale=-1]
\path (0,0) node[rectangle,rounded corners,draw=white] (x1) {\Green{X}$\red{\widehat{Y}}$$\red{\widehat{Z}}$};
\path (2,1) node[rectangle,rounded corners,draw=white] (x2) {\Green{X}$\red{\widehat{Y}}$\Green{Z}};
\path (-2,1) node[rectangle,rounded corners,draw=white] (x3) {\Green{X}\Green{Y}$\red{\widehat{Z}}$};
\path (0,2) node[rectangle,rounded corners,draw=Emerald] (x4) {\Green{X}\Green{Y}\Green{Z}};
\path (0,4) node[rectangle,rounded corners,draw=white] (x5) {$\red{\widehat{X}}$\Green{B}\Green{C}};
\path (4,5) node[rectangle,rounded corners,draw=white] (x6) {$\red{\widehat{Y}}$$\red{\widehat{X}}$\Green{C}};
\path (2,5) node[rectangle,rounded corners,draw=white] (x7) {\Green{Y}$\red{\widehat{B}}$\Green{C}};
\path (-2,5) node[rectangle,rounded corners,draw=white] (x8) {\Green{Z}\Green{B}$\red{\widehat{C}}$};
\path (-4,5) node[rectangle,rounded corners,draw=white] (x9) {$\red{\widehat{Z}}$\Green{B}$\red{\widehat{X}}$};
\path (0,6) node[rectangle,rounded corners,draw=white] (X) {\Green{A}$\red{\widehat{B}}$$\red{\widehat{C}}$};
\path (0,8) node[rectangle,rounded corners,draw=white] (X1) {$\red{\widehat{A}}$\Green{Z}\Green{Y}};
\path (2,9) node[rectangle,rounded corners,draw=white] (X2) {$\red{\widehat{C}}$\Green{Z}$\red{\widehat{Y}}$};
\path (-2,9) node[rectangle,rounded corners,draw=white] (X3) {$\red{\widehat{B}}$$\red{\widehat{Z}}$\Green{Y}};
\path (0,10) node[rectangle,rounded corners,draw=red] (X4) {$\red{\widehat{X}}$$\red{\widehat{Z}}$$\red{\widehat{Y}}$};
\path[-triangle 45, thick, line width=1.5mm]
(x4)edge[Emerald!40!] node[thick, Black]{\bf{2}} (x2)
edge[Emerald!40!] node[thick, Black]{\bf{3}} (x3)
edge[Emerald!40!] node[thick, Black]{\bf{1}} (x5);
\path[-triangle 45, thick, line width=1.5mm]
(x2)edge[Emerald!40!] node[thick, Black]{\bf{3}} (x1);
\path[-triangle 45, thick, line width=1.5mm]
(x5)edge[Emerald!40!] node[thick, Black]{\bf{2}} (x7)
edge[Emerald!40!] node[thick, Black]{\bf{3}} (x8);
\path[-triangle 45, thick, line width=1.5mm]
(x7)edge[Emerald!40!] node[thick, Black]{\bf{1}} (x6)
edge[Emerald!40!] node[thick, Black]{\bf{3}} (X);
\path[-triangle 45, thick, line width=1.5mm]
(X1)edge[Emerald!40!] node[thick, Black]{\bf{3}} (X2)
edge[Emerald!40!] node[thick, Black]{\bf{2}} (X3);
\path[-triangle 45, thick, line width=1.5mm]
(X3)edge[Emerald!40!] node[thick, Black]{\bf{3}} (X4);
\path[-triangle 45, thick, line width=1.5mm]
(x8)edge[Emerald!40!] node[thick, Black]{\bf{1}} (x9);
\path[-triangle 45, thick, line width=1.5mm]
(X) edge[Emerald!40!] node[thick, Black]{\bf{1}} (X1);
\path[thick, gray, dashed] (x3) edge[->,>=latex] (x1);
\path[thick, gray, dashed] (x8) edge[->,>=latex] (X);
\path[thick, gray, dashed] (X2) edge[->,>=latex] (X4);
\path[thin, gray!80!, dashed] (x1) edge[->,>=latex, bend left=11] (X4);
\path[thick, gray, dashed] (x3) edge[->,>=latex] (x9);
\path[thick, gray, dashed] (x9) edge[->,>=latex] (X3);
\path[thick, gray, dashed] (x2) edge[->,>=latex] (x6);
\path[thick, gray, dashed] (x6) edge[->,>=latex] (X2);
\end{tikzpicture}
\caption{The supporting tree of $\EG_Q$ with respect to $c=s_1 s_2 s_3$}
\label{fig:main}
\end{figure}
\begin{example}\label{ex}
Consider an $A_3$ type quiver $Q\colon2 \leftarrow 1 \rightarrow 3$
with $c=s_1s_2s_3$.
We have the tree of $c$-sortable words below.
\[\xymatrix@C=1.5pc{
&s_2\ar[r]&s_2s_3&s_1s_2|s_1\\
e\ar[r]\ar[dr]\ar[ur]&s_1\ar[r]\ar[dr]&s_1s_2\ar[r]\ar[ur]&s_1s_2s_3\ar[r]
&s_1s_2s_3|s_1\ar[dr]\ar[r]&s_1s_2s_3|s_1s_2\ar[r]&s_1s_2s_3|s_1s_2s_3\\
&s_3&s_1|s_3\ar[r]&s_1s_3|s_1&&s_1s_2s_3|s_1s_3
}\]
Moreover, a piece of AR-quiver of $\mathrm{\mathcal{D}}(Q)$ is as follows
\[
\xymatrix@R=1pc@C=1pc{
\red{\widehat{Z}} \ar[dr] && \red{\widehat{B}} \ar[dr] && \green{Y} \ar[dr] && \green{C} \ar[dr] \\
& \red{\widehat{A}} \ar[ur]\ar[dr] && \red{\widehat{X}} \ar[ur]\ar[dr] && \green{A} \ar[ur]\ar[dr] && \green{ X} \\
\red{\widehat{Y}} \ar[ur] && \red{\widehat{C}} \ar[ur] && \green{Z} \ar[ur] && \green{B} \ar[ur]}
\]
where the green vertices are the indecomposables in $\h_Q$
and the red hatted ones are there shift minus one.
Note that $\green{X},\green{Y},\green{Z}$ are the simples $S_1, S_2, S_3$ in $\h_Q$ respectively.
Figure~\ref{fig:main} is the exchange graph $\EG_Q$ (cf. \cite[Figure~1 and 4]{KQ11}).
where we denote a heart $\h_\mathbf{w}$ by the set of its simples $S_1^\mathbf{w} S_2^\mathbf{w} S_3^\mathbf{w}$ (in order).
The green edges are the green mutations in some green mutation sequences
induced from $c$-sortable words.
The number on a green edges indicates where the mutation is at.
Note that the underlying graph of Figure~\ref{fig:main} is the associahedron (of dimension 3).
Further, Table~\ref{table} is a list of correspondences between
$c$-sortable words, hearts (denoted by their simples as in the Figure~\ref{fig:main}),
descents, cover reflection, inversions and (finite) torsion classes.
Note that this table is consistent with \cite[Table 1]{IT09},
in the sense that the objects in the $j$-th row here are precisely
objects in the $j$-th row there.
\end{example}
\begin{table}[t]
\caption{Example:$A_3$}
\label{table}
\begin{tabular}{c|c|c|c|c}
\\
\toprule
$c$-sortable & Heart & Descent & Cover ref.
& Torsion class \\
word $\mathbf{w}$ & $\h_\mathbf{w}$ & $\mathrm{Des}(\mathbf{w})$ & $\mathrm{Cov}(\mathbf{w})$
& $\mathcal{T}_\mathbf{w}$ \\ \midrule
$s_1 s_2 s_3 | s_1 s_2 s_3$
& ${\red{\widehat{X}}\red{\widehat{Z}}\red{\widehat{Y}}}$ & $s_1,s_2,s_3 $ & $t_X, t_Y, t_Z $ & $\green{\underline{XBCAZY}}$ \\ \midrule
$s_1 s_2 s_3| s_1 s_2 $
& ${\red{\widehat{B}}\red{\widehat{Z}}\green{Y}}$ & $s_1,s_2 $ & $t_B, t_Z $ & $\green{X\underline{B}C\underline{AZ}} $ \\ \midrule
$s_1 s_2 s_3| s_1 s_3 $
& ${\red{\widehat{C}}\green{Z}\red{\widehat{Y}}}$ & $s_1,s_3 $ & $t_C, t_Y $ & $\green{XB\underline{CAY}} $ \\ \midrule
$s_2 s_3 $ &
${\green{X}\red{\widehat{Y}}\red{\widehat{Z}}}$ & $s_2,s_3 $ & $t_Y, t_Z $ & $\green{\underline{YZ}} $ \\ \midrule
$s_1 s_2 s_3 $
& ${\green{A}\red{\widehat{B}}\red{\widehat{C}}}$ & $s_2,s_3 $ & $t_B, t_C $ & $\green{X\underline{BC}} $ \\ \midrule
$s_1 s_3| s_1 $
& ${\red{\widehat{Z}}\green{B}\red{\widehat{X}}}$ & $s_1,s_3 $ & $t_Z, t_X $ & $\green{\underline{XCZ}} $ \\ \midrule
$s_1 s_2| s_1 $
& ${\red{\widehat{Y}}\red{\widehat{X}}\green{C}}$ & $s_1,s_2 $ & $t_Y, t_X $ & $\green{\underline{XBY}} $ \\ \midrule
$s_2 $
& ${\green{X}\red{\widehat{Y}}\green{Z}}$ & $s_2 $ & $t_Y $ & $\green{\underline{Y}} $ \\ \midrule
$s_3 $
& ${\green{X}\green{Y}\red{\widehat{Z}}}$ & $s_3 $ & $t_Z $ & $\green{\underline{Z}} $ \\ \midrule
$s_1 s_2 s_3| s_1 $
& ${\red{\widehat{A}}\green{Z}\green{Y}}$ & $s_1 $ & $t_A $ & $\green{XBC\underline{A}} $ \\ \midrule
$s_1 s_2 $
& ${\green{Y}\red{\widehat{B}}\green{C}}$ & $s_2 $ & $t_B $ & $\green{X\underline{B}} $ \\ \midrule
$s_1 s_3 $
& ${\green{Z}\green{B}\red{\widehat{C}}}$ & $s_3 $ & $t_C $ & $\green{X\underline{C}} $ \\ \midrule
$s_1 $
& ${\red{\widehat{X}}\green{B}\green{C}}$ & $s_1 $ & $t_X $ & $\green{\underline{X}} $ \\ \midrule
$e $
& ${\green{X}\green{Y}\green{Z}}$ & $\emptyset $ & $\emptyset $ & $\emptyset$ \\
\bottomrule
\end{tabular}
\[ \]
\textbf{N.B.}$1\colon\; \{t_X,t_Y,t_Z,t_A,t_B,t_C\}
=\{s_1,s_2,s_3,s_2 s_3 s_1 s_3 s_2, s_1 s_2 s_1, s_1 s_3 s_1\}\,$.
\textbf{N.B.}$2\colon$ The underlines objects in $\mathcal{T}_{\w}%{\widetilde{\mathbf{w}}}$
form the wide subcategory $\mathcal{W}_{\w}%{\widetilde{\mathbf{w}}}$.
\end{table}
|
1205.0064
|
\subsection{Preliminaries} \label{Preliminaries} \
\bigskip
\begin{definition}
\label{y3}
For a graph $G$, let ${\rm ch}(G)$, the chromatic number of $G$ be the
minimal cardinal $\chi$ such that there is colouring $\bold c$ of $G$
with $\chi$ colours, that is $\bold c$ is a function from the set of
nodes of $G$ into $\chi$ or just a set of of cardinality $\le \chi$ such
that $\bold c(x) = \bold c(y) \Rightarrow \{x,y\} \notin {\rm edge}(G)$.
\end{definition}
\begin{definition}
\label{y6}
1) We say ``we have $\lambda$-incompactness for the $(< \chi)$-chromatic
number" or ${\rm INC}_{{\rm chr}}(\lambda,< \chi)$ {\underline{when}} \,:
there is a graph $G$ with $\lambda$ nodes, chromatic number $\ge \chi$
but every subgraph with $< \lambda$ nodes has
chromatic number $< \chi$.
\noindent
2) If $\chi = \mu^+$ we may replace $``< \chi"$ by $\mu$;
similarly in \ref{y8}.
\end{definition}
\noindent
We also consider
\begin{definition}
\label{y8}
1) We say ``we have $(\mu,\lambda)$-incompactness for $(< \chi)$-chromatic
number" or ${\rm INC}_{{\rm chr}}(\mu,\lambda,< \chi)$
{\underline{when}} \, there is an increasing continuous sequence $\langle
G_i:i \le \lambda\rangle$ of graphs each with $\le \mu$ nodes, $G_i$ an
induced subgraph of $G_\lambda$ with ${\rm ch}(G_\lambda) \ge \chi$ but $i
< \lambda \Rightarrow {\rm ch}(G_i) < \chi$.
\noindent
2) Replacing (in part (1)) $\chi$ by $\bar\chi = (< \chi_0,\chi_1)$
means ${\rm ch}(G_\lambda)) \ge \chi_1$ and $i < \lambda \rightarrow {\rm ch}(G_i) <
\chi_0$; similarly in \ref{y6} and parts 3),4) below.
\noindent
3) We say we have incompactness for length $\lambda$ for $(< \chi)$-chromatic
(or $\bar\chi$-chromatic) number {\underline{when}} \, we fail to have
$(\mu,\lambda)$-compactness for $(< \chi)$-chromatic (or
$\bar\chi$-chromatic) number for some $\mu$.
\noindent
4) We say we have $[\mu,\lambda]$-incompactness for $(<
\chi)$-chromatic number or ${\rm INC}_{{\rm chr}}[\mu,\lambda,< \chi]$
{\underline{when}} \, there is a graph $G$ with $\mu$ nodes,
${\rm ch}(G) \ge \chi$ but $G^1 \subseteq G \wedge |G^1| < \lambda \Rightarrow
{\rm ch}(G^1) < \chi$.
\noindent
5) Let ${\rm INC}^+_{{\rm chr}}(\mu,\lambda,< \chi)$ be as in part (1) but we
add that even the $c \ell(G_i)$, the colouring number of $G_i$ is $<
\chi$ for $i < \lambda$, see below.
\noindent
6) Let ${\rm INC}^+_{{\rm chr}}[\mu,\lambda,< \chi]$ be as in part (4) but we
add $G^1 \subseteq G \wedge |G^1| < \lambda \Rightarrow c \ell(G^1) <
\chi$.
\noindent
7) If $\chi = \kappa^+$ we may write $\kappa$ instead of ``$< \chi$".
\end{definition}
\begin{definition}
\label{y11}
1) For regular $\lambda > \kappa$ let $S^\lambda_\kappa = \{\delta <
\lambda:{\rm cf}(\delta) = \kappa\}$.
\noindent
2) We say $C$ is a $(\ge \theta)$-closed subset of a set $B$ of
ordinals when: if $\delta = \sup(\delta \cap B) \in B,{\rm cf}(\delta)
\ge \theta$ and $\delta = \sup(C \cap \delta)$ then $\delta \in C$.
\end{definition}
\begin{definition}
\label{y13}
For a graph $G$, the colouring number $c \ell(G)$ is the minimal
$\kappa$ such that there is a list $\langle a_\alpha:\alpha <
\alpha(*)\rangle$ of the nodes of $G$ such that $\alpha < \alpha(*)
\Rightarrow \kappa > |\{\beta < \alpha:\{a_\beta,a_\alpha\} \in {\rm edge}(G)\}$.
\end{definition}
\newpage
\section {From non-reflecting stationary in cofinality $\aleph_0$} \label{From}
\begin{claim}
\label{a3}
There is a graph $G$ with $\lambda$ nodes and chromatic number $>
\kappa$ but every subgraph with $< \lambda$ nodes have
chromatic number $\le \kappa$ {\underline{when}} \,:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus$] $(a) \quad \lambda,\kappa$ are regular cardinals
{\smallskip\noindent}
\item[${{}}$] $(b) \quad \kappa < \lambda = \lambda^\kappa$
{\smallskip\noindent}
\item[${{}}$] $(c) \quad S \subseteq S^\lambda_\kappa$
is stationary, not reflecting.
\end{enumerate}
\end{claim}
\begin{PROOF}{\ref{a3}}
\noindent
\underline{Stage A}: Let $\bar X = \langle X_i:i < \lambda\rangle$ be a
partition of $\lambda$ to sets such that $|X_i| = \lambda$ or just $|X_i|
= |i+2|^\kappa$ and $\min(X_i) \ge i$ and let $X_{<i} = \cup\{X_j:j
< i\}$ and $X_{\le i} = X_{<(i+1)}$.
For $\alpha < \lambda$ let $\bold i(\alpha)$ be the unique ordinal
$i < \lambda$ such that
$\alpha \in X_i$. We choose the set of points = nodes of
$G$ as $Y = \{(\alpha,\beta):\alpha < \beta < \lambda,\bold i(\beta) \in
S$ and $\alpha < \bold i(\beta)\}$ and let $Y_{<i} = \{(\alpha,\beta) \in
Y:\bold i(\beta) < i\}$.
\medskip
\noindent
\underline{Stage B}: Note that if $\lambda = \kappa^+$, the complete
graph with $\lambda$ nodes is an example (no use of the further
information in $\boxplus$). So {\rm without loss of generality} \, $\lambda > \kappa^+$.
Now choose a sequence satisfying the following properties,
exists by \cite[Ch.III]{Sh:g}:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus$] $(a) \quad \bar C = \langle C_\delta:\delta \in
S\rangle$
{\smallskip\noindent}
\item[${{}}$] $(b) \quad C_\delta \subseteq \delta = \sup(C_\delta)$
{\smallskip\noindent}
\item[${{}}$] $(c) \quad {\rm otp}(C_\delta) = \kappa$ such that $(\forall
\beta \in C_\delta)(\beta +1,\beta +2 \notin C_\delta)$
{\smallskip\noindent}
\item[${{}}$] $(d) \quad \bar C$ guesses\footnote{the guessing clubs
are used only in Stage D.}clubs.
\end{enumerate}
{\medskip\noindent}
Let $\langle \alpha^*_{\delta,\varepsilon}:\varepsilon <
\kappa\rangle$ list $C_\delta$ in increasing order.
For $\delta \in S$ let $\Gamma_\delta$ be the set of sequence
$\bar\beta$ such that:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_{\bar\beta}$] $(a) \quad \bar \beta$ has the form
$\langle \beta_\varepsilon:\varepsilon < \kappa \rangle$
{\smallskip\noindent}
\item[${{}}$] $(b) \quad \bar\beta$ is increasing with limit $\delta$
{\smallskip\noindent}
\item[${{}}$] $(c) \quad \alpha^*_{\delta,\varepsilon} <
\beta_{2 \varepsilon +i} < \alpha^*_{\delta,\varepsilon +1}$ for $i <
2,\varepsilon < \kappa$
{\smallskip\noindent}
\item[${{}}$] $(d) \quad \beta_{2 \varepsilon +i} \in
X_{< \alpha^*_{\delta,\varepsilon +1}} \backslash X_{\le
\alpha^*_{\delta,\varepsilon}}$ for $i < 2,\varepsilon < \kappa$
{\smallskip\noindent}
\item[${{}}$] $(e) \quad (\beta_{2 \varepsilon},\beta_{2\varepsilon
+1}) \in Y$ hence $\in Y_{< \alpha^*_{\delta,\varepsilon +1}} \subseteq
Y_{< \delta}$ for each $\varepsilon < \kappa$
\end{enumerate}
{\medskip\noindent}
(can ask less).
So $|\Gamma_\delta| \le |\delta|^\kappa \le |X_\delta| \le \lambda$
hence we can choose a sequence
$\langle \bar\beta_\gamma:\gamma \in X'_\delta \subseteq
X_\delta\rangle$ listing $\Gamma_\delta$.
Now we define the set of edges of $G$: ${\rm edge}(G) =
\{\{(\alpha_1,\alpha_2),(\min(C_\delta),\gamma)\}:\delta \in S,\gamma
\in X'_\delta$ hence the sequence $\bar\beta_\gamma = \langle
\beta_{\gamma,\varepsilon}:\varepsilon < \kappa\rangle$ is well
defined and we demand $(\alpha_1,\alpha_2) \in \{(\beta_{\gamma,2
\varepsilon},\beta_{\gamma,2 \varepsilon +1}):\varepsilon < \kappa\}\}$.
\medskip
\noindent
\underline{Stage C}: Every subgraph of $G$ of cardinality $<
\lambda$ has chromatic number $\le \kappa$.
For this we shall prove that:
{\medskip\noindent}
\begin{enumerate}
\item[$\oplus_1$] ${\rm ch}(G {\restriction} Y_{<i}) \le \kappa$ for every $i <
\lambda$.
\end{enumerate}
{\medskip\noindent}
This suffice as $\lambda$ is regular, hence every subgraph with $<
\lambda$ nodes is included in $Y_{<i}$ for some $i < \lambda$.
For this we shall prove more by induction on $j < \lambda$:
{\medskip\noindent}
\begin{enumerate}
\item[$\oplus_{2,j}$] if $i < j,i \notin S,\bold c_1$ a colouring of
$G {\restriction} Y_{<i},{\rm Rang}(\bold c_1) \subseteq \kappa$ and $u \in
[\kappa]^\kappa$ {\underline{then}} \, there is a colouring $\bold c_2$ of
$G {\restriction} Y_{<j}$ extending $\bold c_1$ such that ${\rm Rang}(\bold c_2 {\restriction}
(Y_{<j} \backslash Y_{<i})) \subseteq u$.
\end{enumerate}
\medskip
\noindent
\underline{Case 1}: $j=0$
Trivial.
\medskip
\noindent
\underline{Case 2}: $j$ successor, $j-1 \notin S$
By the induction hypothesis {\rm without loss of generality} \, $j=i+1$, but then every node
from $Y_j \backslash Y_i$ is an isolated node in $G {\restriction} Y_{<j}$,
because if $\{(\alpha,\beta),(\alpha',\beta')\}$ is an edge of $G
{\restriction} Y_j$ then $\bold i(\beta),\bold i(\beta') \in S$ hence
necessarily $\bold i(\beta) \ne j-1=i,\bold i(\beta') \ne j-1=i$ hence
both $(\alpha,\beta),(\alpha,\beta')$ are from $Y_i$.
\medskip
\noindent
\underline{Case 3}: $j$ successor, $j-1 \in S$
Let $j-1$ be called $\delta$ so $\delta \in S$. But $i \notin S$
by the assumption in $\oplus_{2,j}$ hence $i < \delta$.
Let $\varepsilon(*) < \kappa$ be such that
$\alpha^*_{\delta,\varepsilon(*)} > i$.
Let $\langle u_\varepsilon:\varepsilon \le \kappa\rangle$ be a
sequence of subsets of $u$, a partition of $u$ to sets each of
cardinality $\kappa$; actually the only disjointness used is
that $u_\kappa \cap (\bigcup\limits_{\varepsilon < \kappa}
u_\varepsilon) = \emptyset$.
We let $i_0 = i,i_{1 + \varepsilon} =
\cup\{\alpha^*_{\delta,\varepsilon(*)+1+\zeta} +1:\zeta < 1 +
\varepsilon\},i_\kappa = \delta,i_{\kappa +1} = \delta +1 =j$.
Note that:
{\medskip\noindent}
\begin{enumerate}
\item[$\bullet$] $\varepsilon < \kappa \Rightarrow i_\varepsilon
\notin S_j$.
\end{enumerate}
{\medskip\noindent}
[Why? For $\varepsilon=0$ by the assumption on $i$, for $\varepsilon$
successor $i_\varepsilon$ is a successor ordinal and for $i$ limit clearly
${\rm cf}(i_\varepsilon) = {\rm cf}(\varepsilon) < \kappa$ and $S \subseteq
S^\lambda_\kappa$.]
We now choose $\bold c_{2,\zeta}$ by induction on $\zeta \le \kappa +1$ such
that:
{\medskip\noindent}
\begin{enumerate}
\item[$\bullet$] $\bold c_{2,0} = \bold c_1$
{\smallskip\noindent}
\item[$\bullet$] $\bold c_{2,\zeta}$ is a colouring of $G {\restriction} Y_{<i_\zeta}$
{\smallskip\noindent}
\item[$\bullet$] $\bold c_{2,\zeta}$ is increasing with $\zeta$
{\smallskip\noindent}
\item[$\bullet$] ${\rm Rang}(\bold c_{2,\zeta} {\restriction} (Y_{< i_{\xi +1}}
\backslash Y_{< i_\xi})) \subseteq u_\xi$ for every $\xi < \zeta$.
\end{enumerate}
{\medskip\noindent}
For $\zeta = 0,\bold c_{2,0}$ is $\bold c_1$ so is given.
For $\zeta = \varepsilon +1 < \kappa$: use the induction hypothesis,
possible as necessarily $i_\varepsilon \notin S$.
For $\zeta \le \kappa$ limit: take union.
For $\zeta = \kappa+1$, note that each node $b$ of $Y_{< i_\zeta}
\backslash Y_{< i_\kappa}$ is not connected to any other such node and
if the node $b$ is connected to a node from $Y_{< i_\kappa}$ then the
node $b$ necessarily has the form
$(\min(C_\delta),\gamma),\gamma \in X'_\delta$, hence $\bar\beta_\gamma$
is well defined, so the node $b = (\min(C_\delta),\gamma)$
is connected in $G$, more exactly in $G {\restriction} Y_{\le \delta}$
exactly to the $\kappa$ nodes
$\{(\beta_{\gamma,2 \varepsilon},\beta_{\gamma,2 \varepsilon
+1}):\varepsilon < \kappa\}$, but for every $\varepsilon < \kappa$
large enough, $\bold c_{2,\kappa}((\beta_{\gamma,2
\varepsilon},\beta_{\gamma,2 \varepsilon +1})) \in
u_\varepsilon$ hence $\notin u_\kappa$ and
$|u_\kappa| = \kappa$ so we can choose a colour.
\medskip
\noindent
\underline{Case 4}: $j$ limit
By the assumption of the claim there is a club $e$
of $j$ disjoint to $S$ and {\rm without loss of generality} \, $\min(e) =
i$. Now choose $\bold c_{2,\xi}$ a colouring of $Y_{< \xi}$
by induction on $\xi \in e \cup \{j\}$, increasing with $\xi$ such that
${\rm Rang}(\bold c_{2,\xi} {\restriction} (Y_{< \varepsilon} \backslash Y_{<i}))
\subseteq u$ and $\bold c_{2,0} = \bold c_1$
{\medskip\noindent}
\begin{enumerate}
\item[$\bullet$] For $\xi = \min(e) = i$ the colouring $\bold
c_{2,\xi} = \bold c_{2,i} = \bold c_1$ is given,
{\smallskip\noindent}
\item[$\bullet$] for $\xi$ successor in $e$, i.e. $\in
{\rm nacc}(e) \backslash \{i\}$, use the induction hypothesis with
$\xi,\max(e \cap \xi)$ here playing the role of $j,i$ there recalling
$\max(e \cap \xi) \in e,e \cap S = \emptyset$
{\smallskip\noindent}
\item[$\bullet$] for $\xi = \sup(e \cap \xi)$ take union.
\end{enumerate}
{\medskip\noindent}
Lastly, for $\xi=j$ we are done.
\medskip
\noindent
\underline{Stage D}: ${\rm ch}(G) > \kappa$.
Why? Toward a contradiction, assume $\bold c$ is
a colouring of $G$ with set of colours
$\subseteq \kappa$. For each $\gamma < \lambda$ let
$u_\gamma = \{\bold c((\alpha,\beta)):\gamma < \alpha <
\beta < \lambda$ and $(\alpha,\beta) \in Y\}$. So $\langle u_\gamma:\gamma
< \lambda\rangle$ is $\subseteq$-decreasing sequence of subsets of $\kappa$ and
$\kappa < \lambda = {\rm cf}(\lambda)$,
hence for some $\gamma(*) < \lambda$ and $u_*
\subseteq \kappa$ we have $\gamma \in (\gamma(*),\lambda)
\Rightarrow u_\gamma = u_*$.
Hence $E = \{\delta < \lambda:\delta$ is a limit ordinal $> \gamma(*)$
and $(\forall \alpha < \delta)((\bold i(\alpha) < \delta)$
and for every $\gamma < \delta$ and $i \in u_*$ there are $\alpha <
\beta$ from $(\gamma,\delta)$ such that $(\alpha,\beta) \in Y$ and
$\bold c((\alpha,\beta))=i\}$ is a club of $\lambda$.
Now recall that $\bar C$ guesses clubs hence for some $\delta \in S$ we have
$C_\delta \subseteq E$, so for every $\varepsilon < \kappa$ we can
choose $\beta_{2 \varepsilon} < \beta_{2 \varepsilon +1}$ from
$(\alpha^*_{\delta,\varepsilon},\alpha^*_{\delta,\varepsilon +1})$
such that $(\beta_{2 \varepsilon},\beta_{2 \varepsilon +1}) \in Y$ and
$\varepsilon \in u_* \Rightarrow \bold c((\beta_{2
\varepsilon},\beta_{2 \varepsilon +1})) = \varepsilon$. So $\langle
\beta_\varepsilon:\varepsilon < \kappa\rangle$ is well defined, increasing and
belongs to $\Gamma_\delta$, hence $\bar\beta_\gamma = \langle
\beta_\varepsilon:\varepsilon < \kappa\rangle$ for some $\gamma \in
X_\delta$, hence $(\alpha^*_{\delta,0},\gamma)$ belongs to $Y$ and is
connected in the graph to $(\beta_{2 \varepsilon},\beta_{2 \varepsilon
+1})$ for $\varepsilon < \kappa$. Now if $\varepsilon\in u_*$ then
$\bold c((\beta_{2 \varepsilon},\beta_{2 \varepsilon +1})) =
\varepsilon$ hence $\bold c((\alpha^*_{\delta,0},\gamma)) \ne
\varepsilon$ for every $\varepsilon \in u_*$,
so $\bold c((\alpha^*_{\delta,0},\gamma)) \in \kappa
\backslash u_*$. But $u_* = u_{\alpha^*_{\delta,0}}$ and $\bold
c((\alpha^*_{\delta,0},\gamma)) \in \kappa \backslash u_*$, so we get
contradiction to the definition of $u_{\alpha^*_{\delta,0}}$.
\end{PROOF}
\noindent
Similarly
\begin{claim}
\label{a6}
There is an increasing continuous sequence $\langle G_i:i \le
\lambda \rangle$ of graphs each of cardinality $\lambda^\kappa$ such
that ${\rm ch}(G_\lambda) > \kappa$ and $i < \lambda$ implies ${\rm ch}(G_i) \le
\kappa$ and even $c \ell(G_i) \le \kappa$ {\underline{when}} \,:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus$] $(a) \quad \lambda = {\rm cf}(\lambda)$
{\smallskip\noindent}
\item[${{}}$] $(b) \quad S \subseteq \{\delta < \lambda:{\rm cf}(\delta) =
\kappa\}$ is stationary not reflecting.
\end{enumerate}
\end{claim}
\begin{PROOF}{\ref{a6}}
Like \ref{a3} but the $X_i$ are not necessarily $\subseteq \lambda$ or
use \ref{c3}.
\end{PROOF}
\newpage
\section {From almost free} \label{Fromalmost}
\begin{definition}
\label{c1}
Suppose $\eta_\beta \in {}^\kappa{\rm Ord}$ for every $\beta < \alpha(*)$
and $u \subseteq \alpha(*)$, and $\alpha < \beta < \alpha(*)
\Rightarrow \eta_\alpha \ne \eta_\beta$.
\noindent
1) We say $\{\eta_\alpha:\alpha \in u\}$ is free {\underline{when}} \, there exists a
function $h:u \rightarrow \kappa$ such that $\langle
\{\eta_\alpha(\varepsilon):\varepsilon \in
[h(\alpha),\kappa)\}:\alpha \in u\rangle$ is a sequence of pairwise
disjoint sets.
\noindent
2) We say $\{\eta_\alpha:\alpha \in u\}$ is weakly free {\underline{when}} \, there
exists a sequence $\langle u_{\varepsilon,\zeta}:\varepsilon,\zeta <
\kappa\rangle$ of subsets of $u$ with union $u$, such that the
function $\eta_\zeta \mapsto \eta_\zeta(\varepsilon)$ is a
one-to-one function on $u_{\varepsilon,\zeta}$,
for each $\varepsilon,\zeta < \kappa$.
\end{definition}
\begin{claim}
\label{c3}
1) We have ${\rm INC}_{{\rm chr}}(\mu,\lambda,\kappa)$ and even
${\rm INC}^+_{{\rm chr}}(\mu,\lambda,\kappa)$, see Definition \ref{y8}(1),(5) {\underline{when}} \,:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus$] $(a) \quad \alpha(*) \in [\mu,\mu^+)$ and $\lambda$
is regular $\le \mu$ and $\mu = \mu^\kappa$
{\smallskip\noindent}
\item[${{}}$] $(b) \quad \bar \eta = \langle \eta_\alpha:\alpha <
\alpha(*)\rangle$
{\smallskip\noindent}
\item[${{}}$] $(c) \quad \eta_\alpha \in {}^\kappa \mu$
{\smallskip\noindent}
\item[${{}}$] $(d) \quad \langle u_i:i \le \lambda\rangle$
is a $\subseteq$-increasing continuous sequence of subsets of
$\alpha(*)$
\hskip25pt with $u_\lambda = \alpha(*)$
{\smallskip\noindent}
\item[${{}}$] $(e) \quad \bar \eta {\restriction} u_\alpha$ is free {\underline{iff}} \,
$\alpha < \lambda$ {\underline{iff}} \, $\bar\eta {\restriction} u_\alpha$ is weakly free.
\end{enumerate}
{\medskip\noindent}
2) We have ${\rm INC}_{{\rm chr}}[\mu,\lambda,\kappa]$ and even
${\rm INC}^+_{{\rm chr}}[\mu,\lambda,\kappa]$ , see Definition \ref{y8}(4) {\underline{when}} \,:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_2$] $(a),(b),(c) \quad$ as in $\boxplus$ from \ref{c3}
{\smallskip\noindent}
\item[${{}}$] $(d) \quad \bar\eta$ is not free
{\smallskip\noindent}
\item[${{}}$] $(e) \quad \bar\eta {\restriction} u$ is free when
$u \in [\alpha(*)]^{< \lambda}$.
\end{enumerate}
\end{claim}
\begin{PROOF}{\ref{c3}}
We concentrate on proving part (1); the proof of part (2) is similar.
For ${\mathscr A} \subseteq {}^\kappa{\rm Ord}$, we define $\tau_{{\mathscr A}}$ as the
vocabulary $\{P_\eta:\eta \in {\mathscr A}\} \cup \{F_\varepsilon:\varepsilon <
\kappa\}$ where $P_\eta$ is a unary predicate, $F_\varepsilon$ a unary
function (will be interpreted as possibly partial).
{\rm Without loss of generality} \, for each $i < \lambda,u_i$ is an initial
segment of $\alpha(*)$ and let
${\mathscr A} = \{\eta_\alpha:\alpha < \alpha(*)\}$
and let $<_{{\mathscr A}}$ be the well ordering
$\{(\eta_\alpha,\eta_\beta):\alpha < \beta < \alpha(*)\}$ of ${\mathscr A}$.
We further let $K_{{\mathscr A}}$ be the class of structures $M$ such that
(pedantically, $K_{{\mathscr A}}$ depend also on the sequence $\langle
\eta_\alpha:\alpha < \alpha(*)\rangle$:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_1$] $(a) \quad M = (|M|,F^M_\varepsilon,
P^M_\eta)_{\varepsilon < \kappa,\eta \in {\mathscr A}}$
{\smallskip\noindent}
\item[${{}}$] $(b) \quad \langle P^M_\eta:\eta \in {\mathscr A}\rangle$ is a
partition of $|M|$, so for $a \in M$ let $\eta_a$
\hskip25pt $= \eta^M_a$ be the unique $\eta \in {\mathscr A}$ such that $a \in P^M_\eta$
{\smallskip\noindent}
\item[${{}}$] $(c) \quad$ if $a_\ell \in P^M_{\eta_\ell}$ for
$\ell=1,2$ and $F^M_\varepsilon(a_2) = a_1$ then
\hskip25pt $\eta_1(\varepsilon) = \eta_2(\varepsilon)$ and
$\eta_1 <_{{\mathscr A}} \eta_2$.
\end{enumerate}
{\medskip\noindent}
Let $K^*_{{\mathscr A}}$ be the class of $M$ such that
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_2$] $(a) \quad M \in K_{{\mathscr A}}$
{\smallskip\noindent}
\item[${{}}$] $(b) \quad \|M\| = \mu$
{\smallskip\noindent}
\item[${{}}$] $(c) \quad$ if $\eta \in {\mathscr A},u \subseteq \kappa$ and
$\eta_\varepsilon <_{{\mathscr A}} \eta,\eta_\varepsilon(\varepsilon)
= \eta(\varepsilon)$ and $a_\varepsilon \in P^M_{\eta_\varepsilon}$
\hskip25pt for $\varepsilon \in u$ {\underline{then}} \, for some $a \in P^M_\eta$ we have
$\varepsilon \in u \Rightarrow F^M_\varepsilon(a) = a_\varepsilon$
\hskip25pt and $\varepsilon \in \kappa
\backslash u \Rightarrow F^M_\varepsilon(a)$ not defined.
\end{enumerate}
{\medskip\noindent}
Clearly
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_3$] there is $M \in K^*_{{\mathscr A}}$.
\end{enumerate}
{\medskip\noindent}
[Why? As $\mu = \mu^\kappa$ and $|{\mathscr A}| = \mu$.]
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_4$] for $M \in K_{{\mathscr A}}$ let $G_M$ be the graph with:
{\smallskip\noindent}
\begin{enumerate}
\item[$\bullet$] set of nodes $|M|$
{\smallskip\noindent}
\item[$\bullet$] set of edges $\{\{a,F^M_\varepsilon(a)\}:a \in
|M|,\varepsilon < \kappa$ when $F^M_\varepsilon(a)$ is defined$\}$.
\end{enumerate}
\end{enumerate}
{\medskip\noindent}
Now
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_5$] if $u \subseteq \alpha(*),{\mathscr A}_u =
\{\eta_\alpha:\alpha \in u\} \subseteq {\mathscr A}$ and $\bar\eta {\restriction} u$
is free, and $M \in K_{{\mathscr A}}$ {\underline{then}} \, $G_{M,{\mathscr A}_u} := G_M {\restriction}
(\cup\{P^M_\eta:\eta \in {\mathscr A}_u\})$
has chromatic number $\le \kappa$; moreover has colouring number $\le \kappa$.
\end{enumerate}
{\medskip\noindent}
[Why? Let $h:u \rightarrow \kappa$ witness that $\bar\eta {\restriction} u$ is
free and for $\varepsilon < \kappa$ let ${\mathscr B}_\varepsilon :=
\{\eta_\alpha:\alpha \in u$ and $h(\alpha) = \varepsilon\}$, so ${\mathscr B} =
\cup\{{\mathscr B}_\varepsilon:\varepsilon < \kappa\}$, hence it is enough to
prove for each $\varepsilon < \kappa$ that $G_{\mu,{\mathscr B}_\varepsilon}$
has chromatic number $\le \kappa$. To prove this, by induction on
$\alpha \le \alpha(*)$ we choose $\bold c^\varepsilon_\alpha$ such that:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_{5.1}$] $(a) \quad \bold c^\varepsilon_\alpha$ is a
function
{\smallskip\noindent}
\item[${{}}$] $(b) \quad \langle \bold c_\beta:\beta \le
\alpha\rangle$ is increasing continuous
{\smallskip\noindent}
\item[${{}}$] $(c) \quad {\rm Dom}(\bold c^\varepsilon_\alpha) =
B^\varepsilon_\alpha := \cup\{P^M_{\eta_\beta}:\beta < \alpha$ and
$\eta_\beta \in {\mathscr B}_\varepsilon\}$
{\smallskip\noindent}
\item[${{}}$] $(d) \quad {\rm Rang}(\bold c^\varepsilon_\alpha) \subseteq
\kappa$
{\smallskip\noindent}
\item[${{}}$] $(e) \quad$ if $a,b, \in {\rm Dom}(\bold c_\alpha)$ and
$\{a,b\} \in {\rm edge}(G_M)$ then $\bold c_\alpha(a) \ne \bold
c_\alpha(b)$.
\end{enumerate}
{\medskip\noindent}
Clearly this suffices. Why is this possible?
If $\alpha = 0$ let $\bold c^\varepsilon_\alpha$ be empty, if $\alpha$
is a limit ordinal let $\bold c^\varepsilon_\alpha = \cup\{\bold
c^\varepsilon_\beta:\beta < \alpha\}$ and if $\alpha = \beta +1 \wedge
\alpha(\beta) \ne G$ let $\bold c_\alpha = \bold c_\beta$.
Lastly, if $\alpha = \beta +1 \wedge h(\beta) = \varepsilon$ we define
$\bold c^\varepsilon_\alpha$ as follows for $a \in {\rm Dom}(\bold
c^\varepsilon_\alpha),\bold c^\varepsilon_\alpha(a)$ is:
\bigskip
\noindent
\underline{Case 1}: $a \in B^\varepsilon_\beta$.
Then $\bold c^\varepsilon_\alpha(a) = \bold c^\varepsilon_\beta(a)$.
\bigskip
\noindent
\underline{Case 2}: $a \in B^\varepsilon_\alpha \backslash
B^\varepsilon_\beta$.
Then $\bold c^\varepsilon_\alpha(a) = \min(\kappa
\backslash \{\bold c^\varepsilon_\beta(F^M_\zeta(a)):\zeta < \varepsilon$
and $F^M_\zeta(a) \in {\rm Dom}(\bold c^\varepsilon_\beta)\})$.
This is well defined as:
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_{5.2}$] $(a) \quad B^\varepsilon_\alpha =
B^\varepsilon_\beta \cup P^M_{\eta_\beta}$
{\smallskip\noindent}
\item[${{}}$] $(b) \quad$ if $a \in B^\varepsilon_\beta$ then $\bold
c^\varepsilon_\beta(a)$ is well defined (so case 1 is O.K.)
{\smallskip\noindent}
\item[${{}}$] $(c) \quad$ if $\{a,b\} \in {\rm edge}(G_M),a \in
P^M_{\eta_\beta}$ and $b \in B^\varepsilon_\alpha$ {\underline{then}} \, $b \in
B^\varepsilon_\beta$ and
\hskip25pt $b \in \{F^M_\zeta(a):\zeta < \varepsilon\}$
{\smallskip\noindent}
\item[${{}}$] $(d) \quad \bold c^\varepsilon_\alpha(a)$ is well
defined in Case 2, too
{\smallskip\noindent}
\item[${{}}$] $(e) \quad \bold c^\varepsilon_\alpha$ is a function
from $B^\varepsilon_\alpha$ to $\kappa$
{\smallskip\noindent}
\item[${{}}$] $(f) \quad \bold c^\varepsilon_\alpha$ is a colouring.
\end{enumerate}
{\medskip\noindent}
[Why? Clause (a) by $\boxplus_{5.1}(c)$, clause (b) by the induction
hypothesis and clause (c) by $\boxplus_1(c) + \boxplus_4$. Next,
clause (d) holds as $\{\bold c^\varepsilon_\beta(F^M_\zeta(a)):\zeta <
\varepsilon$ and $F^M_\zeta(a) \in B^\varepsilon_\beta = {\rm Dom}(\bold
c^\varepsilon_\beta )\}$ is a set of cardinality $\le |\varepsilon|
< \kappa$. Clause (e) holds by the choices of the $\bold
c^\varepsilon_\alpha(a)$'s. Lastly, to check that clause (f) holds
assume $(a,b)$ is an edge of $G_M {\restriction} B^\varepsilon_\alpha$, for
some $\zeta < \kappa$ we have $b = F^M_\zeta(a)$, hence $\eta^M_a <_{{\mathscr A}}
\eta^M_b$. If $a,b \in B^\varepsilon_\beta$ use the induction
hypothesis. Otherwise, $\zeta < \varepsilon$ by the definition of
``$h$ witnesses $\bar\eta {\restriction} u$ is free" and the choice of
$B^\varepsilon_\alpha$ in $\boxplus_{5.1}(c)$. Now use the choice of
$\bold c^\varepsilon_\alpha(a)$ in Case 2 above.]
So indeed $\boxplus_5$ holds.]
{\medskip\noindent}
\begin{enumerate}
\item[$\boxplus_6$] ${\rm chr}(G_M) > \kappa$ if $M \in K^*_{{\mathscr A}}$.
\end{enumerate}
{\medskip\noindent}
Why? Toward contradiction assume $\bold c:G_M \rightarrow \kappa$ is
a colouring. For each $\eta \in {\mathscr A}$ and $\varepsilon < \kappa$ let
$\Lambda_{\eta,\varepsilon} = \{\nu:\nu \in {\mathscr A},\nu <_{{\mathscr A}}
\eta,\nu(\varepsilon) = \eta(\varepsilon)$ and for some $a \in
P^M_\nu$ we have $\bold c(a) = \varepsilon\}$.
Let ${\mathscr B}_\varepsilon = \{\eta \in {\mathscr A}:|\Lambda_{\eta,\varepsilon}| <
\kappa\}$. Now if ${\mathscr A} \ne \cup\{{\mathscr B}_\varepsilon:\varepsilon
< \kappa\}$ then pick any $\eta \in
{\mathscr A} \backslash \cup\{{\mathscr B}_\varepsilon:\varepsilon < \kappa\}$ and by
induction on $\varepsilon < \kappa$ choose $\nu_\varepsilon \in
\Lambda_{\eta,\varepsilon} \backslash \{\nu_\zeta:\zeta <
\varepsilon\}$, possible as $\eta \notin {\mathscr B}_\varepsilon$
by the definition of ${\mathscr B}_\varepsilon$. By the definition of
$\Lambda_{\eta,\varepsilon}$ there is $a_\varepsilon \in
P^M_{\nu_\varepsilon}$ such that $\bold c(\nu_\varepsilon) =
\varepsilon$. So as $M \in K^*_{{\mathscr A}}$ there is $a \in P^M_\eta$
such that $\varepsilon < \kappa \Rightarrow
F^M_\varepsilon(a) = a_\varepsilon$, but $\{a,a_\varepsilon\} \in
{\rm edge}(G_M)$ hence $\bold c(a) \ne \bold c(a_\varepsilon) =
\varepsilon$ for every $\varepsilon < \kappa$, contradiction. So ${\mathscr A}
= \cup\{{\mathscr B}_\varepsilon:\varepsilon < \kappa\}$.
For each $\varepsilon < \kappa$ we choose $\zeta_\eta < \kappa$ for
$\eta \in {\mathscr B}_\varepsilon$ by induction on $<_{{\mathscr A}}$ such
that $\zeta_\eta \notin \{\zeta_\nu:\nu \in
\Lambda_{\eta,\varepsilon} \cap {\mathscr B}_\varepsilon\}$. Let
${\mathscr B}_{\varepsilon,\zeta} = \{\eta \in {\mathscr B}_\varepsilon:\zeta_\eta
=\zeta\}$ for $\varepsilon,\zeta < \kappa$ so ${\mathscr A} =
\cup\{{\mathscr B}_{\varepsilon,\zeta}:\varepsilon,\zeta < \kappa\}$
and clearly $\eta \mapsto \eta(\varepsilon)$ is a
one-to-one function with domain ${\mathscr B}_{\varepsilon,\zeta}$,
contradiction to ``$\bar\eta = \bar\eta {\restriction} u_\lambda$ is not weakly free".
\end{PROOF}
\begin{observation}
\label{c6}
1) If ${\mathscr A} \subseteq {}^\kappa \mu$ and $\eta \ne \nu \in {\mathscr A}
\Rightarrow (\forall^\infty \varepsilon < \kappa)(\eta(\varepsilon)
\ne \nu(\varepsilon))$ then ${\mathscr A}$ is free iff ${\mathscr A}$ is weakly
free.
\noindent
2) The assumptions of \ref{c3}(2) hold {\underline{when}} \,: $\mu \ge \lambda > \kappa$ are
regular, $S \subseteq S^\mu_\kappa$ stationary, $\bar\eta =
\langle \eta_\delta:\delta \in S\rangle,\eta_\delta$ an increasing
sequence of ordinals of length $\kappa$ with limit $\delta$ such
that $u \subseteq [\lambda]^{< \lambda} \Rightarrow \langle
{\rm Rang}(\eta_\delta):\eta \in u\rangle$ has a one-to-one choice function.
\end{observation}
\begin{conclusion}
\label{c12}
Assume that for every graph $G$, if $H \subseteq G \wedge |H| <\lambda
\Rightarrow {\rm chr}(H) \le \kappa$ then ${\rm chr}(G) \le \kappa$.
{\underline{Then}} \,:
{\medskip\noindent}
\begin{enumerate}
\item[$(A)$] if $\mu > \kappa = {\rm cf}(\mu)$ and $\mu \ge \lambda$ then
${\rm pp}(\mu) = \mu^+$
{\smallskip\noindent}
\item[$(B)$] if $\mu > {\rm cf}(\mu) \ge \kappa$ and $\mu \ge \lambda$ then
${\rm pp}(\mu) = \mu^+$, i.e. the strong hypothesis
{\smallskip\noindent}
\item[$(C)$] if $\kappa = \aleph_0$ then above $\lambda$ the SCH
holds.
\end{enumerate}
\end{conclusion}
\begin{PROOF}{\ref{c12}}
\noindent
\underline{Clause $(A)$}: By \ref{c3} and \cite[Ch.II]{Sh:g},
\cite[Ch.IX,\S1]{Sh:g}.
\medskip
\noindent
\underline{Clause $(B)$}: Follows from (A) by
\cite[Ch.VIII,\S1]{Sh:g}.
\medskip
\noindent
\underline{Clause $(C)$}: Follows from (B) by \cite[Ch.IX,\S1]{Sh:g}.
\end{PROOF}
|
2007.10973
|
\section{Experiments}
\vspace{-0.2cm}
In this section we show qualitative and quantitative results on the task of auto-encoding and single view reconstruction of 3D shapes with comparison against several state of the art baselines. In addition to these tasks, we also demonstrate several additional features and applications of our approach including latent space interpolation texture mapping, consistent correspondence and shape deformations in the supplementary material.
\vspace{-0.3cm}
\paragraph{Data} We evaluate our approach on the ShapeNet Core dataset \cite{shapenet}, which consists of 3D models across 13 object categories which are preprocessed with \cite{shapenetmanifold} to obtain manifold meshes. We use the training, validation and testing splits provided by \cite{3dr2n2} to be comparable to other baselines. We use rendered views from \cite{3dr2n2}.
\begin{figure}[!!t]
\centering
\includegraphics[width=0.95\linewidth]{Images/AE-chair2.pdf}
\vspace{-0.2cm}
\caption{Auto-encoder: The first row shows mesh geometry along with self-intersections (red) and flipped normals (black). The bottom row shows results from physically based rendering with dielectric and conducting materials. The appearances of the red box, green ball and blue ball are more realistic for NMF than AtlasNet, since the latter suffers from severe self-intersections and flipped normals.}
\label{AE}
\end{figure}
\vspace{-0.3cm}
\paragraph{Evaluation criteria}
We evaluate the predicted shape $\mathcal{M}_P$ for geometric accuracy to the ground truth $\mathcal{M}_T$ as well as for manifoldness. For geometric accuracy, we follow \cite{meshrcnn} and compute the bidirectional Chamfer distance according to \eqref{CD} and normal consistency using \eqref{NC} on 10000 points sampled from each mesh. Since Chamfer distance is sensitive to the size of meshes, we scale the meshes to lie within a unit radius sphere. Chamfer distances are report by multiplying with $10^3$. With $\Tilde{M_P},\Tilde{M_T}$ the point sets sampled from $\mathcal{M}_p,\mathcal{M}_T$ and $\Lambda_{P,Q} = \{(p,argmin_{q} ||p-q||) : p \in P\}$, we define
\begin{equation}
\mathcal{L}_{n} = |\Tilde{M_P}|^{-1} \!\!\!\! \sum_{(p,q)\in \Lambda_{\Tilde{M}_P,\Tilde{M}_T}} \!\!\!\! |u_p \cdot u_q| \: + \: |\Tilde{M_T}|^{-1} \!\!\!\! \sum_{(p,q)\in \Lambda_{\Tilde{M}_T,\Tilde{M}_P}} \!\!\!\! |u_q \cdot u_p| \: - \: 1
\label{NC}
\end{equation}
We detect non-manifold vertices (Fig.~\ref{manifoldness}(b)) and edges (Fig.~\ref{manifoldness}(a)) using \cite{open3d} and report the metrics `NM-vertices', `NM-edges' respectively as the ratio($\times 10^5$) of number of non-manifold vertices and edges to total number of vertices and edges in a mesh. To calculate non-manifold faces, we count number of times adjacent face normals have a negative inner product, then the metric `NM-Faces' is reported as its ratio(\%) to the number of edges in the mesh. To calculate the number of instances of self-intersection, we use \cite{torch-isect} and report the ratio(\%) of number of intersecting triangles to total number of triangles in a mesh. Only the mean over all ShapeNet categories are reported in this paper with category specific details can be found in the supplementary.
For qualitative evaluation, we render predicted meshes via a physically based renderer \cite{mitsuba} with dielectric and metallic materials to highlight artifacts due to non-manifoldness. While we render NMF meshes directly, other methods render poorly due to non-manifoldness and are smoothed prior to rendering to obtain better visualizations. Please see supplementary for further visualizations.
\vspace{-0.3cm}
\paragraph{Baselines}
We compare with official implementations for Pixel2Mesh \cite{meshrcnn,pixel2mesh}, MeshRCNN \cite{meshrcnn} and AtlasNet \cite{AtlasNet}. We use pretrained models for all these baselines motioned in this paper since they share the same dataset split by \cite{3dr2n2}. We use the implementation of Pixel2Mesh provided by MeshRCNN, as it uses a deeper network that outperforms the original implementation. We also consider AtlasNet-O which is a baseline proposed in \cite{AtlasNet} that uses patches sampled from a spherical mesh, making it closer to our own choice of initial template mesh. We also create a baseline of our own called NMF-M, which is similar in architecture to NMF but trained with a larger icosphere of 2520 vertices, leading to slight differences in test time performance. To account for possible variation in manifoldness due to simple post processing techniques, we also report outputs of all mesh generation methods with 3 iterations of Laplacian smoothing. Further iterations of smoothing lead to loss of geometric accuracy without any substantial gain in manifoldness. We also compare with occupancy networks \cite{OccNet}, a state-of-the-art indirect mesh generation method based on implicit surface representation. We compare with several variants of OccNet based on the resolution of Multi Iso-Surface Extraction algorithm \cite{OccNet}. To this end, we create OccNet baselines OccNet-1, OccNet-2 and OccNet-3 with MISE upsampling of 1, 2 and 3 times respectively. For fair comparison to other baselines, we use OccNet's refinement module to output its meshes with 5200 faces.
\begin{table}[!!t]
\resizebox{\textwidth}{!}{\begin{tabular}{|l|l l l|l l l|l l l|l l l|l l l|}
\hline
\multirow{3}{*}{} &
\multicolumn{3}{c|}{Chamfer-L2 ($\downarrow$)} &
\multicolumn{3}{c|}{Normal Consistency ($\uparrow$)} &
\multicolumn{3}{c|}{NM-Faces ($\downarrow$)} &
\multicolumn{3}{c|}{Self-Intersection ($\downarrow$)} \\
&AtNet&AtNet-O&NMF&AtNet&AtNet-O&NMF&AtNet&AtNet-O&NMF&AtNet&AtNet-O&NMF\\
\hline
mean & 4.15 & \textbf{3.50} & 5.54 & 0.815 & 0.816 & \textbf{0.826} & 1.72 & 1.43 & \textbf{0.71} & 24.80 & 6.03 & \textbf{0.10}\\
mean (with Laplace) & 4.59 & \textbf{3.81} & 5.25 & 0.807 & 0.811 & \textbf{0.826} & 0.47 & 0.56 &\textbf{0.38} & 13.26 & 2.02 & \textbf{0.00}\\
\hline
\end{tabular}}
\caption{Auto-encoding performance.}
\vspace{-0.9cm}
\label{AE_table}
\end{table}
\vspace{-0.2cm}
\paragraph{Auto-encoding 3D shapes}
We now evaluate NMF's ability to generate a shape given an input 3D point cloud and compare against AtlasNet \cite{AtlasNet} and AtlasNet-O\cite{AtlasNet} in Table \ref{AE_table}. We note that NMF outperforms AtlasNet in terms of manifoldness with 20 times less self-intersections. NMF generates meshes with a higher normal consistency, leading to more realistic results in simulations and physically-based rendering. All the three methods have manifold vertices. While both NMF and AtlasNet-O have no non-manifold edges, AtlasNet yields a constant value of $7400$ due to its constituent 25 non-manifold open templates. Visualizations in Fig. \ref{AE} show severe self-intersections and flipped normals for AtlasNet baselines which are absent for NMF. This leads to NMF giving more realistic physically based rendering results. Note the reflection of red box and green ball through NMF mesh, which are either distorted or absent for AtlasNet. The blue ball's reflection on conductor's surface is closer to ground truth for NMF due to higher manifoldness.
\vspace{-0.2cm}
\begin{figure}[!!t]
\centering
\includegraphics[width=\linewidth]{Images/SVR/SVR-table.pdf}
\caption{Single View Reconstruction: We compare NMF to other mesh generating baselines for SVR. Top row shows mesh geometry along with self-intersections (red) and flipped normals (black). Physically based renders for dielectric and conductor material are shown in rows 2 and 3 respectively. Notice the reflection of checkerboard floor, occluded part of red box and balls are all visible through NMF render but not with other baselines. This is due to the presence of severe self-intersection and flipped normals. The reflection of blue ball on metallic table is more realistic for NMF than other methods.}
\label{SVR_fig}
\end{figure}
\paragraph{Single-view reconstruction} We evaluate NMF for single-view reconstruction and compare against state-of-the-art methods in Table \ref{SVR_table}. We note significantly lower self-intersections for NMF compared to the best baseline even after smoothing. Our method again results in fewer than 50\% non-manifold faces compared to the best baseline. NMF-M also gets the highest normal consistency performance. Due to the cubify step as part of the MeshRCNN \cite{meshrcnn} pipeline which converts a voxel grid into a mesh, the method has several non-manifold vertices and edges compared to deformation based methods Pixel2Mesh \cite{pixel2mesh, meshrcnn}, AtlasNet-O \cite{AtlasNet} and NMF. AtlasNet suffers from the most number of non manifold edges, almost 100 times that of MeshRCNN. We note that MeshRCNN\cite{meshrcnn} better performance in Chamfer Distance come at a cost of other metrics. We qualitatively show the effects of non-manifoldness in Figure \ref{SVR_fig} and supplementary material. We observe that for dielectric material (second row), NMF is able to transmit background colors closest to the ground truth, whereas other baselines only reflect the white sky due to the presence of flipped normals.
\begin{table}[!!t]
\resizebox{\textwidth}{!}{\begin{tabular}{|l|l l|l l|l l|l l|l l|l l|l l|}
\hline
\multirow{2}{*}{} &
\multicolumn{2}{c|}{Chamfer-L2 ($\downarrow$)} &
\multicolumn{2}{c|}{Normal Consistency ($\uparrow$)} &
\multicolumn{2}{c|}{NM-Vertices ($\downarrow$)} &
\multicolumn{2}{c|}{NM-Edges ($\downarrow$)} &
\multicolumn{2}{c|}{NM-Faces ($\downarrow$)} &
\multicolumn{2}{c|}{Self-Intersection ($\downarrow$)} \\
& & w/ Laplace & & w/ Laplace & & w/ Laplace & & w/ Laplace & & w/ Laplace & & w/ Laplace\\
\hline
MeshRCNN\cite{meshrcnn} & \textbf{4.73} & \textbf{5.96} & 0.698 & 0.758 & 9.32 & 9.32 & 17.88 & 17.88 & 5.18 & 0.86 & 7.07 & 1.41 \\
Pixel2Mesh\cite{pixel2mesh} & 5.48 & 10.79 & 0.706 & 0.720 & 0.00 & 0.00 & 0.00 & 0.00 & 3.33 & 0.88 & 12.29 & 6.52 \\
AtlasNet-25\cite{AtlasNet} & 5.48 & 7.76 & 0.826 & 0.824 & 0.00 & 0.00 & 7400 & 7400 & 1.76 & 0.48 &26.94 & 17.57 \\
AtlasNet-sph\cite{AtlasNet} & 6.67 & 7.35 & 0.838 & 0.836 & 0.00 & 0.00 & 0.00 & 0.00 & 2.19 & 1.08 & 11.07 &5.94 \\
\hline
NMF & 7.82 & 8.64 & 0.829 & 0.837 & 0.00 & 0.00 & 0.00 & 0.00 & 0.83 & 0.45 & 0.12 &\textbf{0.00} \\
NMF-M & 9.05 & 8.73 & \textbf{0.839} & \textbf{0.838} & \textbf{0.00} & \textbf{0.00} & \textbf{0.00} & \textbf{0.00} & \textbf{0.76} & \textbf{0.42} & \textbf{0.11} & \textbf{0.00}\\
\hline
\end{tabular}}
\caption{Single-view reconstruction.}
\label{SVR_table}
\vspace{-0.6cm}
\end{table}
\begin{figure}[!!t]
\centering
\includegraphics[width=0.95\linewidth]{Images/sim3.pdf}
\caption{Qualitative results for soft body simulation. (a) While AtlasNet breaks down into 25 meshlets, (b) AtlasNet-O suffers from severe self-intersections leading to unrealistic simulations. (c) Pixel2Mesh leads to artifacts such as the chair going through the floor due to higher number of non-manifold faces. (d) MeshRCNN has a high degree of non-manifoldness resulting in unrealistic simulation. (e) NMF due to being a manifold mesh, is close to (f) the ground truth.}
\label{sim}
\vspace{-0.6cm}
\end{figure}
\vspace{-0.2cm}
\paragraph{Soft body simulation (watch supplementary video for better understanding)}
To further demonstrate the usefulness of manifoldness, we qualitatively evaluate predicted meshes with soft body simulation in Fig. \ref{sim}. Here we simulate dropping meshes on the floor using Blender \cite{blender}, with settings $pull=0.9, \ push=0.9, \ bending=10$ to represent a rubber-like material. We note that AtlasNet \cite{AtlasNet} breaks into its constituent 25 independent meshlets upon hitting the floor. This behaviour is expected of methods that predict shapes as a set of n-connected components \cite{cvxnet}. Both Pixel2Mesh \cite{pixel2mesh} and AtlasNet-O \cite{AtlasNet} yield unrealistic simulations due to the presence of severe self-intersection artifacts. We note that MeshRCNN \cite{meshrcnn} suffers from \textit{over-bounciness} due to non-manifoldness and poor normal consistency. In contrast, NMF yields simulations with properties that are closest to the ground truth.
\vspace{-0.2cm}
\paragraph{3D printing}
We now show in Fig. \ref{3dprint} a few renders of a 3D printed shape predicted by NMF using image from Figure \ref{teaser}. Since NMF predicts a manifold mesh, we can 3D print the predicted shapes without any post processing or repair efforts, obtaining satisfactory printed products.
\vspace{-0.2cm}
\begin{figure}[!!t]
\centering
\includegraphics[width=\linewidth]{Images/3dprint2.pdf}
\vspace{-0.6cm}
\caption{A few renders of the 3D printing output for a shape generated by NMF.}
\label{3dprint}
\end{figure}
\paragraph{Comparison with implicit representation method}
We evalute NMF against state-of-the-art indirect mesh generation method OccNet \cite{OccNet} for the task of single view reconstruction in Table \ref{OccNet_table}. We observe that NMF outperforms the best baseline OccNet-3 in terms of geometric accuracy. This is primarily because NMF predicts a singly connected mesh object as opposed to OccNet which leads to several disconnected meshes. Moreover, due to the limitations imposed by the marching cubes algorithm discussed in Section \ref{sec:related}, OccNet-1,2,3 have several non-manifold vertices and edges where as by construction, NMF doesn't suffer from such limitation. An example of non-manifold edge is shown in Fig. \ref{OccNet}. For sake of completeness, we also show the mesh generated by MeshRCNN \cite{meshrcnn} that suffers from non-manifold vertices and edges. NMF is also competitive with OccNet in terms of self-intersections since both methods become practically intersection-free with Laplacian smoothing . While OccNet outperforms NMF in terms of non-manifold faces, we argue that this comes at a cost of higher inference time. For reference, the fastest version of OccNet has comparable non-manifold faces and self-intersections but suffers relatively in terms of other metrics.
\vspace{-0.2cm}
\begin{table}[!!t]
\label{SVR}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\toprule
Single View Recon. & Chamfer-L2 ($\downarrow$) & Normal Consistency ($\uparrow$) & NM-Vertices ($\downarrow$) & NM-Edges ($\downarrow$) & NM-Faces ($\downarrow$)& Self-Intersection ($\downarrow$) & Time ($\downarrow$)\\
\midrule
OccNet-1\cite{OccNet} & 8.77 & 0.814 & 1.13 & 0.85 & 0.36 & \textbf{0.00} & 871 \\
OccNet-2\cite{OccNet} & 8.66 & 0.814 & 2.67 & 1.79 & 0.21 & 0.03 & 1637 \\
OccNet-3\cite{OccNet} & 8.33 & 0.814 & 2.79 & 1.90 & \textbf{0.15} & 0.09 & 6652 \\
\midrule
NMF & \textbf{7.82} & \textbf{0.829} & \textbf{0.00} & \textbf{0.00} & 0.83 & 0.12 & \textbf{187}\\
NMF w/ Laplace & 8.64 & 0.837 & 0.00 &0.00 &0.45 & \textbf{0.00} & 292\\
\bottomrule
\end{tabular}
}
\caption{Comparison with implicit representation method OccNet \cite{OccNet} for single view reconstruction.}
\vspace{-0.3cm}
\label{OccNet_table}
\end{table}
\begin{figure}[!!t]
\centering
\includegraphics[width=\linewidth]{Images/SVR/occnet.pdf}
\caption{Implicit methods: OccNet fails to give meshes that are singly connected and MeshRCNN has poor normal consistency along with severe self-intersections. Both OccNet and NMF has non-manifold edges (shown as zoomed out insets). NMF generates meshes that are visually appealing with higher manifoldness.}
\label{OccNet}
\vspace{-0.6cm}
\end{figure}
\section{Introduction}
\vspace{-0.3cm}
\label{sec:intro}
Polygon meshes allow an efficient virtual representation of 3D objects, enabling applications in graphics rendering, simulations, modeling and manufacturing. Consequently, mesh generation or reconstruction from images or point sets has received significant recent attention. While prior approaches have primarily focused on obtaining geometrically accurate reconstructions, we posit that physically-based applications require meshes to also satisfy {\em manifold} properties. Intuitively, a mesh is manifold if it can be physically realized, for example, by 3D printing. Typically, reconstructed meshes are post-processed with humans in the loop for manifoldness, in order to enable ray tracing, slicing or Boolean operations. In contrast, we propose a novel deep network that directly generates manifold meshes (Fig.~\ref{teaser}), alleviating the need for manual post-processing.
A manifold is a topological space that locally resembles Euclidean space in the neighbourhood of each point. A manifold mesh is a discretization of the manifold using a disjoint set of simple 2D polygons, such as triangles, which allows designing simulations, rendering and other manifold calculations. While a mesh data structure can simply be defined as a set $(\mathcal{V},\mathcal{E}, \mathcal{F})$ of vertices $\mathcal{V}$ and corresponding edges $\mathcal{E}$ or face $\mathcal{F}$, not every mesh $(\mathcal{V},\mathcal{E}, \mathcal{F})$ is manifold. Mathematically, we list various constraints on a singly connected mesh with the set $(\mathcal{V},\mathcal{E}, \mathcal{F})$ that enables \textit{manifoldness}\footnote{In the scope of this work, meshes do not exhibit defects like duplicate elements, isolated vertices, degenerate faces and inner surfaces that can also cause a mesh to be \textit{non-manifold}.}.
\vspace{-0.2cm}
\begin{tight_itemize}
\item Each edge $e\in\mathcal{E}$ is common to exactly 2 faces in $\mathcal{F}$ (Fig.~\ref{manifoldness}a)
\item Each vertex $v\in\mathcal{V}$ is shared by exactly one group of connected faces (Fig.~\ref{manifoldness}b)
\item Adjacent faces $F_i, F_j$ have normals oriented in same direction (Fig.~\ref{manifoldness}c)
\end{tight_itemize}{}
The above mentioned constraints on a mesh $(\mathcal{V},\mathcal{E}, \mathcal{F})$ guarantee it to be a manifold in the limit of infinitesimally small discretization. That is not the case when dealing with practical meshes with large and non-uniformly distributed triangles. To ensure physical realizability, we tighten the definition with a fourth practical constraint that no two triangles may {\em intersect} (Fig.~\ref{manifoldness}d).
\begin{figure}[!!t]
\centering
\begin{minipage}{0.1\linewidth}
\centering
\includegraphics[width=\linewidth]{Images/Teaser/inputs.pdf}
\subcaption{Inputs}
\label{fig:env_atk}
\end{minipage}\quad
\begin{minipage}{0.45\linewidth}
\label{manifoldness_overview}
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{llllll}
\toprule
\cmidrule(r){1-2}
& Approach & Vertex & Edge & Face & Non-Int. \\
\midrule
MeshRCNN\cite{meshrcnn} & explicit & \xmark & \xmark & \xmark & \xmark \\
AtlasNet\cite{AtlasNet} & explicit & \cmark & \xmark & \xmark & \xmark \\
AtlasNet-O\cite{AtlasNet} & explicit & \cmark & \cmark & \xmark & \xmark \\
Pixel2Mesh\cite{pixel2mesh} & explicit & \cmark & \cmark & \xmark & \xmark \\
GEOMetrics\cite{geometrics} & explicit & \cmark & \cmark & \xmark & \xmark \\
3D-R2N2\cite{3dr2n2} & implicit & \xmark & \xmark & \cmark & \cmark \\
PSG\cite{PSG} & implicit & \xmark & \xmark & \cmark & \cmark \\
OccNet\cite{OccNet} & implicit & \xmark & \xmark & \cmark & \cmark \\
\cmidrule(r){1-6}
\textbf{NMF (Ours)}& explicit & \textbf{\cmark} & \textbf{\cmark} & \textbf{\cmark} & \textbf{\cmark} \\
\bottomrule
\end{tabular}}
\subcaption{Manifoldness of prior work}
\end{minipage}\quad
\hspace{-0.5cm}
\begin{minipage}{0.4\linewidth}
\centering
\includegraphics[width=\linewidth]{Images/Teaser/appliations_enabled.pdf}
\vspace{-0.2in}
\subcaption{Applications enabled by NMF Manifold Meshes}
\end{minipage}
\vspace{-0.2cm}
\caption{Given an input as either a 2D image or a 3D point cloud (a) Existing methods generate corresponding 3D mesh that fail one or more manifoldness conditions (b) yielding unsatisfactory results for various applications including physically based rendering (c). NeuralMeshFlow generates manifold meshes which can directly be used for high resolution rendering, physics simulations (see supplementary video) and be 3D printed without the need for any prepossessing or repair effort.}
\label{teaser}
\vspace{-0.2cm}
\end{figure}
\begin{figure}[!!t]
\begin{minipage}[c]{0.26\textwidth}
\includegraphics[width=\textwidth]{Images/manifoldness.pdf}
\end{minipage}\hfill
\begin{minipage}[c]{0.70\textwidth}
\caption{Non-manifold geometries for a part of singly connected mesh: (a) An edge that is shared by either exactly one (red) or more than two (red dashed) faces. (b) A vertex (red) shared by more than one group of connected faces. (c) Adjacent faces that have normals (red-arrow) oriented in opposite directions. (d) Faces intersecting other triangles of the same mesh.
}
\label{manifoldness}
\end{minipage}
\vspace{-0.4cm}
\end{figure}
In this work, we pose the task of 3D shape generation as learning a diffeomorphic flow from a template genus-0 manifold mesh to a target mesh. Our key insight is that manifoldness is conserved under a diffeomorphic flow due to their uniqueness \cite{ANODE,NODEapprox} and orientation preserving property \cite{flowsanddiff,flowembed}. In contrast to methods that learn ``deformations'' of a template manifold using an MLP or graph-based network \cite{AtlasNet,pixel2mesh,geometrics}, our approach ensures manifoldness of the generated mesh. We use Neural ODEs \cite{NODE} to model the diffeomorphic flow, however, must overcome their limited capability to represent a wide variety of shapes \cite{ANODE,NODEapprox,dissectingNODE}, which has restricted prior works to single-category representations \cite{occflow,pointflow}.
We propose novel architectural features such as an instance normalization layer that enables generating 3D shapes across multiple categories and a series of diffeomorphic flows to gradually refine the generated mesh. We show quantitative comparisons to prior works and more importantly, compare resulting meshes on physically meaningful tasks such as rendering, simulation and 3D printing to highlight the importance of manifoldness.
\vspace{-0.3cm}
\paragraph{Toy example: regularizer's dilemma}
Consider the task of deforming a template unit spherical mesh $S$ (Fig.~\ref{toy}a) into a target star mesh $T$ (Fig.~\ref{toy}b). We approximate the deformation with a multi-layer perceptron (MLP) $f_{\theta}$ with a unit hidden layer of $256$ neurons with $relu$ and output layer with $tanh$ activation. We train $f_{\theta}$ by minimizing various losses over the points sampled from $S, T$. A conventional approach involves minimizing the Chamfer Distance $L_c$ between $S, T$, leading to accurate point predictions but several edge-intersections (Fig.~\ref{toy}c). By introducing edge length regularization \cite{pixel2mesh} $L_e$, we get fewer edge-intersections (Fig.~\ref{toy}d) but the solution is also geometrically sub-optimal. We can further reduce edge-intersections with Laplacian regularization \cite{pixel2mesh} (Fig.~\ref{toy}e), but this takes a bigger toll on geometric accuracy. Thus, attempting to reduce self-intersections by explicit regularization not only makes the optimization hard, but can also lead to predictions with lower geometric accuracy. In contrast, our proposed use of NODE (with dynamics $f_{\theta}$) is designed by construction \cite{ANODE,NODEapprox} to prevent self-intersections without explicit regularization (Fig.~\ref{toy}f).
\begin{figure}[!!t]
\centering
\includegraphics[width=\linewidth]{Images/toy.pdf}
\caption{2D Toy Example: For the task of deforming a manifold template mesh (a) to a target mesh (b) using explicit mesh regularization (c-e) trades edge-intersections for geometric accuracy . In contrast, a NODE \cite{NODE} (f) is implicitly regularized preventing edge-intersections without loosing geometric accuracy.}
\label{toy}
\vspace{-.3cm}
\end{figure}
In summary, we make the following contributions:
\vspace{-0.2cm}
\begin{tight_itemize}
\item A novel approach to 3D mesh generation, Neural Mesh Flow (NMF), with a series of NODEs that learn to deform a template mesh (ellipsoid) into a target mesh with greater \textit{manifoldness}.
\item Extensive comparisons to state-of-the-art mesh generation methods for physically based rendering and simulation (see supplementary video), highlighting the advantage of NMF's manifoldness.
\item New metrics to evaluate manifoldness of 3D meshes and demonstration of applications to single-view reconstruction, 3D deformation, global parameterization and correspondence.
\end{tight_itemize}{}
\section{NeuralMeshFlow}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Images/pipeline-2.pdf}
\vspace{-0.4cm}
\caption{Neural Mesh Flow consists of three deformation blocks that perform point-wise flow on spherical mesh vertices based on the shape embedding $z$ from target shape $\mathcal{M}_T$. The bottom row shows an actual chair being generated at various stages of NMF. Time instances $0<T_1<T_2<T$ show the deformation of spherical mesh into a coarse chair representation $\mathcal{M}_{p0}$ by the first deformation block. Further deformation blocks perform refinements to yield refined meshes $\mathcal{M}_{p1}, \mathcal{M}_{p2}$.}
\vspace{-0.4cm}
\label{pipeline}
\end{figure}
\section{Neural Mesh Flow}
\vspace{-0.3cm}
We now introduce Neural Mesh Flow (Fig \ref{pipeline}), which learns to auto-encode 3D shapes. NMF broadly consists of four components. First, the target shape $\mathcal{M}_T$ is encoded by uniformly sampling $N$ points from its surface and feeding them to a PointNet \cite{pointnet} encoder to get the global shape embedding $z$ of size $k$. Second, NODE blocks diffeomorphically \textit{flow} the vertices of template sphere towards target shape conditioned on shape embedding $z$. Third, the instance normalization layer performs non-uniform scaling of NODE output to ease cross-category training. Finally, refinement flows provide gradual improvement in quality. We start with a discussion of NODE and its regularizing property followed by details on each component.
\vspace{-0.3cm}
\paragraph{NODE Overview.} A NODE learns a transformation $\phi_T: \mathcal{X} \rightarrow \mathcal{X}$ as solutions for initial value problem (IVP) of a parameterized ODE $x_T = \phi_T(x_0) = x_0 + \int_0^T f_{\Theta}(x_t)dt$. Here $x_0, x_T \in \mathcal{X} \subset \mathbb{R}^n$ respectively represent the input and output from the network with parameters $\Theta$ ($n = 3$ for our case), while $T\in \mathbb{R}$ is a hyper parameter that represents the duration of the \textit{flow} from $x_0$ to $x_T$. For a well behaved dynamics $f_{\Theta}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ that is Lipschitz continuous, any two distinct trajectories in $\mathbb{R}^n$ of NODE with duration $T$ may not intersect due to the existence and uniqueness of IVP solutions \cite{ANODE,NODEapprox}.
Moreover, NODE manifests the orientation preserving property of diffeomorphic flows \cite{flowsanddiff,flowembed}. These lead to strong implicit regularizations against self-intersection and non-manifold faces. There are several other advantages to NODE compared to traditional MLPs such as improved robustness \cite{robustnode}, parameter efficiency \cite{NODE}, ability to learn normalizing flows \cite{occflow,pointflow,ffjord} and homeomorphism \cite{NODEapprox}. We refer the readers to \cite{ANODE,NODEapprox,dissectingNODE} for more details.
\vspace{-0.3cm}
\paragraph{Diffeomorphic Conditional Flow.}
The standard NODE \cite{NODE} formulation cannot be used directly for the task of 3D mesh generation since they lack any means to feed in shape embedding and are therefore restricted to learning a few shape. A naive way would be to concatenate features to point coordinates like is done with traditional MLPs \cite{pixel2mesh,geometrics} but this destroys the shape regularization properties due to several augmented dimensions \cite{ANODE,NODEapprox}. Our key insight is that instead of a fixed NODE dynamics $f_{\Theta}$ we can use a family of dynamics $f_{\Theta|z}$ parameterized by $z$ while still retaining the uniqueness property as long as $z$ is held constant for the purpose of solving IVP with initial conditions $\{x_0,x_T\}$.
The objective of conditional flow (NODE Block) therefore is to learn a mapping $F_{\Theta|z}$ (\ref{forward}) given the shape embedding $z$ and initial values $\{(p^i_I,p^i_O) | p^i_I \in \mathcal{M}_I, p^i_O \in \mathcal{M}_O\}$ where $\mathcal{M}_I,\mathcal{M}_O$ are respectively the input and output point clouds.
\vspace{-0.6cm}
\begin{equation}
p^i_O = F_{\Theta|z}(p^i_I,z) = p^i_I + \int_0^T f_{\Theta|z}(p^i_I,z)dt
\label{forward}
\end{equation}{}
\vspace{-.3cm}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Images/norm.pdf}
\caption{The impact of instance normalization (IN) and refinement flows in NMF. (a) Learning deformation of a template (black) to target shapes of different variances (red and green) require longer non-uniform NODE trajectories making learning difficult. (b) IN allows NODE to learn deformations to an arbitrary variance. (c) This leads to simpler dynamics and can later be scaled back to correct shape variance. (d) A model trained without IN leads to self-intersections and non-manifold faces due to very complex dynamics being learned. (e) A model with IN is smoother and regularized.}
\label{IN}
\vspace{-0.1cm}
\end{figure}
\begin{table}[!h]
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{|l|l|l|l|l|l|}
\toprule
& Chamfer-L2 ($\downarrow$) & Normal Consistency ($\uparrow$) & NM-Faces ($\downarrow$)& Self-Intersection ($\downarrow$) & Time ($\downarrow$)\\
\midrule
No Instance Norm & 6.48 & 0.820 & 2.94 & 3.28 & 183\\
\midrule
0 refinement & 5.00 & 0.818 & 0.39 & 0.03 & \textbf{68}\\
1 refinement & 4.93 & 0.819 & \textbf{0.38} & \textbf{0.03} & 124\\
2 refinement & \textbf{4.65} & \textbf{0.818} & 0.73 & 0.09 & 189\\
\bottomrule
\end{tabular}}
\caption{Ablation for Instance Normalization and refinement}
\label{ablation}
\vspace{-0.8cm}
\end{table}
\paragraph{Instance Normalization.}
Normalizing input and hidden features to zero mean and unit variance is important to reduce co-variate shift in deep networks \cite{norm1,norm2,norm3,norm4,norm5,norm6}. While trying to deform a template sphere to targets with different variances (like a firearm and chair) different parts of the template need to be \textit{flown} by very different amounts to different locations (Fig. \ref{IN}a). This is observed to causes significant \textit{strain} on the NODE which ends up learning more complex dynamics resulting in meshes with poor geometric accuracy and manifoldness (Fig \ref{IN}d and Table \ref{ablation}). Instance normalization separates the task of learning target variances from that of learning target attributes. It gives NODE flexibility to deform the template to a target with arbitrary variance which yields better geometric accuracy(Fig. \ref{IN}b). This is later scaled back to the correct variance by instance normalization layer (Fig. \ref{IN}c) Given an input point cloud $\mathcal{M}\in \mathbb{R}^{N\times n}$ and its shape embedding $z$, the instance normalization calculates the point average $ \mu \leftarrow \frac{1}{|\mathcal{M}|}\sum_{i}p^i , p^i \in \mathcal{M}$ and then applies non-uniform scaling $ \mathcal{M} \leftarrow (\mathcal{M} - \mu )\odot \Delta(z)$ to arrive at correct target variances. Here $\Delta:\mathbb{R}^k \rightarrow \mathbb{R}^n$ is an MLP that regress variance coefficients for the $n$ dimensions based on shape embedding $z$. $\odot$ refers to the element wise multiplication.
\vspace{-0.3cm}
\paragraph{Overall Architecture.}
A single NODE block is often not sufficient to get desired quality of results. We therefore stack up two NODE blocks in a sequence followed by an instance normalization layer and call the collection a deformation block. While a single deformation block is capable of achieving reasonable results (as shown by $\mathcal{M}_{p0}$ in Fig.~\ref{pipeline}) we get further refinement in quality by having two additional deformation blocks. Notice how the $\mathcal{M}_{p1}$ has a better geometric accuracy than $\mathcal{M}_{p0}$ and $\mathcal{M}_{p2}$ is \textit{sharper} compared to $\mathcal{M}_{p1}$ with additional refinement. We report the geometric accuracy, manifoldness and inference time for different amounts of refinement in Table \ref{ablation}. The reported quantities are averaged over the 11 Shapnet categories (this excludes watercraft and lamp where NMF struggles with thin structures). For details on per category ablation, please see the supplementary material.
To summarize, the entire NMF pipeline can be seen as three successive diffeomorphic flows $\{F^0_{\Theta|z},F^1_{\Theta|z},F^2_{\Theta|z}\}$ of the initial spherical mesh to gradually approach the final shape.
\vspace{-.3cm}
\paragraph{Loss Function.}
In order to learn the parameters $\Theta$ it is important to use a loss which meaningfully represents the difference between the predicted $M_P$ and the target $M_T$ meshes. To this end we use the bidirectional Chamfer Distance (\ref{CD}) on the points sampled differentiably \cite{pytorch3d} from predicted $\Tilde{M}_P$ and target $\Tilde{M}_T$ meshes.
\vspace{-0.3cm}
\begin{equation}
\mathcal{L}(\Theta) = \sum_{p\in \Tilde{M}_P} \min_{q \in \Tilde{M}_T} || p - q ||^2 + \sum_{q\in \Tilde{M}_T} \min_{p \in \Tilde{M}_P} || p - q ||^2
\label{CD}
\end{equation}
We compute chamfer distances $\mathcal{L}_{p1}, \mathcal{L}_{p2}$ for meshes after deformation blocks $F^1_{\Theta|z}$ and $F^2_{\Theta|z}$. For meshes generated from $F^0_{\Theta|z}$ we found that computing chamfer distance $\mathcal{L}_{v}$ on the vertices gave better results since it encourages predicted vertices to be more uniformly distributed (like points sampled from target mesh). We thus arrive at the overall loss function to train NMF.
\begin{equation}
\mathcal{L} = w_0\mathcal{L}_v + w_1\mathcal{L}_{p1} + w_2\mathcal{L}_{p2}
\end{equation}
Here we take $w_0 = 0.1, w_1=0.2, w_3=0.7$ so as to enhance mesh prediction after each deformation block. The adjoint sensitivity \cite{adjointsensitivity} method is employed to perform the reverse-mode differentiation through the ODE solver and therefore learn the network parameters $\Theta$ using the standard gradient descent approaches.
\vspace{-0.3cm}
\paragraph{Dynamics Equation.}
The Neural ODE $F_{\theta|z}$ is built around the dynamics equation $f_{\theta|z}$ that is learned by a deep network. Given a point $x\in \mathbb{R}^n$, we first get 512 length point features by applying a linear layer. To condition the NODE on shape embedding, we extract a 512 length shape feature from the shape embedding $z$ and multiply it element wise with the obtained point features to get the \textit{point-shape} features. Thus, \textit{point-shape} features contains both the point features as well as the global instance information. We find that dot multiplication of the shape and point features yields similar performance as their concatenation, albeit requiring less memory. Lastly, we feed the \textit{point-shape} features into two residual MLP blocks each of width 512 and subsequent MLP of width 512 which outputs the predicted point location $y\in\mathbb{R}^n$. Based on the findings of \cite{dissectingNODE,identityresnet} we make use of the $tanh$ activation after adding the residual predictions at each step. This ensures maximum flexibility in the dynamics learned by the deep network. More details about the architecture can be found in supplementary material.
\vspace{-0.3cm}
\paragraph{Implementation Details} For the Neural Mesh Flow architecture, both the mesh vertices and NODE dynamics operate in $n=3$ dimensions. We uniformly sample $N=2520$ from the target mesh and using PointNet \cite{pointnet} encoder, get a shape embedding $z$ of size $k=1000$. During training, the NODE is solved with a tolerance of $1e^{-5}$ and interval of integration set to $t=0.2$ for deforming an icosphere with 622 vertices. The integration time was empirically determined to be large enough for flow to work, but not too large to cause overfitting. At test time, we use an icosphere of 2520 vertices and tolerance of $1e^{-5}$. We train NMF for 125 epochs using Adam \cite{adam} optimizer with a learning rate of $10^{-5}$, weight decay of $0.95$ after every $250$ iterations and a batch size of 250, on 5 NVIDIA 2080Ti GPUs for 2 days. For single view reconstruction, we train an image to point cloud predictor network with pretrained ResNet encoder of latent code 1000 and a fully-connected decoder with size 1000,1000,3072 with relu non-linearities. The point predictor is trained for 125 epochs on the same split as NMF auto-encoder.
\vspace{-0.3cm}
\section{Conclusions}
\vspace{-0.3cm}
In this paper, we have considered the problem of generating manifold 3D meshes using point clouds or images as input. We define manifoldness properties that meshes must satisfy to be physically realizable and usable in practical applications such as rendering and simulations. We demonstrate that while prior works achieve high geometric accuracy, such manifoldness has previously not been sought or achieved. Our key insight is that manifoldness is conserved under a diffeomorphic flow that deforms a template mesh to the target shape, which can be modeled by exploiting properties of Neural ODEs \cite{NODE}. We design a novel architecture, termed Neural Mesh Flow, composed of deformation blocks with instance normalization and refinement flows, to achieve manifold meshes without any post-processing. Our results in the paper and supplementary material demonstrate the significant benefits of NMF for real-world applications.
\section*{Broader Impact}
The broader positive impact of our work would be to inspire methods in computer graphics and associated industries such as gaming and animation, to generate meshes that require significantly less human intervention for rendering and simulation. The proposed NMF method addresses an important need that has not been adequately studied in a vast literature on 3D mesh generation. While NMF is a first step in addressing that need, it tends to produce meshes that are over-smooth (also reflected in other methods sometimes obtaining greater geometric accuracy), which might have potential negative impact in applications such as manufacturing. Our code, models and data will be publicly released to encourage further research in the community.
\section*{Acknowledgement}
We would like to thank Krishna Murthy Jatavallabhula and anonymous reviewers for valuable discussions and feedback. We would also like to thank Pengcheng Cao with UCSD CHEI for providing 3D printed models and Shreyam Natani for helping with Blender.
{\small
\section{Experiments}
\section{Results}
In this section we show qualitative and quantitative results on the task of auto-encoding and single view reconstruction of 3D shapes with comparison against several state of the art baselines. In addition to these tasks, we also demonstrate several additional features and applications of our approach including latent space interpolation texture mapping, consistent correspondence and shape deformations in the supplementary material.
\vspace{-0.2cm}
\paragraph{Data} We evaluate our approach on the ShapeNet Core dataset \cite{shapenet}, which consists of 3D models across 13 object categories. We use the training, validation and testing splits provided by \cite{3dr2n2} to be comparable to other baselines. We use rendered views from \cite{3dr2n2} and sample 3D points using \cite{shapenetmanifold}.
\begin{figure}[!!t]
\centering
\includegraphics[width=\linewidth]{Images/AE/AE.pdf}
\caption{AE results}
\label{AE}
\end{figure}
\vspace{-0.2cm}
\paragraph{Evaluation criteria}
We evaluate the predicted shape $\mathcal{M}_P$ for geometric accuracy to the ground truth $\mathcal{M}_T$ as well as for manifoldness. For geometric accuracy, we follow \cite{meshrcnn} and compute the bidirectional Chamfer distance according to \eqref{CD} and normal consistency using \eqref{NC} on 10000 points sampled from each mesh. Since Chamfer distance is sensitive to the size of meshes, we scale the meshes to lie within a unit radius sphere. With $\Tilde{M_P},\Tilde{M_T}$ the point sets sampled from $\mathcal{M}_p,\mathcal{M}_T$ and $\Lambda_{P,Q} = \{(p,argmin_{q} ||p-q||) : p \in P\}$, we define
\begin{equation}
\mathcal{L}_{n} = |\Tilde{M_P}|^{-1} \!\!\!\! \sum_{(p,q)\in \Lambda_{\Tilde{M}_P,\Tilde{M}_T}} \!\!\!\! |u_p \cdot u_q| \: + \: |\Tilde{M_T}|^{-1} \!\!\!\! \sum_{(p,q)\in \Lambda_{\Tilde{M}_T,\Tilde{M}_P}} \!\!\!\! |u_q \cdot u_p| \: - \: 1
\label{NC}
\end{equation}
We detect non-manifold vertices and edges using \cite{open3d} and report the metrics `NM-vertices', `NM-edges' respectively as the ratio of number of non-manifold vertices and edges to total number of vertices and edges in a mesh. To calculate non-manifold faces, we count number of times adjacent face normals have a negative inner product, then the metric `NM-Faces' is reported as its ratio to the number of edges in the mesh. To calculate the number of instances of self-intersection, we use \cite{torch-isect} and report the ratio of number of intersecting triangles to total number of triangles in a mesh.
\vspace{-0.2cm}
\paragraph{Baselines}
We compare with previously published state-of-the-art shape generation methods. This primarily includes direct mesh generation methods Pixel2Mesh \cite{pixel2mesh,meshrcnn}, MeshRCNN \cite{meshrcnn} and AtlasNet \cite{AtlasNet}. We use pretrained models for all these baselines motioned in this paper since they share the same dataset split by \cite{3dr2n2}. We use the implementation of Pixel2Mesh provided by MeshRCNN, as it uses a deeper network that outperforms the original implementation. We also consider AtlasNet-O which is a baseline proposed in \cite{AtlasNet} that uses patches sampled from a spherical mesh, making it closer to our own choice of initial template mesh. We also create a baseline of our own called NMF-M, which is similar in architecture to NMF but trained with a larger icosphere of 2520 vertices, leading to slight differences in test time performance. To account for possible variation in manifoldness due to simple post processing techniques, we also report outputs of all mesh generation methods with 3 iterations of Laplacian smoothing. Further iterations of smoothing lead to loss of geometric accuracy without any substantial gain in manifoldness.
We also compare with occupancy networks \cite{OccNet}, a state-of-the-art indirect mesh generation method based on implicit surface representation. Since OccNet is several orders of magnitude slower than direct mesh generation methods, we compare with several variants of OccNet based on the resolution of Multi Iso-Surface Extraction algorithm \cite{OccNet}. For fair comparison to other baselines, we use OccNet's refinement module to output its meshes with 5200 faces.
\vspace{-0.2cm}
\paragraph{Auto-encoding 3D shapes}
In this section we evaluate NMF's ability to generate a shape given an input 3D point cloud and compare against AtlasNet\cite{AtlasNet} and AtlasNet-O\cite{AtlasNet}. We evaluate the geometric accuracy and manifoldness of generated meshes. Additionally, we show physically based renderings of generated meshes with dielectric and conductor materials to highlight artifacts due to non-manifoldness. We report the quantitative results for shape generation from point clouds in Table \ref{AE_table}. We report only the weighted mean here and per category results can be found in supplementary material. Notice that NMF outperforms both AtlasNet\cite{AtlasNet} and AtlasNet-O\cite{AtlasNet} in terms of self-intersection with as much as 60 times less instances of self-intersections. Interestingly, even after applying laplacian smoothing, AtlasNet-O\cite{AtlasNet} has 20 times the self-intersections compared to NMF lacking any smoothing. With laplacian smoothing, our method becomes practically intersection free. NMF also outperform the baselines with and without smoothing in terms of non-manifold faces by as much as 1.5 times. Our method generate meshes with a higher normal consistency and therefore they give more realistic results from simulations and physically based rendering applications. It is important to note that AtlasNet\cite{AtlasNet} used 25 mesh templates to construct the final mesh. Since these templates are not closed, the boundary edges are non-manifold (associated to a single face) and hence we get a constant value of 7404 for its non-manifold edges. Both AtlasNet-O and NMF deform a manifold sphere which has manifold edges. All the three methods have manifold vertices. We show visualizations for two samples in Fig. \ref{AE}. In the first row we notice that AtlasNet\cite{AtlasNet} has severe self-intersections (red) and flipped normals (black) whereas they are minimal for NMF. While the overall geometry of the meshes are similar, these non-manifoldness leads to a big difference in physically based rendering quality. The second row shows meshes with dielectric material. Notice the reflection of red box through chair's backrest is quite realistic for NMF and is close to the ground truth. A harder feature is the green ball besides one of the legs. Notice its reflections is visible through the chair for NMF whereas i is scrambled for AtlasNet-O and completely blocked for AtlasNet. The flipped normals are more problemaitc when it comes to rendering conductor material. Notice the reflections of blue ball onto the chair. NMF is again more realistic and closer to ground truth whereas both variants of AtlasNet struggle with huge black spots in their renderings.
\begin{table}
\resizebox{\textwidth}{!}{\begin{tabular}{|l|l l l|l l l|l l l|l l l|l l l|}
\hline
\multirow{3}{*}{} &
\multicolumn{3}{c|}{Chamfer-L2 ($\downarrow$)} &
\multicolumn{3}{c|}{Normal Consistency ($\uparrow$)} &
\multicolumn{3}{c|}{NM-Faces ($\downarrow$)} &
\multicolumn{3}{c|}{Self-Intersection ($\downarrow$)} \\
&AtNet&AtNet-O&NMF&AtNet&AtNet-O&NMF&AtNet&AtNet-O&NMF&AtNet&AtNet-O&NMF\\
\hline
mean & 4.15 & \textbf{3.50} & 5.54 & 0.815 & 0.816 & \textbf{0.826} & 1.72 & 1.43 & \textbf{0.71} & 24.80 & 6.03 & \textbf{0.10}\\
mean (with Laplace) & 4.59 & \textbf{3.81} & 5.25 & 0.807 & 0.811 & \textbf{0.826} & 0.47 & 0.56 &\textbf{0.38} & 13.26 & 2.02 & \textbf{0.00}\\
\hline
\end{tabular}}
\caption{Auto Encoding Performance}
\label{AE_table}
\end{table}
\vspace{-0.2cm}
\paragraph{Single-view reconstruction}
\begin{table}
\resizebox{\textwidth}{!}{\begin{tabular}{|l|l l|l l|l l|l l|l l|l l|l l|}
\hline
\multirow{2}{*}{} &
\multicolumn{2}{c|}{Chamfer-L2 ($\downarrow$)} &
\multicolumn{2}{c|}{Normal Consistency ($\uparrow$)} &
\multicolumn{2}{c|}{NM-Vertices ($\downarrow$)} &
\multicolumn{2}{c|}{NM-Edges ($\downarrow$)} &
\multicolumn{2}{c|}{NM-Faces ($\downarrow$)} &
\multicolumn{2}{c|}{Self-Intersection ($\downarrow$)} \\
& & w/ Laplace & & w/ Laplace & & w/ Laplace & & w/ Laplace & & w/ Laplace & & w/ Laplace\\
\hline
MeshRCNN\cite{meshrcnn} & \textbf{4.73} & \textbf{5.96} & 0.698 & 0.758 & 9.32 & 9.32 & 17.88 & 17.88 & 5.18 & 0.86 & 7.07 & 1.41 \\
Pixel2Mesh\cite{pixel2mesh} & 5.48 & 10.79 & 0.706 & 0.720 & 0.00 & 0.00 & 0.00 & 0.00 & 3.33 & 0.88 & 12.29 & 6.52 \\
AtlasNet-25\cite{AtlasNet} & 5.48 & 7.76 & 0.826 & 0.824 & 0.00 & 0.00 & 7.40 & 7.40 & 1.76 & 0.48 &26.94 & 17.57 \\
AtlasNet-sph\cite{AtlasNet} & 6.67 & 7.35 & 0.838 & 0.836 & 0.00 & 0.00 & 0.00 & 0.00 & 2.19 & 1.08 & 11.07 &5.94 \\
\hline
NMF & 7.82 & 8.64 & 0.829 & 0.837 & 0.00 & 0.00 & 0.00 & 0.00 & 0.83 & 0.45 & 0.12 &\textbf{0.00} \\
NMF-M & 9.05 & 8.73 & \textbf{0.839} & \textbf{0.838} & \textbf{0.00} & \textbf{0.00} & 0.00 & 0.00 & \textbf{0.76} & \textbf{0.42} & \textbf{0.11} & \textbf{0.00}\\
\hline
\end{tabular}}
\caption{Single View Reconstruction}
\label{SVR_table}
\end{table}
We evaluate the potential of our method for single-view reconstruction and compare against state-of-the-art methods AtlasNet\cite{AtlasNet}, AtlasNet-O\cite{AtlasNet}, Pixel2Mesh\cite{meshrcnn,pixel2mesh} and MeshRCNN\cite{meshrcnn}. The quantitative results are reported in Table \ref{SVR_table} where we only show the mean of various metrics over shapenet categories. We notice that self-intersections in NMF are significantly lower (about 60 times) than best baseline and are practically zero with laplace smoothing. Whereas with the possible exception of MeshRCNN\cite{meshrcnn} all other baselines suffer from severe self-intersections even after laplace smoothing. Our method again results in fewer than 50\% non-manifold faces compared to the best baseline. NMF-M also gets the highest normal consistency performance. Due to the \textit{cubify} step as part of the MeshRCNN\cite{meshrcnn} pipeline which converts a voxel grid into mesh, the method has several non-manifold vertices and edges compared to deformation based methods Pixel2Mesh\cite{pixel2mesh}, AtlasNet-O\cite{AtlasNet} and NMF. AtlasNet\cite{AtlasNet} suffers from the most number of non manifold edges almost 100 times that of MeshRCNN\cite{meshrcnn}. We observe that while being poor in terms of manifoldness, MeshRCNN\cite{meshrcnn} outperforms other baselines in terms of chamfer distance. We qualitatively show the effects of non-manifoldness in Fig. \ref{SVR_table}-\ref{SVR_chair}. We observe that for dielectric material (second row) NMF is able to pass on the background colors closest to that of the ground truth whereas other baselines only reflect the white sky due to the presence of flipped normals.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{Images/SVR/SVR-table.pdf}
\caption{SVR results}
\label{SVR_table}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{Images/SVR/SVR-chair.pdf}
\caption{SVR results}
\label{SVR_chair}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Images/SVR/Occnet2.pdf}
\caption{Comparison with Implicit methods}
\label{fig:fig}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Images/ablation.pdf}
\caption{Ablation}
\label{fig:fig}
\end{figure}
\section{Introduction}
Mesh representation for 3D objects are a promising approach for encoding the structure of 3D objects. They provide a cheap and efficient virtual representation of physical entities and are widely used in computer graphics, modelling, simulation and additive manufacturing etc. In order to get desirable results, the mesh representations, in addition to being geometrically accurate, are required to be \textit{manifold}. This prevents ambiguities while using techniques like ray tracing, slicing and mesh boolean operations resulting in physically consistent results (see Fig. \ref{teaser}). A manifold is defined as a topological space that locally resembles Euclidean space in the neighbourhood of each point. A manifold mesh is a discretization of the manifold using a disjoint set of simple 2D polygons like triangles. Such a discretization is simpler to work with when designing simulations, rendering and other manifold calculations. While a mesh data structure can simply be defined as a set $(\mathcal{V},\mathcal{E}, \mathcal{F})$ of vertices $\mathcal{V}$ and corresponding edges $\mathcal{E}$ or face $\mathcal{F}$, not every mesh $(\mathcal{V},\mathcal{E}, \mathcal{F})$ is manifold. Intuitively, a mesh exhibit manifold property only when it can be realized physically (say by 3D printing). Mathematically, we list various constraints on a singly connected mesh with the set $(\mathcal{V},\mathcal{E}, \mathcal{F})$ that enables \textit{manifoldness}.
\begin{figure}
\centering
\includegraphics[width=0.25\linewidth]{Images/manifoldness.pdf}
\caption{Non-manifold geometries for a part of singly connected mesh: (a) An edge that is shared by either exactly one (red) or more than two (red dashed) faces. (b) A vertex (red) shared by more than one group of connected faces. (c) Adjacent faces that have normals (red-arrow) oriented in opposite directions. (d) faces intersecting other triangles of the same mesh. }
\label{manifoldness}
\end{figure}
\begin{itemize}
\item Edge $e\in\mathcal{E}$ is common to exactly 2 faces in $\mathcal{F}$ (Fig \ref{manifoldness} a)
\item Vertex $v\in\mathcal{V}$ shared by exactly one group of connected faces (Fig \ref{manifoldness} b)
\item Adjacent faces $F_i, F_j$ have normals oriented in same direction (Fig \ref{manifoldness} c)
\end{itemize}{}
The above mentioned constraints on a mesh $(\mathcal{V},\mathcal{E}, \mathcal{F})$ guarantees to be a manifold in the limit that the manifold discretization is infinitesimally small. This is not the case when dealing with practical meshes that often have large areas and are non-uniformly spread. Such meshes often have \textit{self-intersections} (Fig \ref{manifoldness} d) which occur when two or more triangles belonging to the same mesh intersect one another. It is easy to observe that a mesh with manifold vertices, edges and faces
can have \textit{self-intersections} failing it to be realized physically. In the scope of this work, we therefore tighten the definition of manifold meshes to exclude those meshes that have \textit{self-intersections} to arrive at a more useful definition for practical meshes.
\begin{figure}[t]
\centering
\begin{minipage}{0.1\textwidth}
\centering
\includegraphics[width=\linewidth]{Images/Teaser/inputs.pdf}
\vspace{-0.2in}
\subcaption{Possible Inputs}
\label{fig:env_atk}
\end{minipage}\quad
\begin{minipage}{0.3\linewidth}
\label{manifoldness_overview}
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{llllll}
\toprule
\cmidrule(r){1-2}
& Approach & Vertex & Edge & Face & Non-Int. \\
\midrule
MeshRCNN\cite{meshrcnn} & explicit & \xmark & \xmark & \xmark & \xmark \\
AtlasNet\cite{AtlasNet} & explicit & \cmark & \xmark & \xmark & \xmark \\
AtlasNet-O\cite{AtlasNet} & explicit & \cmark & \cmark & \xmark & \xmark \\
Pixel2Mesh\cite{pixel2mesh} & explicit & \cmark & \cmark & \xmark & \xmark \\
GEOMetrics\cite{geometrics} & explicit & \cmark & \cmark & \xmark & \xmark \\
3D-R2N2\cite{3dr2n2} & implicit & \xmark & \xmark & \cmark & \cmark \\
PSG\cite{PSG} & implicit & \xmark & \xmark & \cmark & \cmark \\
OccNet\cite{occupancy} & implicit & \xmark & \xmark & \cmark & \cmark \\
\cmidrule(r){1-6}
\textbf{NMF (Ours)}& explicit & \textbf{\cmark} & \textbf{\cmark} & \textbf{\cmark} & \textbf{\cmark} \\
\bottomrule
\end{tabular}}
\subcaption{Manifoldness of prior work}
\end{minipage}\quad
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=\linewidth]{Images/Teaser/appliations_enabled.pdf}
\vspace{-0.2in}
\subcaption{Applications enabled by NMF Manifold Meshes}
\label{teaser}
\end{minipage}
\caption{Given an input as either a 2D image or a 3D point cloud (a) Existing methods generate corresponding 3D mesh that fail one or more manifoldness conditions (b) yielding unsatisfactory results for various applications including physically based rendering (c). NeuralMeshFlow generates manifold meshes which can directly be used for high resolution rendering, physics simulations (see supplementary video) and be 3D printed without the need for any prepossessing or repair effort.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Images/toy.pdf}
\caption{2D Toy Example: For the task of deforming a manifold template mesh (a) to a target mesh (b) using explicit mesh regularization (c-e) trades edge-intersections for geometric accuracy . In contrast, a NODE\cite{NODE} (f) is implicitly regularized preventing edge-intersections without loosing geometric accuracy.}
\label{toy}
\end{figure}
Existing learning based surface generation methods, while giving impressive geometric accuracy, fail one or more \textit{manifoldness} conditions (Fig. \ref{teaser} b). While the indirect approaches \cite{3dr2n2,PSG,OccNet,deepsdf,DeepLevelSets,ImplicitFields} fail because of the non-manifoldness of marching cube algorithm \cite{marching} (refer section 2 for details), direct methods \cite{meshcnn}\cite{pixel2mesh}\cite{AtlasNet}\cite{geometrics} suffer from what we call the \textit{regularizer's dilemma}. The dilemma refers to the trade-off between geometric accuracy and higher manifoldness. We explain this dilemma with the help of a 2D toy example (Fig. \ref{toy}). Consider the task of deforming a template unit spherical mesh $S$ (Fig \ref{toy} (a)) into a target star mesh $T$ (Fig. \ref{toy} b). Let's use a multi layer perceptron $f_{\theta}$ with a unit hidden layer of $256$ neurons with $relu$ and output layer with $tanh$ activation to approximate this deformation. We can train $f_{\theta}$ by minimizing some loss $L$ over the points sampled from $S, T$. A naive approach involves minimizing the Chamfer Distance $L_c$ between $S, T$ which leads to accurate point predictions but has several edge-intersections (Fig. \ref{toy}c). By introducing edge length regularization $L_e$ we get fewer edge-intersections (Fig \ref{toy}d) but the solution is also sub-optimal geometrically. We can further bring down the edge-intersections by including Laplacian regularization (Fig \ref{toy}e) however this takes a bigger toll on geometric accuracy. This demonstrates that an attempt to reduce self-intersections by introducing explicit regularization terms not only makes the optimization hard but often lead to unpleasant predictions with lower geometric accuracy. In this work, we hypothesize that NODE\cite{NODE} (with dynamics $f_{\theta}$) are implicitly regularized by construction to prevent self-intersections \cite{NODEapprox}\cite{ANODE} and thus do not require any explicit regularization which let's us resolve the dilemma.
In this work we exploit the property of NODE\cite{NODE} of being implicit regularizers against self-intersections to do 3D manifold mesh generation. Our proposed model NeuralMeshFlow consists of several NODE blocks that successively learn to deform a spherical template mesh to a target given its encoding.
In summary, we make the following contributions to 3D mesh generation research.
\begin{itemize}
\item We propose a novel approach to 3D mesh generation with NeuralMeshFlow (NMF), which is made up of a series of NODE blocks that learn to deform a template mesh (ellipsoid) into the target shape that encourages \textit{manifoldness} of generated mesh.
\item We carry out extensive evaluations of state-of-the-art mesh generation methods for physically based rendering and simulation (see supplementary video) to highlight that they fail to give realistic results due to absence of manifoldness.
\item We also show applications to single-view reconstruction, 3D deformation, global parameterization and correspondence capabilities of NMF.
\end{itemize}{}
\section{NeuralMeshFlow}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Images/pipeline.pdf}
\caption{NeuralMeshFlow consists of three deformation blocks that perform subsequent point-wise flow on spherical mesh vertices based on the shape embedding $z$ from target shape $\mathcal{M}_T$. The bottom row shows an actual chair being generated at various instants of NMF. Time instances $0<T_1<T_2<T$ show the deformation of spherical mesh into a coarse chair representation $\mathcal{M}_{p0}$ by first deformation block. Further deformation blocks perform subsequent refinements to get $\mathcal{M}_{p1}, \mathcal{M}_{p2}$.}
\label{pipeline}
\end{figure}
\section{NeuralMeshFlow}
In this section we introduce NeuralMeshFlow (Fig \ref{pipeline}), which learns to auto-encode 3D shapes. Given a 3D shape, $M_T$, we uniformly sample $N=2520$ points from its surface and feed to a PointNet\cite{pointnet} encoder which gives us the shape embedding $z$ of length $k=1000$. The vertices of an icosphere $M_S$ are passed through six NODE blocks (conditioned on $z$) across the three deformation blocks yielding vertices of the predicted shape. Since NMF does not alter the mesh topology, we simply use the connectivity information of icosphere to generate shapes $M_{p1}, M_{p2}$ from respective predicted vertices. In this section, we first briefly discuss the implicit shape regularization in NODE. We then explain the dynamics equation $f_{\theta|z}$ which forms the core representation learning part of Neural ODEs. We then describe the deformation flow $F_{\theta|z}$ that does mesh prediction which is then later refined by subsequent flows. Lastly, we discuss about the loss functions used for training NeuralMeshFlow.
\paragraph{NODE Overview.} A NODE learns a transformation $\phi_T: \mathcal{X} \rightarrow \mathcal{X}$ as solutions for initial value problem (IVP) of a parameterized ODE (eq.\ref{ODEeq}). Here $x_0, x_T \in \mathcal{X} \subset \mathbb{R}^n$ represent the input and output from the network respectively, while $T\in \mathbb{R}$ is a hyper parameter which represents the duration of the \textit{flow} from $x_0$ to $x_T$. For a well behaved dynamics $f_{\Theta}$ (Lipschitz continuous) the IVP enforces the existence and uniqueness of solutions \cite{ANODE}\cite{NODEapprox} which implies that any two distinct trajectories of NODE can never intersect. The uniqueness property presents several advantages to NODE compared to traditional MLPs such as improved robustness \cite{robustnode}, parameter efficiency \cite{NODE} and the ability to learn normalizing flows \cite{ffjord}\cite{pointflow}\cite{occflow} and homeomorphism \cite{NODEapprox}. As illustrated by the toy example in figure \ref{toy} we argue that regularization against self-intersection non-manifold faces is yet another consequence of the uniqueness and orientation preserving property of NODE. We refer readers to \cite{ANODE}\cite{NODEapprox}\cite{dissectingNODE} for more details.
\begin{equation}
x_T = \phi_T(x_0) = x_0 + \int_0^T f_{\Theta}(x_t)dt
\label{ODEeq}
\end{equation}{}
\paragraph{Dynamics Equation.}
The Neural ODE $F_{\theta|z}$ is built around the dynamics equation $f_{\theta|z}$ which is learned by a deep network. Given a point $x\in \mathbb{R}^3$ we first get 512 length point features by applying a linear layer. Then we extract a 512 length shape feature from the shape embedding $z$ and multiply it element wise with the obtained point features to get the \textit{point-shape} features. Lastly, we feed the \textit{point-shape} features into two residual MLP blocks each of width 512 and subsequent MLP of width 512 which outputs the predicted point location $y\in\mathbb{R}^3$. Based on the findings of \cite{identityresnet}\cite{dissectingNODE} we make use of the $tanh$ activation after adding the residual predictions at each step. This ensures maximum flexibility in the dynamics learned by the deep network.
\paragraph{Deformation Flow.}
The objective of deformation flow is to learn a conditional mapping $F_{\Theta|z}: \mathcal{M}_I \rightarrow \mathcal{M}_O \subset \mathbb{R}^3$ given the shape embedding $z$. Here $\mathcal{M}_I, \mathcal{M}_O$ refer to the set of input and output points respectively.
\begin{equation}
p^i_O = F_{\Theta|z}(p^i_I,z) = p^i_I + \int_0^T f_{\Theta|z}(p^i_I,z)dt
\label{forward}
\end{equation}{}
For each point $p^i_I\in \mathcal{M}_I$ we solve the \textit{forward flow} (Eq. \ref{forward}) to get the updated point location $p^i_O\in\mathbb{R}^3$ of the output points set. It should be noted that the IVP in eq.\ref{forward} is similar to eq. \ref{ODEeq} in the sense that it also enforces point trajectories $p^i_I \rightarrow p^i_O$ to never intersect. However, here we condition the flow on the shape embedding which allows for different IVPs to be learned for different shapes. We also give the network flexibility to stretch the predicted point cloud along the three axis by means of an instance normalization layer. This is achieved by an MLP with one hidden layer of size 256 that regresses the three scaling factors from the shape embedding $z$.
\paragraph{Refinement Flows.}
We successively apply the deformation flow in Eq. \ref{forward} to get two additional refinement flows. Therefore, the entire NMF can be seen as three successive deformation flows $\{F^0_{\Theta|z},F^1_{\Theta|z},F^2_{\Theta|z}\}$ of the initial ellipsoid to get to the final prediction.
\paragraph{Loss Function.}
In order to learn the parameters $\Theta$ it is important to use a loss which meaningfully represents the difference between the predicted $M_P$ and the target $M_T$ meshes. To this end we use the bidirectional Chamfer Distance (eq.\ref{CD}) on the points sampled differentiably\cite{pytorch3d} from predicted $\Tilde{M}_P$ and target $\Tilde{M}_T$ meshes.
\begin{equation}
\mathcal{L}(\Theta) = \sum_{p\in \Tilde{M}_P} \min_{q \in \Tilde{M}_T} || p - q ||^2 + \sum_{q\in \Tilde{M}_T} \min_{p \in \Tilde{M}_P} || p - q ||^2
\label{CD}
\end{equation}
We compute chamfer distances $\mathcal{L}_{p1}, \mathcal{L}_{p2}$ for meshes after deformation blocks $F^1_{\Theta|z}$ and $F^2_{\Theta|z}$. For meshes generated from $F^0_{\Theta|z}$ we found that computing chamfer distance $\mathcal{L}_{v}$ on the vertices gave better results since it encourages predicted vertices to be more uniformly distributed (like points sampled from target mesh). We thus arrive at the overall loss function to train NMF.
\begin{equation}
\mathcal{L} = w_0\mathcal{L}_v + w_1\mathcal{L}_{p1} + w_2\mathcal{L}_{p2}
\end{equation}
Here we take $w_0 = 0.1, w_1=0.2, w_3=0.7$ so as to enhance mesh prediction after each deformation flow. The adjoint sensitivity \cite{adjointsensitivity} method is employed to perform the reverse-mode differentiation through the ODE solver and therefore learn the network parameters $\Theta$ using the standard gradient descent approaches.
\section{Related Work}
\label{sec:related}
Existing learning based mesh generation methods, while yielding impressive geometric accuracy, do not satisfy one or more \textit{manifoldness} conditions (Fig.~\ref{teaser}b). While indirect approaches \cite{3dr2n2,PSG,OccNet,deepsdf,DeepLevelSets,ImplicitFields} suffer from the non-manifoldness of the marching cube algorithm \cite{marching}, direct methods \cite{meshcnn,pixel2mesh,AtlasNet,geometrics} are faced with the \textit{regularizer's dilemma} on the trade-off between geometric accuracy and higher manifoldness, illustrated in Fig.~\ref{toy} and discussed in Sec.~\ref{sec:intro}.
\subsection{Indirect Mesh Prediction}
Indirect approaches tend to predict the 3D geometry as either a distribution of voxels\cite{voxel4}\cite{voxel5}\cite{voxel6} \cite{voxel1}\cite{voxel2}\cite{voxel3}\cite{HSP}\cite{Octtree}, point clouds\cite{PSG}\cite{foldingnet} or an implicit function representing signed distance from the surface. Both voxel and point set prediction methods struggle to generate high resolution outputs which later makes the iso-surface extraction tools ineffective or noisy\cite{AtlasNet}.
More recently, neural networks have been used to learn a continuous implicit function representing shape\cite{deepsdf}\cite{OccNet}\cite{ImplicitFields}. The idea is simply to feed a neural network with latent code and a query point (x-y-z) to predict the TSDF value \cite{deepsdf} or the binary occupancy of the point \cite{OccNet}\cite{ImplicitFields}. However, this approach is computationally expensive since in order to get a surface from implicit function representation, several thousands of points are required to be sampled which makes the process slow and time consuming at run time. Moreover, for shapes that have thin structures (like chair) implicit methods often fail to produce a singly-connected component.
In spite varied levels of successes in terms of voxel, point cloud and implicit function representations, they all depend on the Marching Cube algorithm \cite{marching} for iso-surface extraction. While Marching Cube can be applied directly to the boundary voxel grids, point clouds first need to regress the iso-surface function using surface normal information. Note that implicit function representation learn to regress TSDF values per voxel instead and then perform extensive query to generate iso-surface based on a threshold $\tau$. Marching cube is essentially a look-up table for different triangle arrangement based on the classification of grid vertices $v_i \in \mathcal{V}$ as `inside' ($TSDF(v_i)\leq \tau$) and `outside' ($TSDF(v_i)\geq \tau$). It should be noted however that marching cube does not guarantee \textit{manifoldness} of generated mesh \cite{marchingextended}\cite{marching33}\cite{marchingdualcountouring}.
\subsection{Direct Mesh Prediction}
A mesh based representation stores the surface information cheaply as list of vertices and faces that respectively define the geometric and topological information. Early methods of mesh generation relied on predicting the parameters of category based mesh models. While these methods output manifold meshes, they work only when parameterized manifold meshes are available for the object category. Recently some success has been made to generate meshes for a wide class of categories using topological priors \cite{AtlasNet} \cite{pixel2mesh}. Deep networks are used to update the vertices of initial mesh to match that of the final mesh. AtlasNet\cite{AtlasNet} uses chamfer distance applied on the vertices for training while Pixel2Mesh \cite{pixel2mesh} uses a coarse to fine deformation approach again using vertex chamfer loss. However, using a point set training scheme for meshes leads to severe topological issues and produced meshes are not manifold. Some recent works have proposed to use mesh regularizers like Laplacian\cite{pixel2mesh}\cite{geometrics}\cite{meshrcnn}\cite{3dn}, edge lengths\cite{meshrcnn}\cite{geometrics} and normal consistency\cite{meshrcnn} to constrain the flexibilty of vertex predictions however they suffer from \textit{regularizer's dilemma} as better geometric accuracy comes at a cost of manifoldness.
In contrast to the aforementioned approaches, our approach leads to high resolution meshes with guaranteed \textit{manifoldness} across a wide variety of shape categories. The idea is similar to previous approaches \cite{AtlasNet}\cite{pixel2mesh}\cite{geometrics} based on deforming an initial ellipsoid by updating its vertices. However, instead of using explicit mesh regularizers, our approach uses NODE blocks to learn the deformation flow which as mentioned in section 2.2 implicitly discourage self-intersections to maintain the topology and therefore, guarantee \textit{manifoldness} of generated shape. The entire pipeline is end-to-end trainable without requiring any post-processing.
\section{Related Work}
\vspace{-0.3cm}
\label{sec:related}
Existing learning based mesh generation methods, while yielding impressive geometric accuracy, do not satisfy one or more \textit{manifoldness} conditions (Fig.~\ref{teaser}b). While indirect approaches \cite{3dr2n2,PSG,OccNet,deepsdf,DeepLevelSets,ImplicitFields} suffer from the non-manifoldness of the marching cube algorithm \cite{marching}, direct methods \cite{meshrcnn,AtlasNet,pixel2mesh,geometrics} are faced with the \textit{regularizer's dilemma} on the trade-off between geometric accuracy and higher manifoldness, illustrated in Fig.~\ref{toy} and discussed in Sec.~\ref{sec:intro}.
\vspace{-0.3cm}
\paragraph{Indirect Mesh Prediction}
Indirect approaches predict the 3D geometry as either a distribution of voxels \cite{voxel4,voxel5,voxel6,voxel1,voxel2,voxel3,HSP,Octtree}, point clouds \cite{PSG,foldingnet} or an implicit function representing signed distance from the surface \cite{OccNet,deepsdf,ImplicitFields}. Both voxel and point set prediction methods struggle to generate high resolution outputs which later makes the iso-surface extraction tools ineffective or noisy \cite{AtlasNet}. Implicit methods feed a neural network with a latent code and a query point, encoding the spatial coordinates \cite{OccNet,deepsdf,ImplicitFields} or local features \cite{localsdf}, to predict the TSDF value \cite{deepsdf} or the binary occupancy of the point \cite{OccNet,ImplicitFields}. However, these approaches are computationally expensive since in order to get a surface from the implicit function representation, several thousands of points must be sampled. Moreover, for shapes such as chairs that have thin structures, implicit methods often fail to produce a single connected component.
All the above methods depend on the marching cube algorithm \cite{marching} for iso-surface extraction. While marching cubes can be applied directly to voxel grids, point clouds first regress the iso-surface using surface normals. Implicit function representations must regress TSDF values per voxel and then perform extensive query to generate iso-surface based on a threshold $\tau$. This is used to classify grid vertices $v_i \in \mathcal{V}$ as `inside' ($TSDF(v_i)\leq \tau$) and `outside' ($TSDF(v_i)\geq \tau$). For each voxel, based on the arrangement of its grid vertices, marching cubes \cite{marching, marching33,marchingdualcountouring,marchingextended} follows a lookup-table to find a triangle arrangement. Since this rasterization of iso-surface is a purely local operation, it often leads to ambiguities \cite{marching33,marchingdualcountouring,marchingextended}, resulting in meshes being \textit{non-manifold}.
\vspace{-0.3cm}
\paragraph{Direct Mesh Prediction}
A mesh based representation stores the surface information cheaply as list of vertices and faces that respectively define the geometric and topological information. Early methods of mesh generation relied on predicting the parameters of category based mesh models \cite{Lions,menagerie,humanshape}. While these methods output manifold meshes, they work only for object category with available parameterized manifold meshes. Recently, meshes have been successfully generated for a wide class of categories using topological priors \cite{AtlasNet,pixel2mesh}. Deep networks are used to update the vertices of initial mesh to match that of the final mesh. AtlasNet \cite{AtlasNet} uses Chamfer distance applied on the vertices for training, while Pixel2Mesh \cite{pixel2mesh} uses a coarse-to-fine deformation approach using vertex Chamfer loss. However, using a point set training scheme for meshes leads to severe topological issues and produced meshes are not manifold. Some recent works have proposed to use mesh regularizers like Laplacian \cite{meshrcnn,pixel2mesh,geometrics,3dn}, edge lengths \cite{meshrcnn,geometrics}, normal consistency \cite{meshrcnn} or pose it as a linear programming problem \cite{LPmeshmanifold} to constrain the flexibilty of vertex predictions. They suffer from the \textit{regularizer's dilemma} discussed in Fig.~\ref{toy}, as better geometric accuracy comes at a cost of manifoldness.
In contrast to the above approaches, the proposed NMF achieves high resolution meshes with a high degree of manifoldness across a wide variety of shape categories. Similar to previous approaches \cite{AtlasNet,pixel2mesh,geometrics}, an initial ellipsoid is deformed by updating its vertices. However, instead of using explicit mesh regularizers, NMF uses NODE blocks to learn the diffeomorphic flow to implicitly discourage self-intersections, maintain the topology and thereby achieve better manifoldness of generated shape. The method is end-to-end trainable without requiring any post-processing.
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1208.3770
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\section{Introduction}
The Economists are interested in monitoring the welfare of the worse-off in
one given population. In this capacity, poverty measures are defined and used to
compare subgroups and to follow the evolution of the poor with respect to
time. A poverty measure is assumed to fulfill a number of axioms since the
pioneering work of Sen (\cite{sen}). Many authors proposed poverty indices
and studied their advantages, like Sen himself, Thon (\cite{thon}), Kakwani
\cite{kakwani}), Clark-Hemming-Ulph ( \cite{clark}), Foster-Greer-Thorbecke
\cite{fgt}), Ray (\cite{ray}), Shorrocks (\cite{shorrocks}). Most of the
required properties for such indices are stated and described in (\cit
{zheng}) along with a broad survey of the available poverty indices.
Asymptotic theories for theses quantities, when they come from random
samplings, have been given in recent years. Dia(\cite{dia}) used point
process theory to give asymptotic normality for the Foster-Greer-Thorbecke
(FGT) index. Sall and Lo(\cite{lo1}) studied an asymptotic theory for the
poverty intensity defined below and further, Sall, Seck and Lo(\cite{lo2})
proved a larger asymptotic normality for a general measure including the
Sen, Kakwani, FGT and Shorrocks ones.
Now our aim, here, is to unify the monetary poverty measurements with
respect as well to Sen's axiomatic approach as to the asymptotic aspects. We
point out that poverty may be studied through aspects other than monetary
ones as well. It can be viewed through the capabilities to meet basic needs
(food, education, health, clothings, lodgings, etc.). In our monetary frame,
the main tools are the poverty indices. We give, here, a general poverty
index denoted as the General Poverty Index (GPI), which is aimed to
summarize all the known and former ones. Let us make some notation in order
to define it.
We consider a population of individuals, each of which having a random
income or expenditure $Y$\ with distribution function $G(y)=P(Y\leq y).$\ An
individual is classified as poor\ whenever his income or expenditure $Y$\
fulfills $Y<Z,$\ where $Z$\ is a specified threshold level (the poverty
line).
Consider now a random sample $Y_{1},Y_{2},...Y_{n}$\ of size $n$\ of
incomes, with empirical distribution function $G_{n}(y)=n^{-1}\#\left\{
Y_{i}\leq y:1\leq i\leq 1\right\} $. The number of poor individuals within
the sub-population is then equal to $Q_{n}=nG_{n}(Z).$
Given these preliminaries, we introduce measurable functions $A(p,q,z)$,
w(t)$, and $d(t)$\ of $p,q\in N,$\ and $z,t\in R$. The meaning of these
functions will be discussed later on. Set $B(Q_{n},n)=\sum_{i=1}^{q}w(i).$
Let now $Y_{1,n}\leq Y_{2,n}\leq ...\leq Y_{n,n}$\ be the order statistics
of the sample $Y_{1},Y_{2},...Y_{n}$\ of $Y$. We consider general\ \
poverty\ indices\ \ of the form
\begin{equation}
GPI_{n}=\frac{A(Q_{n},n,,Z)}{nB(Q_{n},n)}\sum_{j=1}^{Q_{n}}w(\mu _{1}n+\mu
_{2}Q_{n}-\mu _{3}j+\mu _{4})\text{\ }d\left( \frac{Z-Y_{j,n}}{Z}\right) ,
\label{gpi01}
\end{equation
where $\mu _{1},\mu _{2},\mu _{3},\mu _{4}$\ are constants. By
particularizing the functions $A$ and $w$ and by giving fixed values to the
\mu _{i}^{\prime }s,$ we may find almost all the available indices, as we
will do it later on. In the sequel, (\ref{gpi01}) will be called a poverty
index (indices in the plural) or simply a poverty measure according to the
economists terminology.
The poverty line Z is defined by economics specialists or governmental
authorities so that any individual or household with income (say yearly)
less than Z is considered as a poor one. The poverty line determination
raises very difficult questions as mentioned and shown in ( \cite{kak2}).
We suppose here that Z is known, given and justified by the specialists.
\bigskip
Our unified and global approach will permit various research works, as well
in the Statistical Mathematics field as in the Economics one. It happens
that poverty indices are also somewhat closely connected with economic
growth questions. We should find conditions on the functions and the
constants in (\ref{gpi01}) so that any kind of needed requirements are met
and that the hypotheses imposed by the asymptotic normality are also
fulfilled. This may lead to a class of perfect or almost perfect poverty
measures. In this paper, we concentrate on the description of the GPI and on
the asymptotic normality theory. Our best achievement is that (\ref{gpi01}),
is asymptotically normal for a very broad class of underlying distributions.
These results are then specialized for the particular and popular indices.
We then begin to describe all the available indices in the frame of (\re
{gpi01}) in the next section. In section 3, we establish the asymptotic
normality. Related application works to poverty databases can be found in
\cite{lo6} for instance.
\section{How does the GPI include the poverty indices}
We begin by making two remarks. First, for almost all the indices, the
function $\delta (\cdot )$ is the identity one
\begin{equation*}
\forall (u\geq 0),\text{ }\delta (u)=I_{d}(u)=u.
\end{equation*}
We only noticed one exception in the Clark-Hemming-Ulph (CHUT) index.
Secondly, we may divide the poverty indices into non-weighted and weighted
ones. The non weighted measures correspond to those for which the weight is
constant and equal to one :
\begin{equation*}
w(\mu _{1}n+\mu _{2}Q_{n}-\mu _{3}j+\mu _{4})\equiv 1.
\end{equation*
We begin with them.
\subsection{The non-weighted indices.}
First of all, the Foster-Greer-Thorbecke (FGT) index of parameter \cite{fgt}
defined for $\alpha \geq 0,
\begin{equation}
FGT_{n}(\alpha )=\frac{1}{n}\overset{Q_{n}}{\underset{j=1}{\sum }}\left(
\frac{Z-Y_{j,n}}{Z}\right) ^{\alpha }. \label{sall2}
\end{equation}
is obtained from the GPI with
\begin{equation*}
\begin{array}{cccc}
\delta =I_{d}, & w\equiv 1, & d(u)=u^{\alpha }, & B(Q_{n},n)=Q_{n}\text{ }an
\text{ }A(Q_{n},n,Z)=Q_{n}
\end{array
\end{equation*
The Ray index defined by (see \cite{ray}), for $\alpha>0$,
\begin{equation}
R_{R,n}=\frac{g}{nZ}\sum_{i=1}^{Q_{n}}((Z-Y_{j,n})/g)^{\alpha }
\end{equation
where
\begin{equation}
g=\frac{1}{Q_{n}}\sum_{j=1}^{j=Q_{n}}(Z-Y_{j,n})
\end{equation
is derived from the GPI with
\begin{equation*}
\begin{array}{cccc}
\delta =I_{d}, & w\equiv 1, & d(u)=u^{\alpha }, & B(Q_{n},n)=Q_{n}\text{ }an
\text{ }A(Q_{n},n,Z)=Q_{n}(g/Z)^{\alpha -1}
\end{array
\end{equation*
The coefficient $A(Q_{n},n,Z)$\ depends here on the income or the
expenditure. This is quite an exception among the poverty indices. We may
also cite here the Watts index \ (see \cite{watts})
\begin{equation*}
P_{W,n}=\frac{1}{n}\sum_{j=1}^{j=Q_{n}}(\ln Z-lnY_{j,n}).
\end{equation*
But this may be derived from the FGT one as follows. The income Y is
transformed into $\ln Y$\ and, consequently, the poverty line is taken as
\ln Z.$\ \ It follows that
\begin{equation*}
W(Y)=FGT(1,\ln Y)
\end{equation*
for the poverty line $\ln Z$. The case is similar for the Chakravarty index
(see \cite{chakravarty}), $0<\alpha <1,$\
\begin{equation*}
P_{Ch}=\frac{1}{n}\sum_{j=1}^{j=Q_{n}}(1-(\frac{Y_{j,n}}{Z})^{\alpha }).
\end{equation*
We may consider it through the FGT class
\begin{equation*}
W(Y)=FGT(1,Y^{\alpha })
\end{equation*
for the poverty line $Z^{\alpha}$.
\noindent Now, we have that the CHU index is clearly of the GPI form with
\begin{equation*}
\begin{array}{cccc}
\delta =(u)=u^{1/\alpha }, & w\equiv 1, & d(u)=u^{\alpha } & B(Q_{n},n)=Q_{n
\text{ }and\text{ }A(Q_{n},n,Z)=Q_{n}^{\alpha }/n^{\alpha -1}
\end{array
\end{equation*
\noindent Now let us describe the weighted indices.
\subsection{The weighted indices}
First, the Kakwani (\cite{kakwani}) class of poverty measures
\begin{equation}
P_{KAK,n}(k)=\frac{Q_{n}}{n\Phi _{k}(Q_{n})}\overset{Q_{n}}{\underset{j=1}
\sum }}(Q_{n}-j+1)^{k}\left( \frac{Z-Y_{j,n}}{Z}\right) , \label{sall4}
\end{equation
where
\begin{equation*}
\Phi _{k}(Q_{n})=\sum_{j=1}^{j=Q_{n}}j^{k\text{ \ }}=B(Q_{n},n)
\end{equation*
comes from the GPI with
\begin{equation*}
\delta =I_{d},\text{ }w(u)\equiv (u),\text{ }d(u)=u,\text{ }\mu _{1}=0,
\end{equation*
\begin{equation*}
\mu _{2}=1,\text{ }\mu _{3}=-1,\text{ }\mu _{4}=1\text{ }and\text{
A(n,Q_{n},Z)=Q_{n}
\end{equation*
For $k=1,$\ $P_{KAK,n}(1)$\ is nothing else but the Sen poverty measure
\begin{equation}
P_{Sen}=\frac{2}{n(Q_{n}+1)}\overset{Q_{n}}{\underset{j=1}{\sum }
(Q_{n}-j+1)\left( \frac{Z-Y_{j,n}}{Z}\right) . \label{sall3}
\end{equation
As to the Shorrocks (\cite{shorrocks}) index
\begin{equation}
P_{SH,n}=\frac{1}{n^{2}}\overset{Q_{n}}{\underset{j=1}{\sum }
(2n-2j+1)\left( \frac{Z-Y_{j,n}}{Z}\right) , \label{sall5}
\end{equation
it is obtained from the GPI with
\begin{equation*}
B(Q_{n},n)=Q(2n-Q),\text{ }A(n,Q_{n},Z)=Q_{n}(2n-Q_{n})/n
\end{equation*
and\
\begin{equation*}
\begin{array}{cccccc}
\delta =I_{d}, & w(u)\equiv (u), & d(u)=u, & \mu _{1}=2, & \mu _{2}=0, & \mu
_{3}=2
\end{array
\mu _{3}=1.
\end{equation*
Thon (\cite{thon}) proposed\bigskip\ the following measure
\begin{equation*}
P_{Th}=\frac{2}{n(n+1)}\overset{Q_{n}}{\underset{j=1}{\sum }}(n-j+1)\left(
\frac{Z-Y_{j,n}}{Z}\right)
\end{equation*
which belongs to the GPI family for
\begin{equation*}
B(n,Q_{n})=Q_{n}(n-Q_{n}+1)/2,\text{ }A(n,Q_{n},Z)=Q(n-Q+1)/(n+1),
\end{equation*
and
\begin{equation*}
\begin{array}{cccccc}
\delta =I_{d}, & w(u)\equiv u, & d(u)=u, & \mu _{1}=1, & \mu _{2}=0, & \mu
_{3}=1
\end{array
\begin{array}{cc}
\mu _{3}=1. &
\end{array
\end{equation*
\noindent Not all the poverty indices are derived from the GPI. What precedes only
concerns those based on the poverty gaps
\begin{equation*}
(Z-Y_{j}),\text{ }1\leq j\leq Q_{n}.
\end{equation*
We mention one of them in the next subsection.
\subsection{An index not derived from the GPI}
The Takayama (\cite{takayama}) index
\begin{equation*}
P_{TA,n}=1+\frac{1}{n}-\frac{2}{\mu n^{2}}\sum_{j=1}^{Q_{n}}(n-j+1)Y_{j,n},
\end{equation*
where $\mu $\ is the empirical mean of the censored income, cannot be
derived from the GPI. The main reason is that, it is not based on the
poverty gaps $Z-Y_{j,n}.$\ It violates the monotonicity axiom which states
that the poverty measure increases when one poor individual or household
becomes richer.
$\bigskip $
Now we must study the so-called GPI with respect to the axiomatic approach
as well as to the asymptotic theory. We focus in this paper to the general
theory of asymptotic normality. The interest of this unified approach is
based on the fact that we obtain at once the asymptotic behaviors for all
the available poverty indices, as particular cases. Indeed, in the next
section, we will describe apply the general theorem to the particular usual
indices.
\section{Asymptotic normality of the GPI}
Let us write the GPI in the form
\begin{equation}
GPI_{n}=\delta (J_{n}) \label{sall1}
\end{equation
with
\begin{equation}
J_{n}=\frac{1}{n}\overset{Q_{n}}{\underset{j=1}{\sum }}c(n,Q_{n},j)\text{
d\left( \frac{Z-Y_{j,n}}{Z}\right) , \label{sall6}
\end{equation
where $c(n,Q_{n},j)=A(Q_{n},n,,Z)\times w(\mu _{1}n+\mu _{2}Q_{n}-\mu
_{3}j+\mu _{4})$ $/$ $B(Q_{n},n).$\
Since Y is an income or expenditure variable, its lower endpoint $y_{0}$\ is
not negative. This allows \ us to study (\ref{sall1}) via the transform
X=1/(Y-y_{0}).$\ Throughout this paper, the distribution function of $X$\ is
\begin{equation*}
F(\cdot )=1-G(y_{0}+1/\cdot ),
\end{equation*
whose upper endpoint is then +$\infty .$\ Hence (\ref{sall6}) is transformed
as
\begin{equation}
J_{n}=\frac{1}{n}\overset{q}{\underset{j=1}{\sum }}c(n,q,j)\text{ }d\left(
\frac{Z-y_{0}-X_{n-j+1,n}^{-1}}{Z}\right) . \label{sall7}
\end{equation}
\bigskip\
We will need conditions on the function d($\cdot $) and on the weight
c(n,q,j), as in (\cite{lo2}). First assume that
$\medskip $
$(D1)$\ $d(\cdot )$\ admits a continuous derivative on $]0,1)\medskip $.
$(D2)$\ $d^{\prime }(\frac{z-y_{0}}{z})$\ and $d((z-y_{0})/z)$\ are
finite.\bigskip\
\qquad For $A(u)=1/F^{-1}(1-u)$, we assume that:\bigskip\
$(C1)$\ $A(\cdot )$\ is differentiable $(0,1)$\ ( and its derivative is
denoted $A^{\prime }(u)=a(u)$.)
$(C2)$\ $a(\cdot )$\ is continuous on an interval $[a^{\prime },a^{\prime
\prime }]$\ with $0<a^{\prime }<a^{\prime \prime }<1$.\medskip\
$(C3)$\ $\exists \ u_{0}>0,\exists \ \eta >-3/2,\ \forall \ u\in \left(
0,u_{0}\right) ,\left\vert a(u)\right\vert <C_{0}\ u^{\eta }\exp
(\int_{u}^{1}b(t)t^{-1}dt),with$\ $b(t)\rightarrow 0\ as$\ $t\rightarrow 0.$
$\bigskip $
The condition (C3) means that $a(\cdot )$\ bounded by a regularly varying
function \
\begin{equation*}
S(u)=C_{0}\ u^{\eta }\exp (\int_{u}^{1}b(t)t^{-1}dt)
\end{equation*
\ of exponent $\eta >-3/2$. \bigskip As to the function $\delta $, we need
it to be differentiable on $]0,+\infty \lbrack $, precisely :\newline
\bigskip $(E)$\ There is $\kappa >0$\ such that\ $\delta $($\cdot $) is
continuously differentiable on $]0,\kappa ]$.\newline
\bigskip \newline
We also need some conditions on the weight $c(\cdot )$. In order to state
the hypotheses, we introduce further notation. In fact we use in this paper
the representations of the studied random variables $X_{i},$\ $i\geq 1$,\ by
$F^{-1}(1-U_{i}),$\ $i\geq 1$, where $U_{1},$\ $U_{2},...$\ is a sequence of
independent random variables uniformly distributed on $\left( 0,1\right) $.
Now let $U_{n}(\cdot )$\ and $V_{n}(\cdot )$\ be the uniform empirical
distribution and the empirical quantile function based on $U_{i},1\leq i\leq
n$.\ We have
\begin{equation}
j\geq 1,\text{ \ }\frac{j-1}{n}<s\leq \frac{j}{n}\Longrightarrow \frac{j}{n
=U_{n}(V_{n}(s)) \label{sall10}
\end{equation
so that
\begin{equation}
j\geq 1,\text{ \ }\frac{j-1}{n}<s\leq \frac{j}{n}\Longrightarrow
c(n,q,j)=c(n,q,nU_{n}(V_{n}(s))\equiv L_{n}(s). \label{sall11}
\end{equation
Since $U_{n}(V_{n}(s))\rightarrow s,$\ as n$\rightarrow \infty ,$\ our
condition on the weight $c(\cdot )$\ is that the function $L_{n}(\cdot )$\
is uniformly bounded by some constant $D>0$\ and
\begin{equation}
L_{n}(s)\rightarrow L(s),as\text{ }n\rightarrow \infty , \label{sall12}
\end{equation
where $L(\cdot )$\ is a non-negative $C^{1}-$function on $(0,1)$.
$\bigskip $
We further require that, as $n\rightarrow \infty ,$\
\begin{equation}
\underset{0\leq s\leq 1}{\sup }\left\vert \sqrt{n}(L_{n}(s)-L(s))-\gamma (s
\sqrt{n}(G_{n}(Z)-G(Z))\right\vert =o_{p}(1) \label{sall12a}
\end{equation
for some function $\gamma (\cdot )$. Let us finally put
\begin{equation*}
m(s)=L(s)\text{ }d\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right) .
\end{equation*}
\bigskip We are now able to give our general theorem for the GPI.
\begin{theorem}
Suppose that (C1-2-3), (D1-2) and (\ref{sall12a}) hold and let
\begin{equation*}
\mu =\int_{0}^{G(Z)}\gamma (s)d\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right)
\text{ }ds.
\end{equation*
and
\begin{equation*}
D=\int_{0}^{G(Z)}L(s)\text{ }d\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right)
ds,
\end{equation*
Then
\begin{equation*}
{\large \sqrt{n}(J_{n}-D)\rightarrow \mathcal{N}(0,\vartheta ^{2})}
\end{equation*
\ with
\begin{equation*}
\vartheta ^{2}=\theta ^{2}+(m(G(Z))+\mu )^{2}G(Z)(1-G(Z))+\frac
2(m(G(Z))+\mu )}{Z}\int_{0}^{G(Z)}sL(s)h(s)ds
\end{equation*
and with
\begin{equation*}
\theta ^{2}=Z^{-2}\int_{0}^{G(Z)}\int_{0}^{G(Z)}L(u)\text{ }L(v)\text{
h(u)h(v)(u\wedge v-uv)\text{ }du\text{ }dv
\end{equation*
where
\begin{equation*}
h(s)=a(s)\text{ }d^{\prime }(\frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}).
\end{equation*
\bigskip If furthermore (E) holds and $D\in ]0,\kappa \lbrack ,$\ then
\begin{equation*}
{\large \sqrt{n}(GPI}_{n}{\large -\delta (D))\rightarrow }\mathcal{N}{\large
(0,\vartheta }^{2}\delta ^{\prime }(D)^{2})
\end{equation*}
\end{theorem}
$\bigskip $
The interest of this paper resides on the particular applications of the
theorem for the known indices. Before this, we give the guidelines of the
proof.
\section{PROOFS OF THE RESULTS}
All our results will be derived from the lemma below. But, first we place
ourselves on a probability space where one version of the so-called
Hungarian constructions holds. Namely, M. Cs\"{o}rg\H{o} and al. (see \cit
{cchm}) have constructed a probability space holding a sequence of
independent uniform random variables $U_{1},\ U_{2},$\ ... and a sequence of
Brownian bridges $B_{1},B_{2},...$\ such that for each $0<\nu <1/2$, $as$\
n\rightarrow \infty ,
\begin{equation}
\underset{1/n\leq s\leq 1-1/n}{\sup }\frac{\left\vert \beta
_{n}(s)-B_{n}(s)\right\vert }{(s(1-s))^{1/2-\nu }}=O_{p}(n^{-\nu })
\label{sall12c}
\end{equation
and for each $0<\nu <1/4$
\begin{equation}
\underset{1/n\leq s\leq 1-1/n}{\sup }\frac{\left\vert B_{n}(s)-\alpha
_{n}(s)\right\vert }{(s(1-s))^{1/2-\nu }}=O_{p}(n^{-\nu }), \label{sall12d}
\end{equation
where $\left\{ \alpha _{n}(s)=\sqrt{n}\left( U_{n}(s)-s)\right) ,0\leqslant
s\leqslant 1\right\} $\ is the uniform empirical process and $\left\{ \beta
_{n}(s)=\sqrt{n}\left( s-V_{n}(s)\right) ,0\leqslant s\leqslant 1\right\} $\
is the uniform quantile process. (See also \cite{ch} for a shorter and more
direct proof, and \cite{mvz} for dual version, in the sens that, \re
{sall12c} holds for $0<\nu <1/2$ and \ref{sall12d} for $0<\nu <1/4$ in \cit
{ch}, while \ref{sall12c} is established for $0<\nu <1/4$ and \ref{sall12d}
for $0<\nu <1/2$ in \cite{mvz}). Throughout $\nu $\ will be fixed with $0<\nu
<1/4.$\ Now we are able to give the lemma.
\begin{lemma}
Suppose that (C1-2-3) and (D1-2) hold and
\begin{equation}
\underset{0\leq s\leq 1}{\sup }\sqrt{n}\left\vert L_{n}(s)-L(s)\right\vert
=O_{P}(1)\text{ as n}\rightarrow \infty . \label{sall12b}
\end{equation
Let
\begin{equation*}
D=\int_{0}^{G(Z)}L(s)\text{ }d\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right)
ds.
\end{equation*
Then we have the expansion
{\Large
\begin{equation*}
\sqrt{n}(J_{n}-D)=N_{n}(1)+N_{n}(2)
\end{equation*
}
{\Large
\begin{equation*}
+\int_{1/n}^{G(Z)}\sqrt{n}(L_{n}(s)-L(s))d\left( \frac{Z-y_{0}-1/F^{-1}(1-s)
}{Z}\right) ds+o_{P}(1)
\end{equation*
}
\noindent with
\begin{equation}
N_{n}(1)=\frac{1}{Z}\int_{1/n}^{G(Z)}L(s)B_{n}(s)h(s)ds \label{salln1}
\end{equation
and
\begin{equation}
N_{n}(2)=m(G(Z))B_{n}(G(Z)) \label{salln2}
\end{equation
for
\begin{equation*}
m(s)=L(s)\text{ }d\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right) .
\end{equation*}
\end{lemma}
\begin{proof}
This expansion is formulae (4.14) in (\cite{lo2}). Then, we have the
expansion
\begin{equation*}
\sqrt{n}(J_{n}-C_{n})=\frac{1}{Z}\int_{1/n}^{G(Z)}L(s)B_{n}(s)h(s)\text{
ds+n^{-1/2}L_{n}(1/n)\text{ }d\left( \frac{Z-y_{0}-1/F^{-1}(1-U_{1,n})}{Z
\right)
\end{equation*
\begin{equation*}
+\int_{1/n}^{G_{n}(Z)}\sqrt{n}(L_{n}(s)-L(s))\text{ }d\left( \frac
Z-y_{0}-1/F^{-1}(1-V_{n}(s))}{Z}\right) ds
\end{equation*
\begin{equation*}
+\frac{1}{Z}\int_{G(Z)}^{G_{n}(Z)}L(s)B_{n}(s)h(s)ds+\frac{1}{Z
\int_{1/n}^{G_{n}(Z)}L_{n}(s)B_{n}(s)\text{ }(h(\zeta _{n}(s))-h(s))\text{
ds
\end{equation*
\
\begin{equation*}
+\frac{1}{Z}\int_{1/n}^{G_{n}(Z)}L_{n}(s)\text{ }(\beta _{n}\left( s\right)
-B_{n}(s))\text{ }h(\zeta _{n}(s))\text{ }ds
\end{equation*
It is proved in (\cite{lo2}) that
\begin{equation*}
\sqrt{n}(J_{n}-C_{n})=N_{n}(1)+N_{n}(2)
\end{equation*
\begin{equation*}
+\int_{1/n}^{G_{n}(Z)}\sqrt{n}(L_{n}(s)-L(s))\text{ }d\left( \frac
Z-y_{0}-1/F^{-1}(1-V_{n}(s))}{Z}\right) ds+o_{P}(1).
\end{equation*
This gives
\begin{equation*}
\sqrt{n}(J_{n}-C_{n})=N_{n}(1)+N_{n}(2)
\end{equation*
\begin{equation*}
+\int_{1/n}^{G(Z)}\sqrt{n}(L_{n}(s)-L(s))d\left( \frac{Z-y_{0}-1/F^{-1}(1-s)
}{Z}\right) ds
\end{equation*
\begin{equation*}
+\int_{Gn(Z)}^{G(Z)}\sqrt{n}(L_{n}(s)-L(s))\text{ }d\left( \frac
Z-y_{0}-1/F^{-1}(1-V_{n}(s))}{Z}\right) ds+o_{P}(1)
\end{equation*
The condition (\ref{sall12b}) leads to the result.
\end{proof}
$\bigskip $
We are now able to prove the Theorem.
\begin{proof}
$\ $Let($\Omega $,$\Sigma $,P)\ be the probability space on which (\re
{sall12c}) and (\ref{sall12d}) hold. The Lemma together with (\ref{sall12b
), (\ref{sall12a}) and (\ref{salln2}), imply
\begin{equation*}
{\large \sqrt{n}(J_{n}-D)=N}_{n}(1)+N_{n}(3)+o_{P}(1),
\end{equation*
where $N_{n}(1)$\ is defined in (\ref{salln1}) and
\begin{equation}
N_{n}(3)=(m(G(Z)+\mu )\alpha _{n}(G(Z))+o_{P}(1)=(m(G(Z)+\mu
)B_{n}(G(Z))+o_{P}(1). \label{sall21}
\end{equation
The vector $(N_{n}(1),N_{n}(3))$\ is Gaussian and
\begin{equation}
cov(N_{n}(1),N_{n}(3))=\frac{m(G(Z))+\mu }{Z
E\int_{1/n}^{G(Z)}L(s)h(s)B_{n}(G(Z))B_{n}(s)ds \label{salln4}
\end{equation
\begin{equation*}
=\frac{m(G(Z))+\mu }{Z}\int_{1/n}^{G(Z)}s\text{ }L(s)\text{ }h(s)\text{ }ds.
\end{equation*
Then $\sqrt{n}(J_{n}-D)$\ is a linear transform $N_{n}(1)+N_{n}(3)$\ of the
Gaussian vector $(N_{n}(1),N_{n}(3)),$\ plus an $o_{P}(1)$\ term. The
variance of this Gaussian term is easily computed through (\ref{salln4}) and
the conclusion follows, that is $\sqrt{n}(J_{n}-D)$\ is asymptotically a
centered Gaussian random variable with variance (\ref{salln4}).
\end{proof}
\section{$\protect\bigskip $Asymptotic normality of particular indices}
\subsection{The FGT-like class}
This concerns the indices of the form
\begin{equation*}
FGT(\alpha )=\frac{1}{n}\overset{Q_{n}}{\underset{j=1}{\sum }}d\left( \frac
Z-Y_{j,n}}{Z}\right) .
\end{equation*
We have here
\begin{equation*}
L_{n}=1
\end{equation*
so that
\begin{equation*}
\gamma =0
\end{equation*
Then
\begin{equation*}
{\large \sqrt{n}(J}_{n}{\large -D)\rightarrow \mathcal{N}(0,\vartheta }^{2})
\end{equation*
\ with
\begin{equation*}
\vartheta ^{2}=\theta ^{2}+m(G(Z)^{2}G(Z)(1-G(Z))+\frac{2m(G(Z))}{Z
\int_{0}^{G(Z)}sh(s)ds
\end{equation*
and
\begin{equation*}
D=\int_{0}^{G(Z)}\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right) ^{\alpha }ds.
\end{equation*
We should remark that the conditions $(D1-D2)$\ hold for $d(u)=u^{\alpha
},\alpha \geq 0.$
\subsubsection{The statistics nearby the FGT-class}
This concerns the statistics of the form
\begin{equation*}
J_{n}=\delta \biggl(\frac{A(Q_{n},n)}{n}\overset{Q_{n}}{\underset{j=1}{\sum }
d\left( \frac{Z-Y_{j,n}}{Z}\right) \biggl) ,
\end{equation*
where we have a random weight not depending on the rank's statistic. We will
have two sub-cases.
\subsubsection{The case of CHU's index}
Recall
\begin{equation*}
CHU_{n}(\alpha )=\frac{Q_{n}}{nZ}\left\{ \frac{1}{Q_{n}
\sum_{j=1}^{Q_{n}}(Z-Y_{j,n})^{\alpha }\right\} ^{1/\alpha }
\end{equation*
\begin{equation*}
=\left\{ \frac{1}{n}\frac{Q_{n}^{\alpha -1}}{n^{\alpha -1}
\sum_{j=1}^{Q_{n}}(\frac{Z-Y_{j,n}}{Z})^{\alpha }\right\} ^{1/\alpha
}=\delta (J_{n})
\end{equation*
We easily get,
\begin{equation*}
\sqrt{n}((q/n)^{\alpha -1}-G(Z)^{\alpha -1})=(\alpha -1)G(Z)^{\alpha -2
\sqrt{n}(G_{n}(Z)-G(Z))+o_{p}(1)
\end{equation*
\begin{equation*}
=(\alpha -1)G(Z)^{\alpha -2}B_{n}(G(Z))+o_{p}(1).
\end{equation*
By putting
\begin{equation*}
C_{n}=FGT(\alpha )=\frac{1}{n}\sum_{j=1}^{Q_{n}}(\frac{Z-Y_{j,n}}{Z
)^{\alpha }
\end{equation*
and
\begin{equation}
C=\int_{0}^{G(Z)}\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right) ^{\alpha }ds,
\label{chu00}
\end{equation
we have, by the general theorem
\begin{equation*}
\sqrt{n}(C_{n}-C)=N_{n}(1)+N_{n}(2)+o_{p}(1)
\end{equation*
with $L=1$\ . By combining these formulae, we get
\begin{equation*}
\sqrt{n}(J_{n}-G(Z)^{\alpha -1}C)\rightarrow N(0,\zeta ^{2})
\end{equation*
with
\begin{equation*}
\zeta ^{2}=\theta ^{2}+H(1-G(Z))\int_{0}^{G(Z)}s\text{
a(s)ds+H^{2}G(Z)(1-G(Z))/2
\end{equation*
where,
\begin{equation*}
H=C(\alpha -1)+G(Z)m(G(Z))G(Z)^{\alpha -2}.
\end{equation*
Finally, we get
\begin{equation*}
\sqrt{n}(CHU_{n}(\alpha )-\delta (G(Z)^{\alpha -1}C)\rightarrow N(0,(\zeta
\delta ^{\prime }(G(Z)^{\alpha -1}C)^{2}),
\end{equation*
where
\begin{equation*}
\delta ^{\prime }(G(Z)^{\alpha -1}C)^{2}=G(Z)^{-(\alpha -1)^{2}/\alpha
}C^{(1-\alpha )/\alpha }.
\end{equation*}
\subsubsection{The case of Ray's index}
Recall
\begin{equation}
P_{R,n}(\alpha)=\frac{g}{nZ}\sum_{j=1}^{Q_{n}}((Z-Y_{j,n})/g)^{\alpha }
\end{equation
where
\begin{equation}
g=\frac{1}{q}\sum_{j=1}^{j=Q_{n}}(Z-Y_{j,n}).
\end{equation
We have
\begin{equation*}
J_{n}=g^{\alpha -1}\times C_{n}
\end{equation*
with
\begin{equation*}
C_{n}=FGT_{n}(\alpha )
\end{equation*
and
\begin{equation*}
C(\alpha )=\int_{0}^{G(Z)}\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right)
^{\alpha }ds.
\end{equation*
We use the notation for the CHU index and we also get (\ref{chu00}). But
\begin{equation*}
g=\frac{Zn}{Q_{n}}\times \frac{1}{n}\sum_{j=1}^{j=Q_{n}}(\frac{Z-Y_{j,n}}{Z
) \equiv \frac{Zn}{Q_{n}}K_{n}.
\end{equation*
We also have
\begin{equation*}
\sqrt{n}(K_{n}-K)=\frac{1}{Z
\int_{1/n}^{G(Z)}B_{n}(s)a(s)ds+m_{1}(G(Z))B_{n}(G(Z))+o_{p}(1)
\end{equation*
with
\begin{equation*}
K=C(1)=\int_{0}^{G(Z)}\frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}ds
\end{equation*
and
\begin{equation*}
\sqrt{n}(\frac{Zn}{q}-ZG(Z)^{-1})=Z\sqrt{N}(G(Z)-G_{n}(Z))G(Z)^{-2}+o_{p}(1)
\end{equation*
\begin{equation*}
=-Z\sqrt{n}B_{n}(G_{n}(Z))G(Z)^{-2}+o_{p}(1).
\end{equation*
By combining all that precedes, we arrive at
\begin{equation*}
\sqrt{n}(g-KZG(Z)^{-1})=(m_{1}(G)ZG(Z)^{-1}-KZG(Z)^{-2})B_{n}(G(Z))
\end{equation*
\begin{equation*}
+\frac{1}{G(Z)}\int_{1/n}^{G(Z)}B_{n}(s)a(s)ds+o_{p}(1)
\end{equation*
and
\begin{equation*}
\sqrt{n}(g^{\alpha -1}-(KZ/G(Z))^{\alpha -1})=(\alpha -1)(KZ/G(Z))^{\alpha
-2}
\end{equation*
\begin{equation*}
=(\alpha -1)(KZ/G(Z))^{\alpha -2}\text{
(m_{1}(G)ZG(Z)^{-1}-KZG(Z)^{-2})B_{n}(G(Z))
\end{equation*
\begin{equation*}
+\frac{(\alpha -1)(KZ/G(Z))^{\alpha -2}}{G(Z)
\int_{1/n}^{G(Z)}B_{n}(s)a(s)ds+o_{p}(1)
\end{equation*
\begin{equation*}
=R_{1}B_{n}(G(Z))+R_{2}\int_{1/n}^{G(Z)}B_{n}(s)a(s)ds+o_{p}(1).
\end{equation*
Finally
\begin{equation*}
\sqrt{n}(R_{n}-(KZ/G(Z))^{\alpha -1})C)=
\end{equation*
\begin{equation*}
\frac{(KZ/G(Z))^{\alpha -1})}{Z}\int_{1/n}^{G(Z)}B_{n}(s)h(s)ds+(KZ/G(Z))^
\alpha -1})\text{ }m_{\alpha }(G(Z))B_{n}(G(Z))
\end{equation*
\begin{equation*}
+CR_{1}B_{n}(G(Z))+CR_{2}\int_{1/n}^{G(Z)}B_{n}(s)a(s)ds+o_{p}(1)
\end{equation*
\begin{equation*}
=\int_{1/n}^{G(Z)}B_{n}(s)\psi (s)ds+\left\{ (KZ/G(Z))^{\alpha -1})\text{
m_{\alpha }(G(Z))+CR_{1}\right\} B_{n}(G(Z))+o_{P}(1)
\end{equation*}
\begin{equation*}
=\int_{1/n}^{G(Z)}B_{n}(s)a(s)ds+A_{2}B_{n}(G(Z))+o_{P}(1),
\end{equation*
with
\begin{equation*}
\psi (s)=a(s)\left\{ C(\alpha )R_{2}+(KZ/G(Z))^{\alpha -1}Z^{-1}d^{\prime
}(Z^{-1}(Z-y_{0}-1/F^{-1}(1-s)))\right\} .
\end{equation*
Notice that $\int_{1/n}^{G(Z)}B_{n}(s)h(s)ds+A_{1}B_{n}(G(Z))$\ is a normal
centered random variable with variance
{\Large
\begin{multline*}
\xi ^{2}=\int_{0}^{G(Z)}\int_{0}^{G(Z)}\psi (u)\psi (v)(u\wedge v-uv)\text{
du\text{ }dv \\
+A_{1}^{2}G(Z)(1-G(Z))+2A_{1}(1-G(Z))\int_{0}^{G(Z)}s\text{ }\psi (s)ds.
\end{multline*
}
\noindent We conclude that
\begin{equation*}
\sqrt{n}(P_{R,n}(\alpha)-(KZ/G(Z))^{\alpha -1})C)\rightarrow _{d}N(0,\xi ^{2})
\end{equation*
with
\begin{equation*}
m_{\alpha }(u)=(Z^{-1}(Z-y_{0}-1/F^{-1}(1-s))^{\alpha },
\end{equation*
\begin{equation*}
R_{1}=(\alpha -1)(KZ/G(Z))^{\alpha -2}\text{
(m_{1}(G)ZG(Z)^{-1}-KZG(Z)^{-2}),
\end{equation*
\begin{equation*}
R_{2}=(\alpha -1)(KZ/G(Z))^{\alpha -2}G(Z)^{-1},
\end{equation*
and
\begin{equation*}
A_{1}=(KZ/G(Z))^{\alpha -1})\text{ }m_{\alpha }(G(Z))+CR_{1}
\end{equation*}
\subsection{The Shorrocks-like indices}
This concerns the Thon and Shorrocks measures. They both have a similar
asymptotic behavior.
For Shorrocks's index, we have
\begin{equation*}
P_{SH,n}=\frac{1}{n^{2}}\overset{Q_{n}}{\underset{j=1}{\sum }
(2n-2j+1)\left( \frac{Z-Y_{j,n}}{Z}\right) .
\end{equation*
But
\begin{equation}
j\geq 1,\ \frac{j-1}{n}<s\leq \frac{j}{n}\Longrightarrow
L_{n}(s)=c(n,q,j)=(2-2\ast j/n+1/n) \label{sall14}
\end{equation
\begin{equation*}
\rightarrow L(s)=2(1-s),
\end{equation*
and,
\begin{equation*}
\sqrt{n}(L_{n}(s)-L(s))=-2\ast \sqrt{n}(U_{n}(V_{n}(s))-s)+1/\sqrt{n,}
\end{equation*
By\ (\cite{shwell}), $\ $p.151,
\begin{equation*}
\sqrt{n}\underset{0\leq s\leq 1}{\sup }\left\vert L_{n}(s)-L(s)\right\vert
\leq 3/\sqrt{n}.
\end{equation*
and then
\begin{equation*}
\gamma \equiv 0\text{, }h(\cdot )=a(\cdot )
\end{equation*
For the Thon Statistic,
\begin{equation*}
P_{T,n}=\frac{2}{n(n+1)}\overset{Q_{n}}{\underset{j=1}{\sum }}(n-j+1)\left(
\frac{Z-Y_{j,n}}{Z}\right) ,
\end{equation*
we also have
\begin{equation*}
L(s)=2(1-s),\text{ }\gamma \equiv 0,\text{ }a(\cdot )=h(s).
\end{equation*
In both cases, for $P_n=P_{SH,n}$ or $j_n=P_{T,n}$, we get
\begin{equation*}
{\large \sqrt{n}(P}_{n}{\large -D)\rightarrow \mathcal{N}(0,\vartheta }^{2})
\end{equation*
\ with
\begin{equation*}
D=2\int_{0}^{G(Z)}(1-s)\text{ }\left( \frac{Z-y_{0}-1/F^{-1}(1-s)}{Z}\right)
ds,
\end{equation*
\begin{equation*}
\vartheta ^{2}=\theta ^{2}+m(G(Z)G(Z)(1-G(Z))+\frac{4m(G(Z))}{Z
\int_{0}^{G(Z)}s(1-s)a(s)ds
\end{equation*
and with
\begin{equation*}
\theta ^{2}=4Z^{-2}\int_{0}^{G(Z)}\int_{0}^{G(Z)}(1-u)(1-v)\text{
a(u)a(v)(u\wedge v-uv)\text{ }du\text{ }dv.
\end{equation*}
\subsection{The Kakwani-class.}
The Kakwani class
\begin{equation*}
P_{KAK,n}=\frac{Q_{n}}{n\Phi _{k}(Q_{n})}\overset{Q_{n}}{\underset{j=1}{\sum
}}(Q_{n}-j+1)^{k}\left( \frac{Z-Y_{j,n}}{Z}\right) ,
\end{equation*
is introduced with a positive integer. We consider here that k is merely a
non-negative real number. It is proved in (\cite{lo3}) that
\begin{equation*}
L(s)=(k+1)(1-s/G(Z))^{k}
\end{equation*
and that
\begin{equation*}
\gamma (s)=k(k+1)(1-s/G(Z))^{k-1}(s/G(Z)^{2}).{\large \ }
\end{equation*
We remark that $m(G(Z))=0.$\ Then our result is particularized as
\begin{equation*}
{\large \sqrt{n}(P}_{KAK,n}(k){\large -D)\rightarrow \mathcal{N}(0,\vartheta
}^{2})
\end{equation*
\ with
\begin{equation}
\vartheta ^{2}=\theta ^{2}+\mu ^{2}G(Z)(1-G(Z))+\frac{2\mu }{Z
(1-G(Z))\int_{0}^{G(Z)}sL(s)h(s)ds \label{salln3}
\end{equation
and with
\begin{equation*}
\theta ^{2}=Z^{-2}\int_{0}^{G(Z)}\int_{0}^{G(Z)}L(u)\text{ }L(v)\text{
h(u)h(v)(u\wedge v-uv)\text{ }du\text{ }dv.
\end{equation*
for a fixed real number $k\geq 1.$
\bigskip\
We have now finished the poverty indices' review. Some of these results have
been simulated and applied in particular issues with the Senegalese Data.
\section{Conclusion}
The GPI includes most of the poverty indices. We have established here their
asymptotic normality with immediate applications to poor countries data for
finding accurate confidence intervals of the real poverty measurement. In
coming papers, a special study will be devoted to the Takayama statistic.
The GPI is to be thoroughly visited through the poverty axiomatic approach
as well.
\newpage
|
1208.3572
|
\section*{Acknowledgments}
One of the authors (PVP) wishes to thank the Physics Department of King's College London for hospitality during the initial stages of this work. This work is supported by the Grants Spanish MINECO FPA 2011-23596 (JB and PVP), the Generalitat Valenciana PROMETEO - 2008/004 (JB) and by the London Centre for Terauniverse Studies (LCTS), using funding from the European Research
Council via the Advanced Investigator Grant 267352 (NEM).
|
1809.03033
|
\section{Introduction}
In the early twentieth century it was noticed that while the prime-counting function $\pi(x)$
and the logarithmic integral function $\textnormal{li}(x) = \int_0^x dt/\log t$ are satisfyingly close together for all values of $x$ where
both had been computed, $\textnormal{li}(x)$ always seemed to be slightly larger than $\pi(x)$. It was a breakthrough when
Littlewood~\cite{Litt} proved that in fact the sign of $\textnormal{li}(x)-\pi(x)$ changes infinitely often as $x\to\infty$.
We still do not know a specific numerical value of $x$ for which this difference is negative, but the smallest such value
is suspected to be very large, near $1.4\times10^{316}$ (see~\cite{BH} and subsequent refinements). We do know that $\pi(x)<\textnormal{li}(x)$ for all $2\le x\le10^{19}$,
thanks to calculations of B\"uthe~\cite{Bu}.
Another important development concerning this ``race" between $\pi(x)$ and $\textnormal{li}(x)$ was the
paper of Rubinstein and Sarnak~\cite{RubSarn}. Assuming some standard conjectures
about the zeros of the Riemann zeta-function, namely the Riemann hypothesis and
a linear independence hypothesis on the zeros of $\zeta(\frac12+it)$, they showed that the logarithmic density $\delta(\Pi)$ of
the set
\begin{equation} \label{Pi defn}
\Pi := \{x\in{\mathbb R}_{\ge1}\colon \pi(x)>\textnormal{li}(x)\}
\end{equation}
exists and is a positive number
\begin{equation} \label{Delta defn}
\delta(\Pi) = \Delta \mathop{\dot=} 2.6\times10^{-7}.
\end{equation}
Here, given a set $\mathcal M \subset {\mathbb R}_{\ge1}$, the logarithmic density of $\mathcal M$ is defined as usual as
\begin{align*}
\delta(\mathcal M) := \lim_{x\to \infty}\frac{1}{\log x}\int_{t\in \mathcal M\cap [1,x]}\frac{dt}{t},
\end{align*}
provided the limit exists.
Since $\pi(x)$ counts primes, it is natural to consider the actual primes in the race:
What can be said about the set of primes $p$ for which $\pi(p)>\textnormal{li}(p)$?
We define the discrete logarithmic density of a set $\mathcal M \subset {\mathbb R}_{\ge1}$ relative to the prime numbers as
\begin{align*}
\delta'(\mathcal M) := \lim_{x\to \infty}\frac{1}{\log\log x}\sum_{\substack{p \le x\\p\in \mathcal M}}\frac{1}{p},
\end{align*}
if the limit exists.
Due to the partial summation formula
\begin{align*}
\sum_{\substack{p \le x\\p\in \mathcal M}}\frac{1}{p} = \frac{1}{\log x}\sum_{\substack{p \le x\\p\in \mathcal M}}\frac{\log p}{p} + \int_2^x \frac{1}{t\log^2 t}\sum_{\substack{p \le t\\p\in \mathcal M}}\frac{\log p}{p}\,dt,
\end{align*}
we see that if the modified limit
\begin{equation} \label{d^* defn}
\delta^*(\mathcal M) := \lim_{x\to \infty}\frac{1}{\log x}\sum_{\substack{p \le x\\p\in \mathcal M}}\frac{\log p}{p}
\end{equation}
exists, then it is equal to $\delta'(\mathcal M)$. (The converse does not hold in general, since $\delta^*(\mathcal M)$ might not exist even if $\delta'(\mathcal M)$ does. For example, let $\mathcal P_k$ be the set of all primes between $2^{(2k-1)!}$ and $2^{(2k)!}$, and let $\mathcal P=\bigcup_{k\ge1} \mathcal P_k$. Then $\delta'(\mathcal P) = 1/2$ but $\delta^*(\mathcal P)$ does not exist.)
We shall find it more convenient to deal with $\delta^*(\mathcal M)$ in our proofs below. We also let $\overline{\delta}^*$ and $\underline{\delta}^*$ denote the expression on the right-hand side of equation~\eqref{d^* defn} with $\lim$ replaced by $\limsup$ and $\liminf$, respectively.
Our general philosophy is that the primes are reasonably randomly distributed; in particular, there seems to be no reason for the primes to conspire to lie in the set of real numbers $\Pi$ any more or less often than expected.
With the aid of an old theorem of Selberg that most short intervals contain the ``right" number of primes,
we prove that there is no such conspiracy; more precisely we prove, under the same two assumptions as Rubinstein and Sarnak, that $\delta^*(\Pi)=\Delta$ (see Theorem~\ref{thm:pivsli}). Moreover, we prove similar results---comparing the logarithmic density of a set of real numbers to the relative logarithmic density of the primes lying in that set---for a number of other prime races, some of which have not been considered before (see Theorems~\ref{thm:mertens} and~\ref{Zhang primes density thm}). These results resolve some problems from \cite{LP} and so make progress
on the Erd\H os conjecture on primitive sets (see Section \ref{sec:mertens}).
Finally we remark that our approach applies equally to prime races involving residue classes.
To do this one would replace Selberg's theorem on the distribution of primes in almost all short intervals
with a result of Koukoulopoulos \cite[Theorem 1.1]{K} which does the same for
primes in a residue class to a fixed modulus.
\section{A key result}
For a ``naturally occurring'' set $\mathcal M$ of real numbers for which $\delta(\mathcal M)$ exists,
it is natural to wonder how $\delta^*(\mathcal M)$ compares to $\delta(\mathcal M)$.
We prove the two densities are equal in the case of sets of the form
\begin{equation} \label{M_a defn}
\mathcal M_a(f) = \{ x\colon f(x) > a\}
\end{equation}
for functions $f$ that are suitably nice.
\begin{theorem} \label{thm:density}
Consider a function $f\colon{\mathbb R}_{\ge1}\to {\mathbb R}$ satisfying the following two conditions:
\begin{enumerate}
\item[(a)] For all real numbers $a>b$, there exists $x_0=x_0(a,b)$ such that for all $x\ge x_0$, if $f(x)>a$ then $f(z)>b$ for all $z\in [x,x+x^{1/3}]$; and similarly for the function $-f(x)$.
\item[(b)] The function $f$ has a continuous logarithmic distribution function: for all $a\in{\mathbb R}$, the set $\mathcal M_a(f)$ has a well-defined logarithmic density $\delta(\mathcal M_a(f))$, and the map $a\mapsto \delta(\mathcal M_a(f))$ is continuous.
\end{enumerate}
Then for every real number $a$, the relative density $\delta^*(\mathcal M_a(f))$ exists and is equal to $\delta(\mathcal M_a(f))$.
\end{theorem}
\noindent
It is worth noting that the assumptions and conclusion of the theorem imply that the relative density map $a\mapsto \delta^*(\mathcal M_a(f))$ is also continuous; in particular, ``ties have density~$0$'', meaning that $\delta^*(\{x\colon f(x)=a\}) = 0$. Thus there is no difference between considering $f(x)>a$ and considering $f(x)\ge a$ in the situations we investigate.
Recall the Linear Independence hypothesis (LI), which asserts that the sequence of numbers $\gamma_n > 0$ such that $\zeta(\tfrac{1}{2} + i\gamma_n) = 0$ is linearly independent over ${\mathbb Q}$.
\begin{theorem}\label{thm:pivsli}
Let the set $\Pi$ and the number $\Delta=\delta(\Pi)$ be defined as in equations~\eqref{Pi defn} and~\eqref{Delta defn}, respectively. Assuming RH and LI, the discrete logarithmic density of $\Pi$ relative to the primes is $\delta^*(\Pi) = \Delta$.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:pivsli} via Theorem \ref{thm:density}]
Consider the normalized error function
$$
E_{\pi}(x)=\frac{\log x}{\sqrt{x}}(\pi(x) - \textnormal{li}(x)),
$$
and note that $\Pi = \{x\colon \pi(x) > \textnormal{li}(x)\} = \mathcal M_0(E_{\pi})$. It thus suffices to show that $E_{\pi}$ satisfies conditions~(a) and~(b) of Theorem~\ref{thm:density}.
Consider any number $z\in[x,x+x^{1/3}]$. We have $\pi(z)-\pi(x) \le x^{1/3}$ and $\textnormal{li}(z) - \textnormal{li}(x) \le x^{1/3}$ trivially, and hence $|E_{\pi}(z)-E_{\pi}(x)| \le (2\log x)/{x^{1/6}}$. Since the right-hand side tends to $0$, this inequality easily implies condition~(a) of the theorem.
Moreover, condition (b)---namely the fact that $E_{\pi}$ has a continuous limiting logarithmic distribution---is a consequence (under RH and LI) of the work of Rubinstein and Sarnak: first, they establish a formula for the Fourier transform of this limiting logarithmic distribution (see~\cite[equation~(3.4) and the paragraph following]{RubSarn}). They then argue that this Fourier transform is rapidly decaying (see~\cite[Section 2.3]{FM} for a more explicit version of their method). From this they conclude that the distribution itself is continuous (and indeed much more, namely that it corresponds to an analytic density function---see~\cite[Remark 1.3]{RubSarn}).
\end{proof}
The continuity of the limiting logarithmic distribution of $E_{\pi}$ can be deduced from a substantially weakened version of LI: indeed, we only require the imaginary part of one nontrivial zero of $\zeta(s)$ to not be a rational linear combination of other such imaginary parts (see~\cite[Theorem 2.2(2)]{Devin}).
In the next section we prove Theorem \ref{thm:density}, which will complete the proof of Theorem~\ref{thm:pivsli}.
\section{Proof of Theorem \ref{thm:density}}
We begin with some notation. For any interval $I$ of real numbers, let $\pi(I)$ denote the number of primes in~$I$. For any positive real number $y$, define the half-open interval $I(y):=(y,y+y^{1/3}]$. Define an increasing sequence of real numbers recursively by $y_1=1$ and $y_{k+1} = y_k+y_k^{1/3}$ for $k\ge1$, and let $I_k:=I(y_k) = (y_k,y_{k+1}]$. We have thus partitioned ${\mathbb R}_{>1} = \bigcup_{k=1}^\infty I_k$ into a disjoint union of short half-open intervals.
\begin{lemma} \label{yk sum lemma}
For any fixed real number $\alpha>\frac23$, we have $\sum_{k=1}^\infty y_k^{-\alpha} \ll 1$.
\end{lemma}
\begin{proof}
For any $U\ge1$, the number of integers $k$ such that $y_k\in[U,2U)$ is at most $U^{2/3}$, since the length of each corresponding interval $I_k$ is at least $U^{1/3}$. Therefore
\[
\sum_{k=1}^\infty y_k^{-\alpha} = \sum_{j=0}^\infty \sum_{k\colon y_k\in[2^j,2^{j+1})} y_k^{-\alpha} \le \sum_{j=0}^\infty (2^j)^{2/3} (2^j)^{-\alpha} = \frac1{1-2^{2/3-\alpha}},
\]
since we assumed $\alpha>\frac23$.
\end{proof}
Given $\epsilon>0$, we say that an interval $I(y)$ is $\epsilon$-good if
\[
\bigg|\pi(I(y)) - \frac{y^{1/3}}{\log y} \bigg| \le \frac{\epsilon y^{1/3}}{\log y},
\]
and otherwise we say that $I(y)$ is $\epsilon$-bad. Selberg~\cite{Selb} showed
that there exists a set $\mathcal S\subset {\mathbb R}_{\ge1}$ whose natural density equals $1$ for which
\begin{align}
\label{eq:sel}
\pi(y+y^{1/3}) - \pi(y) \sim \frac{y^{1/3}}{\log y} \quad \text{for all } y\in \mathcal S.
\end{align}
This implies that for any $\epsilon>0$, the set of real numbers $y$ for which $I(y)$ is $\epsilon$-bad has density~$0$.
(Selberg \cite{Selb} proved \eqref{eq:sel} where the exponent ``$1/3$" is permitted to be any constant in $(19/77,1]$.
Selberg's
theorem has been subsequently improved: from Huxley \cite{H}, one may take the exponent in \eqref{eq:sel}
as any number in $(1/6,1]$,
see \cite[(1.3)]{K}.)
Our next lemma shows that $\epsilon$-bad intervals among the $I_k$ are also sparse.
\begin{lemma}\label{lemma:bad}
For each $\epsilon > 0$, the union of the $\epsilon$-bad intervals $I_k$ has natural density~$0$, and hence logarithmic density~$0$.
\end{lemma}
\begin{proof}
For every $k\ge1$, define $J_k := (y_k, y_k + \frac\epsilon{14} y_k^{1/3}]$.
Suppose that $k\ge1$ is chosen so that $I_k$ is an $\epsilon$-bad interval. Note that for all $y \in J_k$, the intervals $I(y)$ and $I_k = I(y_k)$ have nearly the same number of primes; more precisely,
\begin{equation} \label{lil pi difference}
\pi(I(y)) - \pi(I_k) = \pi\big((y_{k+1},y + y^{1/3}]\big) - \pi\big((y_k,y]\big),
\end{equation}
since the primes in the larger interval $(y,y_{k+1}]$ cancel in the difference.
By Titchmarsh's inequality~\cite[equation~(1.12)]{MV}, we have $\pi(I) \le 2h/\log h$ for all intervals $I$ of length $h>1$;
and since $2h/\log h$ is an increasing function of $h$ for $h>e$, we deduce that for any interval $I$ of length at most $\tfrac\epsilon{13} y_k^{1/3}$,
\[
\pi(I) \le \frac{2\cdot \tfrac\epsilon{13} y_k^{1/3}}{\log(\tfrac\epsilon{13} y_k^{1/3})} < \frac\epsilon2 \frac{y_k^{1/3}}{\log y_k}
\]
when $k$ is sufficiently large in terms of~$\epsilon$. (This deduction assumed that the length of $I$ exceeds $e$, but the final inequality is trivial for large $k$ when the length of $I$ is at most~$e$.)
In particular, both intervals on the right-hand side of equation~\eqref{lil pi difference} have length at most $\tfrac\epsilon{13} y_k^{1/3}$ when $k$ is sufficiently large, from which we see that $\big| \pi(I(y)) - \pi(I_k) \big| \le \frac\epsilon2 {y_k^{1/3}}/\log y_k$. Consequently, since $I_k$ is $\epsilon$-bad,
we conclude that
\begin{align*}
\bigg|\pi(I(y)) - \frac{y^{1/3}}{\log y} \bigg| &\ge \bigg|\pi(I_k) - \frac{y_k^{1/3}}{\log y_k} \bigg| - \big|\pi(I(y)) - \pi(I_k) \big| - \bigg| \frac{y^{1/3}}{\log y} - \frac{y_k^{1/3}}{\log y_k} \bigg| \\
&\ge \frac{\epsilon y_k^{1/3}}{\log y_k} - \frac\epsilon2 \frac{y_k^{1/3}}{\log y_k} + o(1) > \frac\epsilon3 \frac{y^{1/3}}{\log y}
\end{align*}
when $k$ is sufficiently large (where the mean value theorem was used in the middle inequality).
In other words, we have shown that $I_k$ being $\epsilon$-bad implies that $I(y)$ is $\frac\epsilon3$-bad for all $y\in J_k$.
Let $J$ be the (disjoint) union of all the intervals $J_k$, where $k$ ranges over those positive integers for which $I_k$ is $\epsilon$-bad.
By the result of Selberg described above, the set of $\frac\epsilon3$-bad real numbers (which contains $J$) has density~$0$, so $J\cap[1,x]$ has measure~$o(x)$. But this measure is at least $\frac\epsilon{14}$ times the measure of the union of all $\epsilon$-bad intervals~$I_k$; hence, the union of these intervals below $x$ also has measure~$o(x)$, which completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:density}]
For the sake of simplicity, we abbreviate $\mathcal M_a(f)$ to $\mathcal M_a$ during this proof.
Let $\epsilon$ and $\eta$ be positive parameters, and let $\mathcal B_\epsilon$ denote the union of all $\epsilon$-bad intervals of the form $I_k$, so that $\mathcal B_\epsilon$ has logarithmic density $0$ by Lemma~\ref{lemma:bad}.
Suppose that $I(y)$ is any $\epsilon$-good interval. Since $\int_{I(y)} dt/t = \int_y^{y+y^{1/3}} dt/t = \log(1+y^{-2/3}) = y^{-2/3}+O(y^{-4/3})$, we see that
\begin{equation} \label{good loggy bound}
\sum_{p\in I(y)}\frac{\log p}{p} \le \frac{\log y}{y}\pi(I(y)) \le (1+\epsilon)y^{-2/3} = (1+\epsilon)\int_{I(y)} \frac{dt}{t} + O(y^{-4/3}),
\end{equation}
where the second inequality used the $\epsilon$-goodness of~$I(y)$. On the other hand, even if $I(y)$ is an $\epsilon$-bad interval, Titchmarsh's inequality still yields
\begin{equation} \label{bad loggy bound}
\sum_{p\in I(y)}\frac{\log p}{p} \le \frac{\log y}{y}\pi(I(y)) \ll \frac{\log y}{y} \frac{y^{1/3}}{\log(y^{1/3})} \ll y^{-2/3} \ll \int_{I(y)} \frac{dt}{t}.
\end{equation}
By condition (a) of Theorem \ref{thm:density}, there exists a positive integer $C$ (depending on $a$ and~$\eta$) such that if $p$ is a prime in an interval $I_k$ with $k>C$, then the inequality $f(p)>a$ implies that $f(z) > a-\eta$ for all $z\in I_k$. In particular, every $I_k$ containing a prime $p$ with $f(p)>a$ is either a subset of $\mathcal B_\epsilon$ or else is an $\epsilon$-good interval contained in $\mathcal M_{a-\eta}$, so that
\begin{align*}
\sum_{\substack{p\le x\\f(p)>a}}\frac{\log p}{p}\le \sum_{\substack{I_k\subset \mathcal M_{a-\eta}\cap[1,x] \\ I_k \text{ is $\epsilon$-good}}}\sum_{p\in I_k}\frac{\log p}{p} + \sum_{I_k\subset \mathcal B_\epsilon\cap[1,x]}\sum_{p\in I_k}\frac{\log p}{p}.
\end{align*}
Using equation~\eqref{good loggy bound} for the terms in the first sum and equation~\eqref{bad loggy bound} for the second sum, we obtain the upper bound
\begin{align*}
\sum_{\substack{p\le x\\f(p)>a}}\frac{\log p}{p} &\le (1+\epsilon)\int_{\mathcal M_{a-\eta}\cap[1,x]} \frac{dt}{t} + O\bigg(\sum_{y_k\le x}y_k^{-4/3}\bigg) + O\bigg( \int_{\mathcal B_\epsilon\cap[1,x]} \frac{dt}{t} \bigg) \\
& \le (1+\epsilon)\int_{\mathcal M_{a-\eta}\cap[1,x]} \frac{dt}{t} + O(1) + o(\log x)
\end{align*}
by Lemma~\ref{yk sum lemma} and the fact that $\mathcal B_\epsilon$ has logarithmic density~$0$.
Therefore we have
\begin{align}\label{eq:over}
\overline{\delta}^*(\mathcal M_a) &= \limsup_{x\to\infty}\frac{1}{\log x}\sum_{\substack{p\le x\\f(p)>a}}\frac{\log p}{p} \\
&\le \limsup_{x\to\infty} \bigg( \frac{(1+\epsilon)}{\log x}\int_{\mathcal M_{a-\eta}\cap[1,x]} \frac{dt}{t} + o(1) \bigg) = (1+\epsilon)\delta(\mathcal M_{a-\eta})
\end{align}
since $\delta(\mathcal M_{a-\eta})$ exists by condition~(b) of Theorem \ref{thm:density}.
Similarly, the primes in $\mathcal M_a$ that are contained in $\epsilon$-good intervals $I_k\subset \mathcal M_a$ form a subset of all primes in $\mathcal M_a$. Then for a lower bound, it suffices to consider the $\epsilon$-good intervals in $\mathcal M_a$, which by a simple computation gives the bound $\underline{\delta}^*(\mathcal M_a) \ge (1-\epsilon)\delta(\mathcal M_a)$.
Since these bounds hold for all $\epsilon>0$, we see that
\begin{align}
\delta(\mathcal M_a) \le \underline{\delta}^*(\mathcal M_a) \le \overline{\delta}^*(\mathcal M_a) \le \delta(\mathcal M_{a-\eta}).
\end{align}
Finally, by condition (b) the map $\eta\mapsto \delta(\mathcal M_{a-\eta})$ is continuous, so since $\eta>0$ was arbitrary we conclude that $\underline{\delta}^*(\mathcal M_a) = \overline{\delta}^*(\mathcal M_a)= \delta(\mathcal M_a)$ as desired.
\end{proof}
\section{The Mertens race}
\label{sec:mertens}
In 1874, Mertens proved three remarkable and related
results on the distribution of prime numbers. His third theorem asserts that
\begin{align*}
\prod_{p < x}\Big(1 - \frac{1}{p}\Big)^{-1} \sim e^\gamma \log x \quad\textnormal{as }x\to\infty,
\end{align*}
where $\gamma$ is the Euler--Mascheroni constant. The ``Mertens race'' between $e^\gamma \log x$ and this product of Mertens is mathematically analagous to the race between $\textnormal{li}(x)$ and $\pi(x)$. Recent analysis of Lamzouri \cite{Lamz} implies, conditionally on RH and LI, that the normalized error function
\begin{align*}
E_M(x) = \bigg( \log \prod_{p < x}\Big(1 - \frac{1}{p}\Big)^{-1} - \log\log x -\gamma \bigg) \sqrt{x}\log x
\end{align*}
possesses the exact same limiting distribution as that of
\begin{align*}
-E_{\pi}(x) = \frac{\log x}{\sqrt{x}}\Big(\textnormal{li}(x) - \pi(x) \Big)
\end{align*}
that appeared in the proof of Theorem~\ref{thm:pivsli}.
We say a prime $p$ is \textbf{Mertens} if $E_M(p) > 0$. It can be checked that the first $10^8$ odd primes are Mertens. The first and third authors have shown~\cite[Theorem 1.3]{LP}, assuming RH and LI, that the lower relative logarithmic density of the Mertens primes exceeds $.995$. Applying Theorem \ref{thm:density} to the Mertens race by choosing $f(x) = E_M(x)$ leads immediately to the following improvement.
\begin{theorem}\label{thm:mertens}
Assuming RH and LI, the Mertens primes have relative logarithmic density $1-\Delta$, where $\Delta$ was defined in equation~\eqref{Delta defn}.
\end{theorem}
\begin{proof}
Consider any number $z\in[x,x+x^{1/3}]$. We have $\log\log z - \log \log x \ll 1/(x\log x)$ by the mean value theorem and
\begin{align*}
\sum_{x \le p < z}\log\Big(1 - \frac{1}{p}\Big)^{-1} \ll \sum_{x \le p < z} \frac1p \le \sum_{x\le n< x+x^{1/3}} \frac1n < x^{-2/3},
\end{align*}
and so $|E_M(z)-E_M(x)| \ll (\log x)/x^{1/6}$. The fact that this upper bound tends to $0$ easily implies condition~(a) of Theorem \ref{thm:density}. Finally, condition (b) is satisfied by a similar argument as in the proof of Theorem~\ref{thm:pivsli}, using work of Lamzouri~\cite{Lamz} on the limiting logarithmic distribution of $E_M(x)$.
\end{proof}
Theorem \ref{thm:mertens} has an interesting application to the Erd\H os conjecture for primitive sets.
A subset of the integers larger than $1$ is {\bf primitive} if no member divides another. Erd\H os \cite{E35} proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$
is universally bounded over all choices for $A$. Some years later in a 1988 seminar in Limoges, he asked if this universal bound is
attained for the set of prime numbers.
If we define $f(a)=1/(a\log a)$ and $f(A)=\sum_{a\in A}f(a)$, and let
$\mathcal{P}(A)$ denote the set of primes that divide some member of~$A$, then this conjecture is seen
to be equivalent to the following assertion.
\begin{conjecture}[Erd\H os]\label{conj:Erdos}
For any primitive set $A$, we have
$f(A) \le f(\mathcal{P}(A))$.
\end{conjecture}
The Erd\H{o}s conjecture remains open, but progress has been made in certain cases. Say a
prime $p$ is {\bf Erd\H os-strong} if $f(A)\le f(p)$ for any primitive set $A$ such that each
member of $A$ has $p$ as its least prime factor. By partitioning the elements of $A$ into sets $A'_p$ by their smallest prime factor $p$, it is clear that the Erd\H{o}s conjecture would follow if every
prime $p$ is Erd\H os-strong. The first and third authors~\cite[Corollary 3.0.1]{LP}
proved that every Mertens prime is Erd\H{o}s-strong.
In particular, the Erd\H{o}s conjecture holds for any primitive set $A$ such that, for all $a\in A$, the smallest prime factor of $a$ is Mertens.
In~\cite{LP} it was conjectured that all primes are Erd\H{o}s-strong. Since $2$ is not a Mertens prime, it would be great progress just to be able to prove that $2$ is Erd\H{o}s-strong. Nevertheless, Theorem \ref{thm:mertens} implies the following corollary.
\begin{corollary}
Assuming RH and LI, the lower relative logarithmic density of the Erd\H{o}s-strong primes is at least $1-\Delta$. In particular, the Erd\H os conjecture holds for all primitive sets whose elements have smallest prime factors in a set of primes of lower relative logarithmic density at least $1-\Delta$.
\end{corollary}
\section{The Zhang race}
By the prime number theorem, one has the asymptotic relation
\begin{align*}
\sum_{p\ge x}\frac{1}{p\log p} \sim \frac{1}{\log x}
\end{align*}
as $x\to\infty$,
and by inspection one further has
\begin{align}\label{eq:Zhang}
\sum_{p\ge x}\frac{1}{p\log p}\le \frac{1}{\log x}
\end{align}
for a large range of $x$.
Beyond its aesthetic appeal, this inequality arises quite naturally in the study of primitive sets. Indeed, Z.~Zhang~\cite{Z} used a weakened version of \eqref{eq:Zhang} to prove Conjecture~\ref{conj:Erdos} for all primitive sets whose elements have at most 4 prime factors, which represented the first significant progress in the literature after \cite{E35}.
Call a prime $q$ {\bf Zhang} if the inequality~\eqref{eq:Zhang} holds for $x=q$. From computations in \cite{LP}, the first $10^8$ primes are all Zhang except for $q=2,3$. Following some ideas of earlier work of Erd\H os and Zhang \cite{EZ}, the first and third authors have shown~\cite[Theorem 5.1]{LP} that Conjecture~\ref{conj:Erdos}
holds for any primitive set
$A$ such that every member of $\mathcal{P}(A)$ is Zhang.
We wish to find the density of $\mathcal N$, the set of real numbers for which the Zhang inequality~\eqref{eq:Zhang} holds. Note that $x\in\mathcal N$ if and only if the normalized error
\begin{align} \label{EZ defn}
E_Z(x) := \bigg(\frac{1}{\log x} - \sum_{p\ge x}\frac{1}{p\log p}\bigg)\sqrt{x}\log^2 x
\end{align}
is nonnegative. To show the density of $\mathcal N$ exists we follow the general plan laid out by Lamzouri \cite{Lamz}, who proved analogous results for the Mertens race, with some important modifications.
\subsection{Explicit formula for $E_Z(x)$}
First we relate the sum over primes, $\sum_{p\ge x} 1/(p\log p)$, to the corresponding series over prime powers, $\sum_{n\ge x} \Lambda(n)/(n\log^2 n)$, in the following lemma.
\begin{lemma} \label{lem:Ep}
For all $x>1$,
\[
E_Z(x) = \bigg(\frac{1}{\log x} - \sum_{n\ge x}\frac{\Lambda(n)}{n\log^2 n}\bigg)\sqrt{x}\log^2 x + 1 + O\Big(\frac{1}{\log x}\Big).
\]
\end{lemma}
\begin{proof}
Our first step is to convert the sum over primes to prime powers, via
\begin{align} \label{here's Lambda}
\sum_{p\ge x}\frac{1}{p\log p} & = \sum_{n\ge x}\frac{\Lambda(n)}{n\log^2 n} - \sum_{\substack{p^k> x\\ k\ge2}}\frac{1}{k^2 p^k \log p}.
\end{align}
The prime number theorem gives that $\pi(y) = y/\log y + O(y/\log^2 y)$, so for any $y\ge2$,
\begin{align*}
\sum_{p\ge y}\frac{1}{p^2\log p} &
= -\frac{\pi(y)}{y^2\log y} + \int_{y}^\infty\frac{2\log t + 1}{t^3\log^2 t}\pi(t)\;dt\\
& = -\frac{1}{y\log^2 y} + O\bigg(\frac{1}{y\log^3 y} \bigg) + \int_{y}^\infty\frac{2\log t + 1}{t^2\log^3 t} \bigg( 1 + O\bigg(\frac1{\log t} \bigg)\bigg) \;dt \\
& = -\frac{1}{y\log^2 y} + \frac{2}{y\log^2 y} + O\bigg(\frac{1}{y\log^3 y}\bigg) = \frac{1}{y\log^2 y} + O\bigg(\frac{1}{y\log^3 y}\bigg).
\end{align*}
In particular, taking $y=\sqrt x$,
\begin{align} \label{squares}
\sum_{p^2>x}\frac{1}{4p^2\log p} & = \frac{1}{\sqrt{x}\log^2 x} + O\bigg(\frac{1}{\sqrt{x}\log^3 x}\bigg).
\end{align}
For the larger powers of primes, we have
\begin{align*}
\sum_{p^k> x}\frac{1}{p^k\log p} < \sum_{n> x^{1/k}} \frac1{n^k} < \frac1{\lceil x^{1/k} \rceil^k} + \int_{x^{1/k}}^\infty \frac{dt}{t^k} \le \frac1x + \frac1{(k-1)x^{1-1/k}} \ll x^{-2/3}
\end{align*}
uniformly for $k\ge3$, and thus
\begin{align} \label{higher powers}
\sum_{k\ge3}\sum_{p^k> x}\frac{1}{k^2p^k\log p} \ll \sum_{k\ge3}\frac{x^{-2/3}}{k^2} \ll x^{-2/3}.
\end{align}
Inserting the estimates~\eqref{squares} and~\eqref{higher powers} into equation~\eqref{here's Lambda} then yields
\begin{align*}
\sum_{p\ge x}\frac{1}{p\log p} & = \sum_{n\ge x}\frac{\Lambda(n)}{n\log^2 n} - \frac{1}{\sqrt{x}\log^2 x} + O\bigg(\frac{1}{\sqrt{x}\log^3 x}\bigg),
\end{align*}
which implies the statement of the lemma.
\end{proof}
By integrating twice, we relate our series $\sum \Lambda(n)/n\log^2 n$ to the series $\sum \Lambda(n)/n^a = -\zeta'/\zeta(a)$, which is more amenable to contour integration. This leads to the following explicit formula for $E_Z(x)$ over the zeros of $\zeta(s)$, analogous to \cite[Proposition 2.1]{Lamz}.
\begin{proposition}\label{prop:Ex}
Unconditionally, for any real numbers $x,T\ge 5$,
\begin{align*}
E_Z(x) & = 1 - \sum_{|\Im(\rho)|< T} \frac{x^{\rho-1/2}}{\rho-1} + O\bigg(\frac{1}{\log x} + \frac{\sqrt{x}}{T}\log^2(xT) + \frac{1}{\log x}\sum_{|\Im(\rho)|< T}\frac{x^{\Re(\rho)-1/2}}{\Im(\rho)^2}\bigg),
\end{align*}
where $\rho$ runs over the nontrivial zeros of $\zeta(s)$.
\end{proposition}
\begin{proof}
Our starting point is a tool from Lamzouri, namely \cite[Lemma 2.4]{Lamz}: for any real numbers $a>1$ and $x,T\ge5$,
\begin{align*}
\sum_{n< x}\frac{\Lambda(n)}{n^a} & = -\frac{\zeta'}{\zeta}(a) + \frac{x^{1-a}}{1-a} - \sum_{|\Im(\rho)|< T}\frac{x^{\rho-a}}{\rho-a}\\
& \qquad + O\bigg(x^{-a}\log x + \frac{x^{1-a}}{T}\Big(4^a + \log^2 x + \frac{\log^2 T}{\log x}\Big) + \frac{1}{T}\sum_n\frac{\Lambda(n)}{n^{a+1/\log x}} \bigg).
\end{align*}
Then integration with respect to $a$ gives for any $b>1$,
\begin{align*}
\sum_{n< x}\frac{\Lambda(n)}{n^b\log n} & = \int_b^\infty \sum_{n< x}\frac{\Lambda(n)}{n^a}\;da\\
& = \log\zeta(b) + \int_b^\infty \frac{x^{1-a}}{1-a}\;da - \sum_{|\Im(\rho)|< T}\int_b^\infty \frac{x^{\rho-a}}{\rho-a}da + E_1,
\end{align*}
where
\begin{align*}
E_1 \ll x^{-b} + \frac{x^{1-b}}{T}\Big(\frac{4^b}{\log x} + \log x + \frac{\log^2 T}{\log^2 x}\Big) + \frac{1}{T}\sum_n\frac{\Lambda(n)}{n^{b+1/\log x}\log n}.
\end{align*}
Integrating once again with respect to $b$, we have
\begin{align}\label{eq:doubleint}
\sum_{n< x}\frac{\Lambda(n)}{n\log^2 n} &= \int_1^\infty \sum_{n< x}\frac{\Lambda(n)}{n^b\log n}\;db\nonumber\\
&~ = \int_1^\infty \log\zeta(b)\;db + \int_1^\infty\int_b^\infty \frac{x^{1-a}}{1-a}\;da\;db - \sum_{|\Im(\rho)|< T} \int_1^\infty\int_b^\infty \frac{x^{\rho-a}}{\rho-a}\;da\;db + E_2,
\end{align}
where
\begin{align*}
E_2 & \ll \frac{1}{x\log x} + \frac{1}{T}\Big(\frac{4}{\log^2 x} + 1 + \frac{\log^2 T}{\log^3 x}\Big) + \frac{1}{T}\sum_n\frac{\Lambda(n)}{n^{1+1/\log x}\log^2 n} \\
& \ll \frac{1}{x\log x} + \frac{1}{T}\Big(1 + \frac{\log^2 T}{\log^3 x}\Big) + \frac{1}{T}\sum_n\frac{\Lambda(n)}{n\log^2 n} \ll \frac{1}{x\log x} + \frac{1}{T}\Big(1 + \frac{\log^2 T}{\log^3 x}\Big),
\end{align*}
since $\sum_n \Lambda(n)/(n\log^2n)\ll1$.
The first term on the right-hand side of equation~\eqref{eq:doubleint} can be written as
\begin{align} \label{first term}
\int_1^\infty \log\zeta(b)\;db = \int_1^\infty \sum_n\frac{\Lambda(n)}{n^b\log n}\;db = \sum_n \frac{\Lambda(n)}{n\log^2 n},
\end{align}
where the Fubini--Tonelli theorem justifies the interchange of summation and integration
since all terms are nonnegative.
The second term on the right-hand side of equation~\eqref{eq:doubleint} evaluates to
\begin{align}
\int_1^\infty\int_b^\infty \frac{x^{1-a}}{1-a}\;da\;db &= \int_1^\infty \frac{x^{1-a}}{1-a} \bigg( \int_1^a \;db \bigg) \;da \notag \\
&= -\int_1^\infty x^{1-a}\;da = \frac{x^{1-a}}{\log x}\bigg|_1^\infty =-\frac{1}{\log x}, \label{second term}
\end{align}
where the interchange of integrals is again justified by the Fubini--Tonelli theorem.
The double integral inside the series on the right-hand side of equation~\eqref{eq:doubleint} is evaluated using a similar calculation:
\begin{align*}
-\int_1^\infty\int_b^\infty \frac{x^{\rho-a}}{\rho-a}\;da\;db & = -\int_1^\infty\frac{a-1}{\rho-a}x^{\rho-a}\;da\\
& = \int_1^\infty x^{\rho-a}\;da - (\rho-1)\int_1^\infty\frac{x^{\rho-a}}{\rho-a}\;da.
\end{align*}
The first integral comes out to $x^{\rho-1}/\log x$, while for the second, integrating by parts twice gives
\begin{align*}
(\rho-1)\int_1^\infty\frac{x^{\rho-a}}{\rho-a}\;da = \frac{x^{\rho-1}}{\log x} + \frac{x^{\rho-1}}{(\rho-1)\log^2 x} + \frac{2(\rho-1)}{\log^2 x}\int_1^\infty\frac{x^{\rho-a}}{(\rho-a)^3}\;da.
\end{align*}
Letting $u = (a-1)\log x$, we have $a=1+u/\log x$ so the latter integral becomes
\begin{align*}
\frac{2(\rho-1)}{\log^2 x}\int_1^\infty\frac{x^{\rho-a}}{(\rho-a)^3}\;da =\frac{2 (\rho-1)x^{\rho - 1}}{\log^3x}\int_0^\infty \frac{e^{-u}}{(\rho -1- u/\log x)^3}\;du.
\end{align*}
Note that $|\rho-1 - u/\log x| \ge |\Im(\rho)|$ for all $u\in{\mathbb R}$, so we deduce
\begin{align*}
\bigg|\frac{2(\rho-1)}{\log^2 x}\int_1^\infty\frac{x^{\rho-a}}{(\rho-a)^3}\;da\bigg| \ll \frac{x^{\Re(\rho)-1}}{\Im(\rho)^2\log^3 x}.
\end{align*}
Thus we have
\begin{align}
-\int_1^\infty\int_b^\infty \frac{x^{\rho-a}}{\rho-a}da\;db & = \frac{x^{\rho-1}}{\log x} - \bigg(\frac{x^{\rho-1}}{\log x} + \frac{x^{\rho-1}}{(\rho-1)\log^2 x} + O\bigg(\frac{x^{\Re(\rho)-1}}{\Im(\rho)^2\log^3 x}\bigg) \bigg) \nonumber\\
& = -\frac{x^{\rho-1}}{(\rho-1)\log^2 x} + O\bigg(\frac{x^{\Re(\rho)-1}}{\Im(\rho)^2\log^3 x}\bigg). \label{third term}
\end{align}
The calculations~\eqref{first term}, \eqref{second term}, and~\eqref{third term} transform equation~\eqref{eq:doubleint} into
\begin{align*}
\sum_{n< x}\frac{\Lambda(n)}{n\log^2 n} & = \sum_n \frac{\Lambda(n)}{n\log^2 n} -\frac{1}{\log x} - \frac{1}{\log^2 x}\sum_{|\Im(\rho)|< T}\frac{x^{\rho-1}}{\rho-1}\\
&\qquad\qquad + O\bigg(\frac{1}{x\log x} + \frac1T\bigg(1+\frac{\log^2 T}{\log^3 x}\bigg)+ \frac{1}{\log^3 x}\sum_{|\Im(\rho)|< T}\frac{x^{\Re(\rho)-1}}{\Im(\rho)^2}\bigg)
\end{align*}
and thus
\begin{align*}
\frac1{\log x}- \sum_{n\ge x}\frac{\Lambda(n)}{n\log^2 n} & = - \frac{1}{\log^2 x}\sum_{|\Im(\rho)|< T}\frac{x^{\rho-1}}{\rho-1}\\
&\quad\qquad + O\bigg(\frac{1}{x\log x} + \frac{1+\log^2 T/\log^3 x}{T} + \frac{1}{\log^3 x}\sum_{|\Im(\rho)|< T}\frac{x^{\Re(\rho)-1}}{\Im(\rho)^2}\bigg).
\end{align*}
The proposition now follows upon comparing this formula to Lemma~\ref{lem:Ep}.
\end{proof}
If we assume the Riemann hypothesis we obtain the following corollary, analogous to \cite[Corollary 2.2]{Lamz}.
\begin{corollary}\label{cor:Ex}
Assume RH, and let $\tfrac{1}{2} + i\gamma$ run over the nontrivial zeros of $\zeta(s)$ with $\gamma>0$. Then, for any real numbers $x, T \ge 5$ we have
\begin{align}
E_Z(x) & = 1 - 2\Re\sum_{0<\gamma< T} \frac{x^{i\gamma}}{-1/2+i\gamma} + O\bigg(\frac{1}{\log x} + \frac{\sqrt{x}}{T}\log^2(xT)\bigg),
\end{align}
\end{corollary}
\begin{proof}
By the Riemann--von Mangoldt formula,
\begin{align*}
\bigg|\sum_{|\gamma|< T}\frac{x^{i\gamma}}{\gamma^2}\bigg| \le \sum_{|\gamma|<T}\frac{1}{\gamma^2} \ll 1,
\end{align*}
so the corollary now follows from Proposition~\ref{prop:Ex}.
\end{proof}
\subsection{Density Results}
Since the explicit formula for the Zhang primes in Corollary \ref{cor:Ex} is exactly the same as that of the Mertens primes given by Lamzouri (upon noting a typo in~\cite[Corollary 2.2]{Lamz}, namely, that ``$E_M(x) = 1+ \cdots$'' should read ``$E_M(x) = 1 - \cdots$''), the analysis therein leads to the following results. Recall that $\mathcal N$ is the set of real numbers for which the Zhang inequality~\eqref{eq:Zhang} holds, and that $E_Z(x)$ is defined in equation~\eqref{EZ defn}.
\begin{theorem}\label{thm:E_Z}
Assume RH. Then
\begin{align*}
0 < \underline \delta(\mathcal N) \le \overline \delta(\mathcal N) < 1.
\end{align*}
Moreover, $E_Z(x)$ possesses a limiting distribution $\mu_N$, that is,
\begin{align*}
\lim_{x\to\infty}\frac{1}{\log x}\int_2^x f(E_Z(t))\;dt = \int_{\mathbb R} f(t)\; d\mu_N(t)
\end{align*}
for all bounded continuous functions $f$ on ${\mathbb R}$.
\end{theorem}
\begin{proposition}
Assume RH and LI. Let $X(\gamma)$ be a sequence of independent random variables, indexed by the positive imaginary parts of the non-trivial zeros of $\zeta(s)$, each of which is uniformly distributed on the unit circle. Then $\mu_N$ is the distribution of the random variable
\begin{align}
Y = 1 - 2\Re \sum_{\gamma > 0}\frac{X(\gamma)}{\sqrt{1/4+\gamma^2}}.
\end{align}
\end{proposition}
\begin{theorem} \label{Zhang primes density thm}
Assume RH and LI. Then $\delta(\mathcal N)$ exists and equals $1-\Delta$. Hence by Theorem \ref{thm:density}, the relative logarithmic density of the Zhang primes is $1-\Delta$.
\end{theorem}
These results are completely analogous to Theorems 1.1 and 1.3 and Propositions 4.1 and 4.2 from~\cite{Lamz}.
Before moving on, we note a further consequence of the fact that $E_M(x)$ and $E_Z(x)$ possess the same explicit formula, namely that the symmetric difference of Mertens primes and Zhang primes has relative logarithmic density~$0$.
\begin{corollary}
Assume RH and LI. Then we have $\delta(S) = \delta^*(S) = 0$ for the symmetric difference
$S=S_1\cup S_2$, where
$$
S_1=\{x:E_M(x)>0\ge E_Z(x)\}\quad\text{and}\quad S_2=\{x:E_Z(x)>0\ge E_M(x)\}.
$$
\end{corollary}
\begin{proof}
Take $\eta > 0$.
Combining \cite[Corollary 2.2]{Lamz} with Corollary \ref{cor:Ex} and letting $T$ tend to infinity, we find that
\begin{align}\label{eq:symdiff}
\big|E_M(x) - E_Z(x) \big|= O\Big(\frac{1}{\log x}\Big).
\end{align}
Let $c$ be the implied constant in equation~\eqref{eq:symdiff}. Thus for all $x \ge e^{c/\eta}$, if $E_M(x) > 0$ then $E_Z(x) > -\eta$. This means that
\begin{align*}
\delta^*(S_1) & \le \delta^*(\{x : E_Z(x) > 0\}) - \delta^*(\{x : E_Z(x) > -\eta\})\\
& = \delta(\{x : E_Z(x) > 0\}) - \delta(\{x : E_Z(x) > -\eta\}),
\end{align*}
which tends to 0 as $\eta \to 0$ by continuity, using Theorem \ref{thm:density} and Theorem \ref{thm:E_Z}. Since this holds for all $\eta >0$, we conclude that $\delta^*(S_1) = 0$. Interchanging the roles of $E_M$ and $E_Z$ proves $\delta^*(S_2) = 0$, and thus $\delta^*(S) = \delta^*(S_1\cup S_2)=0$. A similar argument (simpler even, without the appeal to Theorem \ref{thm:density}) shows that $\delta(S)=0$.
\end{proof}
We also remark, however, that the analogous argument does not work for $E_\pi$. This is because the relevant series over nontrivial zeros is $\sum_{\rho}x^{\rho-1}/(\rho-1)$ for $E_M$ and $E_Z$, while for $E_\pi$ it is $\sum_{\rho}x^\rho/\rho$. Assuming RH, this amounts to the observation that the two series $\sum_{\gamma}x^{i\gamma}/(-1/2+i\gamma)$
and $\sum_{\gamma}x^{i\gamma}/(1/2+i\gamma)$ are not readily comparable for a given $x$---even though, by symmetry, both do possess the same limiting distribution, which explains the appearance of $\delta(\Pi)=\Delta$ in results on the Mertens and Zhang races.
The analogous problem of determining the density of the symmetric difference between the Mertens/Zhang primes and the $\textnormal{li}$-beats-$\pi$ primes is an interesting problem for further investigation; it would presumably proceed by examining the two-dimensional limiting distribution of the ordered pair of normalized error terms, and understanding how the correlations of the two functions' summands impacts the two-dimensional limiting distribution.
\section{Other series over prime numbers}
Before concluding our analysis, we remark that similar considerations apply more generally to series of the form $\sum_{p}p^\alpha (\log p)^{k+1}$, where $k\in{\mathbb Z}$ and $\alpha\in{\mathbb R}$. The basic approach is to first relate the sum of interest to the corresponding sum over prime powers via
\begin{align*}
\sum_{p}\frac{\log p}{p^\alpha}(\log p)^k = \sum_{n} \frac{\Lambda(n)}{n^\alpha}(\log n)^k - \sum_{p^m, m\ge 2} \frac{m^k(\log p)^{k+1}}{p^{m\alpha}}.
\end{align*}
The next step is to employ an exact formula relating the sum over prime powers to series over zeros of $\zeta(s)$. For example, von Mangoldt's exact formula states that
\begin{align}
\sum_{n\le x}\Lambda(n) = x - \frac{\zeta'}{\zeta}(0) - \sum_{\rho}\frac{x^{\rho}}{\rho} + \sum_{m\ge1} \frac{x^{-2m}}{-2m}
\end{align}
provided $x$ is not a prime power. The above formula naturally generalizes to any real exponent $\alpha$. Namely, one may prove by Perron's formula (c.f. \cite[Lemma 2.4]{Lamz}) that
\begin{align}
\sum_{n\le x}\frac{\Lambda(n)}{n^\alpha} = \frac{x^{1-\alpha}}{1-\alpha} - \frac{\zeta'}{\zeta}(\alpha) - \sum_{\rho}\frac{x^{\rho-\alpha}}{\rho-\alpha} + \sum_{m\ge1} \frac{x^{-2m-\alpha}}{-2m-\alpha}.
\end{align}
provided $x$ is not a prime power, and $\alpha$ is neither 1 nor a negative even integer. Note that when $\alpha>1$, we have $-(\zeta'/\zeta)(\alpha) = \sum_n \Lambda(n)/n^\alpha$ so we may simplify the above as
\begin{align}
\sum_{n\ge x}\frac{\Lambda(n)}{n^\alpha} = -\frac{x^{1-\alpha}}{1-\alpha} + \sum_{\rho}\frac{x^{\rho-\alpha}}{\rho-\alpha} - \sum_{m\ge1} \frac{x^{-2m-\alpha}}{-2m-\alpha}.
\end{align}
To gain factors of $\log n$ in the numerator, we differentiate with respect to $\alpha$. Specifically, since $d/d\alpha[x^{c-\alpha}/(c-\alpha)] = (-\log x + 1/(c-\alpha))x^{c-\alpha}/(c-\alpha)$ for any $c\in{\mathbb C}$, by induction one may show
$$\frac{d^k}{d\alpha^k}\Big[\frac{x^{c-\alpha}}{c-\alpha}\Big] = (-\log x)^k\bigg(\frac{x^{c-\alpha}}{c-\alpha} + O_k\Big(\frac1{\log x}\Big)\bigg)$$
This implies, for all $k\ge1$,
\begin{align*}
& \sum_{n\ge x}\frac{\Lambda(n)}{n^\alpha}(\log n)^k \\
&~ = (\log x)^k\bigg(-\frac{x^{1-\alpha}}{1-\alpha} + \sum_{\rho}\frac{x^{\rho-\alpha}}{\rho-\alpha} - \sum_{m\ge1} \frac{x^{-2m-\alpha}}{-2m-\alpha}\bigg)\Big( 1 + O_k\Big(\frac{1}{\log x} \Big)\Big).
\end{align*}
Similarly for integration, we have
$$\int_\alpha^\infty \frac{x^{c-\beta}}{c-\beta}\;d\beta = \textnormal{li}(x^{c-\alpha}) = (1 + O(1/\log x))\frac{x^{c-\beta}}{c-\beta},$$
so an induction argument will establish the exact formula for negative integers $k$.
From here, all that remains is to analyze the sum over nontrivial zeta zeros. Assuming RH, it suffices to consider the series $\sum_{\gamma}x^{i\gamma}/(1/2-\alpha+i\gamma)$. Further assuming LI, this series has a limiting distribution, which may be computed explicitly (as in \cite{ANS,RubSarn}).
\section*{Acknowledgments}
We are grateful for a helpful discussion with Dimitris Koukoulopoulos.
The first-named author thanks the office for undergraduate research at Dartmouth College, and is
currently supported by a Churchill Scholarship at the University of Cambridge.
The second-named author was supported in part by a National Sciences and Engineering Research Council of Canada Discovery Grant. The second- and third-named authors are grateful to the Centre de Recherches Math\'ematiques for
their hospitality in May, 2018 when some of the ideas in this paper were discussed.
\bibliographystyle{amsplain}
|
1809.03320
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\section{Introduction}
One of the most important discoveries in cosmology is the fact that the Universe is expanding at an accelerating rate. Evidence of this accelerating expansion has been observed by the astronomical observations such as Cosmic Microwave Background (CMB) radiation \cite{1}, Supernova Type Ia \cite{2} and large scale structure data \cite{3}. According to Einstein's theory of general relativity and Friedmann cosmology, there exists some matter with a negative pressure and its absolute value is somehow comparable to the energy density (with standard units $c=G=1$) \cite{Babichev}. In fact this matter is called dark energy and is being considered as the mainstream to the modifications of Einstein's theory as it will lead to the violation of the Weak Equivalence Principle. However, the physical origin of this energy is still unknown but many astronomical observations suggest that two thirds of the whole energy of the Universe belongs to the dark energy. We model this dark energy using a perfect fluid with the equation of state $p=k e$ where $k$ is the state parameter ($k=-1$) and $e$ is the energy density.
In the literature there is a huge list of proposed models for dark energy. According to recent observations the cosmological term requires very minute value of the energy density in the vacuum which eventually demands a very large parameter to fit in the field theory. For this reason many other models with $k\neq-1$ have been proposed. For instance, we have phantom energy: a matter whose state parameter is negative ($k<-1$) \cite{4,5,6}. Different aspects of phantom cosmology have already been discussed in the literature \cite{7,8}. One of the exceptional hypothesis in cosmology is the Big Rip scenario \cite{5} which predicts that due to the expansion of the Universe all the matter will tear apart at a finite time interval. Hence this scenario could also justify another form of energy called phantom energy.
Accretion is a process by which a gravitating object such as a black hole or a massive star can capture particles from its vicinity which eventually leads to a change in the physical properties of the accreting body \cite{9,10,11,12}. It is considered as one of the most pervasive process in the Universe. In fact the supermassive black holes at the center of galaxies suggest that black holes could have evolved through the accretion process. However, an accretion process does not always increase the mass of the compact object but it could decrease it (a) when the infalling matter is thrown out in the form of jets or cosmic rays or if the infalling gas is a phantom matter~\cite{Babichev2,Jamil}, or (b) if the compact object is a superspinar, say a naked singularity, where the decrease of mass may possibly occur when matter from orbits with negative energy plunges into the singularity as was noticed for the Kerr naked singularity~\cite{super1,super2}. In the literature there is not any substantial contribution regarding exotic matter. Since our Universe is highly dominated by dark energy and phantom energy therefore, it is more interesting to study the accretion of such energies onto black holes and the accretion of ordinary and phantom matter onto phatom black holes.
The first study on accretion process in the Newtonian framework was done by Bondi in 1952 \cite{Bondi}. Through the evolution of Einstein's theory of gravitation, Michel was the one who investigated the gas accretion onto Schwarzschild black hole in relativistic framework \cite{Michel}. Babichev et al. discussed the stationary black hole in the phantom energy dominated Universe \cite{Babichev}. They have found that phantom energy and dark energy will decrease the mass of the black hole. The effects of phantom accretion and Chaplygin gas were investigated onto the charged black hole by Jamil et al. \cite{Jamil}. Debnath further generalizes the above idea and presented a general framework of a static accretion process onto static, spherically symmetric black holes \cite{Debnath}. Moreover, the accretion process of a spherically symmetric spacetime is investigated in a series of our recent papers \cite{A1,A2,A3,A4}. Accretion of rotating fluids onto stationary solutions has been investigated by one of us~\cite{A5}.
In Ref. \cite{13}, the dynamical behavior of phantom energy near a five-dimensional charged black hole has been considered. The authors formulated the equations for steady state, spherically symmetric flow of phantom fluids onto the black hole and concluded that a five-dimensional black hole cannot be transformed into an extremal black hole. In Ref. \cite {14}, it was shown that the size of the black hole decreases due to the phantom energy accretion. Amani et al. investigated the phantom energy accretion onto the Schwarzschild anti de-Sitter black hole with topological defect \cite{Amani}. In this work we will develop a general formalism of spherical accretion of ordinay and phantom matter by ordinay and phantom black holes.
We use the standard geometric units ($c=G=1$) and the chosen metric signature is $(+,-,-,-)$. This paper is organized as follows: In Sec.~\ref{secge} we develop a general formalism for spherical accretion that applies to all static black holes. In Sec.~\ref{sechs} we derive the Hamiltonian system and in Sec.~\ref{seccp} we determine the critical points (CPs). In Sec.~\ref{secemd} we apply our results to the Einstein--Maxwell--dilaton (EMD) black holes. In Sec.~\ref{secif} we specialize to a particular type of fluids i.e. \emph{ordinary} and \emph{phantom} isothermal test fluids and apply our formalism to the generic case as well as to the special cases of ultra-stiff, ultra-relativistic, radiation, and sub-relativistic fluids. In Sec.~\ref{secmar} we discuss the associated mass accretion rate and backreaction. We conclude in Sec.~\ref{seccon}. An Appendix section has been added to complete the discussion of Sec.~\ref{secif}.
\section{General equations\label{secge}}
In this section, we consider a general metric ansatz of the form
\begin{eqnarray}\label{1}
ds^{2}&=&A(r)dt^{2}-\frac{dr^{2}}{B(r)}-C(r)(d\theta^{2}+\sin^{2}\theta d\phi^{2}),
\end{eqnarray}
whose determinant is given by $g=-AC^{2}/B$. Our results will remain valid for all solutions having the line element of the form \eqref{1} and this includes all known static black holes, wormholes and other relevant solutions.
Now we define the governing equations which we need for the process of spherical accretion. For this, we have two basic conservation laws i.e. particle number conservation and energy conservation. We consider the flow of a perfect fluid onto black hole. The four velocity of the particles is given by $u^{\mu}=\frac{dx^{\mu}}{d\tau}$. If $n$ is the particle's density, then the particle's flux is given by $J^{\mu}=nu^{\mu}$. The law of particle conservation states that there will be no change in the number of particles; particles can never be created nor destroyed i.e. the current density of the particles is divergence free,
\begin{eqnarray}\label{2}
\nabla_{\mu}J^{\mu}&=&\nabla_{\mu}(nu^{\mu})=0,
\end{eqnarray}
where $\nabla_{\mu}$ shows the covariant derivative. The energy momentum tensor for a perfect fluid is given by
\begin{eqnarray}\label{3}
T^{\mu\nu}&=&(e+p)u^{\mu}u^{\nu}-pg^{\mu\nu},
\end{eqnarray}
where $e$ and $p$ denotes the energy density and pressure respectively. The law of energy conservation shows that there will be no change in the total energy of the system and is given by
\begin{eqnarray}\label{4}
\nabla_{\mu}T^{\mu\nu}&=&0.
\end{eqnarray}
On the equatorial plane $(\theta=\pi/2)$, the continuity equation given by \ref{2} yields
\begin{equation}\label{5}
C\sqrt{D}nu=C_{1}\quad\text{with}\quad D\equiv \frac{A}{B},
\end{equation}
where $C_{1}$ is a constant of integration, which is negative for an accretion process and positive otherwise, and $u\equiv u^r$. Since the fluid is flowing radially in the equatorial plane therefore both $u^{\theta}$ and $u^{\phi}$ vanishes, and we are only left with $u^{t}$ and $u^{r}=u$ components. By using the normalization condition $(g_{\mu\nu}u^{\nu}u^{\nu}=1)$ we have
\begin{eqnarray}\label{6}
u^{t}&=&\pm \sqrt{\frac{1+B^{-1}u^{2}}{A}},
\end{eqnarray}
where the minus sign corresponds to accretion. Consequently, we obtain
\begin{eqnarray}\label{7}
u_{t}&=&g_{tt}u^{t}=\pm \sqrt{A(1+B^{-1}u^{2})}= \pm \sqrt{A+Du^2}.
\end{eqnarray}
As we have considered the steady state and spherically symmetric line element, so all the physical parameters e.g. particles density, energy density, pressure, four velocity etc. are functions of the radial coordinate $r$ only \cite{A1}.
\par The thermodynamics of the fluid is described by
\begin{eqnarray}\label{8}
dp=n(dh-Tds),\qquad de=hdn+nTds,
\end{eqnarray}
where $T$ is the temperature, $s$ is the entropy and $h$ is the specific enthalpy (enthalpy per particle) given by
\begin{equation}\label{9}
h=\frac{e+p}{n}.
\end{equation}
Since ordinary matter satisfies the constarint $e+p>0$, we have $h>0$ while phantom matter violates it; that is, $e+p<0$ for phantom matter resulting in $h<0$.
In relativistic hydrodynamics, there exists a scalar $hu_{\mu}\xi^{\mu}$ which is conserved along the trajectories of the fluid \cite{Rezolla} i.e.
\begin{eqnarray}\label{10}
u^{\nu}\nabla_{\nu}(hu_{\mu}\xi^{\mu})&=&0,
\end{eqnarray}
where $\xi^{\mu}$ is Killing vector of the spacetime. For instance, if we take $\xi^{\mu}=(1,0,0,0)$ we obtain~\cite{A3}
\begin{eqnarray}\label{11}
\partial_{r}(hu_{t})&=&0~~~~~\text{or}~~~~~h\sqrt{A+Du^2}=C_{2},
\end{eqnarray}
where $C_{2}$ is a constant of integration: $C_{2}>0$ for ordinary matter and $C_{2}<0$ for phantom matter.
Since in the equatorial plane $(\theta=\pi/2)$ the motion is radial, $d\theta=d\phi=0$ and so we can decompose our metric (\ref{1}) as
\begin{eqnarray}\label{12}
ds^{2}=\Big(\sqrt{A}dt^{2}\Big)^{2}-\Big(\sqrt{\frac{1}{B}}dr^{2}\Big)^{2},
\end{eqnarray}
in the standard relativistic way \cite{b1,b2} as seen by a local static observer. We can define the three velocity by
\begin{eqnarray}\label{13}
v&=&\sqrt{\frac{1}{AB}}\frac{dr}{dt}.
\end{eqnarray}
This leads to
\begin{eqnarray}\label{14}
v^{2}&=&\frac{1}{AB}\Big(\frac{u}{u^t}\Big)^{2},
\end{eqnarray}
where $u^{t}=dt/d\tau$ and $u^{r}=u=dr/d\tau$. Using (\ref{6}) and isolating $u^{2}$ we obtain~\cite{A3}
\begin{eqnarray}
\label{15a}u^{2}&=&\frac{Bv^{2}}{1-v^{2}},\\
\label{15}u_{t}^{2}&=&\frac{A}{1-v^{2}},\\
\label{15b}v^2&=&\frac{Du^2}{A+Du^2},
\end{eqnarray}
and hence Eq. (\ref{5}) becomes~\cite{A3}
\begin{eqnarray}\label{16}
\frac{A(Cnv)^{2}}{1-v^{2}}&=&C_{1}^{2}.
\end{eqnarray}
We use these results below in the Hamiltonian analysis.
Based solely on~\eqref{16}, it was concluded in Ref.~\cite{A3} that the behavior of the fluid near the horizon is independent on the form of $A(r)$. More precisely, the fluid reaches the horizon either with $v\to 0$ or $v\to 1$~\cite{A1,A2}.
The constant $C_1^2$ in~\eqref{16} can be written as $A_0C_0^2n_0^2v_0^2/(1-v_0^2)$ where {\textquotedblleft 0\textquotedblright} denotes any reference point ($r_0,\,v_0$) from the phase portrait; this could be a CP, if there is any, spatial infinity ($r_{\infty},\,v_{\infty}$), or any other reference point. We can thus write~\cite{A3}
\begin{equation}\label{6b}
\frac{n^2}{n_0^2}=\frac{A_0C_0^2v_0^2}{1-v_0^2}~\frac{1-v^2}{AC^2v^2}=\frac{C_1^2}{n_0^2}~\frac{1-v^2}{AC^2v^2}.
\end{equation}
\section{Hamiltonian system\label{sechs}}
We have two integrals of motion $C_{1}$ and $C_{2}$ given by Eqs. (\ref{5}) and (\ref{11}) respectively. By Bernoulli's theorem, the square of $C_{2}$ in Eq. (\ref{11}) is proportional to the fluid energy. So by using (\ref{15}) we can define the Hamiltonian as~\cite{A3}
\begin{eqnarray}\label{17}
\mathcal{H}(r,v)&=&\frac{h^{2}(r,v)A(r)}{1-v^{2}},
\end{eqnarray}
where $v$ is the three velocity given by Eq. (\ref{15b}).
\section{Critical points\label{seccp}}
It is well known that a perfect fluid~\eqref{3} is adiabatic; that is, the specific entropy is conserved along the evolution lines of the fluid: $u^{\mu}\nabla_{\mu}s=0$ (See~\cite{A1} for a proof). In the special case we are considering in this work where the fluid motion is radial, stationary (no dependence on time), and it conserves the spherical symmetry of the black hole, the latter equation reduces to $\partial_rs=0$ everywhere, that is, $s\equiv \text{constant}$. Thus, the motion of the fluid is isentropic and equations~\eqref{8} reduce to
\begin{eqnarray}\label{18}
dp&=&ndh,~~~~~~~~~~de=hdn.
\end{eqnarray}
The adiabatic speed of sound is defined by
\begin{eqnarray}\label{19}
a^{2}&=&\frac{dp}{de}=\frac{d\ln h}{d\ln n}.
\end{eqnarray}
Now, with $\mathcal{H}$ given by Eq. (\ref{17}), the dynamical system reads
\begin{eqnarray}\label{20}
\dot{r}&=&\mathcal{H}_{,v},~~~~~~~~~~\dot{v}=-\mathcal{H}_{,r},
\end{eqnarray}
where the dot denotes the $\bar{t}$ derivative where $\bar{t}$ is the time variable of the Hamiltonian dynamical system. Using the results of Ref.~\cite{A3} we obtain
\begin{align}
\label{d3}&\dot{r}=\frac{2h^2A}{v(1-v^2)^2}~(v^2-a^2),\\
\label{d4}&\dot{v}=-\frac{h^2}{1-v^2}\Big[\frac{d A}{d r}-2a^2A~\frac{d \ln(\sqrt{A}C)}{d r}\Big].
\end{align}
Introducing the notation $g_c=g(r)|_{r=r_c}$ and $g_{c,r_c}=g_{,r}|_{r=r_c}$ where $g$ is any function of $r$, the following equations provide a set of CPs that are solutions to $\dot{r}=0$ and $\dot{v}=0$:
\begin{equation}\label{24}
v_c^2=a_c^2\qquad\text{ and }\qquad a_c^2=\frac{C_cA_{c,r_c}}{C_cA_{c,r_c}+2AC_{c,r_c}}=\frac{C^2A_{,r}}{(C^2A)_{,r}}\Big|_{r=r_c},
\end{equation}
where $a_c$ is the three-dimensional speed of sound evaluated at the CP. The first equation states that at a CP the three-velocity of the fluid equals the speed of sound. The second equation determines $r_c$ once an equation of state is known.
\section{Spherical accretion by Einstein--Maxwell--dilaton black holes\label{secemd}}
The Lagrangian for EMD theory \cite{d2,c1,c2} is given by
\begin{eqnarray}\label{27}
\mathcal{L}&=&R-2\eta_{1}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}+\eta_{2}e^{2\lambda\phi}F_{\mu\nu}F^{\mu\nu},
\end{eqnarray}
where $R$ is the Ricci scalar, $F_{\mu\nu}$ is the Maxwell tensor of the electromagnetic field and $\phi$ represents the dilaton field. The parameter $\lambda$ is the
real dilaton-Maxwell coupling constant, and $\eta_1=\pm 1$, $\eta_2=\pm 1$. Normal EMD corresponds to $\eta_2=\eta_1=+1$, while phantom couplings of the dilaton field $\phi$ or/and Maxwell field $F = d A$ are obtained for $\eta_1=-1$ or/and $\eta_2=-1$ yielding the theories EM$\bar{\text{D}}$ ($\eta_2=+1,\,\eta_1=-1$), E$\bar{\text{M}}$D ($\eta_2=-1,\,\eta_1=+1$), and E$\bar{\text{M}}\bar{\text{D}}$ ($\eta_2=-1,\,\eta_1=-1$). For short we call all these theories EMD theory.
A class of spherically symmetric solutions to the field equations associated with the Lagrangian~\eqref{27} are known in the literature~\cite{d2} and are given by
\begin{eqnarray}\label{28}
ds^{2}&=&f_{+}f_{-}^{\gamma}dt^{2}-\frac{dr^{2}}{f_{+}f_{-}^{\gamma}}-r^{2}f_{-}^{1-\gamma}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),
\end{eqnarray}
where we identify ($A,\,B,\,C$)~\eqref{1} by
\begin{equation}\label{id}
A=B=f_{+}f_{-}^{\gamma},\quad C=r^{2}f_{-}^{1-\gamma}.
\end{equation}
The electric and dilaton fields are given by
\begin{eqnarray}\label{29}
F&=&-\frac{Q}{r^2}dr\wedge dt,~~~~~~~~~~e^{-2\lambda\phi}=f_{-}^{1-\gamma},
\end{eqnarray}
whereas,
\begin{eqnarray}\label{30}
f_{\pm}&=&1-\frac{r_{\pm}}{r},~~~~~~~~~~\gamma=\frac{1-\eta_{1}\lambda^{2}}{1+\eta_{1}\lambda^{2}},~~~~~~~~~~\eta_1\lambda^2=\frac{1-\gamma}{1+\gamma},
\end{eqnarray}
and
\begin{equation}\label{30b}
r_+=M+M\sqrt{1-\frac{2\eta_2Q^2}{M^2}~\frac{\ga}{1+\ga}},\qquad r_-=\frac{2\eta_2Q^2}{(1+\ga)r_+}=\frac{M}{\ga}-\frac{M}{\ga}\sqrt{1-\frac{2\eta_2Q^2}{M^2}~\frac{\ga}{1+\ga}},
\end{equation}
where we have used the notation of Refs.~\cite{c1,d3}. Note that~\cite{d3}
\begin{align}
&\gamma \in (-\infty,-1)\cup[1,+\infty) \quad \text{if}\quad \eta_{1}=-1,\nn\\
\label{gamma}&\gamma \in (-1,+1] \quad\text{if}\quad \eta_{1}=+1.
\end{align}
A useful expression for our further investigations is the equation of which $r_+$ is a solution:
\begin{equation}\label{rp}
r_+^2=2Mr_+-\frac{2\eta_2Q^2\gamma}{1+\gamma}.
\end{equation}
The case $\ga=1$ corresponds to normal and phantom Reissner-Nordstr\"om black holes.
A global flow is possible if the fluid elements can reach spatial infinity. Using the expressions~\eqref{id} of $A$ and $C$ in~\eqref{16}, we see that as $r\to\infty$, $nv$ behaves as
\begin{equation}\label{gf1}
nv\sim\frac{1}{r^{2}}\qquad (\text{for all }\gamma).
\end{equation}
As $r\to\infty$, we may distinguish two cases:
\begin{align}
\label{gf2}&\text{(a)}\quad v=v_{\infty}\Big(1-\frac{v_{x_a}}{r^{x_a}}+\cdots\Big)\qquad (\text{with independent term}),\\
\label{gf3}&\text{(b)}\quad v=\frac{v_y}{r^y}\Big(1-\frac{v_{x_b}}{r^{x_b}}+\cdots\Big)\qquad (\text{no independent term}),
\end{align}
where ($v_{\infty},\,v_{x_a},\,v_{x_b},\,v_y$) are constants and $0<y\leq 2$. The corresponding expansions for $n$ are of the form
\begin{align}
\label{gf4}&\text{(a)}\quad n=\frac{n_2}{r^2}\Big(1+\frac{n_{z_a}}{r^{z_a}}+\cdots\Big),\\
\label{gf5}&\text{(b)}\quad n=\frac{n_{2-y}}{r^{2-y}}\Big(1+\frac{n_{z_b}}{r^{z_b}}+\cdots\Big),
\end{align}
where ($n_2,\,n_{z_a},\,n_{z_b},\,n_{2-y}$) are constants. Now, since the series expansion in powers of $1/r$ of the expression $AC^2/r^4$~\eqref{id} has only positive integer powers, we conlude from~\eqref{16} that the exponents (${x_a},\,{x_b},\,{z_a},\,{z_b},\,2y$), and all the exponents inside the parenthesis in~\eqref{gf2} to~\eqref{gf5}, must be positive integers too:
\begin{equation}
({x_a},\,{x_b},\,{z_a},\,{z_b},\,2y)\in \mathbb{N^+}^5.
\end{equation}
Since $0<y\leq 2$, this constraints $y$ to assume the four values
\begin{equation}
y=\frac{1}{2},\,1,\,\frac{3}{2},\,2.
\end{equation}
The case $y=1/2$ corresponds to Keplerian motion.
Substituting~\eqref{gf2} and~\eqref{gf3} into~\eqref{6b} we obtain
\begin{align}
\label{gf6}&\text{(a)}\quad n_2=\frac{C_1\sqrt{1-v_{\infty}^2}}{v_{\infty}},\quad z_a=1,\quad n_{z_a}=\left\{\begin{array}{ll}
M+(1-\gamma)r_-+\frac{v_{x_a}}{1-v_{\infty}^2}, & \hbox{$x_a=1$;} \\
M+(1-\gamma)r_-, & \hbox{$x_a\geq 2$,}
\end{array}
\right. \\
\label{gf7}&\text{(b)}\quad n_{2-y}=\frac{C_1}{v_y},\quad z_b=1,\quad n_{z_b}=\left\{\begin{array}{llll}
M+(1-\gamma)r_-+v_{x_b}-\frac{v_y^2}{2}, & \hbox{$x_b=1,\,y=\frac{1}{2}$;} \\
M+(1-\gamma)r_--\frac{v_y^2}{2}, & \hbox{$x_b\geq 2,\,y=\frac{1}{2}$;} \\
M+(1-\gamma)r_-+v_{x_b}, & \hbox{$x_b=1,\,y\geq 1$;} \\
M+(1-\gamma)r_-, & \hbox{$x_b\geq 2,\,y\geq 1$,}
\end{array}
\right.
\end{align}
where $C_1/v_{\infty}>0$ and $C_1/v_y>0$. Similarly, from the facts that $\mathcal{H}$~\eqref{17} is a constant of motion and that the series expansion of $A/(1-v^2)$ as $r\to\infty$ includes only positive integer powers of $1/r$, we conclude that the series expasion of $h$ too has only positive integer powers. Hence, we write
\begin{equation}\label{gf8}
h=h_{\infty}\Big(1+\frac{h_1}{r}+\cdots\Big),
\end{equation}
resulting, upon substituting into~\eqref{17}, in
\begin{align}
\label{gf9}&\text{(a)}\quad h_1=\left\{\begin{array}{ll}
M+\frac{v_{x_a}v_{\infty}^2}{1-v_{\infty}^2}, & \hbox{$x_a=1$;} \\
M, & \hbox{$x_a\geq 2$,}
\end{array}
\right. \\
\label{gf10}&\text{(b)}\quad h_1=\left\{\begin{array}{ll}
M-\frac{v_y^2}{2}, & \hbox{$y=\frac{1}{2}$;} \\
M, & \hbox{$y\geq 1$.}
\end{array}
\right.
\end{align}
For ordinary matter $h_{\infty}$ is the baryonic mass $m$. A further discussion of the relationships between the different parameters, in the above expressions for ($v,\,n,\,h$), would depend on an equation of state that relates $e$ to $p$~\eqref{3}. As to $n_2$ and $n_{2-y}$, they depend on the nature of the fluid where in many astrophysical applications it is taken as a perfect gas~\cite{inviscid}.
\section{Isothermal-like test fluids\label{secif}}
Isothermal flow of ordinary fluids refers to flow at constant temperature. This model of flow is a generalizarion of the classical formula $p=\rho RT$ where the fluid is assumed to be a perfect gas. Their equation of state is such that the pressure is directly proportional to the energy density: $p=k e$. This model, however, is not suitable for an isothermal flow description of phantom fluids where $e+p<0$, resulting in $k<-1$ and $a^{2}=dp/de=k<0$.
Instead of a direct law of proportionality, we rather assume a linear dependence of $p$ and $e$~\cite{Babichev2}
\begin{equation}\label{if1}
p=\omega (e-e_0),
\end{equation}
where $0<\omega\leq 1$ and $e_0$ are constants. For ordinary fluids we may take $e_0\equiv 0$. For phantom fluids $0<e<e_0$ to ensure that $p<0$; this, however, does not ensure that $e+p<0$, so we assume that
\begin{equation}\label{if2}
e<\frac{\omega}{1+\omega}~e_0,
\end{equation}
where the rhs constitutes an upper limit for the energy density of phantom fluids.
Using~\eqref{19}, we obtain
\begin{eqnarray}\label{26a}
a^{2}=\frac{d\ln h}{d\ln n}=\omega,
\end{eqnarray}
hence
\begin{eqnarray}\label{26b}
h=\text{constant }\times n^{\omega}=\frac{h_{\infty}}{n_{\infty}^{\omega}}~n^{\omega}.
\end{eqnarray}
Now, using~\eqref{6b} and~\eqref{26b} the Hamiltonian~\eqref{17} reduces to~\cite{A3}
\begin{eqnarray}\label{26c}
\mathcal{H}(r,v)&=&\frac{1}{[vC(r)]^{2\omega}}\Big[\frac{A(r)}{1-v^{2}}\Big]^{1-\omega},
\end{eqnarray}
where all factor constants have been removed.
Restricting ourselves to isothermal fluids, the Hamiltonian~(\ref{26c}) for the phantom black hole reduces to
\begin{equation}\label{35}
\mathcal{H}=\frac{(r-r_+)^{1-\omega}(r-r_-)^{\gamma(1+\omega)-2\omega}}{v^{2\omega}r^{(1+\gamma)(1+\omega)}}.
\end{equation}
Implicit solutions to~\eqref{35}, that is, solutions to $\mathcal{H}=\text{constant}$ providing the profile of $v$ versus $r$ are similar to the plots depicted in Figs. 1 and 2 of Ref.~\cite{A1} and Figs. 1 to 4 of Ref.~\cite{A2}, and they will not be produced here. Our main purpose is to demonstrate and depict the new features pertaining to accretion onto EMD black holes.
As we mentioned above, a global flow is possible if the fluid elements can reach spatial infinity. By the last equation we see that as $r\to\infty$
\begin{equation}\label{35b}
\mathcal{H}\sim\frac{1}{v^{2\omega}r^{4\omega}},
\end{equation}
in a way independent of the value of $\gamma$. Since $\mathcal{H}\propto C_2^2$~\eqref{11} is a constant of motion, the three-velocity must behave as
\begin{equation}\label{35c}
v\sim\frac{1}{r^{2}}\qquad (\text{for all }\gamma),
\end{equation}
in the limit $r\to\infty$. As we noticed earlier, the fluid approaches the horizon in a way independent of the form of the metric component $A(r)$, and particularly, of the value of $\gamma$. Thus, the end-behavior (near the horizon or at spatial infinity) of the fluid flow does not dependent on $\gamma$ but the detailed motion of the fluid and the CPs do depend on $\gamma$.
The expression~\eqref{35c} is of the form~\eqref{gf3} with $y=2$ which means that isothermal-like fluids do not follow a Keplerian motion. Replacing $v_y$ by $v_2$ and $n_{2-y}$ by $n_{\infty}$, the expressions of ($v,\,n,\,h$) reduce to
\begin{align}
\label{gi1}&v=\frac{v_2}{r^2}\Big(1-\frac{v_{x_b}}{r^{x_b}}+\cdots\Big),\\
\label{gi2}&n=n_{\infty}\Big(1+\frac{n_{z_b}}{r}+\cdots\Big)=\frac{C_1}{v_2}\Big(1+\frac{M+(1-\gamma)r_-+\delta^1_{x_b}v_{x_b}}{r}+\cdots\Big),\\
\label{gi3}&h=h_{\infty}\Big(1+\frac{M}{r}+\cdots\Big),
\end{align}
where $C_1/v_2>0$ and
\begin{equation}\label{gi4}
M=\omega [M+(1-\gamma)r_-+\delta^1_{x_b}v_{x_b}],
\end{equation}
which results upon substituting~\eqref{gi2} and~\eqref{gi3} into~\eqref{26b}. Here $\delta^1_{x_b}$ is 1 if $x_b=1$ and 0 otherwise.
The expression for the sound speed at the CP~(\ref{24}) leads to the following expression
\begin{equation}\label{34a}
\omega =\frac{(r_+ +\gamma r_-) r_c-(1+\gamma ) r_- r_+}{4 r_c^2-[3 r_+ +(2+\gamma ) r_-] r_c+(1+\gamma ) r_- r_+}.
\end{equation}
Using~\eqref{30b} and the third expression in~\eqref{30}, we obtain
\begin{align*}
&r_+ +\gamma r_-=2 M,\qquad (1+\gamma ) r_- r_+=2\eta_2 Q^2,\\
&3 r_+ +(2+\gamma ) r_-=3 (r_++\gamma r_-)+2 (1-\gamma ) r_-=6 M+\frac{4 \eta_1\eta_2\lambda ^2 Q^2}{r_+}>0.
\end{align*}
Substituting these three relations into~\eqref{34a} we bring it to the form
\begin{equation}\label{34b}
2 \omega r_+ r_c^2-[(1+3 \omega ) M r_+ +2\eta_1\eta_2 Q^2 \lambda ^2 \omega ] r_c+\eta_2(1+\omega ) Q^2 r_+=0,
\end{equation}
yielding
\begin{equation}\label{34c}
r_c=\frac{(1+3 \omega ) M r_+ +2\eta_1\eta_2 Q^2 \lambda ^2 \omega +\sqrt{[(1+3 \omega ) M r_+ +2\eta_1\eta_2 Q^2 \lambda ^2 \omega]^2- 8\eta_2 \omega (1+\omega ) Q^2 r_+^2}}{4 \omega r_+}.
\end{equation}
Here $r_+$ is given by~\eqref{30b} and $\eta_1\lambda^2$ by~\eqref{30}. The other root to~\eqref{34b}, $\bar{r}_c$, is given by a similar expression to~\eqref{34c} with the minus sign in front of the square root. When the two roots are real, $r_c\geq\bar{r}_c$. In the Appendix, we will show that the roots ($r_c,\,\bar{r}_c$) are always real in the \emph{physical case} $M^2\geq Q^2$ for all $0<\omega\leq 1$ and for all values of ($M,\,Q,\,\gamma,\,\eta_2$) that make $r_+$ real.
However, a CP, yielding a critical behavior, exists only if $r_c>r_+$. With five parameters ($M,\,Q,\,\gamma,\,\eta_2,\,\omega$) being free, it is very cumbersome, even in the physical case $M^2\geq Q^2$, to compare the expression~\eqref{34c} of $r_c$ to that of $r_+$~\eqref{30b}. For that purpose we rather follow another path.
\subsection{Physical case \pmb{$M^2\geq Q^2$}}
Let us first determine a power series for $r_c$ around $\omega=1$. Note that in the case $\omega= 1$, $r_+$ is a solution to~\eqref{34b}. In fact, setting $\omega=1$ and replacing $r_c$ by $r_+$ in~\eqref{34b}, we obtain
\begin{equation}\label{pc1}
r_+^2-2 M r_+ -\eta_1\eta_2 Q^2 \lambda ^2+\eta_2 Q^2=0,
\end{equation}
where the l.h.s is zero by~\eqref{rp} and~\eqref{30}. We can determine the power series upon setting $r_c=r_++\alpha$ and $\omega=1-\epsilon$ in~\eqref{34b} to obtain
\begin{equation}\label{pc2}
r_c=r_++\frac{Mr_+-\eta_2Q^2}{r_+^2-\eta_2Q^2}~\frac{r_+}{2}~(1-\omega) +\cdots,
\end{equation}
where we have used~\eqref{rp} and~\eqref{30}. For $\bar{r}_c$ we obtain
\begin{equation}\label{pc3}
\bar{r}_c=\frac{\eta_2Q^2}{r_+}+\frac{\eta_2Q^2(r_+-M)}{2(r_+^2-\eta_2Q^2)}~(1-\omega) +\cdots.
\end{equation}
In the physical case $M^2\geq Q^2$ and $\eta_2=+1$, $r_c$ approaches $r_+$ from above in the limit $\omega\to 1^{-}$, since in this case $r_+>M\geq |Q|$. This remains true in the case $\eta_2=-1$ for all $M^2$ and $Q^2$.
At the other limit, $\omega\to 0^{+}$, $r_c\to +\infty$.
Now, differentiating both sides of~\eqref{34b} with respect to $\omega$ we obtain
\begin{equation}\label{pc4}
\partial_{\omega}r_c=\frac{r_+(Mr_c-\eta_2Q^2)}{-\omega\sqrt{[(1+3 \omega ) M r_+ +2\eta_1\eta_2 Q^2 \lambda ^2 \omega]^2- 8\eta_2 \omega (1+\omega ) Q^2 r_+^2}}.
\end{equation}
For the parameter space ($M,\,Q,\,\gamma,\,\eta_2,\,\omega$), where the square root is real, $\partial_{\omega}r_c$ is certainly negative and so $r_c$ decreases from $+\infty$ to $r_+$ as $\omega$ runs from $0^+$ to 1. This is obvious for $\eta_2=-1$. For $\eta_2=+1$, had we assumed that $r_c$ first decreased, reached a minimum value, then increased again, we would obtain the minimum at $r_c=\eta_2Q^2/M<M<r_+$ for some value of $0<\omega<1$. But this is not possibe since $r_c$ must approach $r_+$ from above as $\omega\to 1^-$. This shows that for the parameter space ($M,\,Q,\,\gamma,\,\eta_2,\,\omega$), where the square root in~\eqref{pc4} and~\eqref{34c} is real, $r_c>r_+$ if $0<\omega<1$. In the same manner we can show that $\bar{r}_c<r_+$.
For $M^2\geq Q^2$, we have shown that a CP always exists for $0<\omega<1$. Thus, the flow of isothermal fluids onto normal or phantom EMD black holes (ultra-relativistic fluids $\omega=1/2$, radiation fluids $\omega=1/3$, and sub-relativistic fluids $\omega=1/4$) is always critical.
As is well known the accretion of ultra stiff fluids ($\omega=1$) onto ordinary matter is non-critical. We have extended this conclusion to accretion of ultra stiff fluids onto normal and phantom EMD black holes, since in this case the CP, $r_c=r_+$, is at the horizon position, so no critical behavior is observed in the outer region of the event horizon.
\subsection{The case \pmb{$M^2<Q^2$}}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.47\textwidth]{Fig1a.eps} \includegraphics[width=0.47\textwidth]{Fig1b.eps} \\
\caption{\footnotesize{Plots of $r_c$~\eqref{34c} (continuous curve) and $r_+$~\eqref{30b} (dashed curve) vs. $\gamma$ for $\eta_1=\eta_2=+1$, $\omega=0.5$, $M=1$, and $M^2<Q^2$. The graph of $r_c$ is made of two disjoint curve segments, the one is monotonically decreasing and the other is monotonically increasing function of $\gamma$. (a): $|Q|=1.7$. (b): $|Q|=3$. It is clear from this figure that $r_+$, which is a monotonically decreasing function of $\gamma$, is not defined on the whole range of $\gamma$}~\eqref{gamma}; rather, it is defined for $-1<\gamma\leq \dfrac{M^2}{2Q^2-M^2}<1$. There are subintervals of $\gamma$ on which either $r_c<r_+$ or $r_c$ is not real. In these cases, the fluid accretion onto the corresponding black holes is non-critical.}\label{Fig1}
\end{figure*}
The EMD theory and its two derivatives E$\bar{\text{M}}$D and E$\bar{\text{M}}\bar{\text{D}}$ do admit black hole solutions with $M^2<Q^2$ (EM$\bar{\text{D}}$ has no black-hole solutions with $M^2<Q^2$). For fixed values of ($M,\,Q,\eta_1,\,\eta_2$) for which $r_+$ is real, the CP $r_c$ may not exist for some values of ($\gamma,\,\omega$) or may turn smaller than $r_+$ as shown in Fig.~\ref{Fig1}. In these cases, the fluid accretion onto the corresponding black holes is non-critical in the sense that~\eqref{d3} is satisfied but~\eqref{d4} is not; that is, as the fluid three velocity $v$ reaches the value $a$ of the three-dimensional speed of sound, during the accretion process, it does not do it in a stationary way so that $\dot{v}$ is nonzero. This means that, during the accretion process, the speed $v$ increases monotonically from $\sim 1/r^2$~\eqref{35c}, at spatial infinity, to 1, at the event horizon $r_+$. While in a critical flow, as $v$ reaches $a$ it remains stationary there for a while, then it increases again.
Figure.~\ref{Fig1} depicts the functions $r_c(\gamma)$ (continuous curve) and $r_+(\gamma)$ (dashed curve) for the other parameters held constant with $M^2<Q^2$. The graph of $r_c(\gamma)$ is made of two disjoint curve segments, the one is monotonically decreasing and the other is monotonically increasing function of $\gamma$. The one-segment graph of $r_+(\gamma)$ decreases monotonically. $r_+(\gamma)$ is not defined on the whole range of $\gamma$~\eqref{gamma}; rather, it is defined for $-1<\gamma\leq \dfrac{M^2}{2Q^2-M^2}<1$.
We have obtained similar figures to Fig.~\ref{Fig1} for all $0<\omega<1$ and $M^2<Q^2$.
\section{Mass accretion rate and backreaction\label{secmar}}
The accretion rate is the change in the black-hole's mass per unit time. This is related to the flux of $T^{\mu\nu}$ by
\begin{equation}\label{mar1}
\dot M=-4\pi C(r) T^{\ r}_{t}(r)\big|_{r=r_+}.
\end{equation}
With $T^{\ r}_{t}=(e+p)uu_t$ it is easy to show, using Eqs.~\eqref{5} to~\eqref{11}, that $T^{\ r}_{t}=C_1C_2/(C\sqrt{D})$, consequently we have
\begin{equation}\label{mar2}
\dot M=-\frac{4\pi C_1C_2}{\sqrt{D(r)}|_{r=r_+}}.
\end{equation}
For most known black holes the function $D(r)\equiv 1$, as is the case with normal and phantom EMD black holes~\eqref{28}, so that the mass accretion rate reduces to
\begin{equation}\label{mar3}
\dot M=-4\pi C_1C_2.
\end{equation}
The values of the constants of motion ($C_1,\,C_2$) depend on the values of the enthalpy, number density and three-speed at spatial infinity. Using the same Eqs.~\eqref{5} to~\eqref{11} we obtain
\begin{align}
\label{mar4}&C_1=\frac{\sqrt{A(r)}C(r)n(r)v(r)}{\sqrt{1-v(r)^2}}\bigg|_{r\to\infty},\\
\label{mar5}&C_1C_2=\frac{A(r)C(r)n(r)h(r)v(r)}{1-v(r)^2}\bigg|_{r\to\infty}.
\end{align}
For an accretion process $v(r)$ and $C_1$ are negative. If the flowing fluid is made of ordinary matter, $h>0$ and thus $\dot M>0$ so that the mass of the black hole increases. Conversely, if the flowing fluid is made of phantom matter, $h<0$ and $\dot M<0$ and the mass of the black hole decreases.
If we momentarily restrict ourselves to normal and phantom EMD black holes and use Eqs.~\eqref{gi1} to~\eqref{gi3}, we obtain
\begin{equation}
\label{marh8}\dot M=4\pi v_{2}n_{\infty}h_{\infty},
\end{equation}
where $v_{2}$ and $n_{\infty}$ are positive and $h_{\infty}>0$ for ordinary fluids and $h_{\infty}<0$ for phantom fluids. As mentioned above, $n_{\infty}$ depends on the nature of the fluid where in many astrophysical applications it is taken as an ideal gas~\cite{inviscid}. The energy flux~\eqref{marh8} does not depend on the parameters of the EMD black hole ($M,\,Q,\,\gamma,\,\eta_2$), however, the whole process of accretion depends on them. For instance, one may seek to impose constraints on these parameters requiring that the accreting matter (the perfect fluid) have the same three-speed (at spatial infinity) to the third order in powers of $1/r$ and same particle density and enthalpy to the first order. This is the case when investigating the accretion of a given fluid and considering a set of different black holes. In this case we will have the following constraints on the parameters of these black holes: $M$ and the term $(1-\gamma)r_-$, which depends on the still-free parameters ($Q,\,\gamma,\,\eta_2$), must have the same values for the black holes.
Now, back to the most general case of accretion~\eqref{mar3}. If
\begin{equation*}
t\ll \tau_0\equiv \frac{M_i}{4\pi |C_1C_2|},
\end{equation*}
where $M_i$ is the initial mass of the black hole, to the first order in $t/\tau_0$ the backreaction effects of the fluid tend to modify the mass of the black hole according to the linear law
\begin{equation}\label{br1}
M=M_i-4\pi C_1C_2t+\cdots =M_i\Big[1-\text{sgn}(C_1C_2)\frac{t}{\tau_0}+\cdots\Big],
\end{equation}
where $\text{sgn}(C_1C_2)$ is the sign of $C_1C_2$. Here $\tau_0$ is a characteristic time of accretion.
Note that both constants $C_1$ and $C_2$ in~\eqref{mar3} and~\eqref{br1} are independent of the mass of the black hole. Now, back to the steps leading to Eq.~\eqref{5}. Since $M$ is assumed constant in those steps one may {\textquotedblleft accidently\textquotedblright} multiply and divide the rhs of~\eqref{5} by $M^{\alpha}$, where $\alpha$ is some parameter (taken equal to 2 in Ref.~\cite{Babichev2}), to obtain
\begin{equation}\label{5b}
C\sqrt{D}nu=M^{\alpha}\mathcal{C}_{1}\quad\text{with}\quad D\equiv \frac{A}{B},
\end{equation}
where now the new {\textquotedblleft constant\textquotedblright} $\mathcal{C}_{1}\equiv C_{1}/M^{\alpha}$ depends on the mass of the black hole. This converts~\eqref{mar3} to
\begin{equation}\label{mar3b}
\dot M=-4\pi \mathcal{C}_{1}C_2~M^{\alpha}.
\end{equation}
As far as $4\pi \mathcal{C}_{1}C_2$ is seen as a constant, one may integrate~\eqref{mar3b}. For instance if $\alpha\neq 1$, one obtains
\begin{equation}\label{br2}
M=M_i\Big(1-\frac{t}{\tau}\Big)^{1/(1-\alpha)},
\end{equation}
where
\begin{equation}\label{br3}
\tau\equiv \frac{1}{4\pi \mathcal{C}_{1}C_2(1-\alpha)M_i^{\alpha-1}},
\end{equation}
could be positive or negative (had we taken $\alpha=1$, we would have obtained a log function in~\eqref{br2} and a different expression for $\tau$). Taking $\alpha=2$ in~\eqref{br2}, this reduces to Eq.~(7) of Ref.~\cite{Babichev2}. As far as backreaction effects are neglected, Eq.~\eqref{5b} is correct; however, when backreaction effects are taken into consideration Eq.~\eqref{mar3b} is correct to first order only since $\mathcal{C}_{1}$ depends on $M$. Thus, Eq.~\eqref{br2} is also valid to first order only and its series expansion in powers of $t$ reduces to the rhs of~\eqref{br1} once we replace $\mathcal{C}_{1}$ by $C_{1}/M^{\alpha}$. Its term in $t^2$, however, depends well on $\alpha$ even after replacing $\mathcal{C}_{1}$ by $C_{1}/M^{\alpha}$ and thus it does not produce the correct expansion term in $t^2/\tau_0^2$.
It is obvious from the analysis made here that the backreaction effects have been evaluated assuming a perturbed metric but a non-perturbed fluid~\cite{Babichev3}. In order to determine the term in $t^2/\tau_0^2$ in~\eqref{br1} one needs to consider a more advanced analysis where both the metric and the fluid are perturbed.
\section{Conclusion\label{seccon}}
The analysis made in this work is general and concerns accretion onto static solutions. It includes some of the spherical accretion work done previously on the Schwarzschild and Reissner-Nordstr\"om black holes. Our model shares all known features with previously investigated accretion works. The distinguished features discovered in this work, which are emphasized in the two plots of Fig.~\ref{Fig1}, are characteristic of accretion on EMD black holes.
As far as the physical condition $M^2\geq Q^2$ is observed, the accretion process of uncharged ordinary isothermal fluids onto ordinary or phantom EMD black holes is characterized by the presence of a critical point $r_c$ through which the process is critical; in that, the three speed of the fluid becomes stationary for a while as it reaches the speed of sound. All accretion processes terminate at the horizon with the limiting speed of 1.
When the condition $M^2\geq Q^2$ is not observed, the accretion process onto some of the EMD black holes becomes non-critical.
In our investigation, we have restricted ourselves to isothermal fluids and we could extend the study, at least numerically, to other fluids. In the literature many authors have considered polytropic fluids but for such systems global solutions do not exist~\cite{global}. One can also assume a cosmological constant, vacuum energy or dark energy but their accretion does not have a substantial physical impact on the black hole.
Although in the literature, there are a lot of studies on this concept but still one can extend the analysis to include spinning black holes with non-adiabatic systems to take into account the terms of heat transport or viscosity.
\section*{Appendix: Reality of the roots of Eq.~\eqref{34b} for \pmb{$M^2\geq Q^2$} \label{secaa}}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\subsection*{Case \pmb{$\eta_2=-1$}}
For $\eta_2=-1$ the discriminant of~\eqref{34b}, which is the expression under the square root in~\eqref{34c}, is manifestly positive and so the roots $r_c$~\eqref{34c} and $\bar{r}_c$ are real.
\subsection*{Case \pmb{$\eta_2=+1$}}
The discriminant of~\eqref{34b} is positive or zero if
\begin{equation}\label{ap1}
(1+3 \omega ) M r_+ +2\eta_1\eta_2 Q^2 \lambda ^2 \omega\geq 2\sqrt{2}\sqrt{\omega(1+\omega)}|Q|r_+\qquad (\eta_2=+1,\ \eta_1=\pm 1).
\end{equation}
\subsubsection*{Subcase $\eta_2=+1$ and $\eta_1=+1$}
For $0<\omega\leq 1$ it is easy to show that $1+3 \omega\geq 2\sqrt{2}\sqrt{\omega(1+\omega)}$. Then, if $M^2\geq Q^2$ we will have $(1+3 \omega ) M r_+\geq 2\sqrt{2}\sqrt{\omega(1+\omega)}|Q|r_+$ and so~\eqref{ap1} is
trivially satisfied for $\eta_1=+1$.
\subsubsection*{Subcase $\eta_2=+1$ and $\eta_1=-1$}
Using~\eqref{rp} to express $Mr_+$ in terms of $r_+^2$ and $Q^2$ and the third expression in~\eqref{30} to express $\lambda^2$ in terms of $\gamma$~\eqref{gamma}, we bring~\eqref{ap1} to the form
\begin{equation}\label{ap2}
(\sqrt{2\omega}~r_+-\sqrt{\omega+1}~|Q|)^2+\dfrac{1-\omega}{2}\Big(r_+^2-\frac{2Q^2}{1+\gamma}\Big)\geq 0.
\end{equation}
Knowing that $0<\omega\leq 1$, this is trivially satisfied (for all $M^2$ and $Q^2$) if $\gamma<-1$~\eqref{gamma}. Now, if $\gamma\geq 1$~\eqref{gamma} and $M^2\geq Q^2$, since $r_+>M$~\eqref{30b}, we will have
\begin{equation}\label{ap3}
r_+^2-\frac{2Q^2}{1+\gamma}>M^2-\frac{2Q^2}{1+\gamma}\geq M^2-Q^2\geq 0,
\end{equation}
and so~\eqref{ap2} and\eqref{ap1} are satisfied.
\section*{Acknowledgments}
We thank Manuel E. Rodrigues for showing interest in an early stage of this work.
|
2003.09467
|
\section{Introduction} \label{sec:introduction}
Graph sampling provides a statistical approach to study real graphs, which represent the structure of many technological, social or biological phenomena of interest. It is based on exploring the variation over all possible subsets of nodes and edges, i.e. \emph{sample graphs}, which can be taken from the given \emph{population graph}, according to a specified method of sampling. \citet{ZhangPatone2017} synthesise the existing graph sampling theory, extending the previous works on this topic by \citet{Frank1971, Frank1980a, Frank1980b, Frank2011}.
A general definition is given for probability sample graphs, in a manner that is similar to general probability samples from a finite population (Neyman, 1934); and the unbiased Horvitz-Thompson (HT) estimator is developed for arbitrary $T$-stage snowball sampling from finite graphs, as in finite population sampling \citep{Horvitz1952}. To this end the observation procedure of graph sampling must be \emph{ancestral} \citep{ZhangPatone2017}, in that one needs to know which other out-of-sample nodes could have led to the observed motifs in the sample graph, had they been selected in the initial sample of nodes. Under $T$-stage snowball sampling, additional stages of sampling are generally needed in order to identify the ancestors of all the motifs observed by the $T$-th stage.
Ancestral observation procedure is a generalisation of the notion of \emph{multiplicity} in indirect sampling \citep{BirnbaumSirken1965}. As an example, patients can be selected via a sample of hospitals. Insofar as each patient may receive treatment from more than one hospital, the patients are not nested in the hospitals like elements do under cluster sampling \citep{Cochran1977}. Therefore, to compute the inclusion probability of a sample patient, one needs to identify all the relevant hospitals including those outside the sample, which constitutes the information on ``multiplicity'' of sources that must be collected in addition to the sample of hospitals and patients. The same requirement exists as well for the other unconventional sampling methods, such as network sampling \citep{Sirken2005}, or adaptive cluster sampling \citep{Thompson1990}.
The information on multiplicity can be made apparent for the sampling methods above, when they are presented as sampling from a special type of graph which we shall refer to as \emph{bipartite incidence graph (BIG)}. The nodes in a BIG are divided in two parts, which represent the sampling units and motifs of interest, respectively. An edge only exists from one node to another if the selection of the former (representing a sampling unit) leads to the observation of the latter (representing a motif), i.e. there are no edges among the nodes representing the sampling units nor among those of the motifs. In the example of indirect sampling above, hospitals are the sampling units and patients the motifs, and an edge exists only between a hospital and a patient that receives treatment there. The information on the multiplicity of a patient is then simply the knowledge of the nodes that are adjacent to the node representing this patient in the BIG.
In this paper we establish the necessary and sufficient conditions for representing any graph sampling from given population graph as \textit{BIG sampling (BIGS)}, and apply it to arbitrary $T$-stage snowball sampling, as well as all the above unconventional sampling methods. Two major advantages provide the motivation.
First, generally speaking, not all the observed motifs in the sample graph can be used for estimation, but only those associated with the knowledge of their ancestors in accordance with the specified sampling method. The matter is similar in adaptive cluster sampling, where \cite{Thompson1990} proposes to use certain units in estimation only if they are observed in a specific way but not otherwise. Under arbitrary graph sampling, the sample motifs eligible for estimation are those whose multiplicities can be identified in the associated BIG. We shall derive appropriate results to substantiate this insight, and apply them to the motifs that are observed by a given stage of snowball sampling, thereby ridding the need of additional sampling for the ineligible motifs. Indeed, as we will demonstrate, applying the same idea to adaptive cluster sampling would yield other unbiased estimators beyond those considered by \cite{Thompson1990}.
Second, in addition to the HT estimator, \citet{BirnbaumSirken1965} propose an unconventional Hansen-Hurwitz (HH) type estimator \citep{HansenHurwitz1943}, which is based on the sampled hospitals and a \emph{constructed measure} for each of them, derived from the related patients. This estimator is unbiased over repeated indirect sampling and easy to compute, including the second-order inclusion probabilities that are necessary for variance estimation, which are given directly by the sampling design of hospitals. Whereas to apply the HT-estimator, one must first derive all the first and second-order inclusion probabilities of the \emph{indirect} sampling design of patients from that of the hospitals. The HH-type estimator has been used in many works on network sampling, as summarised by \citet{Sirken2005}; it was recast as a generalised ``weight share'' method for indirect sampling \citep{Lavallee2007}; a modified version was proposed by \cite{Thompson1990} for adaptive cluster sampling. It has been observed that either the HH-type or HT estimator may be more efficient than the other in different applications \citep[e.g.][]{Thompson2012}. Adopting the BIGS representation, we shall identify for the first time the general condition that governs the relative efficiency between them.
Thus, capitalising on both the advantages, the BIGS representation of graph sampling provides a unified approach to a large number of situations, considerably extending the choices of applicable unbiased estimators. The availability of the various feasible BIGS strategies offers the potentials of efficiency gains in practice.
In the rest of the paper, graph sampling and BIG sampling are described in Section \ref{sec:graphBIGsampling}, and the sufficient and necessary conditions for a feasible BIGS representation of graph sampling are established. In Section \ref{sec:unconventionalsampling}, formal BIGS representations are described for the aforementioned unconventional sampling methods \citep{BirnbaumSirken1965,Lavallee2007,Sirken1970,Sirken2005,Thompson1990}. Some new unbiased estimators for adaptive cluster sampling by its feasible BIGS representations are explained and illustrated. In Section \ref{sec:TSBS}, we develop the BIGS representation of general $T$-stage snowball sampling, including the relevant results for identifying the sample motifs eligible for estimation. In Section \ref{sec:estimation}, the general condition governing the relative efficiency of the HT and HH-type estimators under BIG sampling is presented. In Section \ref{sec:numericalwork}, some numerical results are provided for two-stage adaptive cluster sampling by revisiting the example considered by \citet{Thompson1991}, and for an example of $T$-stage snowball sampling from an arbitrary population graph. Finally, some concluding remarks are given in Section \ref{sec:conclusion}.
\section{Graph, BIG sampling}\label{sec:graphBIGsampling}
\subsection{Graph sampling} \label{sec:graphsampling}
Let $G = (U,A)$ be the population graph, with node set $U$ and edge set $A$. For simplicity of exposition we focus on simple graphs in this paper, such that there can be at most one edge between a pair of nodes $(i,j)$, where $i,j\in U$. Let $a_{ij}= 1$ if edge $(ij) \in A$ and 0 otherwise. By definition $a_{ji} \neq a_{ij}$ if the graph is \emph{directed}, but $a_{ij} \equiv a_{ij}$ if the graph is \emph{undirected}. The theory developed below can be easily adapted to multigraphs, where there can be more than one edge between any pair of nodes.
The measurement units of interest are called the \emph{motifs} in $G$. Denote by $\Omega = \Omega(G)$ the set of all motifs in $G$. For any $k\in \Omega$, let $M_k$ be the nodes involved in the motif $k$, of \emph{order} $| M_k|$. The motif of these $M_k$ nodes is denoted by $[M_k]$, such that for any $k\neq l\in \Omega$, we have $[M_k] = [M_l]$, but $M_k \neq M_l$, nor is it necessary that $|M_k| = |M_l|$. For example, let $G$ be an undirected graph. Let $\Omega$ consist of all the triangles in $G$, such that $\forall k\in \Omega$, we have $|M_k| = 3$, and $\prod_{i\neq j\in M_k} a_{ij} = 1$, by which the motif $[M_k]$ can be defined. As another example, let the motif of interest be the connected components in an undirected graph $G$. Then, $\forall k \in \Omega$ and $i_1\neq i_2\in M_k$, we have either $a_{i_1 i_2} =1$, or there must exist a sequence of nodes, denoted by $j_1, ..., j_q \in M_k \setminus \{ i_1, i_2\}$, such that $a_{i_1 j_1} a_{j_q i_2} \prod_{g=1}^{q-1} a_{j_g j_{g+1}} =1$, by which the motif can be defined. Whereas we need not have $|M_k| = |M_l|$, for any $k\neq l\in \Omega$.
\citet{ZhangPatone2017} give the following general definition of sample graphs from $G$. Let $s_0$ be an initial sample of nodes taken from the sampling frame $F$, where $s_0 \subset F\subseteq U$, according to the sampling distribution $p(s_0)$, and $\sum_{s_0} p(s_0) = 1$ and $\pi_i =\mbox{Pr}(i\in s_0) >0$ for any $i\in F$. Given $s_0$, graph sampling proceeds according to a specified \emph{observation procedure (OP)}, for edges that are \emph{incident} to the nodes in $s_0$. The observed edges, denoted by $A_s$ for $A_s \subseteq A$, are specified using a \emph{reference set} $s_{ref}$, where $s_{ref} \subseteq U\times U$, such that any existing edge $(ij)$ in $A$ is observed if $(ij) \in s_{ref}$. That is, $s_{ref}$ specifies the parts of the adjacency matrix that are observed under the given OP. Denote by $\mbox{Inc}(a_{ij}) = \{i, j\}$ the nodes that are incident to the edge $(ij)$. Let $\mbox{Inc}(A_s) = \cup_{a_{ij} \in A_s} \mbox{Inc}(a_{ij})$ be the set of nodes incident to the edges $A_s$. The \emph{sample graph} is given by
\[
G_s = (U_s, A_s) \quad\text{and}\quad U_s = s_0 \cup \mbox{Inc}(A_s) ~.
\]
The motifs that are observed in the sample graph $G_s$ can now be given as follows: $\forall k\in \Omega$, we have $k\in \Omega_s = \Omega(G_s)$, iff $M_k \times M_k \subseteq s_{ref}$. In particular, notice that $M_k \subseteq A_s$ does not imply $k\in \Omega_s$ in general, but $k\in \Omega_s$ must imply $M_k \in U_s$.
\subsection{BIG sampling}
Graph sampling can be given a BIGS representation, provided the following. Let
\[
\mathcal{B} = (F\cup \Omega; H)
\]
be the BIG associated with the population graph $G$ and the motif set $\Omega = \Omega(G)$, where the edges $H$ exist only between $F$ and $\Omega$ but not between any $i,j\in F$ or $k,l\in \Omega$, and an edge exists from any $i\in F$ to $k\in \Omega$ iff $k\in \Omega_s$ whenever $i\in s_0$, so that graph sampling from $G$ can be represented as sampling from $\mathcal{B}$ by \emph{incident} OP \citep{ZhangPatone2017} given $s_0$. Hence the term ``bipartite incidence graph''.
For the aforementioned example of indirect sampling, we can simply let $F$ be the set of hospitals, and let $\Omega$ be the set patients. We have $(ik)\in H$, or $h_{ik} =1$, iff the patient $k$ receives treatment at hospital $i$. We have $\sum_{i\in F} h_{ik} > 1$, for a patient $k$ receiving treatment from multiple hospitals. Let $\beta_k$ denote the \emph{predecessors} of $k$ in $\mathcal{B}$, where $h_{ik} = 1$ for any $i\in \beta_k$ and $h_{ik} =0$ if $i\not \in \beta_k$. Clearly, we would observe $k$, denoted by $k\in \Omega_s$, if any of the hospitals in $\beta_k$ is selected in the initial sample $s_0$. Indirect sampling can thus be represented as BIG sampling from $\mathcal{B}$.
More generally, for any graph sampling from given $G$, let $\delta_{i,k} =1$ for any $i\in F$ and $k\in \Omega$, iff $k\in \Omega_s$ whenever $i\in s_0$, or $\mbox{Pr}(k\in \Omega_s | i\in s_0) = 1$, according to the graph sampling design, which consists of $p(s_0)$ and the OP given $s_0$. For any $i\in F$, let
\[
\alpha_i = \{ k : k\in \Omega, \delta_{i,k} = 1\} ~,
\]
which contains all the \emph{successors} of $i$ in $\mathcal{B}$; for any $k\in \Omega$, let
\[
\beta_k = \{ i : i\in F, \delta_{i,k} = 1\} ~,
\]
which contains all the predecessors of $k$ in $\mathcal{B}$. In other words, $(ik) \in H$ or $h_{ik}=1$ in $\mathcal{B}$, iff $\delta_{i,k} =1$ for $i\in F$ and $k\in \Omega$. The \emph{sample BIG} is given by
\begin{equation} \label{sampleBIG}
\mathcal{B}_s = (s_0, \Omega_s; H_s)\quad\text{and}\quad \Omega_s = \alpha(s_0) = \cup_{i\in s_0} \alpha_i
\quad\text{and}\quad H_s = H\cap (s\times \Omega_s) ~.
\end{equation}
Finally, to ensure ancestral OP in BIG, we must also observe $\beta(\Omega_s) \setminus s_0$ even though it is not part of $\mathcal{B}_s$, where $\beta(\Omega_s) = \cup_{k\in \Omega_s} \beta_k$.
Below we summarise in Theorem \ref{thm:BIG} the sufficient and necessary conditions, by which one can determine whether such BIGS representation of graph sampling from $G$ is feasible or not.
\begin{theorem} \label{thm:BIG}
Graph sampling from $G =(U, A)$ with associated motifs $\Omega$ of interest, based on $p(s_0)$ and the given OP, can be represented by ancestral BIG sampling from $\mathcal{B}$, iff
\begin{itemize}[leftmargin=10mm]
\item[(i)] $\forall k\in \Omega$ and $i\in F$, $\delta_{i,k} =1$ or 0 in $G$ can be determined given $i\in s_0$ alone;
\item[(ii)] $\forall k\in \Omega$, we have $\beta_k \neq \emptyset$ in $\mathcal{B}$, or equivalently $\mathop{\cup}\limits_{i\in F} \alpha_i = \Omega$ in $\mathcal{B}$;
\item[(iii)] graph sampling OP in $G$ ensures the observation of $\beta\big(\alpha(s_0)\big) \setminus s_0$ in $\mathcal{B}$.
\end{itemize}
\end{theorem}
\begin{proof
Given (i), we can define the edge set $H$ of $\mathcal{B} = (F, \Omega; H)$. Given (ii), BIG sampling covers all the motifs in $\Omega$, since $\mbox{Pr}(k\in \Omega_s)$ is then positive for any $k\in \Omega$. Given (iii), it is possible to calculate the inclusion probability of $k\in \Omega_s$, based on $p(s_0)$ for $s_0 \subset F$. Thus, conditions (i) - (iii) are sufficient. They are also necessary, because removing any of them would render the BIGS representation infeasible.\end{proof}
Let us illustrate the application of Theorem \ref{thm:BIG} with two examples. First, consider \emph{induced} OP given $s_0$, where we have $s_{ref} = s_0 \times s_0$, such that an edge $(ij)$ is observed in $A_s$ only if both $i$ and $j$ are in $s_0$. For example, let $M_k = \{1, 2, 3\}$ form a triangle (motif of interest) in $G$, it is observed under induced OP only if all the three nodes are in $s_0$, but not otherwise. Graph sampling from $G$ is a probability sampling design, as long as the third-order inclusion probability $\mbox{Pr}(1\in s_0, 2\in s_0, 3\in s_0)$ is positive under the given $p(s_0)$, but one would not be able to represent it by BIG sampling since condition (i) is violated, as $\delta_{i,k}$ cannot be determined given $i\in s_0$ alone, for any $i =$ 1, 2 or 3.
Second, let $M_k = \{ 1, 2, 3 \}$ be a connected component (motif) in $G$, where $a_{12}= a_{23} =1$ but $a_{ij} = a_{ji} = 0$ otherwise for $i\in M_k$, $j\in U$. Let the initial sample size be $|s_0| =1$.
\begin{itemize}[leftmargin=6mm]
\item As in the previous example, BIGS representation is infeasible for induced OP in $G$.
\item Suppose \emph{incident reciprocal} OP, where $s_{ref} = s_0 \times U \cup U \times s_0$. We would observe all the three nodes of $M_k$ given $s_0 = \{ 2\}$. But condition (i) is still not satisfied, because we do not have $M_k \times M_k \subseteq s_{ref} = \{ 2\} \times U \cup U \times \{ 2\}$.
\item Suppose \emph{2-stage snowball sampling} with incident reciprocal OP, where $s_{ref} = \big(s_0 \cup \alpha(s_0)\big) \times U \cup U \times \big(s_0 \cup \alpha(s_0)\big)$. Now we would observe $[M_k]$ given $s_0 = \{2\}$, because the second stage snowball observation from $\alpha(s_0) = \{1, 3\}$ obtained in the first stage would confirm that there are no other adjacent nodes to $M_k$. Thus, $h_{2k} =1$ in $\mathcal{B}$, and conditions (i) and (ii) are satisfied. Condition (iii) is also satisfied for this motif, since it is confirmed that no other nodes in $F$ could lead to it, i.e. $\beta_k \setminus \{2\} = \emptyset$ in $\mathcal{B}$. BIGS representation is feasible for this motif.
\end{itemize}
\section{Indirect, network, adaptive cluster sampling} \label{sec:unconventionalsampling}
Below we describe formally BIGS representation as a unified approach to indirect sampling, network sampling and adaptive cluster sampling.
\subsection{Indirect sampling}
Generally for indirect sampling, let $F$ be the sampling frame, and $\Omega$ the set of measurement units of interest, which are accessible via the sampling units in $F$. For instance, $F$ can be the hospitals and $\Omega$ the patients treated by the hospitals in $F$, as in \citet{BirnbaumSirken1965}. Or, $F$ can be all the parents and $\Omega$ the children to the people in $F$, as in \citet{Lavallee2007}. For any $i\in F$ and $k\in \Omega$, we have $(ik)\in H$ or $h_{ik} = 1$ iff $k$ can be reached given $i\in s_0$, denoted by $\delta_{i,k} =1$. This completes the definition of population graph $\mathcal{B} = (F, \Omega; H)$. The knowledge of multiplicity that is collected under indirect sampling ensures then ancestral BIG sampling from $s_0 \subset F$, where the sample BIG is given by \eqref{sampleBIG}, with the associated out-of-sample \emph{ancestors} $\beta\big( \alpha(s_0) \big) \setminus s_0$ in $\mathcal{B}$.
The probability of inclusion in $\Omega_s$ can be derived from the initial sampling distribution $p(s_0)$, for $s_0 \subset F$. The (first-order) inclusion probability of $k\in \Omega_s$ is given by
\begin{equation} \label{eq:pi1motif}
\pi_{(k)} = 1 - \bar{\pi}_{\beta_k} = 1 - \mbox{Pr}\big( \cap_{i\in \beta_k} i\not \in s_0 \big) ~,
\end{equation}
where $\bar{\pi}_{\beta_k}$ is the exclusion probability of $\beta_k$ in $s_0$, i.e. the probability that none of the ancestors of $k$ in $\mathcal{B}$ is included in the initial sample $s_0$. Notice that the knowledge of the out-of-sample ancestors $\beta_k \setminus s_0$ is required to compute $\bar{\pi}_{\beta_k}$. Similarly, the second-order inclusion probabilities of $k\neq l\in \Omega_s$ is given by
\begin{equation} \label{eq:pi2motif}
\pi_{(kl)} = 1 - \big( \bar{\pi}_{\beta_k} + \bar{\pi}_{\beta_l} - \bar{\pi}_{\beta_k \cup \beta_l} \big) ~.
\end{equation}
\subsection{Network sampling}
Sampling of siblings via an initial sample of households provides an example of network sampling \citep{Sirken2005}. Since the siblings may belong to different households, some of which are outside of the initial sample, the ``network'' relationship among the siblings is needed. Network sampling as such can be viewed as a form of indirect sampling, since the sampling unit (household) is not the unit of measurement (siblings), and the latter cannot be sampled directly. Notice that the term ``network'' has a specific meaning here, unlike when network refers to a whole \emph{valued graph} \citep{Frank1980a,Frank1980b}, e.g. an electricity network, where the nodes and edges have associated values that are of interest.
Let $F$ denote the sampling frame, which is the list of households from which the initial sample $s_0$ can be selected.
Provided the OP under network sampling is \emph{exhaustive}, in the sense that all the siblings are observed, if at least one of them belongs to a household in $s_0$, one can treat each network of siblings as a motif of interest, such that $\Omega$ consists of all the networks of siblings. For any $i\in F$ and $k\in \Omega$, let $(ik)\in H$ iff at least one of the siblings in $M_k$ belongs to household $i$. This yields the population graph $\mathcal{B} = (F, \Omega; H)$. Network sampling with observation of multiplicity is then equivalent to ancestral BIG sampling in $\mathcal{B}$, where $\Omega_s =\alpha(s_0)$, with the associated out-of-sample \emph{ancestors} $\beta\big( \alpha(s_0) \big) \setminus s_0$, such that the inclusion probabilities of the motifs can be calculated by \eqref{eq:pi1motif} and \eqref{eq:pi2motif}.
\subsection{Adaptive cluster sampling (ACS)} \label{sec:ACS}
As a standard example of ACS \citep{Thompson1990}, let $F$ consist of a set of spatial grids over a given area. Let $y_i$ be the amount of a species, which can be found in the $i$-th grid. Given $i\in s_0$, one would survey all its neighbour grids (in four directions) if $y_i$ exceeds a threshold value but not otherwise. The OP is repeated for all the neighbour grids, which may or may not generate further grids to be surveyed. The process is terminated, when the last observed grids are all below the threshold. The interest is to estimate the total amount of species (or mean per grid) over the given area.
One can consider each cluster of contiguous grids, where the associated $y_i$'s all exceed the threshold value, as a network. Let a grid with $y_i$ below the threshold value form a singleton network consisting only of itself. The OP is network exhaustive, since all the grids in a network are observed if at least one of them is selected in $s_0$. A singleton network is an \emph{edge} grid, if it is contiguous to a non-singleton network. Observing a non-singleton network will lead one to observe all its edge grids, but not the other way around, due to the adaptive nature of the OP. When an edge grid is selected in $s_0$, but none of the grids in its non-singleton neighbour network (NNN), the inclusion probability of this edge grid cannot be calculated correctly based on the observed sample.
Below we explain how BIGS can be used to represent the approach proposed by \citet{Thompson1990}, and there can exist other feasible BIGS representations to ACS. The alternative strategies will be illustrated using the example of \citet{Thompson1990}.
\subsubsection{Alternative strategies of feasible BIGS representation} \label{sec:alternativestrategies}
One can represent ACS as BIG sampling from $\mathcal{B}$, where the grids are both the sampling units of $F$ and the motifs of $\Omega$. Let $h_{ik} =1$ if $k$ is observed under ACS whenever $i \in s_0$, for $i\in F$ and $k\in \Omega$. However, the OP of ACS is not ancestral when an edge grid is selected in $s_0$, but none of the grids in its NNN, in which case one would not observe its NNN that are its ancestors in this $\mathcal{B}$ and its inclusion probability cannot be calculated. \citet{Thompson1990} proposes to make an edge grid \emph{eligible} for estimation only if it is selected in $s_0$ directly, the probability of which is known, but not when it is observed via its NNN. Denote this strategy by $(\mathcal{B}, t_{HT}^*)$, with the modified HT estimator $ t_{HT}^*$.
Another strategy is to adopt a feasible BIGS representation, by which the OP of ACS \emph{is} ancestral and one can use the unmodified HT estimator. Two examples are given below.
\begin{itemize}[leftmargin=6mm]
\item $(\mathcal{B}^*, t_{HT})$: An edge grid $k$ is ineligible if it is only observed via its NNN but itself is not selected in $s_0$. That is, set $h_{ik} = 0$ in $\mathcal{B}^*$, where grid $i$ belongs to the NNN of $k$, such that $k$ is eligible for estimation only when it is selected in $s_0$ directly.
\item $(\mathcal{B}^{\dagger}, t_{HT})$: An edge grid $k$ is ineligible if itself is selected in $s_0$ but not its NNN. That is, set $h_{kk} =0$ for an edge grid but keep $h_{ik} = 1$ if grid $i$ belongs to the NNN of $k$, such that $k$ is eligible for estimation only if it is observed via its NNN.
\end{itemize}
It should be noticed that while strategy $(\mathcal{B}^{\dagger}, t_{HT})$ is feasible in the example of \citet{Thompson1990} considered below, it would be infeasible generally provided there exists some edge grid $k$ that is contiguous to more than one NNN, since not all of them will necessarily be included in $s_0$. Moreover, strategy $(\mathcal{B}^*, t_{HT})$ is likely to be more efficient than $(\mathcal{B}^{\dagger}, t_{HT})$, because the inclusion probability of an edge grid tends to be lower under the former, and an edge grid by definition has an associated $y$-value below the threshold. As a matter of fact one obtains the same estimate under either the strategy $(\mathcal{B}, t_{HT}^*)$ or $(\mathcal{B}^*, t_{HT})$. However, $ t_{HT}$ is unmodified under $(\mathcal{B}^*, t_{HT})$, so that it is unchanged by the Rao-Blackwell method; whereas, being a modified HT estimator, $t_{HT}^*$ under $(\mathcal{B}, t_{HT}^*)$ differs generally to its Rao-Blackwellised version, denoted by $t_{RBHT}^*$.
\subsubsection{Example of \citet{Thompson1990}} \label{sec:exampleThompson}
The population consists of 5 grids, with $y$-values $\{ 1, 0, 2, 10,1000\}$. Each grid has either one or two neighbours which are adjacent in the given sequence, as when they are 5 grids beside one another along the west-east axis. The threshold value is 5, such that only the two grids with values 10 and 1000 will lead on to their neighbours. The initial sample of size 2 is by simple random sampling (SRS) without replacement. Let $\Omega$ consist of the same 5 grids. Under strategy $(\mathcal{B}^*, t_{HT})$, $\mathcal{B}^*$ has the following incidence edges:
\[
H^* = \{ (1,1), (0,0), (2,2), (10,10), (10,1000), (1000,10), (1000, 1000)\} ~,
\]
where as in \citet{Thompson1990} we simply denote each grid by its $y$-value. The two grids $\{ 10, 1000\}$ form an NNN to the edge grid 2. The incidence edges $(10, 2)$ and $(1000, 2)$, which are in $\mathcal{B}$ of the strategy $(\mathcal{B}, t_{HT}^*)$, are removed to ensure ancestral OP and feasible BIGS representation. For instance, given $s_0 = \{ 0, 2 \}$, the OP of ACS is not ancestral in $\mathcal{B}$ since the grids $10$ and $1000$ will not be observed, but it is ancestral in $\mathcal{B}^*$, since the grid $2$ has only itself as the ancestor in $\mathcal{B}^*$. Under $(\mathcal{B}^{\dagger}, t_{HT})$, we have
\[
H^{\dagger} = \{ (1,1), (0,0), (10,2), (10,10), (10,1000), (1000, 2), (1000,10), (1000, 1000)\} ~,
\]
where the grid $2$ is only eligible for estimation when observed via its NNN $\{ 10, 1000\}$, but not when itself is selected in $s_0$, now that $(2,2) \not \in H^{\dagger}$. The OP of ACS is ancestral in $\mathcal{B}^{\dagger}$, since both $10$ and $1000$ are observed whenever $2$ is observed.
\begin{table}[ht]
\begin{center}
\caption{Strategies by BIGS representation of ACS from $\{ 1, 0, 2, 10, 1000\}$.}
\begin{tabular}{l | lr | lr | lr} \hline \hline
& \multicolumn{2}{c}{$(\mathcal{B}, t_{HT}^*)$} & \multicolumn{2}{|c|}{$(\mathcal{B}^*, t_{HT})$}
& \multicolumn{2}{c}{$(\mathcal{B}^{\dagger}, t_{HT})$} \\
$s_0$ & $\Omega_s$ & $t_{HT}^*$ & $\Omega_s$ & $t_{HT}$ & $\Omega_s$ & $t_{HT}$ \\ \hline
1,0 & 1,0 & 0.500 & 1,0 & 0.500 & 1,0 & 0.500 \\
1,2 & 1,2 & 1.500 & 1,2 & 1.500 & 1 & 0.500 \\
0,2 & 0,2 & 1.000 & 0,2 & 1.000 & 0 & 0.000 \\
1,10 & 1,10,\emph{2},1000 & 289.071 & 1,10,1000 & 289.071 & 1,10,2,1000 & 289.643 \\
1,1000 & 1,1000,\emph{2},10 & 289.071 & 1,1000,10 & 289.071 & 1,1000,2,10 & 289.643 \\
0,10 & 0,10,\emph{2},1000 & 288.571 & 0,10,1000 & 288.571 & 0,10,2,1000 & 289.143 \\
0,1000 & 0,1000,\emph{2},10 & 288.571 & 0,1000,10 & 288.571 & 0,1000,2,10 & 289.143 \\
2,10 & 2,10,1000 & 289.571 & 2,10,1000 & 289.571 & 2,10,1000 & 289.143 \\
2,1000 & 2,1000,10 & 289.571 & 2,1000,10 & 289.571 & 2,1000,10 & 289.143 \\
10,1000 & 10,1000,\emph{2} & 288.571 & 10,1000 & 288.571 & 10,1000,2 & 289.143 \\ \hline
\multicolumn{1}{l}{Variance} & \multicolumn{2}{|r|}{17418.4} & \multicolumn{2}{r}{17418.4} & \multicolumn{2}{|r}{17533.7} \\ \hline \hline
\end{tabular} \label{tab-ACS} \end{center} \end{table}
Table \ref{tab-ACS} lists the details of the three strategies by BIGS representation of ACS in this case. The respective observed sample $\Omega_s$ is given in addition to the initial sample $s_0$. The strategy $(\mathcal{B}, t_{HT}^*)$ is proposed in \citet{Thompson1990}, where $2$ is given in italic in the 5 samples where it is observed but unused for estimation. The probability that it is eligible is $2/5$, which is the same as its sample inclusion probability under $(\mathcal{B}^*, t_{HT})$.
Apart from the $2$'s in italics, the observed sample $\Omega_s$ is always the same under both the strategies $(\mathcal{B}, t_{HT}^*)$ and $(\mathcal{B}^*, t_{HT})$. Hence, the estimate is the same by both. Nevertheless, as explained before, the two differ regarding the Rao-Blackwell method. In this case, the difference hinges on the last sample $s_0 = \{ 10, 1000\}$. Under $(\mathcal{B}, t_{HT}^*)$, the same sample (including $2$) is also observed from $s_0 = \{ 2, 10\}$ or $\{ 2, 1000\}$, but the estimate $t_{HT}^*$ differs because $2$ is unused when $s_0 = \{ 10, 1000\}$. The Rao-Blackwell method yields $t_{RBHT}^* = 289.238$ given $\Omega_s =\{ 2, 10, 1000\}$. In contrast, under the strategy $(\mathcal{B}^*, t_{HT})$ the estimate $t_{HT}$ is unchanged by Rao-Blackwellisation, because the observed sample $\Omega_s$ from $s_0 = \{ 10, 1000\}$ differs to that from $s_0 = \{ 2, 10\}$ or $\{ 2, 1000\}$.
Under the strategy $(\mathcal{B}^{\dagger}, t_{HT})$, the grid $2$ is not included in $\Omega_s$ given $s_0 = \{ 1,2\}$ or $\{ 0, 2\}$, yielding different $t_{HT}$ to that under $(\mathcal{B}^*, t_{HT})$. Otherwise, the inclusion probability of $2$ is raised to $7/10$, i.e. the same as $10$ or $1000$, which is not a good choice because $2$ is much smaller than 10 or 1000. The variance of $t_{HT}$ is larger than under the strategy $(\mathcal{B}^*, t_{HT})$, although the relative efficiency 0.993 is not of a great concern here.
\section{$T$-stage snowball sampling ($T$-SBS)} \label{sec:TSBS}
\citet{Goodman1961} considers \emph{snowball sampling (SBS)} on a special directed graph, where each node has one and only one out-edge. \citet{Frank1977a} and \citet{FrankSnijders1994} consider one-stage SBS from arbitrary population graphs. \citet{ZhangPatone2017} derive the HT-estimator for general $T$-stage snowball sampling ($T$-SBS). Additional stages of sampling are generally needed in order to identify the ancestors of all the motifs observed under $T$-SBS though. The matter can be illustrated using ACS as follows.
Let $G = (U,A)$ be an undirected simple graph, where $U$ consists of all the grids over a given area, and $(ij)\in A$ iff grids $i$ and $j$ are neighbours and they \emph{both} have values above the threshold. Each grid with $y$-value below the threshold is an isolated node in $G$. Let $F = U$, yielding an initial sample of \emph{seeds} $s_0$ according to $p(s_0)$, where $s_0 \subset U$. Propagation of the sample is only possible from those nodes that are not isolated in $G$; an isolated node with value above the threshold can only be observed if it is selected in $s_0$. Let $s_1 = s_0 \cup \alpha(s_0)$ be the sample of nodes observed after the first stage, where $s_1 \setminus s_0$ is the first-wave snowball sample, which are the \emph{seeds} for the second stage snowball sample, and so on. Denote by $s_T$ the observed sample of nodes after $T$ stages. Under ACS, one would eventually observe all the networks of grids, treated as the motifs of interest, which have at least one node in $s_0$. However, under $T$-SBS the sampling is terminated after $T$ stages, by which time $s_T$ may have only covered a part of a network.
Similarly, for any population graph $G$, a motif in $\Omega(G)$ may be unobserved under $T$-SBS, even though it is observable under SBS with an infinite number of stages. Moreover, not all the observed motifs after $T$ stages are eligible for estimation, and additional stages of sampling may be required in order to observe all the ancestors that could have led to an observed motif by $T$-SBS. However, more motifs of interest may be observed during the additional sampling, which again may or may not be eligible for estimation. Below we develop BIGS representation of $T$-SBS from arbitrarily given population graph, by which this conundrum of ancestral observation can be resolved.
\subsection{Observation distance to motif $k$ from within $M_k$}
For any $k\in \Omega$ and $i\neq j \in M_k$, let $\nu_{ij}$ be the length of the \emph{geodesic} from $i$ to $j$ in $G$, which is the shortest path from $i$ to $j$ in $G$. Since the shortest path from $i$ to $j$ varies with the OP, let us assume incident reciprocal observation for simplicity of exposition here. For example, let $M_k = \{ 1, 2, 3\}$, where $a_{23}= 1$ and $a_{ij} = 0$ for $i\in M_k$ and $j\in U$ otherwise. We have $\nu_{23}= \nu_{32} = 1$ and $\nu_{12} = \nu_{13} = \infty$. Or, for the same $M_k = \{ 1, 2, 3\}$, let $a_{12} = 1$ in addition to $a_{23} = 1$, in which case we have $\nu_{21} = \nu_{23} = 1$ and $\nu_{13}= 2$.
Starting from $i\in s_0 \cap M_k$, the number of stages required to observe all the nodes in $M_k$ by SBS is $\max_{j\in M_k} \nu_{ij}$.
Next, for any $k\in \Omega$ and $i\in M_k$, let $d_{i,k}$ be the SBS \emph{observation distance} from $i$ to $k$, which is the minimum number of stages required to observe $k\in \Omega_s$ under SBS from $G$, when starting from $i$. For the above two examples of $M_k = \{ 1, 2, 3\}$, if only $a_{23}= 1$, then $d_{1,k} = \infty$ and $d_{2,k} = d_{3,k} = 2$; whereas with $a_{12}= a_{23}= 1$, we have $d_{1,k} = d_{2,k} = d_{3,k} = 2$. Generally, we have $d_{i,k} \leq 1+ \max_{j\in M_k} \nu_{ij}$, since we must have $M_k\times M_k \subseteq s_{t+1,ref}$ if $M_k \subseteq s_t$, where $s_{t+1,ref}$ is the reference set of $(t+1)$-stage SBS from $G$. A more detailed result for connected $M_k$ can be given as follows.
\begin{lemma} \label{lemma:dik}
$\forall k\in \Omega$ and $i\in M_k$, if the nodes $M_k$ are connected in $G$, then
\[
d_{i,k} = \begin{cases} \max_{j\in M_k} \nu_{ij} & \text{if } | \arg \max_{j\in M_k} \nu_{ij} | = 1 \\
1 + \max_{j\in M_k} \nu_{ij} & \text{otherwise} \end{cases} ~,
\]
or if there exists a single node other than $i$ which is unconnected to $i$ in $G$, then
\[
d_{i,k} = 1 + \max_{j\in M_{k;i}} \nu_{ij}
\]
where $M_{k;i}$ consists of the nodes in $M_k$ that are connect to $i$ in $G$.
\end{lemma}
\begin{proof
Starting from $i$, it is impossible under $T$-SBS to observe whether there are edges or not among the nodes that are unconnected to $i$, if there are two or more of them. This leaves one with the two possibilities listed above. Let the nodes of $M_k$ be connected, if there is only one node (denoted by $j_0$) which requires the maximum no. steps from $i$, then all the other nodes are observed before $j_0$, which allows one to observe any edge between them and $j_0$ by the last step; whereas if there are more than one node like $j_0$, then an additional step is need to observe the edges among them. Similarly, such an additional step is needed, when there is a single node that is unconnected to $i$.
\end{proof}
\begin{corollary}\label{cor:notBIG}
If there exists $k\in \Omega$, where there are at least two nodes, $i\neq j \in M_k$, such that $d_{i,k} = d_{j,k} = \infty$ in $G$, then BIGS representation of $T$-SBS from $G$ is infeasible.
\end{corollary}
\begin{proof
There is no edge in $\mathcal{B}$ for such a motif $k$, from any $i\in M_k$. Starting from any $i\not \in M_k$, one can reach at most one of the connected components of $M_k$ in $G$. Hence, there are no edges from $F\setminus M_k$ to $k$ either, and condition (ii) of Theorem 1 is violated.
\end{proof}
\subsection{Observation distance to motif $k$ from outside $M_k$}
To clarify the observation distance to motif $k$ from outside of $M_k$, we introduce graph $G_h = (U_h, A_h)$ transformed from $G= (U,A)$ via a \emph{hypernode} $\mathbf{h}$ consisting of the nodes $i_1, ..., i_q$ in $G$. Let $U_{h} = \{\mathbf{h}\} \cup U_{\mathbf{h}}^c$, where $U_{\mathbf{h}}^c = U\setminus U_{\mathbf{h}}$ and $U_{\mathbf{h}} = \{ i_1, ..., i_q \}$, on replacing the $q$ nodes with the hypernode $\mathbf{h}$. Partition the edge set of $G$ as
\[
A = \Big( A \cap (U_{\mathbf{h}}\times U_{\mathbf{h}}) \Big) \cup \Big( A\cap(U_{\mathbf{h}}^c\times U_{\mathbf{h}}^c) \Big)
\cup \Big( A\cap \big( U_{\mathbf{h}} \times U_{\mathbf{h}}^c \cup U_{\mathbf{h}}^c \times U_{\mathbf{h}} \big) \Big) ~.
\]
Remove all the edges $A \cap (U_{\mathbf{h}}\times U_{\mathbf{h}})$ in $G$, which are among the $q$ nodes themselves. Keep in $A_h$ all the edges $A\cap(U_{\mathbf{h}}^c\times U_{\mathbf{h}}^c)$ in $G$, which are not incident to nodes in $U_{\mathbf{h}}$. Regarding $A\cap \big( U_{\mathbf{h}} \times U_{\mathbf{h}}^c \cup U_{\mathbf{h}}^c \times U_{\mathbf{h}} \big)$: for each $j\in U_{\mathbf{h}}^c$, replace all the edges $\{ (ij) : i\in U_{\mathbf{h}}, (ij) \in A\}$ by a single edge $(\mathbf{h}j)$ in $A_h$, and replace all the edges $\{ (ji) : i\in U_{\mathbf{h}}, (ji) \in A\}$ by a single edge $(j\mathbf{h})$ in $A_h$. This yields the transformed graph $G_h =(U_h, A_h)$ via hypernode $\mathbf{h}$.
For any $k\in \Omega$ and $i\not \in M_k$, we have $d_{i,k} = \infty$ if $\nu_{ij} = \infty$ for all $j\in M_k$, in which case motif $k$ cannot be reached from $i$. Otherwise, the nodes in $M_k$ can be partitioned according to $\nu_{ij}$ for each $j\in M_k$. Let $U_{\mathbf{h}} = \{ j : j\in M_k, \nu_{ij} = h\}$ contain the nodes in $M_k$ with geodesic length $h$ to $i$. It takes one more step to observe $U_{\mathbf{h}}\times U_{\mathbf{h}}$ starting from $U_{\mathbf{h}}$. Let $G_h$ be transformed from $G$ via the hypernode ${\mathbf{h}}$, if $U_{\mathbf{h}} \neq \emptyset$. Let
$M_k^h = \{ \mathbf{h}\} \cup (M_k \setminus U_{\mathbf{h}})$. The observation distance from $\mathbf{h}$ to motif $[M_k^h]$ in $G_h$, denoted by $d_{\mathbf{h},k}$, can be calculated as that from any $i$ to $[M_k]$ in $G$ for $i\in M_k$, where the minimum value is 1, including when $M_k = U_{\mathbf{h}}$. Thus, it takes $d_{\mathbf{h},k}$ stages to observe motif $k$ in G, starting from all the nodes in $U_{\mathbf{h}}$ at once, from which we obtain the following result.
\begin{lemma} \label{lemma:hypernode}
$\forall k\in \Omega$ and $i\not \in M_k$, we have $d_{i,k} = \min_{h\geq 1} (h + d_{{\mathbf{h}},k})$, where
$d_{\mathbf{h},k}$ is the observation distance from hypernode $\mathbf{h}$ to motif $[M_k^h]$ in $G_h$, transformed from $G$ via the hypernode $\mathbf{h}$ consisting of nodes $U_{\mathbf{h}} = \{ j : j\in M_k, \nu_{ij} = h\}$.
\end{lemma}
\subsection{Estimation using all the motifs observed under $T$-SBS}
The sample graph observed under $T$-SBS from $G$ has an associated matrix of geodesic distances, which is of dimension $|s_T| \times |s_T|$, where the $(i,j)$-th element is the geodesic distance from $i$ to $j$ in the sample graph $G_s$, denoted by $\nu_{ij}(G_s)$. For instance, we have $\nu_{ij}(G_s) = 1$ iff $(ij)\in A_s$, in which case we have $\nu_{ij}(G) = 1$ in $G$ as well. For non-adjacent nodes $i$ and $j$ in $G$, we have $\nu_{ij}(G_s) = \nu_{ij}(G) < \infty$, provided the connected component containing them in $G$ is fully observed in $G_s$, but not otherwise. Thus, the geodesic-distance matrix based on the sample graph $G_s$ is generally not the same as that of the population graph $G$. Additional sampling in $G$ is then necessary, in order to identify the ancestors of any observed motif in $\Omega_s$, as specified below.
\begin{lemma}\label{lemma:ancestralT}
For any $k\in \Omega_s$, if $|M_k| >1$ then one needs at most $T-1$ stages of additional SBS from $M_k$ to observe all the ancestors of sample motif $k$ under $T$-SBS from $G$, if $|M_k| =1$ then one needs at most $T$ stages of additional SBS from $M_k$.
\end{lemma}
\begin{proof
$(T-1)$-SBS from all the nodes $M_k$ in $G$ is the same as $(T-1)$-SBS from the hypernode $\mathbf{h}$ with $U_{\mathbf{h}} = M_k$ in the graph transformed from $G$ via the hypernode $\mathbf{h}$. This identifies all the nodes in $G$, which can lead to the observation of at least one node in $M_k$ after $T-1$ stages at most. Since $\min_{i\in M_k} d_{i,k} >1$ if $|M_k|>1$, any node that is unobserved after the additional $T-1$ stages cannot be the ancestor of $k$ under $T$-SBS from $G$.
\end{proof}
Suppose $T$-SBS from $G$ is a probability sampling design for $\Omega(G)$ that is of interest.
For BIGS representation of $T$-SBS from $G$, let $F = U$ and $\Omega = \Omega(G)$. By Theorem \ref{thm:BIG}, one needs to set $h_{ik} = 1$ for any $i$ that is the ancestor of motif $k$ under $T$-SBS from $G$. One can set $h_{ik} =1$ in the sample graph $\mathcal{B}_s$ directly, provided $k\in \Omega_s$ can be observed in $G_s$ starting from $i\in s_0$. Moreover, having identified all the ancestors of each observed motif $k\in \Omega_s$ by additional sampling, as guaranteed under Lemma \ref{lemma:ancestralT}, one can set $h_{ik} =1$ for all the out-of-$s_0$ ancestors of $k$ under $T$-SBS from $G$. In this way, ancestral observation is achieved for all the motifs in $\Omega_s$, such that they all can be used for estimation.
\subsection{BIGS representation for eligible motifs under $T$-SBS}
Not all the motifs observed under $T$-SBS are eligible for estimation due to the requirement of ancestral observation. Using the same idea that is expounded for ACS in Section \ref{sec:alternativestrategies}, we develop below strategies of BIGS representation that are feasible based on the eligible motifs observed under $T$-SBS, without additional sampling for ineligible motifs.
Let $\mathcal{B}$ be the population BIG representing $T$-SBS from $G$, where all the observed motifs can be used for estimation. For each $k\in \Omega$ with ancestors $\beta_k$ in $\mathcal{B}$, let $\beta_k^*$ be a non-empty subset of $\beta_k$, where $\emptyset \neq \beta_k^* \subseteq \beta_k$. Consider BIG sampling with \emph{restricted ancestors} from $\mathcal{B}^* = (F, \Omega; H^*)$, where $H^*$ contains only the edges from $\beta_k^*$ to $k$, for each $k\in \Omega$. Since $\beta_k^*$ is non-empty for every $k\in \Omega$, conditions (i) and (ii) of Theorem \ref{thm:BIG} remain satisfied under BIG sampling from $\mathcal{B}^*$. A motif $k$ is observed in the sample $\mathcal{B}_s^*$, iff $s_0$ contains at least one of the nodes in $\beta_k^*$, regardless of the nodes in $\beta_k\setminus \beta_k^*$. Condition (iii) of Theorem \ref{thm:BIG} is satisfied provided the knowledge of $\beta_k^*$, given which the inclusion probabilities can be calculated by \eqref{eq:pi1motif} and \eqref{eq:pi2motif} on replacing $\beta_k$ and $\beta_l$ by $\beta_k^*$ and $\beta_l^*$, respectively.
To ensure that BIG sampling from $\mathcal{B}^*$ is a feasible representation of $T$-SBS from $G$, we need to define $\beta_k^*$ appropriately for the observed eligible motifs. By Corollary \ref{cor:notBIG}, BIGS representation is feasible for any motif consisting of connected nodes. Let the \emph{observation diameter} of a motif $k\in \Omega(G)$ be
\[
\phi_k = \max_{i\in M_k}~ d_{i,k}
\]
which is finite for any motif of connected nodes with $|M_k| < \infty$. Then, by definition, an observed motif with finite $\phi_k$ is eligible for estimation under $\phi_k$-SBS from $G$, provided we restrict its ancestors to $\beta_k^* = M_k$. The result below follows.
\begin{theorem}\label{thm:feasibleBIG}
Provided finite observation diameter $\phi_k$ of all $k\in \Omega$, BIG sampling from $\mathcal{B}^*$ is a feasible representation for $T$-SBS from $G$, where $\beta_k^* = M_k$ and $T = \max_{k\in \Omega} \phi_k$.
\end{theorem}
\begin{proof
Conditions (i) and (ii) of Theorem \ref{thm:BIG} are satisfied, provided $\phi_k < \infty$ for any $k\in \Omega$. Given $T = \max_{k\in \Omega} \phi_k$, all the nodes in $M_k$ are observed under $T$-SBS if $M_k \cap s_0 \neq \emptyset$, such that $\beta_k^* = M_k$ is identified for every $k$ in $\mathcal{B}_s^*$. Therefore, condition (iii) is satisfied as well, and all the motifs in $\mathcal{B}_s^*$ are eligible for estimation.
\end{proof}
Additional sampling is not needed based on BIGS from $\mathcal{B}^*$ with restricted ancestors as a feasible representation of $T$-SBS from $G$. But fewer observed motifs are used compared to BIGS representation with $\mathcal{B}$, which would generally require additional sampling. So there is a trade-off between statistical efficiency and operational cost. In case the uncertainty is too large to be acceptable, based on the eligible motifs in $\mathcal{B}_s^*$ under $T$-SBS with $T = \max_{k\in \Omega} \phi_k$, additional SBS may be administered. This raises the need to update the BIGS representation for $T'$-SBS, where $T' > T$.
Let $\beta^t(M_k)$ contain all the nodes outside of $M_k$, which have maximum geodesic distance $t$ to $M_k$. That is, starting from any node in $\beta^t(M_k)$, it takes at most $t$ stages of SBS to observe at least one of the nodes $M_k$. Under SBS beyond $T = \max_{k\in \Omega} \phi_k$, the nodes in $\beta^t(M_k)$ may be identified as ancestors of eligible motifs, for $t=1, 2,$ ... Let the \emph{diameter} of motif $k$ be given by
\[
\lambda_k = \max_{i,j\in M_k} \nu_{ij}
\]
By Lemma \ref{lemma:dik}, we have $\phi_k \leq 1 + \lambda_k$ given finite $\phi_k$. The result below follows.
\begin{theorem} \label{thm:feasibleBIGStar}
Provided finite observation diameter $\phi_k$ of all $k\in \Omega$, BIG sampling from $\mathcal{B}^*$ is a feasible representation for $T$-SBS from $G$, where $\beta_k^* = M_k\cup\beta^t(M_k)$ with $t\geq1$, and $T=\max_{k\in \Omega} T_k$ with $T_k = \lambda_k + 2t$ for each $k\in \Omega$.
\end{theorem}
\begin{proof}
Any motif $k$ is observed after at most $\phi_k+t$ stages starting from any node in $\beta_k^*$. The first two conditions of Theorem \ref{thm:BIG} are therefore satisfied. If $\phi_k = 1+ \lambda_k$, then all the nodes in $M_k$ must have been observed at stage $\phi_k + t-1$, so that all the nodes in $\beta(M_k)$ are already observed after $\phi_k + t = \lambda_k + t +1$ stages. It remains only to observe all the nodes $\beta^t(M_k) \setminus \beta(M_k)$ in $G$ starting from $\beta(M_k)$, which requires at most $t-1$ stages. Whereas, if $\phi_k = \lambda_k$, then there is at least one node $j\in M_k$, which is first observed at stage $\phi_k$, starting from any node in $M_k$. Another $t$ stages may be needed to observe all the nodes in $G$ which can lead to $j$ in $t$ stages from outside $M_k$. Thus, in either case, $\beta_k^*$ is identified for every $k$ in $\mathcal{B}_s^*$ given $T = \max_{k\in \Omega} T_k$, such that condition (iii) of Theorem \ref{thm:BIG} is also satisfied and all the motifs in $\mathcal{B}_s^*$ are eligible for estimation.
\end{proof}
\section{Two unbiased estimators under BIG sampling}\label{sec:estimation}
For each motif $k\in \Omega(G)$, let $y_k$ be an associated value, which is considered as an unknown constant. Let the target of estimation be the total of $y_k$ over $\Omega$, denoted by
\[
\theta = \sum_{k\in \Omega} y_k ~.
\]
In the case of $y_k\equiv 1$, $\theta$ is simply the total number of motifs in $\Omega$, which is called a \emph{graph total} \citep{ZhangPatone2017}; more generally, $\theta$ is a total over $\Omega$ in a valued graph.
The two unbiased estimators of \citet{BirnbaumSirken1965} can be applied to any graph sampling from $G$, provided a feasible BIGS representation of it satisfying conditions (i) - (iii) of Theorem \ref{thm:BIG}. For simplicity of exposition below, we always denote the population BIG by $\mathcal{B}$, without distinguishing in notation whether restricted ancestors $\mathcal{B}^*$ are used for the eligible motifs. The HT estimator based on $\Omega_s = \Omega(\mathcal{B}_s)$ is given by
\begin{equation}\label{eq:Yhat}
\hat{\theta}_y = \sum_{k\in \Omega_s} y_k/\pi_{(k)} = \sum_{k\in \Omega} \delta_k y_k/\pi_{(k)} ~,
\end{equation}
where $\delta_k =1$ if $k\in \Omega_s$ and 0 otherwise, and $\pi_{(k)}$ is given by \eqref{eq:pi1motif}, for any $k\in \Omega_s$. Generally, to calculate the inclusion probabilities $\pi_{(k)}$ and $\pi_{(kl)}$, we need to know $\beta_k$ for each $k\in \Omega_s$. In the special case of SRS of $s_0$, we only need the cardinality of $\beta_k$ to calculate $\pi_{(k)}$.
The HH-type estimator based on the initial sample $s_0$ is given by
\begin{equation} \label{eq:Zhat}
\hat{\theta}_z = \sum_{i\in s_0} z_i/\pi_i = \sum_{i\in F} \delta_i z_i/\pi_i \quad\text{and}\quad z_i = \sum_{k\in \alpha_i} \omega_{ik} y_k
\quad\text{and}\quad \sum_{i\in \beta_k} \omega_{ik} = 1 ~,
\end{equation}
where $\delta_i =1$ if $i\in s_0$ and 0 otherwise, and $\pi_i$ is the inclusion probability of $i\in s_0$ under $p(s_0)$, and the $\omega_{ik}$'s are constants of sampling, by which $\{ y_k : k\in \Omega \}$ are transformed to the constructed measures $\{ z_i : i\in F\}$. We let $\omega_{ik} =0$ if $i\not \in \beta_k$ or $k\not \in \alpha_i$ in $\mathcal{B}$. As noted by \citet{BirnbaumSirken1965}, the estimator \eqref{eq:Zhat} is unbiased for $\theta$ since
\[
\theta = \sum_{k\in \Omega} y_k = \sum_{k\in \Omega} y_k \big( \sum_{i\in \beta_k} \omega_{ik} \big)
= \sum_{i\in F} \big( \sum_{k\in \alpha_i} \omega_{ik} y_k \big) = \sum_{i\in F} z_i ~.
\]
Notice that in the special case of $|\beta_k| =1$ for all $k\in \Omega$, there exits only one-one or one-many relationship between the sampling units in $F$ and the motifs in $\Omega$, just like when the $|M_k|$ elements are clustered in the sampling unit $i$ under cluster sampling. The two estimators $\hat{\theta}_y$ and $\hat\theta_y$ are then identical. More generally, different choices of $\omega_{ik}$'s would give rise to different estimates, such that $\hat{\theta}_z$ by \eqref{eq:Zhat} defines in fact a family of unbiased estimators. \citet{BirnbaumSirken1965} consider the \emph{equal-share} weights $\omega_{ik} = |\beta_k|^{-1}$. Under BIG sampling, this estimator and the HT-estimator have the same ancestral observation requirement. \citet{Patone2020} proposes unequal weights $\omega_{ik}\propto|\alpha_i|^{-1}$. Additional sampling is generally needed to calculate these weights. For the feasible BIGS representation in Theorem \ref{thm:feasibleBIGStar}, one may need upto $\phi_k+t$ extra stages to observe $\alpha_i$ for any $i\in \beta^t(M_k)$. Both the HH-type estimators will be illustrated in Section \ref{sec:numericalwork}.
\begin{theorem} \label{thm:vardiff}
For $\hat{\theta}_y$ by \eqref{eq:Yhat} and $\hat{\theta}_z$ by \eqref{eq:Zhat} under BIG sampling from $\mathcal{B}$, we have
\[
V(\hat{\theta}_z) - V(\hat{\theta}_y) = \sum_{k\in \Omega} \sum_{l\in \Omega} \Delta_{kl} y_k y_l
\quad\text{where}\quad
\Delta_{kl} = \sum_{i\in \beta_k} \sum_{j\in \beta_l} \frac{\pi_{ij}}{\pi_i \pi_j} \omega_{ik} \omega_{jl} - \frac{\pi_{(kl)}}{\pi_{(k)} \pi_{(l)}} ~.
\]
\end{theorem}
\begin{proof
Since $\hat{\theta}_y$ is unbiased for $\theta$, we have
\[
V(\hat{\theta}_y) = E\big( \hat{\theta}_y^2 \big) - E\big( \hat{\theta}_y \big)^2
= \sum_{k\in \Omega} \sum_{l\in \Omega} \frac{\pi_{(kl)}}{\pi_{(k)}\pi_{(l)}} y_k y_l - \theta^2 ~.
\]
Rewrite $\hat{\theta}_z$ by \eqref{eq:Zhat} as $\hat{\theta}_z = \sum_{k\in \Omega_s} W_k y_k = \sum_{k\in \Omega} \delta_k W_k y_k$, where $W_k = \sum_{i\in \beta_k} \delta_i \omega_{ik}/\pi_i$. Again, since $\hat{\theta}_z$ is unbiased for $\theta$, we have $V(\hat{\theta}_z) = E\big( \hat{\theta}_z^2 \big) - \theta^2$, where
\begin{align*}
E\big( \hat{\theta}_z^2 \big) & = \sum_{k\in \Omega} \sum_{l\in \Omega} y_k y_l\, E\big(\delta_k W_k \delta_l W_l\big) \\
&= \sum_{k\in \Omega} \sum_{l\in \Omega} y_k y_l \,
E\Big(\delta_k \delta_l \sum_{i\in \beta_k} \sum_{j\in \beta_l} \frac{\delta_i \delta_j}{\pi_i \pi_j} \omega_{ik} \omega_{jl} \Big) \\
&= \sum_{k\in \Omega} \sum_{l\in \Omega} y_k y_l\, \mbox{Pr}\big(\delta_k\delta_l=1\big)
\Big(\sum_{i\in \beta_k} \sum_{j\in \beta_l} \frac{E(\delta_i \delta_j|\delta_k\delta_l=1)}{\pi_i \pi_j} \omega_{ik} \omega_{jl} \Big) \\
&= \sum_{k\in \Omega} \sum_{l\in \Omega} y_k y_l \, \mbox{Pr}\big(\delta_k\delta_l=1\big)
\Big(\mbox{Pr}(\delta_k\delta_l=1)^{-1}\sum_{i\in \beta_k} \sum_{j\in \beta_l} \frac{\mbox{Pr}(\delta_i \delta_j=1)}{\pi_i \pi_j} \omega_{ik} \omega_{jl} \Big) \\
&= \sum_{k\in \Omega} \sum_{l\in \Omega} y_k y_l \, \Big(\sum_{i\in \beta_k} \sum_{j\in \beta_l} \frac{\pi_{ij}}{\pi_i \pi_j} \omega_{ik} \omega_{jl}\Big)
\end{align*}
since $\delta_i \delta_j =1$ implies $\delta_k \delta_l =1$ under BIG sampling, for any $i\in \beta_k$ and $j\in \beta_l$. The result follows now from taking the difference $V(\hat{\theta}_z) - V(\hat{\theta}_y)$.
\end{proof}
Theorem \ref{thm:vardiff} is a general result regarding the relative efficiency between $\hat{\theta}_y$ by \eqref{eq:Yhat} and $\hat{\theta}_z$ by \eqref{eq:Zhat}, which applies to all situations where BIG sampling from $\mathcal{B}$ provides a feasible representation of the original graph sampling from $G$. In the special case where $s_0$ is selected by SRS without replacement, we have
\[
\Delta_{kl} = \frac{N}{n} \sum_{i\in \beta_k\cap\beta_l} \omega_{ik} \omega_{il}
+ \frac{N(n-1)}{n (N-1)} \sum_{i \in \beta_k} \sum_{j\neq i\in \beta_l} \omega_{ik} \omega_{jl} - \frac{\pi_{(kl)}}{\pi_{(k)} \pi_{(l)}} ~.
\]
Moreover, for equal-share weights $\omega_{ik}=|\beta_k|^{-1}$, we have
\[
\Delta_{kl} = \frac{N^2}{n(N-1)} (1-\frac{n}{N}) \frac{m_{kl}}{m_k m_l} + \frac{N(n-1)}{n (N-1)} - \frac{\pi_{(kl)}}{\pi_{(k)} \pi_{(l)}}
\]
where $m_k=|\beta_k|$, $m_l=|\beta_l|$ and $m_{kl}=|\beta_k\cap\beta_l|$, since
$\sum_{i\in\beta_k}\omega_{ik} = \sum_{j\in\beta_l}\omega_{jl}=1$.
\section{Numerical examples} \label{sec:numericalwork}
Below we apply first BIGS representation to an example of two-stage ACS \citep{Thompson1991}. Next, we illustrate BIGS representation of SBS from an arbitrarily generated population graph. Unbiased estimators are compared in terms of efficiency.
\subsection{Two-stage ACS}
The left plot in Figure \ref{figure:ACS_Thompson} gives the population graph of two-stage ACS in \citet[][p.1104]{Thompson1991}. Each strip is a primary sampling unit, and each grid a secondary sampling unit. Given a strip selected at the first stage, all the grids belonging to it are searched for the species. Next, neighbouring grids to those with species are searched, and so on, i.e. ACS is applied to the grids at the second stage, which is terminated once no more non-empty grids (i.e. those with species) are found in this way. An edge grid is an empty grid (i.e. one without species) which is contiguous to one or more non-empty grids.
The right plot in Figure \ref{figure:ACS_Thompson} gives a feasible BIGS representation of two-stage ACS above. Let $F$ consist of the strips, and $\Omega$ the grids. Each big node marked by a capital letter denotes a strip, the small nodes denote the grids. There are 10 star-like subgraphs, where a strip is adjacent to its 20 empty grids, which are observed in $\mathcal{B}_s$ only if this strip is selected in $s_0$. The small nodes (of grids) that are adjacent to four big nodes (of strips) form a cluster of non-empty grids, which are all observed if any of the four strips are selected in $s_0$. There are three such clusters of non-empty grids. Finally, each of the 10 strips that contains non-empty grids is also adjacent to the rest of its empty grids. In particular, there is no edge between an edge grid and its neighbour strip, despite the former can be observed via the latter, similarly to $\mathcal{B}^*$ in Section \ref{sec:alternativestrategies}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{ACS}
\caption{Two-stage ACS: population graph, left; BIGS representation, right.}
\label{figure:ACS_Thompson}
\end{figure}
Notice that, in this case, one could have used the BIG with unrestricted ancestors, where an edge grid is adjacent to its neighbour strip. This is because the $y$-value is 0 of any empty grid, so it does not contribute to the $y$-total estimator, whether or not one is able to calculate its inclusion probability. But the representation adopted here is equally applicable, when the singleton networks are associated with non-zero $y$-values, even though they can be small and below a threshold in the context of ACS.
\begin{table}[ht]
\def~{\hphantom{0}}
\centering
\caption{Standard errors given $n =|s_0|$}
\label{table:ACS}
\renewcommand{\tabcolsep}{0.5cm}
\begin{tabular}{rrrrr} \hline
$n$ & $\hat{\theta}$ & $\hat{\theta}_{y}$ & $\hat{\theta}_{z\beta}$ & $\hat{\theta}_{z\alpha}$ \\ \hline
1& 457& 356& 356& 329\\
2& 315& 236& 245& 226\\
3& 250& 179& 194& 180\\
4& 210& 142& 163& 151\\
5& 182& 116& 141& 131\\
6& 160& 95& 125& 115\\
7& 143& 78& 111& 103\\
8& 128& 63& 100& 92\\
9& 116& 51& 90& 83\\
10& 105& 40& 82& 75\\ \hline
\end{tabular}
\end{table}
The total of interest is $\theta = \sum_{i \in F} y_i=326$, summed over the strips. Let $n=|s_0|$. Four estimators are considered. The first one, denoted by $\hat{\theta}$, is only based on the grids belong to the $n$ stripes in $s_0$, without ACS at the second stage. The second estimator $\hat{\theta}_{y}$ given by \eqref{eq:Yhat} is based on the adopted BIGS representation. The inclusion probability of a non-empty grid is $\pi_{(k)}=1-\mathcal{C}(N-4,n)/\mathcal{C}(N,n)$ by \eqref{eq:pi1motif}, where $N = |F| = 20$, and $\mathcal{C}(a,b) = a!/\big(b!(a-b)!\big)$. The third and fourth estimators given by \eqref{eq:Zhat} are, respectively, $\hat{\theta}_{z\beta}$ with equal-share weights $\omega_{ik} = |\beta_k|^{-1}$ and $\hat{\theta}_{z\alpha}$ with weights $\omega_{ik}=|\alpha_i|^{-1}/\sum_{i\in\beta_k}|\alpha_i|^{-1}$, where $|\alpha_i|=2$ for strips C and D, and $|\alpha_i|=1$ for the rest. Notice that additional effort is needed for $\hat{\theta}_{z\alpha}$, where one must search for possible non-empty grids belonging to each strip encountered during ACS, but one does not need to survey these additional non-empty grids and obtain the associated $y$-values. We notice that the first three estimators are the same as those considered by \citet{Thompson1991}.
The standard errors are presented in Table \ref{table:ACS}, for sample sizes $n = 1, ..., 10$. The estimator $\hat{\theta}$ without ACS has the largest standard error for every sample size. The HT-estimator $\hat{\theta}_y$ is more efficient than both $\hat{\theta}_{z\beta}$ and $\hat{\theta}_{z\alpha}$ as $n$ increases. The estimator $\hat{\theta}_{z\alpha}$ with unequal weights is the most efficient for $n<3$, and is always more efficient than $\hat{\theta}_{z\beta}$ with equal-share weights. There are infinite ways of constructing the weights $\omega_{ik}$ for $\hat{\theta}_z$ by \eqref{eq:Zhat}, which does not require additional effort, just like $\hat{\theta}_{z\beta}$. Some of them may be more efficient than $\hat{\theta}_y$ in light of Theorem \ref{thm:vardiff}. Thus, the BIGS representation of ACS offers greater potential of efficiency gains, beyond the existing methods for ACS.
\subsection{$T$-SBS from an arbitrary population graph} \label{sec:randomgraph}
Figure \ref{figure:randomgraph} shows a population graph $G$ of $40$ nodes and $72$ edges. Let the motifs of interest be connected components of order $|M_k| \leq 4$, including node ($\mathcal{K}_1$), 2-clique (dyad, $\mathcal{K}_2$), 2-star ($\mathcal{S}_2$), 3-clique (triangle, $\mathcal{K}_3$), 4-clique ($\mathcal{K}_4$), 4-cycle ($\mathcal{C}_4$), 3-star ($\mathcal{S}_3$) and 3-path ($\mathcal{P}_3$). The 40 nodes are all known. The totals of the other motifs (illustrated in Figure \ref{figure:motifsofinterest}) are
\[
(\theta_{\mathcal{K}_2},\theta_{\mathcal{S}_2},\theta_{\mathcal{K}_3},\theta_{\mathcal{K}_4},\theta_{\mathcal{C}_4},\theta_{\mathcal{S}_3},\theta_{\mathcal{P}_3}) =(179,72,19,3,7,141,408) ~.
\]
\begin{figure}[ht]
\centering
\includegraphics[scale=0.35]{randomgraph}
\vspace{-1cm}
\caption{A population graph with $|U|=40$ and $|A|=72$.}
\label{figure:randomgraph}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{motifsofinterest}
\vspace{-3.5cm}
\caption{Motifs of interest}
\label{figure:motifsofinterest}
\end{figure}
\begin{table}[ht]
\def~{\hphantom{0}}
\centering
\caption{Diameter and observation diameter of motifs. Number of stages ($T$) required for SBS with feasible BIGS representation using restricted ancestors $\beta_k^*$.} \label{table:Tstages}
\renewcommand{\tabcolsep}{0.20cm}
\begin{tabular}{lccccccccc} \hline
& & $\mathcal{K}_1$ & $\mathcal{K}_2$ & $\mathcal{S}_2$ & $\mathcal{K}_3$ & $\mathcal{K}_4$ & $\mathcal{C}_4$ & $\mathcal{S}_3$ & $\mathcal{P}_3$\\ \cline{2-10}
& $\lambda_k$ & 0 & 1 & 2 & 1 & 1 & 2 & 2 & 3 \\
& $\phi_k$ & 0 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\ \hline
\multirow{2}{*}{$\beta_k^*=M_k$} & $T$ for $\hat{\theta}_{z\beta}$ & 0 & 1 & 2 & 2 & 2 & 2 & 3 & 3 \\
& $T$ for $\hat{\theta}_{z\alpha}$ & 0 & 2 & 4 & 3 & 3 & 4 & 5 & 6 \\ \hline
\multirow{2}{*}{$\beta_k^*=M_k\cup\beta(M_k)$} & $T$ for $\hat{\theta}_{z\beta}$ & 2 & 3 & 4 & 3 & 3 & 4 & 4 & 5 \\
& $T$ for $\hat{\theta}_{z\alpha}$ & 3 & 5 & 7 & 6 & 6 & 7 & 8 & 9 \\ \hline
\multirow{2}{*}{$\beta_k^*=M_k\cup\beta^2(M_k)$} & $T$ for $\hat{\theta}_{z\beta}$ & 4 & 5 & 6 & 5 & 5 & 6 & 6 & 7 \\
& $T$ for $\hat{\theta}_{z\alpha}$ & 6 & 8 & 10~ & 9 & 9 & 10~ & 11~ & 12~ \\ \hline
\end{tabular}
\end{table}
The diameters ($\lambda_k$) and observation diameters ($\phi_k$) of the motifs are given at the top of Table \ref{table:Tstages}. Next, for feasible BIGS representation with restricted ancestors $\beta_k^* = M_k$, the number of SBS stages required for the HT-estimator $\hat{\theta}_y$ by \eqref{eq:Yhat} and the HH-type estimator $\hat{\theta}_{z\beta}$ by \eqref{eq:Zhat} with weights $\omega_{ik} = |\beta_k|^{-1}$ is given by Theorem \ref{thm:feasibleBIG}, i.e. $T = \phi_k$, whereas one may need up to $\phi_k$ additional stages for the estimator $\hat{\theta}_{z\alpha}$ using weights $\omega_{ik} \propto |\alpha_i|^{-1}$. Moreover, for BIGS representation with restricted ancestors $\beta_k^* = M_k \cup \beta(M_k)$, the number of stages required for $\hat{\theta}_y$ and $\hat{\theta}_{z\beta}$ is given by Theorem \ref{thm:feasibleBIGStar}, i.e. $T = \lambda_k +2t$ and $t=1$, whereas up to $\phi_k + t = \phi_k +1$ additional SBS stages may be needed for $\hat{\theta}_{z\alpha}$. Similarly in the case of $\beta_k^* = M_k \cup \beta^2(M_k)$ with $t=2$.
\subsubsection{An example: $s_0=\{3,12\}$ and $[M_k]=\mathcal{C}_4$}
To illustrate some of the computational details, let 4-cycle $\mathcal{C}_4$ be the motif of interest. The graph total of $\mathcal{C}_4$ is 7 in the population graph $G$ (Figure \ref{figure:randomgraph}). In Figure \ref{figure:samplegraphsatT}, the initial sample $s_0=\{3,12\}$ by SRS is marked as $T$-SBS with $T=0$; the sample graphs observed under $T$-SBS from this $s_0$ are given for $T=1, 2, 3, 4$. The sample graph by 4-SBS includes all the nodes in $G$ but not all the edges.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{samplegraphsatT}
\caption{Initial sample $s_0 =\{3,12\}$, sample graphs by T-SBS from $s_0$ for $T=1,...,4$.}
\label{figure:samplegraphsatT}
\end{figure}
According to Theorem \ref{thm:feasibleBIG}, BIGS with $\beta_k^* = M_k$ is a feasible representation of 2-SBS (i.e. $T=2$) for motif $\mathcal{C}_4$, where $\phi_k = 2$ for any $[M_k] = \mathcal{C}_4$; see also Table \ref{table:Tstages}. The details required for computing the HT-estimator \eqref{eq:Yhat} and the two HH-type estimators \eqref{eq:Zhat} are given in the upper part of Table \ref{table:cycle4}, where $\beta_k^* = M_k$. Motif $A$ with nodes $M_A=\{3,8,21,22\}$ is observed from node $3$, and motifs $\{ B, C\}$ from node 12, where $M_B=\{12,13,18,31\}$ and $M_C=\{12,15,18,32\}$. Two more stages of SBS are needed to apply \eqref{eq:Zhat} with the weights $\omega_{ik} \propto |\alpha_i|^{-1}$; the relevant $|\alpha_i |$'s are given in the last column of Table \ref{table:cycle4}. For instance, we have $|\alpha_{\beta_B^*}| =\{ |\alpha_{12}|, |\alpha_{13}|, |\alpha_{18}|, |\alpha_{31}| \} = \{ 2, 2, 3, 1 \}$ for motif $B$.
\begin{table}[hpb!]
\def~{\hphantom{0}}
\centering
\caption{BIGS representation of $T$-SBS for $\mathcal{C}_4$ with initial $s_0 = \{ 3, 12\}$.}
\label{table:cycle4}
\renewcommand{\tabcolsep}{0.25cm}
\begin{tabular}{lccccc} \hline
& $i\in s_0$ & $k\in\alpha_i$ & $M_k$ & $|\beta_k^*|$ & $|\alpha_{\beta_k^*}|$ \\ \hline
\multirow{3}{*}{$T=2$, $\beta_k^* =M_k$} & ~3 & A & ~~$\{3,8,21,22\}$ & 4 & $\{1, 1, 1, 1\}$ \\ \cline{2-3}
& \multirow{2}{*}{12} & B & $\{12,13,18,31\}$ & 4 & $\{2,2,3,1\}$ \\
& & C & $\{12,15,18,32\}$ & 4 & $\{2,1,3,2\}$ \\ \hline
\multirow{4}{*}{$T=4$, $\beta_k^* =M_k\cup\beta(M_k)$} & ~3 & A & ~~$\{3,8,21,22\}$ & 15 & -- \\ \cline{2-3}
& \multirow{3}{*}{12} & B & $\{12,13,18,31\}$ & 16 & -- \\
& & C & $\{12,15,18,32\}$ & 14 & -- \\
& & D & $\{13,18,29,32\}$ & 12 & -- \\ \hline
\end{tabular}
\end{table}
Under SRS of $s_0$, the inclusion probability \eqref{eq:pi1motif} is $\pi_{(k)} = 1- \mathcal{C}(N-|\beta_k^*|, n)/\mathcal{C}(N, n) \equiv 0.1923$, where $N =40$, $n = 2$ and $|\beta_k^*| \equiv 4$ for $\mathcal{C}_4$. By \eqref{eq:Yhat}, we have $\hat{\theta}_y= 3 / 0.1923=15.6$. For \eqref{eq:Zhat} with $\omega_{ik} = |\beta_k^*|^{-1}$, we have $z_3 = 1/4$ from $\alpha_3 = \{ A\}$ and $z_{12} = 1/4 + 1/4$ from $\alpha_{12} = \{ B, C\}$, such that $\hat{\theta}_{z\beta} = (3/4)/(2/40) =15$, where $\pi_i \equiv 2/40$ for $i = 3, 12$. Finally, for \eqref{eq:Zhat} with $\omega_{ik} \propto |\alpha_i|^{-1}$, we have $z_3 = 1/4$ from $\alpha_3 = \{ A\}$ and $z_{12} = 0.5/2.33 + 0.5/2.33 = 0.43$ from $\alpha_{12} = \{ B, C\}$, such that $\hat{\theta}_{z\alpha} = 13.6$.
By Theorem \ref{thm:feasibleBIGStar}, BIGS with $\beta_k^* = M_k\cup \beta(M_k)$ and $t=1$ is a feasible representation of 4-SBS (i.e. $T=4$) for $\mathcal{C}_4$, where $\lambda_k = 2$ for any $[M_k] = \mathcal{C}_4$; see also Table \ref{table:Tstages}. The details are given in the lower part of Table \ref{table:cycle4}. Motif $A$ is observed from node $3$, and motifs $\{ B, C, D\}$ from node 12 where, compared to 2-SBS above, the extra motif $D$ with $M_D=\{13,18,29,32\}$ is observed via node 18 that is observed at the 1st stage (Figure \ref{figure:samplegraphsatT}). All these motifs eligible for estimation are observed by the 3rd stage; however, since $\lambda_k = \phi_k$ for motif $\mathcal{C}_4$, another stage is needed to ensure ancestral OP, yielding $\lambda_k + 2t = 4$; see also the proof of Theorem \ref{thm:feasibleBIGStar}. The cardinality of the ancestor set $\beta_k^*$ is given in Table \ref{table:Tstages}, which is $15, 16, 14, 12$ for $k=A, B, C, D$, respectively. More stages of SBS are needed to apply \eqref{eq:Zhat} with the weights $\omega_{ik} \propto |\alpha_i|^{-1}$, so that $\hat{\theta}_{z\alpha}$ is not feasible with maximum 4 stages of SBS that is illustrated here and the relevant $|\alpha_{\beta_k^*}|$ omitted in Table \ref{table:Tstages}.
The inclusion probability $\pi_{(k)} =1- \mathcal{C}(N-|\beta_k^*|, n)/\mathcal{C}(N, n)$ is $0.6154$, $0.6462$, $0.5833$ and $0.5154$ for $k = A, B, C$ and $D$, which are much higher than $\pi_{(k)} \equiv 0.1923$ under 2-SBS above. By \eqref{eq:Yhat}, we have $\hat{\theta}_y= 6.83$. For $\hat{\theta}_{z\beta}$ by \eqref{eq:Zhat}, we have $z_3 = 1/15$ from $\alpha_3 = \{ A\}$ and $z_{12} = 1/16 + 1/14 + 1/12$ from $\alpha_{12} = \{ B, C, D\}$, such that $\hat{\theta}_{z\beta} = 5.68$. Both the two estimates are much closer to the graph total $\theta_{\mathcal{C}_4} = 7$ than by 2-SBS above, with only one extra motif $D$. Contrasting SBS with $T=4$ or $T=2$, the inclusion probabilities $\pi_{(k)}$ and the weights $\omega_{ik}$ matter more to estimation than the number of observed motifs.
\subsubsection{Results} \label{sec:resultsrandomgraph}
Consider SBS of maximum 4 stages following SRS of $s_0$ with $|s_0| =2$. Since the diameter of the population graph $G$ is six here, a large part of it may already have been observed by 4-SBS, as in the case of $s_0 = \{ 3, 12\}$ above; indeed, $G$ is fully observed from 215 out of 780 possible initial samples. In addition, we consider induced OP following SRS of $s$, for which $s_{ref}=s\times s$. The size of $s$ is set to be the expected number of observed nodes under $T=1$ and $T=2$, which are 9 and 21, respectively. Denote by $\hat{\theta}$ the resulting HT-estimator.
\begin{table}[ht]
\def~{\hphantom{0}}
\centering
\caption{Mean squared errors of graph total estimators under induced OP from SRS of size $n=9$ or 21, and SBS of maximum 4 stages from SRS of initial sample of size 2.}
\label{table:mse}
\renewcommand{\tabcolsep}{0.10cm}
\begin{tabular}{lcccccccc} \hline
& Estimator & $\mathcal{K}_2$ & $\mathcal{S}_2$ & $\mathcal{K}_3$ & $\mathcal{K}_4$ & $\mathcal{C}_4$ & $\mathcal{S}_3$ & $\mathcal{P}_3$ \\ \hline
Induced OP, $n=9$ &$\hat{\theta}$ & 1\,263& 47\,134& 2\,869& 2\,167& 5\,168& 231\,805& 797\,578 \\
Induced OP, $n=21$ & $\hat{\theta}$ & ~152 & ~4\,533 & ~198 & ~~41 & ~116 & ~11\,523 & ~52\,488 \\ \hline
\multirow{3}{*}{$\beta_k^*=M_k$} &$\hat{\theta}_y$ & ~471 & ~5\,269 & ~193 & ~~10 & ~~38 & ~~5\,092 & ~27\,717 \\
&$\hat{\theta}_{z\beta}$ & ~475 & ~5\,447 & ~199 & ~~10 & ~~39 & ~~5\,368 & ~29\,441 \\
&$\hat{\theta}_{z\alpha}$ & ~116 & ~~613 & ~160 & ~~10 & ~~28 & ~~-- & ~~-- \\ \hline
\multirow{3}{*}{$\beta_k^*=M_k\cup\beta(M_k)$} &$\hat{\theta}_y$ & ~306 & ~1\,614 & ~92 & ~~4 & ~~7 & ~~1\,382 & ~~-- \\
&$\hat{\theta}_{z\beta}$ & ~281 & ~1\,485 & ~98 & ~~5 & ~~7 & ~~1\,403 & ~~-- \\ \hline
\end{tabular}
\end{table}
In Table \ref{table:mse}, we present the mean squared errors (MSEs) of the different estimators. Feasible BIGS representation is used for estimation under $T$-SBS. In case an estimator is not feasible for a certain motif using maximum 4-SBS, the result will be unavailable in the table. Induced OP is understandably much less efficient than incident OP, as the order of the motif of interest increases; compare e.g. the results for SRS of size 21 and $2$-SBS, where both have the same expected number of nodes in the sample graph.
Under $T$-SBS from the population graph in Figure \ref{figure:randomgraph}, the HT-estimator $\hat{\theta}_y$ and the HH-type estimator $\hat{\theta}_{z\beta}$ are about equally efficient for the motifs considered here. The HH-type estimator $\hat{\theta}_{z\alpha}$ can be much more efficient, especially for the lower-order motifs $\mathcal{K}_2$ and $\mathcal{S}_2$. Under SRS of $s_0$, the variance of the HH-type estimator \eqref{eq:Zhat} is minimised, if the constructed $z_i$'s happen to be constant across the sampling units. With unequal-share weights, $z_i$ is proportional to $|\alpha_i|$. Setting $\omega_{ik} \propto |\alpha_i|^{-1}$ tends to even out the $z_i$'s, since a sampling unit with many successors will receive relatively little share from each motif observed from it, although its $z$-value is based on more motifs than another sampling unit with fewer successors. We refer to \citet{Patone2020} for more discussions of $\hat{\theta}_{z\alpha}$.
\section{Conclusion} \label{sec:conclusion}
Graph sampling \citep{ZhangPatone2017} provides a general statistical approach to study real graphs, which can be of interest in numerous investigations. We develop feasible BIGS representation that is applicable to a large number of graph sampling situations, which are based on different incident observation procedures. It avoids the recursive computations that are needed to calculate the inclusion probabilities of the sample motifs under $T$-stage snowball sampling \citep{ZhangPatone2017}. It enables one to identify the motifs that are eligible to estimation in a given sample, which generalises a related idea originally proposed for adaptive cluster sampling \citep{Thompson1990}. It allows one to extend the scope of HH-type estimators developed for indirect sampling \citep{BirnbaumSirken1965}, providing a unified framework for achieving efficiency gains beyond the standard HT-estimator. A current topic of research is developing BIGS methods for general ``representative sample graphs'', as defined by \citet{ZhangPatone2017}, which can be applied to various problems of `graph compression'.
\bibliographystyle{chicago}
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1807.09830
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\section{Introduction}
The study of deep neural network architectures evidenced their efficiency to approximate certain families of complex functions of the input domain\cite{delalleau2011shallow,pascanu2013number}. This characteristic is contrasted with the difficulty of training these neworks using the standard back-propagation algorithm\cite{he2016deep}. We present a model that aims to maintain the flexibility of the deep neural networks without producing these problems. Our model increases the expressive power of a recurrent neural network in a way that is comparable with a deep model with the simplicity of training a shallow model.
Our work aims to introduce a modification in the structure of the LSTM cell block, theoretically-driven by the study of subsequent evaluations of repeating the evaluation of the LSTM cell, updating the hidden state while keeping the input and cell state constant. \\
We argue that this iterative process defines a dynamic system that accounts for the evolution of the hidden state of the network, driving it to compact regions where the information is presumably retained better. The theory behind the dynamic system studied will be explained in the next sections.
The principal motivation for the proposed modification is the theoretical evidence that the dynamic systems could retain information in the form of an analog state vector. This vector is confined to the basin of attraction corresponding to one of the possible states that are relevant to the resolution of the task\cite{bengio1994learning}. On the other hand, nonlinear dynamic systems, as the one defined by our model, are capable of defining a complex behavior over the phase plane that is useful to flatten the hidden class manifolds.
To decide the number of iterations to perform we found that the usage of a simple logistic regression over the LSTM cell block variables, thresholded to select whether or not to modify the current hidden state by an additional iteration, could serve as a controller that is properly optimized to select whether or not to perform an additional iteration. This simple model avoid the need to select the number of iterations as an hyper-parameter of the model.\\
This controller is inspired by the gating mechanisms already present in LSTM cells and its goal is to weight the importance of the iterations in the final result.\\
We provide an open-source implementation\footnote{\texttt{https://github.com/PalmaLeandro/iterativeLSTM}} of the LSTM cell network modified as is proposed, along with empirical evidence of the improvements in the performance of the modified models compared to its former baseline on the task of language modeling.
\section{Related work}
One clear example of an architecture of NN that introduces the iterative scheme is LoopyNN presented in \cite{caswellloopy}. This recurrent model performs subsequent iterations using the last result as input for the next iteration.\\
While this behavior is similar to our proposal, differences can be drawn on the method and on the theory behind it.
Nevertheless, much of the characteristics of our model are noted by Caswell and his colleagues. We note that the amount of unrolls of its architecture is related to the amount of iterations performed by our model.
An iterative scheme similar to the proposed one is studied by \cite{liao2016bridging}. That work covers the improvements in the performance of a fully connected NN that performs several evaluations after every new information is exposed to the network. Moreover, such work relates the iterative scheme with recurrent structures found in the human brain.
The concept of \textit{readout time} presented in that work is related to the number of iterations performed by our model. The relation among these concepts is supported by the similar enhancements found in model's performance while increasing their values.
The analysis performed by \cite{laurent2016recurrent} explores the chaotic properties of LSTM cells as dynamic systems. Even when the conditions over which its LSTM networks are evaluated differ from real applications such work motivated the study of the non-linear components of our model in order to avoid the chaotic behavior exhibited and which would prevent the network state to converge towards stable configurations.
\section{Iterative LSTM cell}
We considered how to fold models that share the weights and state of several layers of LSTM cells to yield a more compact representation that executes a fixed number evaluations.
In order to optimize the model parameters using gradient based methods the state exposed by the network has to be a real-valued vector. Thus, the latching of information in the system is accomplished by the evolution of such vector towards stable configurations\cite{bengio1994learning}. This evolution is governed by the dynamics of a non-autonomous system which is defined by the model's formulation and is parameterized by the input values at every time-step\cite{pascanu2013difficulty}.
For the iterative scheme proposed, the evolution of the network state at every time-step is governed by an autonomous system which would last as long as the model performs additional iterations.\\
The complete evolution of the network state is the aggregation of the several state changes achieved at every time-step.
Such representations are easier to classify by adjacent components due to the sharp frontiers with the set of states that belongs to other attraction basins, and therefore different information.\\
We note that a model whose state evolution is subject to the conditions mentioned has to organize its fixed points to flatten manifolds into the proper configurations as well as connect or maintain these domains across subsequent time-steps in order to unambiguously recall information.
One important observation about dynamic systems derived from the existence and uniqueness theorem is that the trajectories cannot intersect each other and, over convergent conditions, this shall produce the contraction of the volume of states being attracted. Hence, the contracted volume of states exhibits a simpler surface.
This behavior yields sparse and compact regions as more flattened manifolds to be classified by higher layers.
Under the stated conditions we expect an increase in the performance of the model modified as is proposed, with respect to its original version. This is supported by the fact that the diffuse or nonlinear edges of latent manifolds jeopardize the model's performance because of the limitations exhibited by recurrent neural models\cite{elman1990finding}.
In order to achieve the desired convergent conditions we propose a modification to the LSTM structure. This modification aims to induce an autonomous dynamic system whose state, given by the hidden state $h$ of the LSTM block, converges to a confined domain closer to the attractor of the basin in which the initial conditions reside. These initial conditions are given by the value of the hidden state at previous time-step.
The model resulting from applying the proposed modification to a LSTM network could be summarized by the following ecuations calculated in the given order, for every iteration $\tau$.\\
\begin{equation*}
\begin{split}
j(\tau) &= tanh(W_{rec, j} h +W_{in, j} x(t) + b_{j} ) = tanh(W_{rec, j} h + C_j)\\
i(\tau) &= \sigma(W_{rec, i} h + W_{in, j} x(t) + b_{j} ) = \sigma(W_{rec, i} h + C_i)\\
f(\tau) &= \sigma(W_{rec, f} h + W_{in, j} x(t) + b_{j} ) = \sigma(W_{rec, f} h + C_f)\\
c(\tau) &= f(\tau)c(t-1) + i(\tau)j(\tau) = f(\tau) c_0 + i(\tau)j(\tau) \\
o(\tau)& = \sigma(W_{rec, o} h + W_{in, j} x(t) + b_{j} ) = \sigma(W_{rec, o} h + C_o)
\end{split}
\end{equation*}
\begin{equation}\label{iterative_LSTM_cell_formulation}
\begin{split}
h(\tau) &= o(\tau) \, tanh(c(\tau))\\
y(t) &= h(\tau) + x(t)
\end{split}
\end{equation}
As a result, the evolution of the state $h$ through the iterations performed describes a dynamic system. Its phase plane at any time-step $t$ of the input sequence is defined by the former internal cell state $c_0$ and the input values $x(t)$ since the constants $C_\varphi \, , \, \forall \, \varphi \, \in \, \{i, j, f, o\}$ depend on such information.
An iteration activation gate $p$ is introduced with the idea of selecting whether to perform an additional iteration and modify the hidden state or expose it to the next layers. This gate consists of a logistic regression of the inputs, the recently calculated state $h$ and the internal gate variables $i$, $j$ and $f$. Its output is then compared with a threshold which is a parameter of the iteration and varies to more restrictive values to reduce the overall iterations made without constraining the cell to an arbitrary limit.
Another modification introduced in our model is the resolution of a residual mapping of the inputs\cite{he2016deep}. This customization of the model is supported by the results found in \cite{he2016deep} where the addition of a direct connection of the input allows to reference the result of a deep neural network inference with respect to its input.\\
Moreover, \cite{liao2016bridging} stated the similarity of the networks that share weights across the depth dimension with residual networks.\\
Additionally, in \cite{caswellloopy} the residual mapping is tested experimentally and evidence is found to support its application on neural networks with shared parameters.
\section{Convergence of Iterative LSTM cells}
The formulation of RNNs produces a well-known dynamic system studied by \cite{pascanu2013difficulty} and others \cite{bengio1994learning}. Is in the publication of Pascanu et. al. that the effect of the non-autonomous components of the system is explored to expose the challenges that it represents to gradient based methods. Its effect has to be considered in order to produce the intended convergence of the state to the corresponding attractors.\\
In the proposed iterative scheme this is achieved by maintaining a constant input and cell state values through all the iterations performed by the model within a time step of the input sequence. This constraint yields an autonomous dynamic system whose dynamics are determined by the inputs and cell state values, at a particular time-step $t$ of the input sequence.
Then, considering subsequent evaluations of an LSTM network over a constant input $x$ and cell state $c$ values while varying its hidden state $h(\tau)$ at discrete steps $\tau$ yields the folowing dynamic system
\begin{equation}\label{iterative_LSTM_cell_dynamic_formulation}
\frac{\partial h(\tau)}{\partial \tau} = h(\tau + 1) - h(\tau) = LSTM(x(t), c(t), h(\tau)) - h(\tau) = g_t(h(\tau)) - h(\tau)
\end{equation}
where the $LSTM$ function corresponds to the calculations required to update the hidden state of the vanilla LSTM network.\\
This system has been studied by \cite{laurent2016recurrent}, exposing its behavior for more than 200 iterations over a null value of the inputs. The results extracted from that analysis were that the sensibility of the system to variations in the initial conditions produce chaotic trajectories of states. \\
This implies that sightly different initial states could produce a completely different final state by projecting the behavior of the model long enough. Consequently, the predictions of the model for similar inputs may as well be different.
The following result is intended to provide sufficient conditions to bound the system's Liapunov coefficients to a subset that leads to a coherent behavior of the model for variations in the initial conditions\cite{StrogatzBook}. By meeting these conditions the difference on the predictions made for slightly different initial states shall be bounded.
\begin{theorem}
\label{iterative_LSTM_cell_convergence_theorem}
Let $f(x(t), h(t-1), c(t-1))$ be a model consisting of a LSTM cell block such that
\begin{equation}
\begin{split}
h(t) &= f(x(t), c(t-1), h(t-1)) = LSTM(x(t), c(t-1), h(t-1))
\end{split}
\end{equation}
where $x(t)$ is the value of the input sequence at the time-step $t$. $c(t-1)$ and $h(t-1)$ are vectors with the values of the internal and exposed state at the previous time-step, respectively.\\
The subsequent evaluations of $f$ implicitly defines the dynamic system
\begin{equation}\label{iterative_LSTM_cell_convergence_theorem_dynamic_system}
\begin{split}
\dfrac{\partial h(\tau)}{\partial \tau} &= h(\tau + 1) - h(\tau) = f(x(t), c(t-1), h(\tau)) - h(\tau) = g_t(h(\tau)) - h(\tau)
\end{split}
\end{equation}
where $g_t(h(\tau))$ is a function analog to $f(x(t), c(t-1), h(t-1))$ where $x(t)$ and $c(t-1)$ are kept constant. Under these conditions the following implication holds
\begin{equation}
\sigma_j + \frac{1}{4} \sigma_i + \frac{1}{4} \sigma_f + \frac{1}{4} \sigma_o < 1 \implies \lambda_i < 0 \, , \, \forall \, i \in 0,...,n-1
\end{equation}
where $\sigma_j$, $\sigma_i$, $\sigma_f$ and $\sigma_o$ are the principal singular values of the matrices that weights the recurrent connections of the $j$, $i$, $f$ and $o$ gates of the LSTM cell block, respectively. $\lambda_i$ is the Liapunov exponent of the dynamic system defined by (\ref{iterative_LSTM_cell_convergence_theorem_dynamic_system}) at the $i$-th dimension. $n$ is the number of cell units.
\end{theorem}
\begin{proof}
See appendix.
\end{proof}
The principal implication of the theorem \ref{iterative_LSTM_cell_convergence_theorem} is that the evolution of the hidden state over the iterative scheme proposed is not chaotic for a particular set of the model's parameters.
An initial configuration of network parameters that matches the conditions of theorem \ref{iterative_LSTM_cell_convergence_theorem} could be achieved following the suggestions for initialization in \cite{zilly2016recurrent} derived from the Geršgorin circle theorem. Moreover, as mentioned in the publication of Zilly et. al., the L1 and L2 regularization techniques could be used to enforce the conditions required for the application of the theorem \ref{iterative_LSTM_cell_convergence_theorem}.
We believe that the presented analysis could be extended to RNNs in general providing more evidence that the iterative scheme proposed enhances the capabilities of other recurrent models as well.
\section{Experiments}
Following the configurations in \cite{RegularizationZaremba} several experiments were performed training different architectures of RNNs as language models.
We used the implementation of the publication of Zaremba et. al. that was released with the TensorFlow library\footnote{Code available at\\
\parbox[t]{10em}{https://github.com/tensorflow/models/tree/master/tutorials/rnn/ptb}}, as the base for our experiments.\\
The corpus used to train the model is the Penn Treebank \cite{PennTreebankMarcus}\footnote{Corpus available at \texttt{http://www.fit.vutbr.cz/~imikolov/rnnlm/simple-examples.tgz}.}. \\
The base arquitecture used for the recurrent network is the corresponding with the `medium' size model presented in \cite{RegularizationZaremba}.
The medium size model consist of the embedding projection layer, 2 layers of LSTM blocks containing 650 units each and a final softmax layer. The large model architecture is the same but each layer of LSTM cells contained 1500 units and its performance is reported as in its original publication.\\
The embedding projection layer, the layers of the RNN and the softmax layer are connected through regularization connections that applied the Dropout\cite{srivastava2014dropout} method with a probability of keeping the connection of $0.5$.
The parameters were optimized using minibatch gradient descent with a batch size of $20$. These parameters were initialized by a random uniform distribution at the interval $[-0.5, 0.5]$.
The gradients calculated to reduce the loss function are clipped to a norm value of $5$.
The training regime consisted of 39 epochs where for the first 6 epochs a learning rate of $1.0$ was set and then this value was reduced by a factor of 1.2 for the remaining epochs.
Figure \ref{ppl_vs_iterations_figure} presents the results of the experiments performed fixing the amount of iterations executed to incremental values.
\begin{figure}
\centering
\includegraphics[scale=0.2150]{pplVsIterations.png}
\caption{Perplexity per word as function of the iterations executed at every time-step.}
\label{ppl_vs_iterations_figure}
\end{figure}
The results of the experiments shows that there is an enhancement in the proposed model's performance as the number of iterations performed increases.
As is visible in the Figure \ref{ppl_vs_iterations_figure} the reduced model constituted by one layer of iterative LSTM cells consistently outperforms its augmented version as the number of iterations increase. We attribute this difference in the performance of the models to the overfitting suffered by the augmented model whose expressive capabilities were improved over the iterative regime and the inclusion of an additional layer.\\
Such observation supports the conjecture that the iterative scheme proposed improves the expressiveness of the model.
Note that in the case of the execution of a single iteration, where the modified network is comparable to its original form, the performance of the smaller model is worse than the augmented model and this effect is inverted as more iterations are executed. This indicates that the improvements over the models' performance are not caused by the modifications made to its structure, namely the iteration activation gate and the residual mapping chosen, rather than by the application of the iterative scheme proposed.
Table 1 presents the performance of the trained models in terms of perplexity per word, where lower is better.
\begin{center}
\label{models_experiment_performance_table}
Table 1: Perplexity per word on Penn Treebank corpus.
\begin{tabular}[t]{| c | c | c |}
\hline
Model & Size & Perplexity on Test Set\\
\hline
\hline
LSTM & 16M & 84.48\\
\hline
Iterative LSTM & 16M & 78.46\\
\hline
\hline
LSTM(2 layers) & 20M & 83.25\\
\hline
Iterative LSTM(2 layers) & 20M & 78.60\\
\hline
\hline
LSTM(2 layers)& 50M & \textbf{78.29}\\
\hline
\end{tabular}
\end{center}
The results reported in table 1 expose that the performance achieved by the model that applies the proposed modification is comparable with larger versions with more than 3 times the total amount of parameters of our model.
\section{Conclusions}
In this work we studied a modification over the traditional LSTM structure that produces an iterative scheme where the inference is done incrementally.
We presented theoretical evidence to support the proposed scheme based on the study of the dynamic system defined by the iterative evaluation of the recurrent network.\\
The results of the experiments executed to expose the effect of the proposed modification supports the theoretical motivation that lead to the development of the presented model.\\
A comparison of the performance achieved by our model showed a capacity comparable to its largest original version, augmented up to 3 times in terms of the total amount of parameters.
\section{Appendix}
\subsection{Proof of theorem 1} \label{iterative_LSTM_cell_convergence}
\begin{proof}
The execution of subsequent evaluations of the proposed model yields the dynamic system defined by (\ref{iterative_LSTM_cell_dynamic_formulation})
\begin{equation*}
\frac{\partial h(\tau)}{\partial \tau} = h(\tau + 1) - h(\tau) = LSTM(x(t), c(t), h(\tau)) - h(\tau) = g_t(h(\tau)) - h(\tau)
\end{equation*}
Then, considering an infinitesimal small variation on the initial conditions $\delta_0$ we look for the variations $\delta_\tau$ on the result of the system after $\tau$ evaluations which would be
\begin{equation*}
|\delta_\tau| = |\delta_0| e ^{\tau\lambda} \iff \frac{|\delta_\tau|}{|\delta_0|} = e ^{\tau\lambda} \iff \lambda = \dfrac{1}{\tau} \ln\left\lvert\dfrac{\delta_\tau}{\delta_0}\right\rvert
\end{equation*}
Where $\lambda$ is a vector such that it's coordinates $\lambda_i$ are the Liapunov exponents of the dynamic system at the $i$-th dimension of the vector state $h(\tau)$.
\begin{equation*}
\begin{split}
\lambda = \dfrac{1}{\tau} \ln\left\lvert\dfrac{\delta_\tau}{\delta_0}\right\rvert \iff \lambda = \dfrac{1}{\tau} \ln \left\lvert \dfrac{g^\tau(h(t-1) + \delta_0) - g^\tau(h(t-1))}{\delta_0} \right\rvert \iff \lambda = \dfrac{1}{\tau} \ln | (g^\tau)' |
\end{split}
\end{equation*}
$g^\tau(h)$ is the application of the $g$ function, defined at (\ref{iterative_LSTM_cell_dynamic_formulation}), $\tau$ times over the state vector $h$.\\
Then, requiring the Liapunov coefficients to have values that imply the convergence of the sequence of state values yields
\begin{equation*}
\dfrac{1}{\tau} \ln | (g^\tau)' | < 0 \iff | (g^\tau)' | < 1 \iff \lambda_i < 0 \, , \, \forall \, i \in 0,...,n-1\\
\end{equation*}
Where $\lambda_i$ is the Liapunov exponent of the dynamic system defined by (\ref{iterative_LSTM_cell_dynamic_formulation}) at the $i$-th dimension. $n$ is the number of cell units.
Next, it is possible to derive $(g^\tau)'$ applying the chain rule to obtain
\begin{equation*}
(g^\tau)' = \prod_{i = 0}^{\tau}g'(h(i))
\end{equation*}
Replacing this identity on the conditions required for the convergence of the dynamic system in every dimension yields
\begin{equation*}
\begin{split}
| (g^\tau)' | < 1 \iff \left\lvert \prod_{i = 0}^{\tau}g'(h(i)) \right\rvert < 1 &\iff |g'(h(i))| < 1 \quad i \in 0, ... , \tau\\
&\iff \lambda_i < 0 \, , \, \forall \, i \in 0,...,n-1
\end{split}
\end{equation*}
Meanwhile, resolving the derivative of the variables $j, i, f, o$ and $g$, defined at (\ref{iterative_LSTM_cell_formulation}), with respect to a generic state vector $h$ yields
\begin{equation*}
\begin{split}
\dfrac{\partial j(h)}{\partial h} = & \, (1- tanh^2(W_{rec,j} \, h + C_j))W_{rec,j}\\
\dfrac{\partial i(h)}{\partial h} = & \, \sigma(W_{rec,i} \, h + C_i) (1-\sigma(W_{rec,i} \, h + C_i)) W_{rec,i}\\
\dfrac{\partial f(h)}{\partial h} = & \, \sigma(W_{rec,f} \, h + C_f) (1-\sigma(W_{rec,f} \, h + C_f)) W_{rec,f}\\
\dfrac{\partial o(h)}{\partial h} = & \, \sigma(W_{rec,o} \, h + C_o) (1-\sigma(W_{rec,o} \, h + C_o)) W_{rec,o}\\
\dfrac{\partial g(h)}{\partial h} = & \, \dfrac{\partial o(h)}{\partial h} \, tanh( \, c(h) \, ) + o(h) \, (1 - tanh( \, c(h) \, )^2) \, \dfrac{\partial f(h)}{\partial h} \, c_0\\
&+ o(h) \, (1 - tanh( \, c(h) \, )^2) \, \dfrac{\partial i(h)}{\partial h} \, j(h)+ o(h) \, (1 - tanh( \, c(h) \, )^2) \, \dfrac{\partial j(h)}{\partial h} \, i(h)
\end{split}
\end{equation*}
where $\frac{\partial tanh(x)}{\partial x} = 1 - tanh(x)^2$ and $\frac{\partial \sigma(x)}{\partial x} = \sigma(x) (1-\sigma(x))$ are diagonal matrices whose coefficients $\frac{\partial tanh(x)}{\partial x}_{i,i}$ and $\frac{\partial tanh(x)}{\partial x}_{i,i}$ corresponds to the evaluation of the functions $tanh'(x_i)$ and $\sigma'(x_i)$ over the $i$-th coordinate of the vector $x$, respectively. Therefore, replacing the definition of $g'(h)$ obtained from the conditions that Liapunov coefficients have to hold in order to produce the intended convergence yields the inequity
\begin{equation*}
\begin{split}
\lvert \dfrac{\partial o(h)}{\partial h} \, tanh( c(h) \, ) + o(h) \, (1 - tanh( \, c(h) \, )^2) \, \dfrac{\partial f(h)}{\partial h} \, c_0+\quad\quad&\\
o(h) \, (1 - tanh( \, c(h) \, )^2) \, \dfrac{\partial j(h)}{\partial h} \, i(h) \rvert < 1 \iff &\lambda_i < 0\\
\left\lvert \dfrac{\partial o(h)}{\partial h} \right\rvert \, \underbrace{| tanh( c(h) \, )|}_{\le1} + \underbrace{| o(h) |}_{< 1}\,\underbrace{| (1 - tanh( \, c(h) \, )^2) |}_{<1} \, \left\lvert \dfrac{\partial f(h)}{\partial h} \right\rvert \, \underbrace{| c_0|}_{<1} +&\underbrace{| o(h) |}_{< 1}\,\underbrace{| (1 - tanh( \, c(h) \, )^2) |}_{<1} \\
\left\lvert \dfrac{\partial i(h)}{\partial h} \right\rvert \, \underbrace{| j(h)|}_{<1} + \underbrace{| o(h) |}_{< 1}\,\underbrace{| (1 - tanh( \, c(h) \, )^2) |}_{<1} \, \left\lvert \dfrac{\partial j(h)}{\partial h} \right\rvert \, \underbrace{| i(h)| }_{<1}< 1 &\implies \lambda_i < 0\\
\left\lvert \dfrac{\partial o(h)}{\partial h} \right\rvert + \left\lvert \dfrac{\partial f(h)}{\partial h} \right\rvert + \left\lvert \dfrac{\partial i(h)}{\partial h} \right\rvert + \left\lvert \dfrac{\partial j(h)}{\partial h} \right\rvert <1 \implies \lambda_i < 0 &\, , \, \forall \, i \in 0,...,n-1
\end{split}
\end{equation*}
Finally, replacing the derivatives of the variables $j, i, f, o$ and $g$ with respect to the vector state $h$ yields
\begin{equation*}
\begin{split}
&\underbrace{|\sigma(W_{rec,0} \, h + C_i) (1-\sigma(W_{rec,o} \, h + C_o)) |}_{\le \frac{1}{4}} \, |W_{rec,o}|+ \underbrace{|\sigma(W_{rec,f} \, h + C_i) (1-\sigma(W_{rec,f} \, h + C_f)) |}_{\le \frac{1}{4}} \, |W_{rec,f}|+\\
&\underbrace{|(1- tanh^2(W_{rec,j} \, h + C_j))|}_{\le 1} |W_{rec,j}| + \underbrace{|\sigma(W_{rec,i} \, h + C_i) (1-\sigma(W_{rec,i} \, h + C_i)) |}_{\le \frac{1}{4}} \, |W_{rec,i}| <1\\
&\implies \frac{1}{4} \, |W_{rec,o}| + \frac{1}{4} \, |W_{rec,f}| + |W_{rec,j}| + \frac{1}{4} \, |W_{rec,i}| <1 \implies \lambda_i < 0 \, , \, \forall \, i \in 0,...,n-1
\end{split}
\end{equation*}
Hence, if the following condition holds for the matrices $W_{rec,j}, W_{rec,i}, W_{rec,f}, W_{rec,o}$ the dynamic system, produced by subsequent evaluations over the same input and cell state values while varying the hidden state, produces convergent results for similar initial conditions
\begin{equation*}
\sigma_j + \frac{1}{4} \sigma_i + \frac{1}{4} \sigma_f + \frac{1}{4} \sigma_o <1 \implies \lambda_i < 0 \, , \, \forall \, i \in 0,...,n-1
\end{equation*}
where $\sigma_j , \sigma_i , \sigma_f , \sigma_o $ are principal singular values of the matrices $W_{rec,j}, W_{rec,i}, W_{rec,f}, W_{rec,o}$ respectively.
\end{proof}
\bibliographystyle{splncs03}
|
2004.08034
|
\section{Introduction}
Improving the brightness of space-charge dominated electron sources will unlock a wealth of next generation accelerator physics applications.
For example, the largest unit cell that may be studied with single shot ultrafast electron diffraction (UED) is limited by the beam's transverse coherence length, which is determined by transverse emittance, at a high enough bunch charge to mitigate the effects of shot noise in data collection.
The study of protein dynamics with UED requires producing $>1$ nm scale coherence lengths at more than $10^5$ electrons and sub-picosecond pulse lengths at the sample location \cite{siwick_femtosecond_2004, dwyer_femtosecond_2006}.
In another example, the intensity of coherent radiation available to the users of free electron lasers (FELs) is, in part, limited by beam brightness.
Beam brightness affects the efficiency, radiated power, gain length, and photon energy reach of FELs \cite{hyder_emittance_1988, di_mitri_estimate_2014}.
Photoinjectors equipped with low intrinsic emittance photocathodes are among the brightest electron sources in use today.
Peak brightness at the source is limited by two factors: the electric field at the cathode and the photocathode's transverse momentum spread.
Several short-pulse Child-Langmuir-like charge density limits have been derived for the photoemission regimes of relevance to practical photoinjectors.
These current density extraction limits make explicit the dependence of peak brightness on photocathode parameters and the electric field. \cite{bazarov_maximum_2009, filippetto_maximum_2014, shamuilov_child-langmuir_2018}.
Depending on the aspect ratio of the bunch, the brightness limit is super-linear in the electric field and motivates the push towards high accelerating gradient photoinjectors.
Contemporary DC, normal-conducting RF (NCRF), and superconducting RF (SRF) photoelectron guns have peak accelerating fields of order 10 MV/m \cite{pinayev_high-gradient_2015, dunham_performance_2007, dowell_status_2006, arnold_overview_2011} with very high repetition rates (well above 1 MHz).
At the cost of duty factor, state of the art NCRF electron guns can offer even higher fields of order 100 MV/m \cite{ferrario_homdyn_2000} and recent experimental results suggest the possibility of pushing peak fields to nearly 500 MV/m for cryogenically cooled accelerating structures \cite{rosenzweig_ultra-high_2018, cahill_high_2018, wang_experimental_1995, rosenzweig_next_2019, mceuen_high-power_1985, schwettman_low_1967, nordlund_defect_2012, fortgang_cryogenic_1987, descoeudres_dc_2009, grudiev_new_2009, dolgashev_geometric_2010, marsh_x_2011}.
In this work, we characterize the intrinsic emittance at the photocathode source via the Mean Transverse Energy (MTE):
\begin{equation}
\varepsilon_{\text{C}} = \sigma_x \sqrt{\frac{\text{MTE}}{m c^2}},
\label{eq:initial_emittance}
\end{equation}
where $\sigma_x$ is the laser spot size, and $m$ is the mass of the electron.
Here, it is clear that MTE plays the role of an effective temperature of emission.
\begin{figure*}[htp]
\centering
\includegraphics[width=\linewidth]{Figures/pingu0.png}
\caption{The on-axis electric and magnetic field as seen by a reference particle in the center of the electron bunch. In each sub-figure, the cavity and magnet parameters are taken from an individual along the 0 meV Pareto front of the respective beamline. Fields are output directly from General Particle Tracer and computed from ASTRA.}
\label{fig:fields}
\end{figure*}
Great progress is being made in the discovery of low MTE photocathodes which are expected to improve the usable brightness of photoinjectors.
Due to the practical tradeoffs involved with photocathode choice, most photoinjectors today use materials with an MTE of around 150 meV \cite{weathersby_mega-electron-volt_2015, yang_100-femtosecond_2009, ding_measurements_2009, maxson_direct_2017}.
At the cost of QE, this MTE may be reduced by tuning the driving laser's wavelength.
For example, in Cs$_3$Sb and Cs:GaAs photocathodes, the lowest MTE that may achieved via wavelength tuning at room temperature is nearly 35 meV and 25 meV respectively, but at $10^{-6}$ - $10^{-5}$ QE \cite{pastuszka_transverse_1997, cultrera_cold_2015, musumeci_advances_2018}.
Recent work has shown that the cryogenic cooling of photocathodes emitting at threshold can reduce MTE even further, potentially down to single digit meV MTEs \cite{karkare_<10_2018}.
However, a natural question arises amidst this progress in MTE reduction: in modern space-charge-dominated applications, to what extent does MTE reduction actually improve the final emittance?
Even in the case of linear transport, 3D space-charge effects lead to a transverse position-angle correlation which varies along the longitudinal coordinate and leads to an inflation of projected emittance that requires compensation \cite{carlsten_new_1988, floettmann_emittance_2017, qiu_demonstration_1996, serafini_envelope_1997}.
The residual emittance after compensation is due to non-linear forces, either from space-charge or beamline elements.
Scaling laws exist to help estimate their effects \cite{carlsten_space-charge-induced_1995, bazarov_comparison_2011}.
In some cases, non-linearity can cause phase space wave-breaking in unevenly distributed beams that is a source of irreversible emittance growth \cite{anderson_internal_1987, anderson_nonequilibrium_2000}.
Another irreversible cause of emittance growth is disorder induced heating (DIH) and other Coulomb scattering effects which are expected to become important in the cold dense beams of future accelerator applications \cite{maxson_fundamental_2013}.
Avoiding these emittance growth mechanisms requires the advanced design and tuning of photoinjector systems.
Multi-objective genetic algorithm (MOGA) optimization is a popular technique for the design and tuning of realistic photoinjectors \cite{baptiste_status_2009, panofski_multi-objective_2017, ineichen_massively_2013, panofski_multi-objective_2018, qian_s-band_2016, emery_global_2005, papadopoulos_multiobjective_2010}.
Photoinjectors often have to balance several key design parameters or objectives that determine the usefulness of the system for a given application.
MOGA is a derivative free method for computing the Pareto front, or family of highest performing solutions, in a parallel and sample efficient manner \cite{deb_fast_2002}.
Elitist genetic algorithms are known to converge to the global optima of sufficiently well-behaved fitness functions given enough evaluations \cite{rudolph_convergence_1996}.
This makes them well suited for problems involving many local extrema.
Practical problems often require optimizations to be performed over a constrained search space and there exist techniques of incorporating these constraints into existing genetic algorithms without sacrificing efficiency \cite{bazarov_multivariate_2005}.
In this work, we examine the limits beam transport places on the ability of photoinjectors to take advantage of low MTE photocathodes in a diverse set of realistic simulated photoinjectors that have been tuned by a MOGA for ultimate performance.
This article begins with a discussion of our results involving the simulations of beamlines with idealized zero emittance photocathodes.
These simulations are performed on three important examples of high brightness electron beam applications: high repetition rate FELs, as well as single-shot DC and RF-based UED devices.
Using zero cathode emittance simulations, we introduce a new metric called the characteristic MTE to help understand the scale of photocathode MTE which is relevant to final beam quality.
It is shown that, depending on the properties of the beamline, system parameters need to be re-optimized to take full advantage of photocathode improvements.
We present a method of estimating when re-optimization needs to be performed and the magnitude of its effect on final emittance.
Finally, we set the scale for the magnitude of emittance growth due to point-to-point Coulomb interactions using a stochastic space-charge algorithm.
\section{Optimizations with a 0 \MakeLowercase{me}V MTE Photocathode}
To understand the contribution of photocathode MTE towards the final emittance of high brightness photoinjectors, we directly compare injector performance with a contemporary $\sim$150 meV MTE photocathode to what would be achievable with a perfect 0 meV MTE counterpart.
To cover the wide range of existing and near future accelerator technologies, we chose three realistic beamlines with significantly different energies as a representative set of high brightness photoinjector applications.
A DC and NCRF electron gun based single shot UED beamline reflect the two predominant energy scales of electron diffraction with single nanometer scale emittance at 10 - 100 fC bunch charge: order of magnitude 100 keV and 1 MeV.
At higher bunch charge, we select an SRF photoinjector under development at KEK expected to be capable of sub-$\mu$m scale emittance at 100 pC bunch charge for simulations representative of FEL driver applications.
The ultimate performance of each system is evaluated on the basis of the particle tracking codes General Particle Tracer \cite{van_der_geer_applications_1997} and ASTRA \cite{floettmann_astra:_2017} with optimization carried out in the framework of MOGA.
Children were generated with simulated binary crossover and polynomial mutation \cite{deb_multi-objective_2001}.
Selection was performed with SPEA-II \cite{zitzler_spea2:_2001} in the case of both UED examples and with NSGA-II \cite{deb_fast_2002} in the case of the FEL example.
Emittance preservation is known to depend strongly on the initial transverse and longitudinal distribution of the beam.
To this end, the optimizer is given the power to change parameters controlling the initial particle distribution using the same method described in \cite{bazarov_comparison_2011}.
The DC UED beamline is modeled after a similar system under development at Cornell University using the cryogenically cooled photoemission source described in \cite{lee_cryogenically_2018}.
The performance of this system under different conditions than presently considered is discussed in \cite{gulliford_multiobjective_2016}
where a detailed description of the layout and simulation methodology is also provided.
On-axis fields for this beamline are shown in Fig. \ref{fig:fields}a.
The beamline consists of two solenoids that surround an NCRF single cell bunching cavity and aid in transporting the high brightness beam to the sample located at $s = 1$ m.
The optimizer is given control over all magnet and cavity settings to minimize the RMS emittance at the sample while maximizing bunch charge.
Only solutions that keep the final spot size smaller than 100 \si{\micro\meter} RMS and the final beam length less than 1 ps RMS are considered.
These constraints were chosen based on common sample sizes used in diffraction \cite{weathersby_mega-electron-volt_2015} and the timescale of lattice vibration dynamics \cite{ligges_observation_2009, stern_mapping_2018}.
For a complete description of the decisions, objectives, and constraints used for this system, refer to Tab. \ref{tab:dc}.
The high gradient NCRF UED beamline is driven by a 1.6 cell 2.856 GHz gun capable of 100 MV/m and based on a design currently in use at a number of labs \cite{weathersby_mega-electron-volt_2015, musumeci_time_2009, zhu_femtosecond_2015, zhu_dynamic_2013, filippetto_design_2016}.
Samples are located at $s = 2.75$ m and the optimizer is given full control over two solenoids which surround a nine cell bunching cavity that is modeled after the first cell of the SLAC linac described in \cite{neal_stanford_1968}.
A discussion of our previous optimization experience with this beamline under a different set of constraints can be found in \cite{gulliford_multiobjective_2017}.
As in the case of the DC UED beamline, the optimizer was configured to minimize final RMS emittance while maximizing delivered bunch charge under the constraint of keeping the final spot size less than 100 \si{\micro\meter} RMS and the final length shorter than 1 ps RMS.
The decisions, objectives, and constraints of this optimization are detailed in Tab. \ref{tab:ncrf} and an example of the on-axis fields from an optimized individual is shown in Fig. \ref{fig:fields}b.
Our FEL driver example includes a 1.5 cell 1.3 GHz SRF gun in development at KEK for use in a CW ERL light source coupled with a photoinjector lattice aimed at use in the LCLS-II HE upgrade \cite{konomi_development_2019}.
The gun energy is controlled by the optimizer, but is in the range 1.5 - 3.5 MeV.
Immediately after the gun is a 1.3 GHz 9 cell capture cavity surrounded by two solenoids.
The remaining cavities, of the same design as the capture cavity, are shown in the plot of external fields in Fig. \ref{fig:fields}c and accelerate the beam to its final energy of roughly 100 MeV.
Accelerating cavity number three was kept off during optimization as a planned backup for cavity failure in the real machine.
The bunch charge was fixed to 100 pC, and optimizations were performed to minimize both RMS emittance and bunch length at the end of the injector system.
Energy constraints were tailored for the injector's use in the LCLS-II HE upgrade, and so we required valid solutions to have an energy greater than 90 MeV, an energy spread below 200 keV, and a higher order energy spread less than 5 keV.
The full set of decisions, objectives, and constraints is compiled in Tab. \ref{tab:fel}.
\begin{table}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Decision} & {\bf Range} \\
\hline\hline
Bunch Charge & 0 - 160 fC\\
Initial RMS Beam Size & 0 - 1 mm\\
Intitial RMS Beam Length & 0 - 50 ps\\
MTE & 0, 150 meV\\
Gun Voltage & 225 kV\\
Solenoid Current 1 and 2 & 0 - 4 A\\
Buncher Voltage & 0 - 60 kV\\
Buncher Phase & 90 degrees\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Objective} & {\bf Goal} \\
\hline\hline
RMS Emittance & Minimize\\
Delivered Bunch Charge & Maximize\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Constraint} & {\bf Value} \\
\hline\hline
Final RMS Spot Size & $<$ 100 \si{\micro\meter}\\
Final RMS Bunch Length & $<$ 1 ps\\
\hline
\end{tabular}
\caption{Optimizer configuration for the DC gun UED beamline}
\label{tab:dc}
\end{table}
\begin{table}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Decision} & {\bf Range} \\
\hline
\hline
Bunch Charge & 0 - 300 fC\\
Initial RMS Beam Size & 0 - 50 \si{\micro\meter}\\
Intitial RMS Beam Length & 0 - 50 ps\\
MTE & 0, 150 meV\\
Gun Phase & -90 - 90 degrees\\
Peak Gun Field & 100 MV/m\\
Beam Energy & 4.5 MeV\\
Solenoid Current 1 and 2 & 0 - 4 A\\
Buncher Peak Power & 0 - 25 MW\\
Buncher Phase & 90 degrees\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Objective} & {\bf Goal} \\
\hline\hline
RMS Emittance & Minimize\\
Delivered Bunch Charge & Maximize\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Constraint} & {\bf Value} \\
\hline\hline
Final RMS Spot Size & $<$ 100 \si{\micro\meter}\\
Final RMS Bunch Length & $<$ 1 ps\\
\hline
\end{tabular}
\caption{Optimizer configuration for the NCRF UED beamline}
\label{tab:ncrf}
\end{table}
\begin{table}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Decision} & {\bf Range} \\
\hline
\hline
Bunch Charge & 100 pC\\
Initial RMS Beam Size & 0.05 - 10 mm\\
Initial RMS Beam Length & 5 - 70 ps\\
MTE & 0, 130 meV\\
Gun Gradient & 20-50 MV/m\\
Gun Phase & -60 - 60 degrees\\
Gun Energy * & 1.5-3.5 MeV\\
Solenoid 1 Field & 0 - 0.4 T\\
Capture Cavity Gradient & 0 - 32 MV/m\\
Capture Cavity Phase & -180 - 180 degrees\\
Capture Cavity Offset & 0 - 2 m\\
Solenoid 2 Field & 0 - 0.3 T\\
Solenoid 2 Offset & 0 - 2 m\\
Cryomodule Offset & 0 - 3 m\\
Accel. Cavity 1, 2, and 4 Field & 0 - 32 MV/m\\
Accel. Cavity 1, 2, and 4 Phase & -90 - 90 degrees\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Objective} & {\bf Goal} \\
\hline\hline
RMS Emittance & Minimize\\
Final RMS Bunch Length & Minimize\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Constraint} & {\bf Value} \\
\hline\hline
Final Energy & $>$ 90 MeV\\
Energy Spread & $<$ 200 keV\\
Higher Order Energy Spread & $<$ 5 keV\\
\hline
\end{tabular}
\caption{Optimizer configuration for the KEK gun FEL driver example. (*) gun energy is computed from gradient and phase and not directly controlled by optimizer.}
\label{tab:fel}
\end{table}
Initial generations of the genetic optimization were evaluated with a small number of macroparticles to develop a good approximation of the global optima before moving on to the more accurate simulations involving $10^5$ macroparticles for the UED examples and $10^4$ macroparticles for the FEL driver.
The optimization stopping condition was that improvement of the Pareto front with each successive generation fell below a threshold of approximately 10\% relative change.
The products of these optimizations are shown in Fig. \ref{fig:pareto_and_mte}.
\begin{figure}[htp]
\centering
\centering
\includegraphics[width=\linewidth]{Figures/pingu6.png}
\caption{The distribution of initial spot sizes among the optimized individuals. The three example beamlines are labeled by color and individuals from the $\sim$150 meV fronts are in dashed lines while the individuals from the 0 meV fronts are represented by solid lines.}
\label{fig:spot_size}
\end{figure}
Both UED beamlines show a factor of between 10 and 100 improvement in brightness when the 150 meV photocathode is replaced by its 0 meV counterpart.
The degree of improvement is slightly greater in the case of the DC gun UED beamline.
As seen in Fig. \ref{fig:spot_size}, the optimizer chooses a smaller initial spot size for the NCRF gun individuals than for the DC gun individuals.
We conjecture that this is enabled by the higher accelerating gradient of the NCRF gun limiting the effects of space-charge emittance growth.
Due to the fact that initial emittance depends on both the photocathode's MTE and the initial spot size, a smaller initial spot size can mitigate the effects of a high thermal emittance photocathode.
The NCRF beamline also outperforms the DC beamline for emittance in absolute terms at similar bunch charges further suggesting a benefit with higher gradients on the cathode.
There is a sharp rise in slice emittance while the beam is still inside the gun and at low energy seen in Fig \ref{fig:example_individuals}a and \ref{fig:example_individuals}b.
This suggests that non-linear space-charge forces play a role in the residual emittance and the higher gradient and energy of the NCRF example could explain why it outperforms the DC example.
We observed that the brightness improvement from the 0 meV photocathode was limited to a factor of ten in the case of the FEL driver.
The higher bunch charge of this application is expected to increase the role of space-charge forces in transport and could be a cause of this more modest improvement.
\begin{figure*}
\centering
\includegraphics{Figures/pingu1.png}
\caption{The Pareto fronts of each beamline for the $\sim$150 meV and 0 meV MTE photocathodes and their characteristic MTE. The UED examples show between a factor of 10 and 100 improvement in brightness between the two Pareto fronts. The characteristic MTE calculated from a simulation including the effects of Coulomb scattering is included for the DC and NCRF Gun UED examples as a yellow cross.}
\label{fig:pareto_and_mte}
\end{figure*}
\begin{figure*}
\centering
\includegraphics{Figures/pingu2.png}
\caption{Emittance and beam sizes for an individual along the 0 meV Pareto front of each example. The projected emittance is the typical RMS normalized transverse emittance and the slice emittance is the average of the emittance evaluated over 100 longitudinal slices. Beam width and length are also plotted for reference. The total projected emittance in Fig. a is clipped at 500 pm for clarity.}
\label{fig:example_individuals}
\end{figure*}
\section{The Characteristic MTE Metric}
As long as the beam dynamics of the system do not change significantly with the introduction of a new photocathode, we can use the heuristic relationship that non-zero initial emittance will add roughly in quadrature with the emittance due to beam transport and the final emittance will be
\begin{equation}
\varepsilon^2 \approx \varepsilon_{\text{T}}^2 + \sigma_{x,i}^2\frac{MTE}{mc^2},
\end{equation}
where $\varepsilon_{\text{T}}$ is the emittance gained in beam transport, $\sigma_{x,i}$ is the initial spot size, and $\varepsilon_{\text{C}} = \sigma_{x,i}\sqrt{\frac{MTE}{mc^2}}$ is the initial emittance due to the photocathode and initial spot size.
To understand when the photocathode's MTE is important in the final emittance, we define a \textit{characteristic MTE} that would result in the emittance contribution of the photocathode and beam transport being equal as
\begin{equation}
\text{MTE}_{C} = mc^2 \left(\frac{\varepsilon_{\text{T}}}{\sigma_{x,i}}\right)^2.
\label{eq:effective_mte}
\end{equation}
The characteristic MTE is a beamline specific quantity that sets the scale for when photocathodes play a significant role in determining the final emittance of a photoinjector.
Photocathode improvements down to the characteristic MTE are likely to translate into increased usable brightness.
The characteristic MTE of each example is shown in Fig. \ref{fig:pareto_and_mte}.
Photocathode improvements down to the level of single meV MTE do affect the final emittance of each photoinjector application studied here.
The characteristic MTE of both the NCRF UED and FEL driver examples increases to roughly 50 meV at high bunch charge and short bunch length respectively.
The larger characteristic MTE of the NCRF UED example is likely due to the smaller initial spot size of the individuals.
This can be seen in Fig. \ref{fig:spot_size}.
That smaller spot size will increase the characteristic MTE for the same emittance because the initial emittance is less sensitive to photocathode parameters.
Characteristic MTE at short bunch lengths in the FEL example are primarily limited by large emittance growth in beam transport.
To test the validity of the heuristic argument that initial and transport emittance should add in quadrature, we simulated each individual from the 0 meV Pareto fronts with a photocathode whose MTE is the characteristic MTE.
The final emittance is expected to grow by a factor of $\sqrt{2}$ and we observe the ratio to be close but slightly larger than that value.
The frequency of ratios for each beamline is plotted in Fig. \ref{fig:ratio}.
For our investigation, we assume that the insertion of a new photocathode does not significantly change beam transport.
However, this condition will be violated to some extent and could explain why the ratio observed is slightly larger than $\sqrt{2}$.
\begin{figure}
\centering
\centering
\includegraphics[width=\linewidth]{Figures/pingu5.png}
\caption{Individuals from the 0 meV beamline were re-simulated with a photocathode MTE equal to their characteristic MTE. The frequency of the ratio of the new final emittance to the original final emittance is plotted.}
\label{fig:ratio}
\end{figure}
\subsection{Re-Optimization for New Photocathodes}
Our optimization experience showed that taking full advantage of the initial emittance improvements afforded by a new low MTE photocathode required the re-optimization of beamline parameters.
In particular, when individuals from the 150 meV Pareto fronts of the UED beamlines are re-simulated with a 0 meV photocathode and no changes to beamline parameters, their emittance is more than fifty percent larger than the emittance of individuals in the 0 meV Pareto front at comparable bunch charge.
This can be understood by considering the sensitivity of the transport emittance optimum to small changes in the initial spot size.
The characteristic MTE analysis does not take into account the fact that if shrinking the initial spot size from its optimal value reduces the initial emittance more than it increases emittance growth in transport, then the overall emittance will still go down.
The initial emittance, as in equation \ref{eq:initial_emittance}, can be reduced by using a smaller initial spot size.
However, if the system was already at the initial spot size which minimizes emittance growth in transport, as is the case of individuals along the 0 meV Pareto front, then changing it will negatively affect beamline performance.
Since the final emittance is roughly the quadrature sum of the initial emittance and the growth during transport, there will be a trade-off in minimizing both the initial emittance and emittance growth.
If the system was previously optimized with a high MTE photocathode, then the optimal spot size will not be at the minimum transport emittance possible and
new low MTE photocathodes can unlock strategies the optimizer avoided due to their larger spot sizes which increase initial emittance.
In this case, re-optimization will be required upon the insertion of a new low MTE photocathode.
\begin{figure}
\centering
\centering
\includegraphics[width=\linewidth]{Figures/pingu3.png}
\caption{An illustration of how re-optimization may be required upon insertion of a new photocathode. In black is the emittance due to transport ($\varepsilon_{\text{T}}$) as a function of the initial spot size. Around the optimal spot size, $\sigma_{x,i,0}$, this is approximately quadratic. The sensitivity in this example is roughly $x\approx 0.001$. The solid lines represent the initial emittance ($\varepsilon_{\text{C}}$) for three different thermal emittances. The dashed lines are the final emittance ($\varepsilon_{\text{F}}$), or the quadrature sums of initial and transport emittance. The optimal spot size with the 150 meV photocathode is significantly smaller than with a 0 meV or even 1 meV photocathode.}
\label{fig:sensitivity}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{Figures/pingu4.png}
\caption{The RMS and core emittance of an individual with $10^5$ electrons per bunch from the DC gun UED and NCRF gun UED 0 meV MTE Pareto fronts. In the row labeled "Beam Dynamics", the yellow lines were computed with the point-to-point space-charge algorithm and the blue lines with smooth space-charge. The solid lines are the RMS normalized emittance and the dashed lines are the core emittance. Below, are plots of the beam's transverse phase space at the sample location computed with the smooth and point-to-point methods. Linear $x$-$p_x$ correlation have been removed and the ellipse of phase space second moments is plotted in addition to the particle density.}
\label{fig:DIH}
\end{figure*}
This trade-off is represented graphically in Fig. \ref{fig:sensitivity} by plotting emittance as a function of initial spot size.
Initial emittance is linear in the initial spot size and is represented by a line whose slope depends on photocathode MTE.
Close to the optimum, the emittance due to transport may be expressed as a polynomial expansion in $\sigma_{x,i}$ which, to lowest order, is quadratic.
The final emittance is roughly the quadrature sum of both terms and has an optima at a smaller spot size than for transport emittance alone.
Characteristic MTE can also be represented in this plot since the initial emittance for a photocathode with an MTE equal to the characteristic MTE will pass through the vertex of the transport emittance parabola.
By using the second order expansion of beam transport's contribution to the emittance ($\varepsilon_{\text{T}}$) as a function of initial spot size around the optimum,
\begin{equation}
\varepsilon_{\text{T}}(\sigma_{x,i}) = A(\sigma_{x,i} - \sigma_{x,i,0})^2 + \varepsilon_{\text{T},0},
\label{eq:optimal_emittance}
\end{equation}
we can find the new optimal emittance with non-zero MTE.
To simplify our discussion, we consider the case of optima that are highly sensitive to changes in initial spot size.
Define the unitless parameter $x = \varepsilon_{\text{T},0}/(A\sigma_{x,i,0}^2)$ to measure the optimum's sensitivity.
In the limit of sensitive optima ($x \ll 1$) the new smallest emittance when the initial spot size is allowed to vary is
\begin{equation}
\varepsilon^2_{\text{opt}} = \varepsilon_{\text{T},0}^2 + \varepsilon_{\text{C}}^2\left[1 - \frac{x}{2}\frac{\text{MTE}}{\text{MTE}_{C}}\right]\ \ \ (x \ll 1).
\label{eq:corrected_mte}
\end{equation}
The new optimal initial spot size will be smaller for the non-zero MTE photocathode and, in the limit of small $x$, is approximately
\begin{equation}
\sigma_{x,i,opt}^2 = \sigma_{x,i,0}^2\left[1 - x\frac{\text{MTE}}{\text{MTE}_{C}}\right]\ \ \ (x \ll 1).
\label{eq:corrected_sigma}
\end{equation}
In practice, we observe the tendency of the optimizer to choose smaller initial spot sizes for beamlines with non-zero photocathode MTE.
In Fig. \ref{fig:spot_size} we plot the frequency of initial spot sizes from the 0 meV and $\sim$150 meV Pareto fronts of each beamline.
For the UED examples, the initial spot sizes for individuals in the 150 meV Pareto front are universally smaller than for those in the 0 meV Pareto front.
There is less of an impact on the FEL example, which could be due to the optima being highly sensitive to changes in initial spot size.
Systems with insensitive optima (large $x$) will tolerate higher MTE photocathodes than the original characteristic MTE metric implies.
Likewise, systems where the emittance grows rapidly for small changes in $\sigma_{x,i}$ (small $x$) cannot afford to decrease the initial spot size to compensate for any increase in the photocathode MTE.
The second term in the square brackets of Eq. \ref{eq:corrected_mte} is the relative scale for how much changing the initial spot size can improve emittance and can provide a rough guide to experimentalists for determining when a new photocathode technology requires re-optimization of the beamline.
The MTE for which the transport and photocathode contributions to the final emittance are the same even when allowing the initial spot size to vary is
\begin{equation}
\text{MTE}^{\prime}_{C} = \text{MTE}_{C}\left[1 + \frac{x}{2}\right]\ \ \ (x \ll 1).
\end{equation}
Although analytical formulas for the optimal emittance and spot size which are accurate to all order in $x$ may be found, they do not lend themselves to efficient analysis and numerical methods may be better suited for investigating the properties of systems with insensitive optima.
For each system, we can use the Pareto fronts obtained for the 0 meV and $\sim$150 meV MTE photocathodes to estimate the sensitivity parameter $x$ and calculate the correction to the characteristic MTE.
These Pareto fronts give us a value of the optimal emittance from Eq. \ref{eq:corrected_mte} for two different values of $\varepsilon_{\text{C}}$ and from there we can solve for $x$.
This operation was performed on each system and the sensitivity parameter was used to calculate the corrected characteristic MTE.
The correction in all cases was at the single percent level indicating that our optima are sensitive to initial spot size.
Consequently, the \textit{uncorrected} characteristic MTE, for the three realistic photoinjectors studied here, does a good job at predicting the scale at which photocathode improvements no longer improve brightness.
\section{Stochastic Space Charge}
Disorder induced heating (DIH) is known to play a role in degrading the emittance of cold and dense electron beams.
When the distance between particles falls below the Debye length of the one component plasma, inter-particle collisions can become important and can affect the momentum distribution of the bunch in a stochastic manner.
This effect will show up prominently when the average kinetic energy of particles in the transverse direction is of the same scale as the potential energy due to the Coulomb repulsion of the particle's neighbors.
The result is that the nascent momentum spread grows above its initial value by an amount $\Delta kT [eV] = 1.04\times 10^{-9} (n_0 [m^{-3}])^{1/3}$ \cite{maxson_fundamental_2013, van_der_geer_simulated_2007}.
Using the electron number density ($n_0$) at the beginning of each optimized example, the scale of DIH expected for all three beamlines is 1 meV.
Beyond DIH, Coulomb scattering after the cathode can lead to continuous irreversible emittance growth, but these effects are difficult to estimate analytically.
We expect DIH to be important in our simulations with 0 meV MTE photocathode due to the cold dense beams inside the guns.
To determine how much of an effect Coulomb scattering has on final emittance in our systems, one example from each of the DC and NCRF UED 0 meV Pareto fronts was chosen and simulated using a stochastic space-charge model.
The new algorithm for efficiently computing the effects of stochastic space-charge is based off of the Barnes-Hut tree method and will be discussed in detail in a forthcoming publication by M. Gordon, J. Maxson, et al.
Both the NCRF and DC UED individuals had a bunch charge of 10 fC.
Simulations were performed with GPT's smooth space-charge model discussed in \cite{bock_fast_2004} and with the tree-code method.
The RMS projected and core emittance \cite{bazarov_synchrotron_2012} along each beamline and with each space-charge model are shown in Fig. \ref{fig:DIH}.
Coulomb scattering contributes a factor of two increase in final emittance for both cases.
\section{Conclusion}
We have shown that characteristic MTE can be a useful tool in understanding the scale of MTE at which photocathode improvements translate to an increase in usable brightness.
These beamlines, which are representative of high brightness photoinjector applications, have characteristic MTEs on the scale of single to tens of meV, well below the 150 meV MTE of today's commonly used photocathodes.
Improvements in photocathode technology down to the level of 1 meV and below stands to improve the brightness of practical photoinjectors by an impressive two orders of magnitude.
However, it is not enough to simply insert a low MTE photocathode into an electron gun to achieve low final emittance.
To achieve this level of photoinjector performance, advanced optimization techniques like MOGA will need to be integrated into the design and tuning of future accelerators.
With the use of new photocathode technologies, further optimization may be required to take full advantage of low MTE.
The sensitivity of the optima to changes in initial spot size provides a guide for when it is necessary to re-optimize.
In addition, when in the regime of single meV photocathodes, existing models of smooth space-charge break down and the effects of Coulomb scattering become important in determining ultimate brightness.
Although the results of the present work are not affected by this problem because we are only concerned with order of magnitude changes in emittance, design tools for future accelerators may need to move to high performance point-to-point space-charge models to obtain good agreement with reality.
With the continued improvement of photocathode based electron sources and, in particular, the reduction of MTE in photocathode materials, bright beams will open up new possibilities for accelerator physics applications.
Notably, an increase in brightness would enable the time resolved characterization of biological macromolecules with UED \cite{sciaini_femtosecond_2011} as well as benefit X-ray FELs with a corresponding increase in total pulse energy benefiting a wide variety of x-ray scattering experiments in fields ranging from condensed matter physics, to chemistry, to biology \cite{abbamonte_new_2015}.
Work is already underway in understanding and beating the effects which limit photocathode MTE and in making existing low MTE photocathodes more practical for accelerator facility use \cite{karkare_effect_2011, jones_transverse_2018, feng_thermal_2015}.
Additionally, structured particle emitters have already been predicted to mitigate the emittance growth observed from disorder induced heating in the present simulations \cite{murphy_increasing_2015}.
If these photocathode improvements can be realized, then their results could provide as much as to two order of magnitude improvement in the final brightness of realistic modern photoinjectors.
\section{Acknowledgements}
This work was supported by the U.S. National Science Foundation under Award PHY-1549132, the Center for Bright Beams. We thank the US-Japan Science and Technology Cooperation Program in High Energy Physics for providing additional travel funding.
\bibliographystyle{bst/apsrev4-2}
\section{Introduction}
Improving the brightness of space-charge dominated electron sources will unlock a wealth of next generation accelerator physics applications.
For example, the largest unit cell that may be studied with single shot ultrafast electron diffraction (UED) is limited by the beam's transverse coherence length, which is determined by transverse emittance, at a high enough bunch charge to mitigate the effects of shot noise in data collection.
The study of protein dynamics with UED requires producing $>1$ nm scale coherence lengths at more than $10^5$ electrons and sub-picosecond pulse lengths at the sample location \cite{siwick_femtosecond_2004, dwyer_femtosecond_2006}.
In another example, the intensity of coherent radiation available to the users of free electron lasers (FELs) is, in part, limited by beam brightness.
Beam brightness affects the efficiency, radiated power, gain length, and photon energy reach of FELs \cite{hyder_emittance_1988, di_mitri_estimate_2014}.
Photoinjectors equipped with low intrinsic emittance photocathodes are among the brightest electron sources in use today.
Peak brightness at the source is limited by two factors: the electric field at the cathode and the photocathode's transverse momentum spread.
Several short-pulse Child-Langmuir-like charge density limits have been derived for the photoemission regimes of relevance to practical photoinjectors.
These current density extraction limits make explicit the dependence of peak brightness on photocathode parameters and the electric field. \cite{bazarov_maximum_2009, filippetto_maximum_2014, shamuilov_child-langmuir_2018}.
Depending on the aspect ratio of the bunch, the brightness limit is super-linear in the electric field and motivates the push towards high accelerating gradient photoinjectors.
Contemporary DC, normal-conducting RF (NCRF), and superconducting RF (SRF) photoelectron guns have peak accelerating fields of order 10 MV/m \cite{pinayev_high-gradient_2015, dunham_performance_2007, dowell_status_2006, arnold_overview_2011} with very high repetition rates (well above 1 MHz).
At the cost of duty factor, state of the art NCRF electron guns can offer even higher fields of order 100 MV/m \cite{ferrario_homdyn_2000} and recent experimental results suggest the possibility of pushing peak fields to nearly 500 MV/m for cryogenically cooled accelerating structures \cite{rosenzweig_ultra-high_2018, cahill_high_2018, wang_experimental_1995, rosenzweig_next_2019, mceuen_high-power_1985, schwettman_low_1967, nordlund_defect_2012, fortgang_cryogenic_1987, descoeudres_dc_2009, grudiev_new_2009, dolgashev_geometric_2010, marsh_x_2011}.
In this work, we characterize the intrinsic emittance at the photocathode source via the Mean Transverse Energy (MTE):
\begin{equation}
\varepsilon_{\text{C}} = \sigma_x \sqrt{\frac{\text{MTE}}{m c^2}},
\label{eq:initial_emittance}
\end{equation}
where $\sigma_x$ is the laser spot size, and $m$ is the mass of the electron.
Here, it is clear that MTE plays the role of an effective temperature of emission.
\begin{figure*}[htp]
\centering
\includegraphics[width=\linewidth]{Figures/pingu0.png}
\caption{The on-axis electric and magnetic field as seen by a reference particle in the center of the electron bunch. In each sub-figure, the cavity and magnet parameters are taken from an individual along the 0 meV Pareto front of the respective beamline. Fields are output directly from General Particle Tracer and computed from ASTRA.}
\label{fig:fields}
\end{figure*}
Great progress is being made in the discovery of low MTE photocathodes which are expected to improve the usable brightness of photoinjectors.
Due to the practical tradeoffs involved with photocathode choice, most photoinjectors today use materials with an MTE of around 150 meV \cite{weathersby_mega-electron-volt_2015, yang_100-femtosecond_2009, ding_measurements_2009, maxson_direct_2017}.
At the cost of QE, this MTE may be reduced by tuning the driving laser's wavelength.
For example, in Cs$_3$Sb and Cs:GaAs photocathodes, the lowest MTE that may achieved via wavelength tuning at room temperature is nearly 35 meV and 25 meV respectively, but at $10^{-6}$ - $10^{-5}$ QE \cite{pastuszka_transverse_1997, cultrera_cold_2015, musumeci_advances_2018}.
Recent work has shown that the cryogenic cooling of photocathodes emitting at threshold can reduce MTE even further, potentially down to single digit meV MTEs \cite{karkare_<10_2018}.
However, a natural question arises amidst this progress in MTE reduction: in modern space-charge-dominated applications, to what extent does MTE reduction actually improve the final emittance?
Even in the case of linear transport, 3D space-charge effects lead to a transverse position-angle correlation which varies along the longitudinal coordinate and leads to an inflation of projected emittance that requires compensation \cite{carlsten_new_1988, floettmann_emittance_2017, qiu_demonstration_1996, serafini_envelope_1997}.
The residual emittance after compensation is due to non-linear forces, either from space-charge or beamline elements.
Scaling laws exist to help estimate their effects \cite{carlsten_space-charge-induced_1995, bazarov_comparison_2011}.
In some cases, non-linearity can cause phase space wave-breaking in unevenly distributed beams that is a source of irreversible emittance growth \cite{anderson_internal_1987, anderson_nonequilibrium_2000}.
Another irreversible cause of emittance growth is disorder induced heating (DIH) and other Coulomb scattering effects which are expected to become important in the cold dense beams of future accelerator applications \cite{maxson_fundamental_2013}.
Avoiding these emittance growth mechanisms requires the advanced design and tuning of photoinjector systems.
Multi-objective genetic algorithm (MOGA) optimization is a popular technique for the design and tuning of realistic photoinjectors \cite{baptiste_status_2009, panofski_multi-objective_2017, ineichen_massively_2013, panofski_multi-objective_2018, qian_s-band_2016, emery_global_2005, papadopoulos_multiobjective_2010}.
Photoinjectors often have to balance several key design parameters or objectives that determine the usefulness of the system for a given application.
MOGA is a derivative free method for computing the Pareto front, or family of highest performing solutions, in a parallel and sample efficient manner \cite{deb_fast_2002}.
Elitist genetic algorithms are known to converge to the global optima of sufficiently well-behaved fitness functions given enough evaluations \cite{rudolph_convergence_1996}.
This makes them well suited for problems involving many local extrema.
Practical problems often require optimizations to be performed over a constrained search space and there exist techniques of incorporating these constraints into existing genetic algorithms without sacrificing efficiency \cite{bazarov_multivariate_2005}.
In this work, we examine the limits beam transport places on the ability of photoinjectors to take advantage of low MTE photocathodes in a diverse set of realistic simulated photoinjectors that have been tuned by a MOGA for ultimate performance.
This article begins with a discussion of our results involving the simulations of beamlines with idealized zero emittance photocathodes.
These simulations are performed on three important examples of high brightness electron beam applications: high repetition rate FELs, as well as single-shot DC and RF-based UED devices.
Using zero cathode emittance simulations, we introduce a new metric called the characteristic MTE to help understand the scale of photocathode MTE which is relevant to final beam quality.
It is shown that, depending on the properties of the beamline, system parameters need to be re-optimized to take full advantage of photocathode improvements.
We present a method of estimating when re-optimization needs to be performed and the magnitude of its effect on final emittance.
Finally, we set the scale for the magnitude of emittance growth due to point-to-point Coulomb interactions using a stochastic space-charge algorithm.
\section{Optimizations with a 0 \MakeLowercase{me}V MTE Photocathode}
To understand the contribution of photocathode MTE towards the final emittance of high brightness photoinjectors, we directly compare injector performance with a contemporary $\sim$150 meV MTE photocathode to what would be achievable with a perfect 0 meV MTE counterpart.
To cover the wide range of existing and near future accelerator technologies, we chose three realistic beamlines with significantly different energies as a representative set of high brightness photoinjector applications.
A DC and NCRF electron gun based single shot UED beamline reflect the two predominant energy scales of electron diffraction with single nanometer scale emittance at 10 - 100 fC bunch charge: order of magnitude 100 keV and 1 MeV.
At higher bunch charge, we select an SRF photoinjector under development at KEK expected to be capable of sub-$\mu$m scale emittance at 100 pC bunch charge for simulations representative of FEL driver applications.
The ultimate performance of each system is evaluated on the basis of the particle tracking codes General Particle Tracer \cite{van_der_geer_applications_1997} and ASTRA \cite{floettmann_astra:_2017} with optimization carried out in the framework of MOGA.
Children were generated with simulated binary crossover and polynomial mutation \cite{deb_multi-objective_2001}.
Selection was performed with SPEA-II \cite{zitzler_spea2:_2001} in the case of both UED examples and with NSGA-II \cite{deb_fast_2002} in the case of the FEL example.
Emittance preservation is known to depend strongly on the initial transverse and longitudinal distribution of the beam.
To this end, the optimizer is given the power to change parameters controlling the initial particle distribution using the same method described in \cite{bazarov_comparison_2011}.
The DC UED beamline is modeled after a similar system under development at Cornell University using the cryogenically cooled photoemission source described in \cite{lee_cryogenically_2018}.
The performance of this system under different conditions than presently considered is discussed in \cite{gulliford_multiobjective_2016}
where a detailed description of the layout and simulation methodology is also provided.
On-axis fields for this beamline are shown in Fig. \ref{fig:fields}a.
The beamline consists of two solenoids that surround an NCRF single cell bunching cavity and aid in transporting the high brightness beam to the sample located at $s = 1$ m.
The optimizer is given control over all magnet and cavity settings to minimize the RMS emittance at the sample while maximizing bunch charge.
Only solutions that keep the final spot size smaller than 100 \si{\micro\meter} RMS and the final beam length less than 1 ps RMS are considered.
These constraints were chosen based on common sample sizes used in diffraction \cite{weathersby_mega-electron-volt_2015} and the timescale of lattice vibration dynamics \cite{ligges_observation_2009, stern_mapping_2018}.
For a complete description of the decisions, objectives, and constraints used for this system, refer to Tab. \ref{tab:dc}.
The high gradient NCRF UED beamline is driven by a 1.6 cell 2.856 GHz gun capable of 100 MV/m and based on a design currently in use at a number of labs \cite{weathersby_mega-electron-volt_2015, musumeci_time_2009, zhu_femtosecond_2015, zhu_dynamic_2013, filippetto_design_2016}.
Samples are located at $s = 2.75$ m and the optimizer is given full control over two solenoids which surround a nine cell bunching cavity that is modeled after the first cell of the SLAC linac described in \cite{neal_stanford_1968}.
A discussion of our previous optimization experience with this beamline under a different set of constraints can be found in \cite{gulliford_multiobjective_2017}.
As in the case of the DC UED beamline, the optimizer was configured to minimize final RMS emittance while maximizing delivered bunch charge under the constraint of keeping the final spot size less than 100 \si{\micro\meter} RMS and the final length shorter than 1 ps RMS.
The decisions, objectives, and constraints of this optimization are detailed in Tab. \ref{tab:ncrf} and an example of the on-axis fields from an optimized individual is shown in Fig. \ref{fig:fields}b.
Our FEL driver example includes a 1.5 cell 1.3 GHz SRF gun in development at KEK for use in a CW ERL light source coupled with a photoinjector lattice aimed at use in the LCLS-II HE upgrade \cite{konomi_development_2019}.
The gun energy is controlled by the optimizer, but is in the range 1.5 - 3.5 MeV.
Immediately after the gun is a 1.3 GHz 9 cell capture cavity surrounded by two solenoids.
The remaining cavities, of the same design as the capture cavity, are shown in the plot of external fields in Fig. \ref{fig:fields}c and accelerate the beam to its final energy of roughly 100 MeV.
Accelerating cavity number three was kept off during optimization as a planned backup for cavity failure in the real machine.
The bunch charge was fixed to 100 pC, and optimizations were performed to minimize both RMS emittance and bunch length at the end of the injector system.
Energy constraints were tailored for the injector's use in the LCLS-II HE upgrade, and so we required valid solutions to have an energy greater than 90 MeV, an energy spread below 200 keV, and a higher order energy spread less than 5 keV.
The full set of decisions, objectives, and constraints is compiled in Tab. \ref{tab:fel}.
\begin{table}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Decision} & {\bf Range} \\
\hline\hline
Bunch Charge & 0 - 160 fC\\
Initial RMS Beam Size & 0 - 1 mm\\
Intitial RMS Beam Length & 0 - 50 ps\\
MTE & 0, 150 meV\\
Gun Voltage & 225 kV\\
Solenoid Current 1 and 2 & 0 - 4 A\\
Buncher Voltage & 0 - 60 kV\\
Buncher Phase & 90 degrees\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Objective} & {\bf Goal} \\
\hline\hline
RMS Emittance & Minimize\\
Delivered Bunch Charge & Maximize\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Constraint} & {\bf Value} \\
\hline\hline
Final RMS Spot Size & $<$ 100 \si{\micro\meter}\\
Final RMS Bunch Length & $<$ 1 ps\\
\hline
\end{tabular}
\caption{Optimizer configuration for the DC gun UED beamline}
\label{tab:dc}
\end{table}
\begin{table}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Decision} & {\bf Range} \\
\hline
\hline
Bunch Charge & 0 - 300 fC\\
Initial RMS Beam Size & 0 - 50 \si{\micro\meter}\\
Intitial RMS Beam Length & 0 - 50 ps\\
MTE & 0, 150 meV\\
Gun Phase & -90 - 90 degrees\\
Peak Gun Field & 100 MV/m\\
Beam Energy & 4.5 MeV\\
Solenoid Current 1 and 2 & 0 - 4 A\\
Buncher Peak Power & 0 - 25 MW\\
Buncher Phase & 90 degrees\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Objective} & {\bf Goal} \\
\hline\hline
RMS Emittance & Minimize\\
Delivered Bunch Charge & Maximize\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Constraint} & {\bf Value} \\
\hline\hline
Final RMS Spot Size & $<$ 100 \si{\micro\meter}\\
Final RMS Bunch Length & $<$ 1 ps\\
\hline
\end{tabular}
\caption{Optimizer configuration for the NCRF UED beamline}
\label{tab:ncrf}
\end{table}
\begin{table}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Decision} & {\bf Range} \\
\hline
\hline
Bunch Charge & 100 pC\\
Initial RMS Beam Size & 0.05 - 10 mm\\
Initial RMS Beam Length & 5 - 70 ps\\
MTE & 0, 130 meV\\
Gun Gradient & 20-50 MV/m\\
Gun Phase & -60 - 60 degrees\\
Gun Energy * & 1.5-3.5 MeV\\
Solenoid 1 Field & 0 - 0.4 T\\
Capture Cavity Gradient & 0 - 32 MV/m\\
Capture Cavity Phase & -180 - 180 degrees\\
Capture Cavity Offset & 0 - 2 m\\
Solenoid 2 Field & 0 - 0.3 T\\
Solenoid 2 Offset & 0 - 2 m\\
Cryomodule Offset & 0 - 3 m\\
Accel. Cavity 1, 2, and 4 Field & 0 - 32 MV/m\\
Accel. Cavity 1, 2, and 4 Phase & -90 - 90 degrees\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Objective} & {\bf Goal} \\
\hline\hline
RMS Emittance & Minimize\\
Final RMS Bunch Length & Minimize\\
\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{|y{145pt}|y{75pt}|}
\hline
{\bf Constraint} & {\bf Value} \\
\hline\hline
Final Energy & $>$ 90 MeV\\
Energy Spread & $<$ 200 keV\\
Higher Order Energy Spread & $<$ 5 keV\\
\hline
\end{tabular}
\caption{Optimizer configuration for the KEK gun FEL driver example. (*) gun energy is computed from gradient and phase and not directly controlled by optimizer.}
\label{tab:fel}
\end{table}
Initial generations of the genetic optimization were evaluated with a small number of macroparticles to develop a good approximation of the global optima before moving on to the more accurate simulations involving $10^5$ macroparticles for the UED examples and $10^4$ macroparticles for the FEL driver.
The optimization stopping condition was that improvement of the Pareto front with each successive generation fell below a threshold of approximately 10\% relative change.
The products of these optimizations are shown in Fig. \ref{fig:pareto_and_mte}.
\begin{figure}[htp]
\centering
\centering
\includegraphics[width=\linewidth]{Figures/pingu6.png}
\caption{The distribution of initial spot sizes among the optimized individuals. The three example beamlines are labeled by color and individuals from the $\sim$150 meV fronts are in dashed lines while the individuals from the 0 meV fronts are represented by solid lines.}
\label{fig:spot_size}
\end{figure}
Both UED beamlines show a factor of between 10 and 100 improvement in brightness when the 150 meV photocathode is replaced by its 0 meV counterpart.
The degree of improvement is slightly greater in the case of the DC gun UED beamline.
As seen in Fig. \ref{fig:spot_size}, the optimizer chooses a smaller initial spot size for the NCRF gun individuals than for the DC gun individuals.
We conjecture that this is enabled by the higher accelerating gradient of the NCRF gun limiting the effects of space-charge emittance growth.
Due to the fact that initial emittance depends on both the photocathode's MTE and the initial spot size, a smaller initial spot size can mitigate the effects of a high thermal emittance photocathode.
The NCRF beamline also outperforms the DC beamline for emittance in absolute terms at similar bunch charges further suggesting a benefit with higher gradients on the cathode.
There is a sharp rise in slice emittance while the beam is still inside the gun and at low energy seen in Fig \ref{fig:example_individuals}a and \ref{fig:example_individuals}b.
This suggests that non-linear space-charge forces play a role in the residual emittance and the higher gradient and energy of the NCRF example could explain why it outperforms the DC example.
We observed that the brightness improvement from the 0 meV photocathode was limited to a factor of ten in the case of the FEL driver.
The higher bunch charge of this application is expected to increase the role of space-charge forces in transport and could be a cause of this more modest improvement.
\begin{figure*}
\centering
\includegraphics{Figures/pingu1.png}
\caption{The Pareto fronts of each beamline for the $\sim$150 meV and 0 meV MTE photocathodes and their characteristic MTE. The UED examples show between a factor of 10 and 100 improvement in brightness between the two Pareto fronts. The characteristic MTE calculated from a simulation including the effects of Coulomb scattering is included for the DC and NCRF Gun UED examples as a yellow cross.}
\label{fig:pareto_and_mte}
\end{figure*}
\begin{figure*}
\centering
\includegraphics{Figures/pingu2.png}
\caption{Emittance and beam sizes for an individual along the 0 meV Pareto front of each example. The projected emittance is the typical RMS normalized transverse emittance and the slice emittance is the average of the emittance evaluated over 100 longitudinal slices. Beam width and length are also plotted for reference. The total projected emittance in Fig. a is clipped at 500 pm for clarity.}
\label{fig:example_individuals}
\end{figure*}
\section{The Characteristic MTE Metric}
As long as the beam dynamics of the system do not change significantly with the introduction of a new photocathode, we can use the heuristic relationship that non-zero initial emittance will add roughly in quadrature with the emittance due to beam transport and the final emittance will be
\begin{equation}
\varepsilon^2 \approx \varepsilon_{\text{T}}^2 + \sigma_{x,i}^2\frac{MTE}{mc^2},
\end{equation}
where $\varepsilon_{\text{T}}$ is the emittance gained in beam transport, $\sigma_{x,i}$ is the initial spot size, and $\varepsilon_{\text{C}} = \sigma_{x,i}\sqrt{\frac{MTE}{mc^2}}$ is the initial emittance due to the photocathode and initial spot size.
To understand when the photocathode's MTE is important in the final emittance, we define a \textit{characteristic MTE} that would result in the emittance contribution of the photocathode and beam transport being equal as
\begin{equation}
\text{MTE}_{C} = mc^2 \left(\frac{\varepsilon_{\text{T}}}{\sigma_{x,i}}\right)^2.
\label{eq:effective_mte}
\end{equation}
The characteristic MTE is a beamline specific quantity that sets the scale for when photocathodes play a significant role in determining the final emittance of a photoinjector.
Photocathode improvements down to the characteristic MTE are likely to translate into increased usable brightness.
The characteristic MTE of each example is shown in Fig. \ref{fig:pareto_and_mte}.
Photocathode improvements down to the level of single meV MTE do affect the final emittance of each photoinjector application studied here.
The characteristic MTE of both the NCRF UED and FEL driver examples increases to roughly 50 meV at high bunch charge and short bunch length respectively.
The larger characteristic MTE of the NCRF UED example is likely due to the smaller initial spot size of the individuals.
This can be seen in Fig. \ref{fig:spot_size}.
That smaller spot size will increase the characteristic MTE for the same emittance because the initial emittance is less sensitive to photocathode parameters.
Characteristic MTE at short bunch lengths in the FEL example are primarily limited by large emittance growth in beam transport.
To test the validity of the heuristic argument that initial and transport emittance should add in quadrature, we simulated each individual from the 0 meV Pareto fronts with a photocathode whose MTE is the characteristic MTE.
The final emittance is expected to grow by a factor of $\sqrt{2}$ and we observe the ratio to be close but slightly larger than that value.
The frequency of ratios for each beamline is plotted in Fig. \ref{fig:ratio}.
For our investigation, we assume that the insertion of a new photocathode does not significantly change beam transport.
However, this condition will be violated to some extent and could explain why the ratio observed is slightly larger than $\sqrt{2}$.
\begin{figure}
\centering
\centering
\includegraphics[width=\linewidth]{Figures/pingu5.png}
\caption{Individuals from the 0 meV beamline were re-simulated with a photocathode MTE equal to their characteristic MTE. The frequency of the ratio of the new final emittance to the original final emittance is plotted.}
\label{fig:ratio}
\end{figure}
\subsection{Re-Optimization for New Photocathodes}
Our optimization experience showed that taking full advantage of the initial emittance improvements afforded by a new low MTE photocathode required the re-optimization of beamline parameters.
In particular, when individuals from the 150 meV Pareto fronts of the UED beamlines are re-simulated with a 0 meV photocathode and no changes to beamline parameters, their emittance is more than fifty percent larger than the emittance of individuals in the 0 meV Pareto front at comparable bunch charge.
This can be understood by considering the sensitivity of the transport emittance optimum to small changes in the initial spot size.
The characteristic MTE analysis does not take into account the fact that if shrinking the initial spot size from its optimal value reduces the initial emittance more than it increases emittance growth in transport, then the overall emittance will still go down.
The initial emittance, as in equation \ref{eq:initial_emittance}, can be reduced by using a smaller initial spot size.
However, if the system was already at the initial spot size which minimizes emittance growth in transport, as is the case of individuals along the 0 meV Pareto front, then changing it will negatively affect beamline performance.
Since the final emittance is roughly the quadrature sum of the initial emittance and the growth during transport, there will be a trade-off in minimizing both the initial emittance and emittance growth.
If the system was previously optimized with a high MTE photocathode, then the optimal spot size will not be at the minimum transport emittance possible and
new low MTE photocathodes can unlock strategies the optimizer avoided due to their larger spot sizes which increase initial emittance.
In this case, re-optimization will be required upon the insertion of a new low MTE photocathode.
\begin{figure}
\centering
\centering
\includegraphics[width=\linewidth]{Figures/pingu3.png}
\caption{An illustration of how re-optimization may be required upon insertion of a new photocathode. In black is the emittance due to transport ($\varepsilon_{\text{T}}$) as a function of the initial spot size. Around the optimal spot size, $\sigma_{x,i,0}$, this is approximately quadratic. The sensitivity in this example is roughly $x\approx 0.001$. The solid lines represent the initial emittance ($\varepsilon_{\text{C}}$) for three different thermal emittances. The dashed lines are the final emittance ($\varepsilon_{\text{F}}$), or the quadrature sums of initial and transport emittance. The optimal spot size with the 150 meV photocathode is significantly smaller than with a 0 meV or even 1 meV photocathode.}
\label{fig:sensitivity}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{Figures/pingu4.png}
\caption{The RMS and core emittance of an individual with $10^5$ electrons per bunch from the DC gun UED and NCRF gun UED 0 meV MTE Pareto fronts. In the row labeled "Beam Dynamics", the yellow lines were computed with the point-to-point space-charge algorithm and the blue lines with smooth space-charge. The solid lines are the RMS normalized emittance and the dashed lines are the core emittance. Below, are plots of the beam's transverse phase space at the sample location computed with the smooth and point-to-point methods. Linear $x$-$p_x$ correlation have been removed and the ellipse of phase space second moments is plotted in addition to the particle density.}
\label{fig:DIH}
\end{figure*}
This trade-off is represented graphically in Fig. \ref{fig:sensitivity} by plotting emittance as a function of initial spot size.
Initial emittance is linear in the initial spot size and is represented by a line whose slope depends on photocathode MTE.
Close to the optimum, the emittance due to transport may be expressed as a polynomial expansion in $\sigma_{x,i}$ which, to lowest order, is quadratic.
The final emittance is roughly the quadrature sum of both terms and has an optima at a smaller spot size than for transport emittance alone.
Characteristic MTE can also be represented in this plot since the initial emittance for a photocathode with an MTE equal to the characteristic MTE will pass through the vertex of the transport emittance parabola.
By using the second order expansion of beam transport's contribution to the emittance ($\varepsilon_{\text{T}}$) as a function of initial spot size around the optimum,
\begin{equation}
\varepsilon_{\text{T}}(\sigma_{x,i}) = A(\sigma_{x,i} - \sigma_{x,i,0})^2 + \varepsilon_{\text{T},0},
\label{eq:optimal_emittance}
\end{equation}
we can find the new optimal emittance with non-zero MTE.
To simplify our discussion, we consider the case of optima that are highly sensitive to changes in initial spot size.
Define the unitless parameter $x = \varepsilon_{\text{T},0}/(A\sigma_{x,i,0}^2)$ to measure the optimum's sensitivity.
In the limit of sensitive optima ($x \ll 1$) the new smallest emittance when the initial spot size is allowed to vary is
\begin{equation}
\varepsilon^2_{\text{opt}} = \varepsilon_{\text{T},0}^2 + \varepsilon_{\text{C}}^2\left[1 - \frac{x}{2}\frac{\text{MTE}}{\text{MTE}_{C}}\right]\ \ \ (x \ll 1).
\label{eq:corrected_mte}
\end{equation}
The new optimal initial spot size will be smaller for the non-zero MTE photocathode and, in the limit of small $x$, is approximately
\begin{equation}
\sigma_{x,i,opt}^2 = \sigma_{x,i,0}^2\left[1 - x\frac{\text{MTE}}{\text{MTE}_{C}}\right]\ \ \ (x \ll 1).
\label{eq:corrected_sigma}
\end{equation}
In practice, we observe the tendency of the optimizer to choose smaller initial spot sizes for beamlines with non-zero photocathode MTE.
In Fig. \ref{fig:spot_size} we plot the frequency of initial spot sizes from the 0 meV and $\sim$150 meV Pareto fronts of each beamline.
For the UED examples, the initial spot sizes for individuals in the 150 meV Pareto front are universally smaller than for those in the 0 meV Pareto front.
There is less of an impact on the FEL example, which could be due to the optima being highly sensitive to changes in initial spot size.
Systems with insensitive optima (large $x$) will tolerate higher MTE photocathodes than the original characteristic MTE metric implies.
Likewise, systems where the emittance grows rapidly for small changes in $\sigma_{x,i}$ (small $x$) cannot afford to decrease the initial spot size to compensate for any increase in the photocathode MTE.
The second term in the square brackets of Eq. \ref{eq:corrected_mte} is the relative scale for how much changing the initial spot size can improve emittance and can provide a rough guide to experimentalists for determining when a new photocathode technology requires re-optimization of the beamline.
The MTE for which the transport and photocathode contributions to the final emittance are the same even when allowing the initial spot size to vary is
\begin{equation}
\text{MTE}^{\prime}_{C} = \text{MTE}_{C}\left[1 + \frac{x}{2}\right]\ \ \ (x \ll 1).
\end{equation}
Although analytical formulas for the optimal emittance and spot size which are accurate to all order in $x$ may be found, they do not lend themselves to efficient analysis and numerical methods may be better suited for investigating the properties of systems with insensitive optima.
For each system, we can use the Pareto fronts obtained for the 0 meV and $\sim$150 meV MTE photocathodes to estimate the sensitivity parameter $x$ and calculate the correction to the characteristic MTE.
These Pareto fronts give us a value of the optimal emittance from Eq. \ref{eq:corrected_mte} for two different values of $\varepsilon_{\text{C}}$ and from there we can solve for $x$.
This operation was performed on each system and the sensitivity parameter was used to calculate the corrected characteristic MTE.
The correction in all cases was at the single percent level indicating that our optima are sensitive to initial spot size.
Consequently, the \textit{uncorrected} characteristic MTE, for the three realistic photoinjectors studied here, does a good job at predicting the scale at which photocathode improvements no longer improve brightness.
\section{Stochastic Space Charge}
Disorder induced heating (DIH) is known to play a role in degrading the emittance of cold and dense electron beams.
When the distance between particles falls below the Debye length of the one component plasma, inter-particle collisions can become important and can affect the momentum distribution of the bunch in a stochastic manner.
This effect will show up prominently when the average kinetic energy of particles in the transverse direction is of the same scale as the potential energy due to the Coulomb repulsion of the particle's neighbors.
The result is that the nascent momentum spread grows above its initial value by an amount $\Delta kT [eV] = 1.04\times 10^{-9} (n_0 [m^{-3}])^{1/3}$ \cite{maxson_fundamental_2013, van_der_geer_simulated_2007}.
Using the electron number density ($n_0$) at the beginning of each optimized example, the scale of DIH expected for all three beamlines is 1 meV.
Beyond DIH, Coulomb scattering after the cathode can lead to continuous irreversible emittance growth, but these effects are difficult to estimate analytically.
We expect DIH to be important in our simulations with 0 meV MTE photocathode due to the cold dense beams inside the guns.
To determine how much of an effect Coulomb scattering has on final emittance in our systems, one example from each of the DC and NCRF UED 0 meV Pareto fronts was chosen and simulated using a stochastic space-charge model.
The new algorithm for efficiently computing the effects of stochastic space-charge is based off of the Barnes-Hut tree method and will be discussed in detail in a forthcoming publication by M. Gordon, J. Maxson, et al.
Both the NCRF and DC UED individuals had a bunch charge of 10 fC.
Simulations were performed with GPT's smooth space-charge model discussed in \cite{bock_fast_2004} and with the tree-code method.
The RMS projected and core emittance \cite{bazarov_synchrotron_2012} along each beamline and with each space-charge model are shown in Fig. \ref{fig:DIH}.
Coulomb scattering contributes a factor of two increase in final emittance for both cases.
\section{Conclusion}
We have shown that characteristic MTE can be a useful tool in understanding the scale of MTE at which photocathode improvements translate to an increase in usable brightness.
These beamlines, which are representative of high brightness photoinjector applications, have characteristic MTEs on the scale of single to tens of meV, well below the 150 meV MTE of today's commonly used photocathodes.
Improvements in photocathode technology down to the level of 1 meV and below stands to improve the brightness of practical photoinjectors by an impressive two orders of magnitude.
However, it is not enough to simply insert a low MTE photocathode into an electron gun to achieve low final emittance.
To achieve this level of photoinjector performance, advanced optimization techniques like MOGA will need to be integrated into the design and tuning of future accelerators.
With the use of new photocathode technologies, further optimization may be required to take full advantage of low MTE.
The sensitivity of the optima to changes in initial spot size provides a guide for when it is necessary to re-optimize.
In addition, when in the regime of single meV photocathodes, existing models of smooth space-charge break down and the effects of Coulomb scattering become important in determining ultimate brightness.
Although the results of the present work are not affected by this problem because we are only concerned with order of magnitude changes in emittance, design tools for future accelerators may need to move to high performance point-to-point space-charge models to obtain good agreement with reality.
With the continued improvement of photocathode based electron sources and, in particular, the reduction of MTE in photocathode materials, bright beams will open up new possibilities for accelerator physics applications.
Notably, an increase in brightness would enable the time resolved characterization of biological macromolecules with UED \cite{sciaini_femtosecond_2011} as well as benefit X-ray FELs with a corresponding increase in total pulse energy benefiting a wide variety of x-ray scattering experiments in fields ranging from condensed matter physics, to chemistry, to biology \cite{abbamonte_new_2015}.
Work is already underway in understanding and beating the effects which limit photocathode MTE and in making existing low MTE photocathodes more practical for accelerator facility use \cite{karkare_effect_2011, jones_transverse_2018, feng_thermal_2015}.
Additionally, structured particle emitters have already been predicted to mitigate the emittance growth observed from disorder induced heating in the present simulations \cite{murphy_increasing_2015}.
If these photocathode improvements can be realized, then their results could provide as much as to two order of magnitude improvement in the final brightness of realistic modern photoinjectors.
\section{Acknowledgements}
This work was supported by the U.S. National Science Foundation under Award PHY-1549132, the Center for Bright Beams. We thank the US-Japan Science and Technology Cooperation Program in High Energy Physics for providing additional travel funding.
\bibliographystyle{bst/apsrev4-2}
|
2207.01758
|
\section{Introduction}
\label{section:sec1}
The Coronavirus Disease 2019 SARS-CoV-2 (COVID-19) is a highly infectious disease, which emerged in December, 2019 \cite{WHO}. Early detection based on chest CT scans is important to the timely treatment of patients and the slowdown of viral transmission.
However, a volumn of CT scans contains hundreds of slices, which requires a heavy workload on radiologists.
Recently, deep learning approaches have achieved excellent performance in fighting against COVID-19. They have been widely applied to many aspects, including the lung and infection region segmentation \cite{weakly,li2020artificial,Chen2020.02.25.20021568,tsinghua2020fourweek} as well as the clinical diagnosis and assessment \cite{wang2020automatically,wang2020a,song2020deep,HOU2021108005}.
In this paper, we present deep learning based solution for the 2nd COV19D Competition of the Workshop ``AI-enabled Medical Image Analysis – Digital Pathology \& Radiology/COVID19 (AIMIA)'', which occurs in conjunction with the European Conference on Computer Vision (ECCV) 2022.
The competition includes two challenges, namely COVID-19 Detection Challenge and COVID-19 Severity Detection Challenge. COVID-19 detection aims to identify COVID from non-COVID cases. Each CT scan is manually annotated with respect to COVID-19 and non-COVID-19 categories. The severity of COVID-19 can be further divided into four stages, including Mild, Moderate, Severe, and Critical. Each severity category is determined by the presence of ground glass opacities and the pulmonary
parenchymal involvement.
A natural way to diagnose COVID-19 based on 3D CT images is to use 3D networks. To address both challenges, we employ the advanced 3D contrastive mixup classification network (CMC-COV19D) in our previous work \cite{hou2021cmc}, which won the first price in the ICCV 2021 COVID-19 Diagnosis Competition of AI-enabled Medical Image Analysis Workshop \cite{kollias2021mia}. Our CMC-COV19D framework introduces contrastive representation learning to discover more discriminative representations of COVID-19 cases. Besides, we use a joint training loss that combines the classification loss, mixup loss, and contrastive loss. We further employ an inflated 3D ImageNet pre-trained ResNest50 \cite{zhang2020resnest} as a strong feature extractor to boost more accurate COVID-19 diagnostic performance. Experimental results on both challenges show that our approach significantly surpasses the baseline model provided by organizers.
\section{Dataset} \label{section:dataset}
COV19-CT-DB \cite{kollias2022ai} includes 3D chest CT scans annotated for existence of COVID-19. It consists of 1,650 COVID and 6,100 non-COVID chest CT scan series, which correspond to a high number of patients (more than 1150) and subjects (more than 2600). In total, 724,273 slices correspond to the CT scans of the COVID-19 category and 1,775,727 slices correspond to the non COVID-19 category. Each of the 3D scans includes different number of slices, ranging from 50 to 700.
The database has been split in training, validation and testing sets. The training set contains 1992 3D CT scans. The validation set consists of 494 3D CT scans. A further split of the COVID-19 cases has been implemented, based on the severity of COVID-19, in the range from 1 to 4. In particular, parts of the COV19-CT-DB COVID-19 training and validation datasets have been accordingly split for severity classification in training, validation and testing sets. The training set contains, in total, 258 3D CT scans. The validation set consists of 61 3D CT scans.
\section{Methodology} \label{section:method}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{cmc.pdf}
\caption{Overview of our CMC-COV19D Network.}
\label{fig:cmc}
\end{figure}
As shown in Fig. \ref{fig:cmc}, Our CMC-COV19D network is composed of contrastive representation learning (CRL) and mixup classification.
\subsubsection{Contrastive Representation Learning.}
Our CMC-COV19D network employs the contrastive representation learning (CRL) as an auxiliary task to learn more discriminative representations of COVID-19. The CRL is comprised of the following components. 1) A stochastic data augmentation module $A(\cdot)$, which transforms an input CT $x$ into a randomly augmented sample $\tilde{x}$. We generate two augmented volumes from each input CT. 2) A base encoder $E(\cdot)$, which maps the augmented CT sample $\tilde{x}$ to a representation vector $r=E(\tilde{x})\in \mathbb{R}^{d_e}$. 3) A projection network $P(\cdot)$, which is used to map the representation vector $r$ to a relative low-dimension vector $z=P(r)\in \mathbb{R}^{d_p}$. 4) A classifier $C(\cdot)$, which classifies the vector $r\in \mathbb{R}^{d_e}$ to the final prediction.
Given a minibatch of $N$ CT images and their labels $\{(x_k,y_k)\}_{k=1,\dots,N}$, we can generate a minibatch of $2N$ samples after data augmentations.
Inspired by the supervised contrastive loss \cite{2020Supervisedcon}, we define the positives as any augmented CT samples from the same category, whereas the CT samples from different classes are considered as negative pairs. Let $i\in \{1,\dots,2N\}$ be the index of an arbitrary augmented sample, the contrastive loss function is defined as:
\begin{equation}
\mathcal{L}_{con}^i=\frac{-1}{2N_{\tilde{y}_i}-1}\sum_{j=1}^{2N}\mathbbm{1}_{i\ne j}\cdot\mathbbm{1}_{\tilde{y}_i=\tilde{y}_j}\cdot\log\frac{\exp(z_i^T\cdot z_j/\tau)}{\sum_{k=1}^{2N}\mathbbm{1}_{i\ne k}\cdot\exp(z_i^T\cdot z_k/\tau)},
\label{infonce}
\end{equation}
where $\mathbbm{1}\in\{0,1\}$ is an indicator function, and $\tau >0$ denotes a scalar temperature hyper-parameter. $N_{\tilde{y}_i}$ is the total number of samples in a minibatch that have the same label $\tilde{y}_i$.
\subsubsection{Mixup classification}
We adopt the mixup \cite{zhang2017mixup} strategy during training to further boost the generalization ability of the model. For each augmented CT sample $\tilde{x}_i$, we generate the mixup sample and it label as:
\begin{equation}
\tilde{x}^{mix}_i = \lambda\tilde{x}_i + (1-\lambda)\tilde{x}_{p},
~\tilde{y}^{mix}_i = \lambda\tilde{y}_i + (1-\lambda)\tilde{y}_{p},
\end{equation}
where $p$ is randomly selected indice. The mixup loss is defined as the cross entropy loss of mixup samples:
\begin{equation}
\mathcal{L}^i_{mix} = \mathrm{CrossEntropy}(\tilde{x}^{mix}_i,\tilde{y}^{mix}_i).
\end{equation}
Different from the original design \cite{zhang2017mixup} where they replaced the classification loss with the mixup loss, we merge the mixup loss with the classification loss to enhance the classification ability on both raw samples and mixup samples. The classification loss $\mathcal{L}_{ce}$ on raw samples is defined as:
\begin{equation}
\mathcal{L}_{clf}^i=-\tilde{y}_i^T\log\hat{y}_i,
\end{equation}
where $\tilde{y}_i$ denotes the one-hot vector of ground truth label, and $\hat{y}_i$ is predicted probability of the sample $x_i$ $(i=1,\dots,2N)$.
Finally, we merge the CRL loss, mixup loss, and classification loss into a combined objective function:
\begin{equation}
\mathcal{L}=\frac{1}{2N}\sum_{i=1}^{2N}(w_1\mathcal{L}^i_{con}+w_2\mathcal{L}^i_{mix}+w_3\mathcal{L}^i_{clf}),
\end{equation}
where $w_1, w_2, w_3$ denote the balance weights.
\section{Experimental Results} \label{section:experiments}
\subsection{Implementation Details}
For the COVID-19 detection, each CT volume is resized from $(N,512,512)$ to $(128,256,256)$, where $N$ denotes the number of slices. Data augmentation includes RandomResizedCrop, random crop on the z-axis to 64, random contrast changes. Other augmentations such as flip and rotation are also tried, but no significant improvement is yielded.
For the COVID-19 severity detection, we resample each CT volume into $(64,256,256)$. The same augmentations are applied, except for random crop on the z-axis.
We employ inflated 3D ResNest50 and Uniformer-S as the backbones in our experiments. The networks are trained for 100 epochs.
We optimize the networks using the Adam algorithm with a weight decay of $10^{-5}$. The initial learning rate is set to $10^{-5}$ and then divided by 10 at $30\%$ and $80\%$ of the total number of training epochs. Our methods are implemented in PyTorch and run on four NVIDIA Tesla V100 GPUs.
We adopt the Macro F1 score as the evaluation metric. The score is defined as the unweighted average of the class-wise/label-wise F1 Scores.
\subsection{Results on the COVID-19 detection challenge}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{The results on the validation set of COVID-19 detection challenge.* indicates adaptive mixup and cutmix strategy.}
\label{table:covid detection}
\begin{tabular}{llcccc}
\hline\noalign{\smallskip}
ID & Backbone & Pretrain & CRL & Mixup & macro F1 score\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
1& CNN-RNN \cite{kollias2022ai} & -& - & - & 0.770\\
2& ResNest50 & ImageNet & & & 0.897\\
3& ResNest50 & ImageNet & $\checkmark$ & $\checkmark$ & 0.924\\
4& Uniformer-S & ImageNet & & & 0.915\\
5& Uniformer-S & ImageNet & $\checkmark$ & $\checkmark$ & 0.920 \\
6& Uniformer-S & k400 & $\checkmark$ & $\checkmark$ & 0.925 \\
7& Uniformer-S & k400\_16x8 & $\checkmark$ & $\checkmark$ & 0.927 \\
8& Uniformer-S* & k400\_16x8 & $\checkmark$ & $\checkmark$ & 0.933 \\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
Table \ref{table:covid detection} shows the results of the baseline model and our methods on the validation set of COVID-19 detection challenge. The baseline CNN-RNN approach follows the work~\cite{kollias2020deep,kollias2020transparent,kollias2018deep} on developing deep neural architectures for predicting COVID-19. It achieves 0.77 macro F1 score. Our 3D CMC-COV19D models with different backbones obtain significant improvements compared with the baseline. The effectiveness of CRL and Mixup modules are also verified.
\subsection{Results on the COVID-19 severity detection challenge}
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{The results on the validation set of COVID-19 severity detection challenge.}
\label{table: severity}
\begin{tabular}{llcccc}
\hline\noalign{\smallskip}
ID& Methods & CRL & Mixup & Macro F1 score\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
1& CNN-RNN \cite{kollias2022ai}& -& - & 0.630\\
2& ResNest50 & & & 0.655\\
3& ResNest50 & & $\checkmark$ & 0.673\\
4& ResNest50 & $\checkmark$ & $\checkmark$ & 0.719\\
5& ResNest50+Lesion & $\checkmark$ & $\checkmark$ & 0.770\\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
Table \ref{table: severity} shows the results of the baseline model and our methods on the validation set of COVID-19 severity detection challenge. The baseline CNN-RNN approach achieves 0.63 macro F1 score. As can be seen from the 2nd to 4th rows of Table \ref{table: severity}, our 3D CMC-COV19D models also achieve better performance.
\section{Conclusions} \label{section:conclusion}
In this paper, we present our solution for the 2nd COVID-19 Competition on two challenges: COVID-19 detection and COVID-19 severity detection.
Our network is composed of contrastive representation learning and mixup classification for more accurate COVID-19 diagnosis. We achieve 0.933 and 0.770 macro F1 score on the the validation set of COVID-19 detection and severity detection, respectively.
\bibliographystyle{splncs04}
|
1204.6691
|
\section{Introduction}
With the global advent of cloud, grid, cluster computing and increasing needs
for large data centers to run these services, the environmental impact of
large-scale computing paradigms is becoming a global problem. The energy
produced to power the ICT industry (and data centers constitute its major part)
is responsible for 2\% of all the carbon dioxide equivalent ($CO_{2}e$\ -- greenhouse
gases normalized to carbon dioxide by their environmental impact) emissions
\cite{_gartner_????-1}, thus accelerating global warming \cite{_ipcc_????}.
Cloud computing facilitate users to buy computing resources from a cloud
provider and specify the exact amount of each resource (such as storage space,
number of cores etc.) that they expect through a Service Level Agreement
(SLA)\footnote{We consider the traditional business model where the desired
specifications are set in advance, as is still the case in most
infrastructure-as-a-service clouds.}. The cloud provider then honors this
agreement by providing the promised resources to avoid agreement violation
penalties (and to keep the customer satisfied to continue doing business).
However, cloud providers are usually faced with the challenge of satisfying
promised SLAs and at the same time not wasting their resources as a user very
rarely utilizes computing resources to the maximum
\cite{beauvisage_computer_2009}.
In order to fight global warming the Kyoto protocol was established by the
United Nations Framework Convention on Climate Change (UNFCCC or FCCC). The goal
is to achieve global stabilisation of greenhouse gas concentrations in the
atmosphere at a level that would prevent dangerous anthropogenic interference
with the climate system \cite{_kyoto_????}. The protocol defines control mechanisms to reduce $CO_{2}e$\
emissions by basically setting a market price for such emissions. Currently, flexible models
for $CO_{2}e$\ trading are developed at different organizational and political level
as for example at the level of a country, industry branch, or a company.
As a result, keeping track of and reducing $CO_{2}e$\ emissions is becoming more and
more relevant after the ratification of the Kyoto protocol.
Energy efficiency has often been a target for research. On the one
hand, there is large body of work done in facilitating energy efficient
management of data centers as for example in \cite{_green_????-1} where current
state of formal energy efficiency control in cloud computing relies on monitoring power usage efficiency (PUE) and the
related family of metrics developed by the Green Grid Consortium. Another
example is discussed in \cite{beloglazov_adaptive_2010} where economic
incentives are presented to promote greener cloud computing policies. On the
other hand, there are several mature models for trading $CO_{2}e$\ obligations in
various industrial branches, as for example in the oil industry
\cite{ellerman_european_????}. Surprisingly, to the best of our knowledge there
exists no related work about the application of the Kyoto protocol to energy
efficient modeling of data centers and cloud infrastructures.
In this paper we propose a $CO_{2}e$-trading model for transparent scheduling of
resources in cloud computing adhering to the Kyoto protocol guidelines
\cite{ellerman_european_????}. First, we present a conceptual model for $CO_{2}e$\
trading compliant to the Kyoto protocol's emission trading scheme. We consider
an \emph{emission trading market (ETM)} where \emph{credits for emission
reduction (CERs)} are traded between data centers. Based on the positive or
negative \emph{CERs} of the data center, a cost is set for the environmental
impact of the energy used by applications. Thereby, a successful application scheduing decission can
be brought after considering the (i) energy costs,
(ii) $CO_{2}e$\ costs and (iii) SLA violation costs. Second, we propose a
\emph{wastage-penalty} model that can be used as a basis for the
implementation of Kyoto protocol-compliant scheduling and pricing models.
Finally, we discuss potential uses of the model as an optimisation heuristic in
the resource scheduler.
The main contribution of the paper are (1) definition of the conceptual
\emph{emission trading market (ETM)} for the application of Kyoto protocol for
the energy efficiency management in Clouds (2) definition of a \emph{wastage -
penalty} model for trading of \emph{credits for emission reduction (CERs)} (3)
discussion on how the presented \emph{wastage-penalty} model can be used for
the implementation of next generation Kyoto protocol compliant energy efficient
schedulers and pricing models.
The paper is structured as follows: Section \ref{sec:related} discusses related
work. Section \ref{sec:kyoto} gives some background as to why cloud computing
might become subject to the Kyoto protocol. Section \ref{sec:model} presents our
model in a general $CO_{2}e$-trading cloud scenario, we then go on to define a formal
model of individual costs to find a theoretical balance and discuss the
usefulness of such a model as a scheduling heuristic. Section
\ref{sec:conclusion} concludes the paper and identifies possible future research
directions.
\section{Related Work}
\label{sec:related}
As our aim is to enable energy efficiency control in the cloud resource
scheduling domain, there are two groups of work related to ours that deal with
the problem:
\begin{enumerate}
\item scheduling algorithms - resource allocation techniques, from which energy cost optimisation is starting to evolve
\item energy efficiency legislation - existing rules, regulations and best behaviour suggestions that are slowly moving from optimising the whole data center efficiency towards optimising its constituting parts
\end{enumerate}
We will examine each of these two groups separately now.
\subsection{Scheduling Algorithms}
There already exist cloud computing energy efficient scheduling solutions, such
as \cite{maurer_enacting_2011,maurer_simulating_2010} which try to minimize
energy consumption, but they lack a strict quantitative model similar to PUE
that would be convenient as a legislative control measure to express exactly how
much they alter $CO_{2}e$\ emission levels. From the $CO_{2}e$\ management perspective,
these methods work more in a best-effort manner, attempting first and foremost
to satisfy SLA constrains.
In \cite{coutinho_workflow_2011} the HGreen heuristic is proposed to schedule
batch jobs on the greenest resource first, based on prior energy efficiency
benchmarking of all the nodes, but not how to optimize a job once it is
allocated to a node - how much of its resources is it allowed to consume. A
similar multiple-node-oriented scheduling algorithm is presented in
\cite{wang_energy-efficient_2011}.
The work described in \cite{beloglazov_adaptive_2010} has the most similarities
with ours, since it also balances SLA and energy constraints and even describes
energy consumption using a similar, linear model motivated by dynamic voltage
scaling, but no consideration of $CO_{2}e$\ management was made inside the model.
A good overview of cloud computing and sustainability is given in
\cite{garg_environment-conscious_2011}, with explanations of where cloud
computing stands in regard to $CO_{2}e$\ emissions. Green policies for scheduling are
proposed that, if accepted by the user, could greatly increase the efficiency of
cloud computing and reduce $CO_{2}e$\ emissions. Reducing emissions is not treated as
a source of profit and a possible way to balance SLA violations, though, but
more of a general guideline for running the data center to stay below a certain
threshold.
\subsection{Energy Efficiency Legislation}
Measures of controlling energy efficiency in data centers do exist -- metrics
such as power usage efficiency (PUE) \cite{_green_????-2}, carbon usage
efficiency (CUE), water usage efficiency (WUE) \cite{_green_????} and others
have basically become the industry standards through the joint efforts of policy
makers and cloud providers gathered behind The Green Grid consortium
\cite{_green_????-1}. The problem with these metrics, though, is that they only
focus on the infrastructure efficiency -- turn as much energy as possible into
computing inside the IT equipment. Once the power gets to the IT equipment,
though, all formal energy efficiency regulation stops, making it more of a
black-box approach. For this reason, an attempt is made in our work to bring
energy efficiency control to the interior operation of clouds -- resource
scheduling.
So far, the measurement and control of even such a basic metric as PUE is not
mandatory. It is considered a best practice, though, and agencies such as the
U.S. Environmental Protection Agency (EPA) encourage data centers to measure it
by rewarding the best data centers with the Energy Star award
\cite{_energy_????}.
\section{Applying the Kyoto Protocol to Clouds}
\label{sec:kyoto}
The Kyoto protocol \cite{grubb_kyoto_1999} commits involved countries to
stabilize their greenhouse gas (GHG) emissions by adhering to the measures
developed by the United Nations Framework Convention on Climate Change (UNFCCC)
\cite{_kyoto_????-1}. These measures are commonly known as \emph{the
cap-and-trade system}. It is based on setting national emission boundaries --
caps, and establishing international emission markets for trading emission
surpluses and emission deficits. This is known as \emph{certified emission
reductions} or \emph{credits for emission reduction} (CERs). Such a trading
system rewards countries which succeeded in reaching their goal with profits
from selling CERs and forces those who did not to make up for it financially by
buying CERs. The European Union Emission Trading System (EU ETS) is an example
implementation of an emission trading market \cite{_eu_????}. Through such
markets, CERs converge towards a relatively constant market price, same as all
the other tradable goods.
Individual countries control emissions among their own large polluters
(individual companies such as power generation facilities, factories\ldots) by
distributing the available caps among them. In the current implementation,
though, emission caps are only set for entities which are responsible for more than 25
Mt$CO_{2}e$/year \cite{_environment_????}. This excludes individual data centers
which have a carbon footprint in the kt$CO_{2}e$/year range \cite{_data_????}.
It is highly possible, though, that the Kyoto protocol will expand to smaller
entities such as cloud providers to cover a larger percentage of polluters and
to increase the chance of global improvement. One such reason is that currently
energy producers take most of the weight of the protocol as they cannot pass the
responsibilities on to their clients (some of which are quite large, such as
data centers). In 2009, three companies in the EU ETS with the largest shortage
of carbon allowances were electricity producers \cite{_eu_????-1}. Another
indicator of the justification of this forecast is that some cloud providers,
such as Google already participate in emission trading markets to achieve carbon
neutrality \cite{_googles_????}.
For this reason, we hypothesize in this paper that cloud providers are indeed
part of an emission trading scheme and that $CO_{2}e$\ emissions have a market price.
\section{Wastage-Penalty Balance in a Kyoto-Compliant Cloud}
\label{sec:model}
In this section we present our $CO_{2}e$-trading model that is to be integrated with
cloud computing. We show how an economical balance can be found in it. Lastly,
we give some discussion as to how such information might be integrated into a
scheduler to make it more energy and cost efficient.
\subsection{The $CO_{2}e$-Trading Model}
The goal of our model is to integrate the Kyoto protocol's $CO_{2}e$\ trading
mechanism with the existing cloud computing service-oriented paradigm. At the
same time we want to use these two aspects of cloud computing to express an
economical balance function that can help us make better decisions in the
scheduling process.
\begin{figure}[h!]
\includegraphics[width=0.95\textwidth]{kyoto_in_clouds}
\caption{Cloud computing integrated with the Kyoto protocol's emission trading scheme}
\label{fig:model}
\end{figure}
The model diagram in Fig. \ref{fig:model} shows the entities in our model and
their relations. A cloud offers some computing resources as services to its
users and they in turn pay the cloud provider for these services. Now, a cloud
is basically some software running on machines in a data center. To operate, a
data center uses electrical energy that is bought from an energy producer. The
energy producers are polluters as they emit $CO_{2}e$\ into the atmosphere. As
previously explained, to mitigate this pollution, energy producers are bound by
the Kyoto protocol to keep their $CO_{2}e$\ emissions bellow a certain threshold and
buy CERs for all the excess emissions from other entities that did not reach
their caps yet over the emission trading market (ETM). This is illustrated by
getting negative CERs (-CERs) for $CO_{2}e$\ responsibilities and having to buy the
same amount of positive CERs (+CERs) over the ETM. It does not make any real
difference for our model if an entity reaches its cap or not, as it can sell the
remaining $CO_{2}e$\ allowance as CERs to someone else over the ETM. Most
importantly, this means that \emph{$CO_{2}e$\ emissions an entity is responsible for
have a price}.
The other important thing to state in our model is that
\emph{$CO_{2}e$\ emission responsibilities for the energy that was bought is
transferred from the energy producer to the cloud provider}. This is shown in
Fig. \ref{fig:model} by energy producers passing some amount of -CERs to the
cloud provider along with the energy that was bought. The cloud provider then
has to buy the same amount of +CERs via the ETM (or he will be able to sell them
if he does not surpass his cap making them equally valuable).
The consequences of introducing this model are that three prices influence the
cloud provider: (1) energy cost; (2) $CO_{2}e$\ cost; (3) service cost. To maximize
profit, the cloud provider is motivated to decrease energy and $CO_{2}e$\ costs and
maximize earnings from selling his service. Since the former is achieved by
minimizing resource usage to save energy and the latter by having enough
resources to satisfy the users' needs, they are conflicting constraints. Therefore, an
economical balance is needed to find exactly how much resources to provide.
The service costs are much bigger than both of the other two combined (that is
the current market state at least, otherwise cloud providers would not operate),
so they cannot be directly compared. There are different ways a service can
be delivered, though, depending on how the cloud schedules resources. The aim of a
profit-seeking cloud provider is to deliver just enough resources to the user so
that his needs are fullfilled and that the energy wastage stays minimal. If a
user happens to be tricked out of too much of the resources initially sold to
him, a service violation occurs and the cloud provider has to pay a penalty
price. This means that we are comparing the energy wastage price with the
occasional violation penalty. This comparison is the core of our wastage-penalty
model and we will now explain how can a wastage-penalty economical balance be
calculated.
\subsection{The Wastage-Penalty Model for Resource Balancing}
As was briefly sketched in the introduction, the main idea is to push cloud
providers to follow their users' demands more closely, avoiding too much
resource over-provisioning, thus saving energy. We do this by introducing additional cost
factors that the cloud provider has to pay if he wastes too much resources --
the energy and $CO_{2}e$\ costs shown in Fig. \ref{fig:model}, encouraging him to
breach the agreed service agreements and only provide what is actually needed.
Of course, the cloud provider will not breach the agreement too much, as that
could cause too many violation detections (by a user demanding what cannot be
provided at the moment) and causing penalty costs. We will now expand our model
with some formal definitions in the cloud-user interface from Fig.
\ref{fig:model} to be able to explicitly express the wastage-penalty balance in
it.
We assume a situation with one cloud provider
and one cloud user. The cloud provides the user with a single, abstract resource
that constitutes its service (it can be the amount of available data storage
expressed in GB, for example). To provide a certain amount of this resource to
the user in a unit of time, a proportional amount of energy is consumed and indirectly a
proportional amount of $CO_{2}e$\ is emitted. An example resource scheduling scenario
is shown in Fig. \ref{fig:agreed-provisioned}. An SLA was signed that binds the
cloud provider to provide the user a constant resource amount, $r_{agreed}$. The
cloud provider was paid for this service in advance. A user uses different
resource amounts over time. At any moment the $R_{demand}$ variable is the
amount required by the user. To avoid over-provisioning the provider
does not actually provision the promised resource amount all the time, but instead adapts this
value dynamically, $r_{provisioned}$ is the resource amount allocated to the
user in a time unit. This can be seen in Fig. \ref{fig:agreed-provisioned} as
$r_{provisioned}$ increases from $t_1$ to $t_2$ to adapt to a sudden rise in
$R_{demand}$.
\begin{figure
\vspace{0cm}
\includegraphics[width=1\textwidth]{agreed-provisioned}
\caption{Changes in the \emph{provisioned} and \emph{demand} resource amounts over time}
\label{fig:agreed-provisioned}
\end{figure}
As we can not know how the user's demand changes over time, we will think of
$R_{demand}$ as a random variable. To express $R_{demand}$ in an explicit way,
some statistical method would be required and research of users' behaviour
similar to that in \cite{beauvisage_computer_2009} to gather real-life data
regarding cloud computing resource demand. To stay on a high level of
abstraction, though, we assume that it conforms to some statistical distribution
and that we can calculate its mean $\overline{R}_{demand}$ and its maximum
$max(R_{demand})$. To use this solution in the real world, an appropriate
distribution should be input (or better yet -- one of several possible
distributions should be chosen at runtime that corresponds to the current user
or application profile). We know the random variable's expected value E and
variance V for typical statistical distributions and we can express
$\overline{R}_{demand}$ as the expected value $E(R)$ and $max(R_{demand})$ as
the sum of $E(R)+V(R)$ with a limited error.
\subsubsection{Wastage Costs}
Let us see how these variables can be used to model resource wastage costs. We
denote the energy price to provision the whole $r_{agreed}$ resource amount per time unit $c_{en}$ and similarly the $CO_{2}e$\ price $c_{co_2}$. By only using
the infrastructure to provision an amount that is estimated the user will
require, not the whole amount, we save energy that would have otherwise been
wasted and we denote this evaded wastage cost $c_{wastage}$. Since $c_{wastage}$
is a fraction of $c_{en}+c_{co_2}$, we can use a percentage $w$ to state the
percentage that is wasted:
\begin{equation}
\label{eq:cw}
c_{wastage} = w * (c_{en}+c_{co_2})
\end{equation}
We know what the extreme cases for $w$ should be -- $0\%$ for provisioning
approximately what is needed, $\overline{R}_{demand}$; and the percentage
equivalent to the ratio of the distance between $\overline{R}_{demand}$ and
$r_{agreed}$ to the total amount $r_{agreed}$ if we provision $r_{agreed}$:
\begin{equation}
w =
\begin{cases}
1-\frac{\overline{R}_{demand}}{r_{agreed}} , & \textrm{if } r_{provisioned}=r_{agreed} \\
0, & \textrm{if } r_{provisioned}=\overline{R}_{demand}
\end{cases}
\end{equation}
We model the distribution of $w$ between these extreme values using linear
interpolation:
average resource utilization - a ratio of the average provisioned resource
amount ($\overline{r}_{provisioned}$) and the promised resource amount
($r_{promise}$):
\begin{equation}
\label{eq:w}
w = \frac{r_{provisioned} - \overline{R}_{demand}}{r_{agreed}}
\end{equation}
If we apply \ref{eq:w} to \ref{eq:cw} we get an expression for the wastage cost.:
\begin{equation}
\label{eq:wastage}
c_{wastage} = \frac{r_{provisioned} - \overline{R}_{demand}}{r_{agreed}} * (c_{en}+c_{co_2})
\end{equation}
\subsubsection{Penalty Costs}
Let us now use a similar approach to model penalty costs. If a user demands more
resources than the provider has provisioned, an SLA violation occurs. The user gets only the provisioned amount of resources in this
case and the provider has to pay the penalty cost $C_{penal}$. While $c_{en}$
and $c_{co_2}$ can be considered constant for our needs, $C_{penal}$ is a random
variable, because it depends on the user's behaviour which we can not predict
with 100\% accuracy, so we will be working with $E(C_{penal})$, its expected
value.
$E(C_{penal})$, the expected value of $C_{penal}$ can be calculated as:
\begin{equation}
\label{eq:penalty}
E(C_{penal}) = p_{viol} * c_{viol}
\end{equation}
where $c_{viol}$ is the constant cost of a single violation (although in reality
probably not all kinds of violations would be priced the same) and $p_{viol}$ is
the probability of a violation occurring. This probability can be expressed as a
function of $r_{provisioned}$, $r_{agreed}$ and $R_{demand}$, the random
variable representing the user's behaviour:
\begin{equation}
p_{viol} = f(\overline{r}_{provisioned}, r_{promise}, R_{demand})
\end{equation}
Again, same as for $c_{wastage}$, we know the extreme values we want for
$p_{viol}$. If 0 is provisioned, we have 100\% violations and if
$max(R_{demand})$ is provisioned, we have 0\% violations:
\begin{equation}
p_{viol} =
\begin{cases}
100\%, & \textrm{if } r_{provisioned}=0 \\%TODO: this may not always be 100%
0\%, & \text{if } r_{provisioned}=max(R_{demand})
\end{cases}
\end{equation}
and if we assume a linear distribution in between we get an
expression for the probability of violations occuring, which is needed for
calculating the penalty costs:
\begin{equation}
\label{eq:violation}
p_{viol}= 1 - \frac{r_{provisioned}}{max(R_{demand})}
\end{equation}
\subsubsection{Combining the Two}
Now that we have identified the individual costs, we can state our goal
function. If the cloud provider provisions too much resources the $c_{wastage}$
wastage cost is too high. If on the other hand he provisions too little
resources, tightens the grip on the user too much, the $E(C_{penal})$ penalty
cost will be too high. The economical balance occurs when the penalty and
wastage costs are equal - it is profitable for the cloud provider to breach the
SLA only up to the point where penalty costs exceed wastage savings. We can
express this economical balance with the following equation:
\begin{equation}
c_{wastage} = E(C_{penal}) + \text{[customer satisfaction factor]}
\end{equation}
The \emph{[customer satisfaction factor]} could be used to model how our
promised-provisioned manipulations affect the user's happiness with the quality
of service and would be dependant of the service cost (because it might
influence if the user would be willing to pay for it again in the future). For simplicity's sake we
will say that this factor equals 0, getting:
\begin{equation}
\label{eq:costs}
c_{wastage} = E(C_{penal})
\end{equation}
Now, we can combine equations \ref{eq:wastage}, \ref{eq:costs}, \ref{eq:penalty}
and \ref{eq:violation} to get a final expression for $r_{provisioned}$:
\begin{equation}
\label{eq:equilibrium}
r_{provisioned} =\frac{max(R_{demand})*\left[\overline{R}_{demand}*(c_{en}+c_{co_2}) + r_{agreed}*c_{viol}\right]}{max(R_{demand})*(c_{en}+c_{co_2})+r_{agreed}*c_{viol}}
\end{equation}
This formula is basically \emph{the economical wastage-penalty balance}. All the
parameters it depends on are constant as long as the demand statistic stays the
same. It shows how much on average should a cloud provider breach the
promised resource amounts when provisioning resources to users so that the
statistically expected costs for SLA violation penalties do not surpass the
gains from energy savings. Vice versa also holds -- if a cloud provider
provisions more resources than this wastage-penalty balance, he pays more for
the energy wastage (energy and $CO_{2}e$\ price), than what he saves on SLA violations.
\subsection{Heuristics for Scheduling Optimisation with Integrated Emission
Management}
In this section we discuss a possible application of our wastage-penalty model
for the implementation of a future-generation data center. Knowing the
economical wastage-penalty balance, heuristic functions can be used to optimize
resource allocation to maximize the cloud provider's profit by integrating both
service and violation penalty prices and energy and $CO_{2}e$\ costs. This is useful,
because it helps in the decision-making process when there are so many
contradicting costs and constraints involved.
A heuristic might state: ``try not to provision more than $\pm x\%$ resources
than the economical wastage-penalty balance''. This heuristic could easily be
integrated into existing scheduling algorithms, such as
\cite{maurer_enacting_2011,maurer_simulating_2010} so that the cloud provider
does not stray too far away from the statistically profitable zone without
deeper knowledge about resource demand profiles. The benefits of using our
wastage-penalty model are:
\begin{itemize}
\item a new, expanded cost model covers all of the influences from Fig. \ref{fig:model}
\item $CO_{2}e$-trading schema-readiness makes it easier to take part in emission trading
\item a Kyoto-compliant scheduler module can be adapted for use in resource
scheduling and allocation solutions
\item the model is valid even without Kyoto-compliance by setting the $CO_{2}e$\
price $c_{co_2}$ to 0, meaning it can be used in traditional ways by weighing
only the energy wastage costs against service violation penalties.
\end{itemize}
The wastage-penalty balance in \ref{eq:equilibrium} is a function of significant
costs and the demand profile's statistical properties:
\begin{equation}
r_{provisioned} =g(max(R_{demand}),\overline{R}_{demand},r_{agreed},c_{en},c_{co_2},c_{viol})
\end{equation}
This function enables the input of various statistical models for user or
application demand profiles ($max(R_{demand})$ and $\overline{R}_{demand}$) and
energy ($c_{en}$), $CO_{2}e$\ ($c_{co_2}$) and SLA violation market prices
($c_{viol}$). With different input parameters, output results such as energy
savings, environmental impact and SLA violation frequency can be compared. This
would allow cloud providers and governing decision-makers to simulate the
effects of different scenarios and measure the influence of individual
parameters, helping them choose the right strategy.
\section{Conclusion}
\label{sec:conclusion}
In this paper we presented a novel approach for Kyoto protocol-compliant
modeling of data centers. We presented a conceptual model for $CO_{2}e$\
trading compliant with the Kyoto protocol's emission trading scheme. We
consider an \emph{emission trading market (ETM)} where $CO_{2}e$\ obligations are
forwarded to data centers, involving them in the trade of credits for emission
reduction (CERs). Such measures would ensure a $CO_{2}e$\ equilibrium and encourage
more careful resource allocation inside data centers.
To aid decission making inside this $CO_{2}e$-trading system, we proposed a
\emph{wastage-penalty} model that can be used as a basis for the
implementation of Kyoto protocol-compliant scheduling and pricing models. In the
future we plan to implement prototype scheduling algorithms for the ETM
considering self-adaptable Cloud infrastructures.
\
\subsubsection{Acknowledgements} The work described in this paper was
funded by the Vienna Science and Technology Fund (WWTF) through project
ICT08-018 and by the TU Vienna funded HALEY project (Holistic Energy Efficient Management of Hybrid Clouds).
\bibliographystyle{splncs03}
|
1511.08340
|
\section{Introduction and an Overview of the Topic}
Properties of collective excitations in physical systems are determined by
the interplay of several fundamental ingredients, \textit{viz}., spatial
dimension, external potential acting on the corresponding physical fields\
(or wave functions), the number of independent components of the fields, the
underlying dispersion relation for linear excitations, and, finally, the
character of the nonlinear interactions of the fields. In particular, the
shape of the external potentials determines the system's symmetry, two most
common types of which correspond to periodic (alias \textit{lattice})
potentials and double-well potentials (DWPs), the latter featuring the
symmetry between two wells separated by a potential barrier. The well are
coupled by the tunneling of fields across the barrier, which is an
essentially linear effect.
One of fundamental principles of quantum mechanics (that, by itself, is a
strictly linear theory) is that the ground state (GS) of the quantum system
exactly follows the symmetry of the potential applied to the system. On the
other hand, excited states may realize other representations of the same
symmetry \cite{LL}. In particular, the GS wave function for a quantum
particle trapped in the one-dimensional DWP is symmetric, i.e., even, with
respect to the double-well structure, while the first excited state always
features the opposite parity, being represented by an antisymmetric
(spatially odd) function. A similarly feature of Bloch wave functions
supported by periodic potentials is that the state at the bottom of the
corresponding lowest Bloch band features the same periodicity, while generic
Bloch functions are quasi-periodic ones, with the quasi-periodicity
determined by the quasi-momentum of the excited states.
While the quantum-mechanical Schr\"{o}dinger equation is linear for the
single particle, the description of ultracold rarefied gases formed by
bosonic particles (i.e., the Bose-Einstein condensate, BECs) is provided by
the Gross-Pitaevskii equation (GPE), which, in the framework of the
mean-field approximation, takes into regard effects of collisions between
the particles, by means of an cubic term added, to the Schr\"{o}dinger
equation for the single-particle wave function \cite{BEC}. The repulsive or
attractive forces between the colliding particles are accounted for by,
respectively, the self-defocusing, alias self-repulsive, or self-focusing,
i.e., self-attractive, cubic term in the GPE.\ Similarly, the nonlinear Schr%
\"{o}dinger equation (NLSE) with the self-focusing or defocusing cubic term
(alias the Kerr or anti-Kerr one, respectively) models the transmission of
electromagnetic waves in nonlinear optical media \cite{NLS}.
As well as their linear counterparts, the GPE and NLSE include external
potentials, which often feature the DWP symmetry. However, the symmetry of
the GS in models with the self-focusing nonlinearity (i.e., the state
minimizing the energy at a fixed number of particles in the bosonic gas, or
fixed total power of the optical beam in the photonic medium---in both
cases, these are represented by a fixed norm of the respective wave
function) follows the symmetry of the underlying potential structure only as
long as the nonlinearity remains weak enough. A generic effect, which sets
in with the increase of the strength of the nonlinearity, i.e., effectively,
with the increase of the norm, is \textit{spontaneous symmetry breaking}
(SSB). In its simplest form, the SSB in terms of the BEC implies that the
probability to find the boson in one well of the trapping DWP structure is
larger than in the other. This, incidentally, implies that another basic
principle of quantum mechanics, according to which the GS cannot be
degenerate, is no longer valid in nonlinear models of the quantum origin,
such as the GPE: obviously, the SSB which takes place in the presence of the
DWP gives rise to a degenerate pair of two mutually symmetric ground states,
with the maximum of the wave function observed in either potential well. In
terms of optics, the SSB makes the light power trapped in either core of the
DWP-shaped dual-core waveguide larger than in the mate core. Thus, the SSB
is a fundamental effect common to diverse models of the quantum and
classical origin alike, which combine the wave propagation, nonlinear
self-focusing, and symmetric trapping potentials.
It should be stressed that the same nonlinear system with the DWP potential
always admits a symmetric state coexisting with the asymmetric ones;
however, past the onset of the SSB, the symmetric state no longer represents
the GS, being unstable against small symmetry-breaking fluctuations.
Accordingly, in the course of the spontaneous transition from the unstable
symmetric state to a stable asymmetric one, the choice between the two
mutually degenerate asymmetric states is governed by perturbations, which
``push" the self-attractive system to place, at random, the
maximum of the wave function in the left or right potential well.
In systems with the self-defocusing nonlinearity, the ground state is always
symmetric and stable. In this case, the SSB manifests itself in the form of
the spontaneous breaking of the \textit{antisymmetry} of the first excited
state (the spatially odd one, which has exactly one zero of the wave
function, at the central point, in the one-dimensional geometry). The state
with the spontaneously broken antisymmetry also features a zero, which is
shifted from the central position to the left or right, the sign of the
shift being randomly selected by initial perturbations.
Historically speaking, the SSB concept for nonlinear systems of the NLSE
type was, probably, first proposed by E. B. Davies in 1979 \cite{Davies},
although in a rather abstract mathematical form. In that work, a nonlinear
extension of the Schr\"{o}dinger equation for a pair of quantum particles,
interacting via a three-dimensional isotropic potential, was addressed, and
the SSB was predicted in the form of the spontaneous breaking of the GS
rotational symmetry. Another early prediction of the SSB was reported in the
\textit{self-trapping model}, which is based on a system of linearly coupled
ordinary differential equations with self-attractive cubic terms \cite{Scott}%
. The latter publication had brought the concept of the SSB to the attention
of the broad research community.
An important contribution to theoretical studies of the SSB was made by work
\cite{Snyder}, which addressed this effect in the model for the propagation
of CW (continuous-wave) optical beams in dual-core nonlinear optical fibers
(alias nonlinear directional couplers), with the underlying symmetry between
the linearly coupled cores . In the scaled form, the corresponding system of
propagation equations for CW amplitudes $u_{1}$ and $u_{2}$ in the two cores
is
\begin{eqnarray}
&&i\frac{du_{1}}{dz}+f\left( \left\vert u_{1}\right\vert ^{2}\right)
u_{1}+\kappa u_{2}=0, \notag \\
&& \label{u1u2} \\
&&i\frac{du_{2}}{dz}+f\left( \left\vert u_{2}\right\vert ^{2}\right)
u_{2}+\kappa u_{1}=0, \notag
\end{eqnarray}%
(in this case, ``CW" implies that the amplitudes do not
depend on the temporal variable), where $z$ is the propagation distance, $%
\kappa $ the coefficient accounting for the inter-core linear coupling
through the mutual penetration of evanescent fields from each core into the
mate one, and $f\left( |u_{1,2}|^{2}\right) $ is a function of the intensity
of the light in each core which represents its intrinsic nonlinearity. In
the simplest case of the Kerr (cubic) self-focusing nonlinearity, which
corresponds to%
\begin{equation}
f\left( u^{2}\right) =|u|^{2}, \label{Kerr}
\end{equation}%
this system gives rise to the symmetry-breaking \textit{bifurcation} of the
\textit{supercritical} type \cite{bif}. This bifurcation destabilizes the
symmetric state and, simultaneously, gives rise to a pair of stable
asymmetric ones, which are mirror images of each other, corresponding to
interchange $u_{1}\rightleftarrows u_{2}$, as shown in Fig. \ref{fig1}(a).
In the figure, the asymmetry and the total norm, which characterizes the
strength of the nonlinearity, are defined as%
\begin{equation}
\nu \equiv \left( \left\vert u_{1}\right\vert ^{2}-\left\vert
u_{2}\right\vert ^{2}\right) /\left( \left\vert u_{1}\right\vert
^{2}+\left\vert u_{2}\right\vert ^{2}\right) ,~N\equiv \left( \left\vert
u_{1}\right\vert ^{2}+\left\vert u_{2}\right\vert ^{2}\right) . \label{nu}
\end{equation}%
On the other hand, the \textit{saturable} nonlinearity, in the form of $%
f\left( \left\vert u\right\vert ^{2}\right) =\left\vert u\right\vert
^{2}/\left( I_{0}+\left\vert u\right\vert ^{2}\right) $, where $I_{0}>0$ is
a constant which determines the intensity-saturation level, gives rise to a
\textit{subcritical} symmetry-breaking bifurcation. In the latter case, the
branches of asymmetric states, which originate at the point of the stability
loss of the symmetric mode, originally evolve \textit{backward} (in terms of
the total power, $\left\vert u_{1}\right\vert ^{2}+\left\vert
u_{2}\right\vert ^{2}$), being unstable, and then turn forward, getting the
stable at the turning point, see Fig. \ref{fig1}(b). This SSB scenario
implies that the pair of stable asymmetric states emerge \textit{%
subcritically}, at a value of the total power smaller than the one at which
the symmetric mode becomes unstable. In terms of statistical physics, the
super- and subcritical bifurcations may be classified as phase transitions
of the second and first kinds, respectively.
\begin{figure}[tbp]
\includegraphics[width=5.0in]{Fig1.pdf}
\caption{{}(Color online) Diagrams for standard supercritical (a) and
subcritical (b) spontaneous-symmetry-breaking bifurcations, as per Ref.
\protect\cite{NatPhot}. Continuous and dashed lines depict, respectively,
stable and unstable solution branches. Total norm $N$ and asymmetry
parameter $\protect\nu $ (see Eq. (\protect\ref{nu})) are shown in arbitrary
units (a.u.).}
\label{fig1}
\end{figure}
The next step in the studies of the SSB phenomenology in models of dual-core
nonlinear optical fibers and similar systems was the consideration of the
fields depending on the temporal variable, $\tau $. In that case, assuming
the anomalous sign of the group-velocity dispersion in each core of the
fiber, Eqs. (\ref{u1u2}) are replaced by a system of NLSEs with the linear
coupling:%
\begin{eqnarray}
&&i\frac{\partial u_{1}}{\partial z}+\frac{1}{2}\frac{\partial ^{2}u_{1}}{%
\partial \tau ^{2}}+f\left( \left\vert u_{1}\right\vert ^{2}\right)
u_{1}+\kappa u_{2}=0, \notag \\
&& \label{u1u2tau} \\
&&i\frac{\partial u_{2}}{\partial z}+\frac{1}{2}\frac{\partial ^{2}u_{2}}{%
\partial \tau ^{2}}+f\left( \left\vert u_{2}\right\vert ^{2}\right)
u_{2}+\kappa u_{1}=0. \notag
\end{eqnarray}%
The same system, with variable $\tau $ replaced by transverse coordinate $x$%
, models the spatial-domain evolution of electromagnetic fields in dual-core
planar waveguides, in which case the second derivatives represent the
paraxial diffraction, instead of the group-velocity dispersion.
The uncoupled NLSEs with the Kerr self-focusing nonlinearity (\ref{Kerr})
give rise to commonly known solitons \cite{NLS}. The corresponding SSB
bifurcation may destabilize obvious symmetric soliton solutions of system (%
\ref{u1u2tau}),%
\begin{equation}
u_{1}=u_{2}=\eta ~\mathrm{sech}\left( \eta \tau \right) \exp \left( \left(
\frac{1}{2}\eta ^{2}+\kappa \right) z\right) , \label{soliton}
\end{equation}%
where $\eta $ is an arbitrary real amplitude of the soliton. The bifurcation
replaces the symmetric soliton mode (\ref{soliton}) by asymmetric
two-component modes. The critical value of the soliton's peak power, $\eta
^{2}$, at which the SSB instability of the symmetric solitons sets in under
the action of the Kerr nonlinearity was found in an exact form, $\eta _{%
\mathrm{crit}}^{2}=4/3$, in Ref. \cite{Wabnitz}. The transition to
asymmetric solitons, following the instability onset, was first predicted,
by means of the variational approximation, in Refs. \cite{Pare} and \cite%
{Maim}. Then, it was found that, on the contrary to the supercritical
bifurcation of the CW states in system (\ref{u1u2}) with the Kerr
self-focusing nonlinearity, the SSB bifurcation of the symmetric soliton in
system (\ref{u1u2tau}) is subcritical \cite{Akhmed,Pak}.
An independent line of the analysis of the SSB had originated from the
studies of GPE-based models for atomic BECs trapped in DWP structures. The
scaled form of the corresponding GPE for the mean-field wave function, $\psi
\left( x,t\right) $, is%
\begin{equation}
i\frac{\partial \psi }{\partial t}=-\frac{1}{2}\frac{\partial ^{2}\psi _{1}}{%
\partial x^{2}}-g\left\vert \psi \right\vert ^{2}\psi +U(x)\psi ,
\label{GPE}
\end{equation}%
where $g<0$ and $g>0$ correspond to the repulsive and attractive
collision-induced nonlinearity, respectively. The DWP can be taken, for
instance, as%
\begin{equation}
U(x)=U_{0}\left( x^{2}-a^{2}\right) ^{2}, \label{DWP}
\end{equation}%
with positive constants $U_{0}$ and $a^{2}$.
The GPE (\ref{GPE})\ can be reduced to the two-mode system, similar to the
system of Eqs. (\ref{u1u2}) (with $z$ replaced by $t$), by means of the
tight-binding approximation \cite{tight}, which adopts $\psi (x,t)$ in the
form of a superposition of two stationary wave functions, $\phi $,
corresponding to the states trapped separately in the two potential wells,
centered at $x=\pm a$:%
\begin{equation}
\psi \left( x,t\right) =u_{1}(t)\phi \left( x-a\right) +u_{2}\phi \left(
x+a\right) . \label{12}
\end{equation}%
In particular, this approximation implies that the nonlinearity acts on each
amplitude $u_{1}$ and $u_{2}$ also separately, while the coupling between
them is linear.
The analysis of the SSB in the models based on the GPE (\ref{GPE}) was
initiated in Refs. \cite{Milburn} and \cite{Smerzi}. Most often, the BEC\
nonlinearity (on the contrary to the self-focusing Kerr terms in optics) is
self-repulsive, which, as mentioned above, gives rise to the spontaneous
breaking of the antisymmetry of the odd states, with $\psi (-x)=-\psi (x)$,
while the GS remains symmetric. Further, the GPE may be extended by adding
an extra spatial coordinate, on which the DWP does not depend, i.e., one
arrives at a two-dimensional GPE with a quasi-one-dimensional\textit{\
double-trough potential}, which is displayed in Fig. \ref{fig2}. In the
latter case, the self-attractive nonlinearity (which, although being less
typical in BEC, is possible too) gives rise to bright matter-wave solitons,
which self-trap in the free direction \cite{soliton}. Accordingly, bright
symmetric solitons are possible in the double-trough potential, and they are
replaced, via a subcritical SSB bifurcation, by stable asymmetric ones at a
critical value of the total norm of the mean-field wave function (which
determines the effective strength of the self-attractive nonlinearity) \cite%
{Warsaw}.
\begin{figure}[tbp]
\includegraphics[width=3.2in]{Fig2.pdf}
\caption{{}An example of the quasi-one-dimensional double-trough potential,
built of two parallel potential toughs with the rectangular profile, as per
Ref. \protect\cite{Warsaw}.}
\label{fig2}
\end{figure}
The above discussion was focused on static symmetric and asymmetric modes in
the nonlinear systems featuring the DWP\ structure. The analysis of
dynamical regimes, usually in the form of oscillations of the norm of the
wave function between two wells of the DWP, i.e., roughly speaking, between
the two equivalent asymmetric states existing above the SSB point, has been
developed too. Following the straightforward analogy with Josephson
oscillations of the wave function in tunnel-coupled superconductors \cite%
{superconductor,Ustinov}, the possibility of the matter-wave oscillations in
\textit{bosonic Josephson junctions} was predicted \cite{junction}.
Similar to the situation in many other areas of nonlinear science, the
variety of theoretically predicted results concerning the SSB phenomenology
by far exceeds the number of experimental findings. Nevertheless, some
manifestations of the SSB have been reported in experiments. In particular,
the self-trapping of a macroscopically asymmetric state of the atomic
condensate of $^{87}$Rb atoms with repulsive interactions between them,
loaded into the DWP (which may be considered, as mentioned above, as a
spontaneous breaking of the antisymmetry of the lowest excited state, above
the symmetric GS) and Josephson oscillations in the same system, were
reported in Ref. \cite{Markus}. On the other hand, the SSB of laser beams
coupled into an effective transverse DWP created in a self-focusing
photorefractive medium (where the nonlinearity is saturable, rather than
strictly cubic) has been demonstrated in Ref. \cite{photo}. Still another
experimental observation of the SSB effect in nonlinear optics was the
spontaneously established asymmetric regime of operation of a symmetric pair
of coupled lasers \cite{lasers}. More recently, symmetry breaking was
experimentally demonstrated in a symmetric pair of nanolaser cavities
embedded into a photonic crystal \cite{France} (although the latter system
is a dissipative one, hence its model is essentially different from those
outlined above, cf. Ref. \cite{Sigler}, where the SSB effect for dissipative
solitons was formulated in terms of linearly-coupled complex Ginzburg-Landau
equations with the cubic-quintic nonlinearity). An observation of a related
effect of the spontaneous breaking of the chiral symmetry in metamaterials
was reported in Ref. \cite{Kivshar}.
Many results for the SSB phenomenology and related Josephson oscillations,
chiefly theoretical ones, but also experimental, obtained in various areas
of physics (nonlinear optics and plasmonics, BECs, superconductivity, and
others) are represented by a collection of articles published in topical
volume \cite{book}.
\section{A Simple Model for the Spontaneous Symmetry Breaking (SSB) in a
Double-Well Potential (DWP)}
\subsection{Formulation of the model}
The objective of this section is to introduce what may be the simplest model
which admits the SSB in a system combining the self-attractive nonlinearity
and a DWP structure. In a sketchy form, the model was mentioned in Ref. \cite%
{NatPhot}, but it was not elaborated there. The account given here is not
complete either, as only approximate analytical results are included. A full
presentation, including relevant numerical results, will be given elsewhere.
The model is schematically shown in Fig. \ref{fig3}. It is built as an
infinitely deep potential box, with the DWP structure created by means of
the delta-functional barrier created in the center, cf. the cross section of
the double-trough potential displayed in Fig. \ref{fig2}. The respective
scaled form of the GPE is given by Eq. (\ref{GPE}) with $g>0$ and $%
U(x)=\varepsilon \delta (x)$, where the delta-functional potential
corresponds to $U_{b}\rightarrow \infty ,$ $a\rightarrow 0$, while the
strength of the barrier, $\varepsilon \equiv U_{b}a,$ is kept fixed. The
edges of the potential box at points $x=\pm 1/2$ are represented by the
boundary conditions (b.c.)%
\begin{equation}
\psi \left( x=\pm \frac{1}{2}\right) =0. \label{bc}
\end{equation}%
Stationary states with chemical potential $\mu $ are looked for as $\psi
(x,t)=e^{-i\mu t}\phi (x),$ with real function $\phi (x)$ obeying the
following stationary equation with the respective b.c.:
\begin{equation}
\mu \phi =-\frac{1}{2}\frac{d^{2}\phi }{dx^{2}}-g\phi ^{3}+\varepsilon
\delta (x)\phi ,~\phi \left( x=\pm \frac{1}{2}\right) =0. \label{NLSE}
\end{equation}%
The delta-functional barrier at $x=0$ implies that $\phi (x)$ is continuous
at this point, while its derivative features a jump:%
\begin{equation}
\frac{d\phi }{dx}|_{x=+0}-\frac{d\phi }{dx}|_{x=-0}=2\varepsilon \phi (x=0).
\label{jump}
\end{equation}%
\begin{figure}[tbp]
\includegraphics[width=3.2in]{Fig3.pdf}
\caption{{}(Color online) A sketch of the double-well-potential (DWP)
structure under the consideration (as per Ref. \protect\cite{NatPhot}): an
infinitely deep potential box ($U_{0}\rightarrow \infty $), of width $%
L\equiv 1$, is split in the middle by a narrow tall barrier, $\protect%
\varepsilon \protect\delta (x)$, see Eq. (\protect\ref{NLSE}). Even and odd
wave functions of the ground and first excited states, in the absence of the
spontaneous symmetry breaking, are shown by the continuoys and dashed
curves, respectively.}
\label{fig3}
\end{figure}
It is possible to fix $g\equiv 1$ in Eqs. (\ref{GPE}) and (\ref{NLSE}) by
means of scaling, but it is more convenient, for the sake of the subsequent
analysis, to keep $g>0$ as a free parameter. The strength of the
nonlinearity is determined by product $gN$, where the total norm of the wave
function is defined as the sum of the norms trapped in the left and right
potential wells (cf. Eq. (\ref{nu})):%
\begin{equation}
N=\left( \int_{-1/2}^{0}+\int_{0}^{+1/2}\right) \phi ^{2}(x)dx\equiv
N_{-}+N_{+}, \label{N}
\end{equation}
Before proceeding to the analysis of the SSB in the nonlinear model, it is
relevant to briefly discuss its linear counterpart, with $g=0$ in Eq. (\ref%
{NLSE}). Spatially symmetric (even) solutions of this equation are looked for
as%
\begin{equation}
\phi _{\mathrm{even}}^{(\mathrm{lin})}(x)=A\sin \left( \sqrt{2\mu }\left(
\frac{1}{2}-|x|\right) \right) , \label{linear}
\end{equation}%
where $A$ is an arbitrary amplitude, and eigenvalue $\mu $ is determined by
the equation following from the jump condition (\ref{jump}):%
\begin{equation}
\tan \left( \sqrt{\mu /2}\right) =-\sqrt{2\mu }/\varepsilon . \label{eigen}
\end{equation}%
It is easy to see that, with the increase of $\varepsilon $ from $0$ to $%
\infty $, $\mu _{0}$ the lowest eigenvalue $\mu _{0}$, corresponding to the
GS of the linear model, monotonously grows from $\mu _{0}\left( \varepsilon
=0\right) =\pi ^{2}/2$ to
\begin{equation}
\mu _{0}\left( \varepsilon =\infty \right) =2\pi ^{2}. \label{muGS}
\end{equation}%
Similarly, the eigenvalue of the first excited symmetric state, $\mu
_{2}\left( \varepsilon \right) $, monotonously grows from to $\mu _{2}\left(
\varepsilon =0\right) =9\pi ^{2}/2$ to $\mu _{2}\left( \varepsilon =\infty
\right) =8\pi ^{2}$. Located between eigenvalues $\mu _{0}$ and $\mu _{2}$,
is $\mu _{1}=2\pi ^{2},$ which corresponds to the lowest excited state,
i.e., the first antisymmetric (spatially odd) eigenfunction, $\phi _{\mathrm{%
odd}}^{(\mathrm{lin})}(x)=A\sin \left( \sqrt{2\mu _{1}}x\right) $.
Naturally, $\mu _{1}$ coincides with the limit value (\ref{muGS}) of $\mu
_{0}$, and it does not depend on $\varepsilon $, as the odd eigenfunction
vanishes at $x=0$, where the $\delta $-function is placed.
\subsection{An analytical solution for the SSB point in the strongly-split
DWP (large $\protect\varepsilon $)}
The main objective of the analysis is to predict the critical norm which
gives rise to the SSB, through the competition between the self-focusing,
which favors the spontaneous accumulation of the wave function in one well,
and the linear coupling between the wells, which tends to distribute the
wave function evenly between them. An approximate analytical solution to
this problem can be obtained in the case of weakly coupled potential wells,
which corresponds to large $\varepsilon $ (a very tall central barrier). In
this case, weak nonlinearity, i.e., a small amplitude of the wave function,
is sufficient to induce the SSB. In turn, the small amplitude implies that
solutions to Eq. (\ref{NLSE}) vanishing at $x=\pm 1/2$ are close to
eigenmodes (\ref{linear}) of the linearized version of the same equation,
i.e., the approximate solutions may be sought for as%
\begin{equation}
\phi (x)=A_{\pm }\sin \left( k_{\pm }\left( \frac{1}{2}-|x|\right) \right) ,
\label{lin}
\end{equation}%
where signs $\pm $ pertain to the regions of $x<0$ and $x>0$, respectively, $%
k_{\pm }$ being appropriate wavenumbers. The substitution of \textit{ansatz}
(\ref{lin}) into the condition of the continuity of the wave function at $x=0
$, and the jump condition (\ref{jump}) for the first derivatives, yields the
following relations between amplitudes $A_{\pm }$ and the wavenumbers:%
\begin{gather}
A_{+}\sin \left( \frac{1}{2}k_{+}\right) =A_{-}\sin \left( \frac{1}{2}%
k_{-}\right) , \label{bc1} \\
A_{-}k_{-}\cos \left( \frac{1}{2}k_{-}\right) -A_{+}k_{+}\cos \left( \frac{1%
}{2}k_{+}\right) =4\varepsilon A_{\pm }\sin \left( \frac{1}{2}k_{\pm
}\right) . \label{bc2}
\end{gather}%
Further, in the same small-amplitude limit, the cubic term in Eq. (\ref{NLSE}%
) may be approximated by the neglecting the third harmonic contained in it:%
\begin{equation}
\left[ A_{\pm }\sin \left( k_{\pm }\left( \frac{1}{2}-|x|\right) \right) %
\right] ^{3}\approx \frac{3}{4}A_{\pm }^{3}\sin \left( k_{\pm }\left( \frac{1%
}{2}-|x|\right) \right) , \label{3/4}
\end{equation}%
which, in turn, implies an effective shift of the chemical potential in Eq. (%
\ref{NLSE}) and determines the corresponding wavenumbers in Eq. (\ref{lin}):%
\begin{equation}
k_{\pm }=\sqrt{2\left( \mu +\frac{3}{4}gA_{\pm }^{2}\right) }. \label{k}
\end{equation}
In the limit of $\varepsilon \rightarrow \infty $, wave functions (\ref{lin}%
) must vanish at $x=0$, hence the respective GS corresponds to $k_{\pm
}=2\pi $, i.e., to the above-mentioned value (\ref{muGS}) of the chemical
potential. In the same limit, the norm (\ref{N}) of the GS is
\begin{equation}
N=\left( A_{-}^{2}+A_{+}^{2}\right) /4.
\end{equation}%
At large but finite $\varepsilon $, the GS chemical potential is sought for
as%
\begin{equation}
\mu =2\pi ^{2}-\delta \mu ,~\mathrm{with~~}\delta \mu \ll 2\pi ^{2}.
\label{delta-mu}
\end{equation}%
Next, the substitution of this expression into Eq. (\ref{k}), expanding it
for small $\delta \mu $ and $A_{\pm }^{2}$, and inserting the result into
Eqs. (\ref{bc1}) and (\ref{bc2}) leads to equations which take a relatively
simple form at the point of the onset of the SSB bifurcation, i.e., in the
limit of $A_{+}-A_{-}\rightarrow 0$ (the vanishingly small factor\ $\left(
A_{+}-A_{-}\right) $ then factorizes out and cancels in in the expanded
version of Eq. (\ref{bc1})):%
\begin{equation}
N_{\mathrm{cr}}=\frac{8\pi ^{2}}{3g}\varepsilon ^{-1},~\left( A_{\pm
}^{2}\right) _{\mathrm{cr}}=2N_{\mathrm{cr}},~\delta \mu =12\pi
^{2}\varepsilon ^{-1}. \label{cr}
\end{equation}%
This result was mentioned, without the derivation, in Ref. \cite{NatPhot}.
Thus, as expected, the critical value of the norm at the SSB point decays ($%
\sim \varepsilon ^{-1}$) with the increase of $\varepsilon $. The
substitution of $\delta \mu $ from Eq. (\ref{cr}) into Eq. (\ref{delta-mu})
suggests that this asymptotic solution is actually valid for $\varepsilon
\gg 6$.
\subsection{An analytical solution for the SSB in the weakly-split DWP
(small $\protect\varepsilon $): the soliton approximation}
The case of small $\varepsilon $, opposite to that considered above, implies
that the central barrier splitting the confined box into the two potential
wells is weak, hence the effective coupling between the wells is strong.
According to the general principles of the SSB theory \cite{book}, strong
nonlinearity, i.e., large norm $N$, is necessary to complete with the strong
coupling. Large $N$, in turn, implies that the wave field self-traps into a
narrow NLSE\ soliton \cite{NLS},%
\begin{equation}
\phi _{\mathrm{sol}}\left( x-\xi \right) =\frac{1}{2}\sqrt{g}N\mathrm{sech}%
\left( \frac{g}{2}N\left( x-\xi \right) \right) , \label{sol}
\end{equation}%
where $\xi $ is the coordinate of the soliton's center, the respective
chemical potential is%
\begin{equation}
\mu _{\mathrm{sol}}=-\left( gN\right) ^{2}/8. \label{mu-sol}
\end{equation}%
and it is assumed that $N$ is large enough to make the soliton's width much
smaller than the size of the box ($L\equiv 1$ in Fig. \ref{fig3}), i.e.,
\begin{equation}
gN\gg 1. \label{>>}
\end{equation}
The soliton is repelled from the edges of the potential box. To comply with
the b.c. in Eq. (\ref{NLSE}), this may be interpreted as the repulsive
interaction with two \textit{ghost solitons} generated as mirror images
(with opposite signs) of the real one (\ref{sol}) with respect to the edges
of the box:%
\begin{equation}
\phi _{\mathrm{ghost}}(x)=-\sqrt{g}\left( N/2\right) \left[ \mathrm{sech}%
\left( \frac{g}{2}N\left( x-\frac{1}{2}+\xi \right) \right) +\mathrm{sech}%
\left( \frac{g}{2}N\left( x+1+\xi \right) \right) \right] . \label{ghost}
\end{equation}%
The potential of the interaction between two far separated NLSE solitons is
well known \cite{sol-sol1}-\cite{sol-sol4}. In the present case, the sum of
the two interaction potentials, corresponding to the pair of the ghosts,
amounts to the following effective potential accounting for the repulsion of
the real soliton from edges of the box in which it is confined:%
\begin{equation}
U_{\mathrm{box}}(\xi )=g^{2}N^{3}\exp \left( -\frac{g}{2}N\right) \cosh
\left( Ng\xi \right) . \label{box}
\end{equation}
On the other hand, the soliton is repelled by the central barrier, the
respective potential being \cite{RMP}%
\begin{equation}
U_{\mathrm{barrier}}(\xi )=\varepsilon \phi _{\mathrm{sol}}^{2}\left( \xi
=0\right) =\frac{\varepsilon g}{4}N^{2}\mathrm{sech}^{2}\left( \frac{g}{2}%
N\xi \right) , \label{barrier}
\end{equation}%
where the latter expression was obtained neglecting the deformation of the
soliton's shape. A straightforward analysis of the total effective
potential, $U(\xi )=U_{\mathrm{box}}(\xi )+U_{\mathrm{barrier}}(\xi )$,
demonstrates that the position of the soliton placed at $\xi =0$, which
represents the symmetric state in the present case, is stable, i.e., it
corresponds to a minimum of the net potential, at%
\begin{equation}
8gN\exp \left( -\frac{g}{2}N\right) >\varepsilon , \label{>}
\end{equation}%
or, in other words, at%
\begin{equation}
N>N_{\mathrm{cr}}\approx \left( 2/g\right) \ln \left( 16/\varepsilon \right)
\label{Ncr}
\end{equation}%
(the underlying assumption that $\varepsilon $ is small was used to derive
Eq. (\ref{Ncr}) from Eq. (\ref{>})). With the increase of $N,$ the SSB
bifurcation takes place at $N=N_{\mathrm{cr}}$, when the potential minimum
at $\xi =0$ turns into a local maximum. At $0<\left( N-N_{\mathrm{cr}%
}\right) /N_{\mathrm{cr}}\ll 1$, the center of the soliton spontaneously
shifts to either of two asymmetric positions, which correspond to a pair of
emerging potential minima at $\xi \neq 0$:%
\begin{equation}
\xi =\pm \sqrt{\left( N-N_{\mathrm{cr}}\right) /\left( gN_{\mathrm{cr}%
}^{2}\right) }. \label{xi}
\end{equation}%
The latter result explicitly describes the SSB bifurcation of the
supercritical type, which occurs in the present setting.
\subsection{The variational approximation for the SSB in the generic DWP}
A possibility to develop a more comprehensive, albeit coarser, analytical
approximation for solutions of Eq. (\ref{NLSE}) in the generic case (when the
strength of the splitting barrier, $\varepsilon $, is neither specifically
large nor small) is suggested by the variational approximation (VA) \cite%
{Progress}. To this end, note that Eq. (\ref{NLSE}) can be derived from the
Lagrangian,%
\begin{equation}
L=\int_{-1/2}^{+1/2}\mathcal{L}(x)dx,~\mathcal{L}=\frac{1}{2}\left( \frac{%
d\phi }{dx}\right) ^{2}-\mu \phi ^{2}-\frac{g}{2}\phi ^{4}+\varepsilon
\delta (x)\phi ^{2}. \label{L}
\end{equation}
Aiming to apply the VA for detecting the SSB onset point, one can adopt the
following \textit{ansatz} for the GS wave function:
\begin{equation}
\phi (x)=a\cos (\pi x)+b\sin (2\pi x)+c\cos (3\pi x), \label{ans}
\end{equation}%
with each term satisfying b.c. in Eq. (\ref{NLSE}). Real constants $a,$ $c,$
and $b$ must be predicted by the VA. The SSB is accounted for by terms $\sim
b$ in ansatz (\ref{ans}), hence the onset of the SSB is heralded by the
emergence of a solution with infinitesimal $b$, branching off from from the
symmetric solution with $b=0$, similar to how the onset of the SSB\
bifurcation in terms of ansatz (\ref{lin}) is signaled by the emergence of
infinitesimal $\left( A_{+}-A_{\_}\right) $ in the above analysis.
The total norm (\ref{N}) of ansatz (\ref{ans}) is $N=\left( 1/2\right)
\left( a^{2}+c^{2}+b^{2}\right) $, while its asymmetry at $b\neq 0$ may be
quantified by%
\begin{equation}
\Theta \equiv \frac{N_{+}-N_{-}}{N}=\frac{16}{15\pi }\frac{b\left(
5a-3c\right) }{a^{2}+c^{2}+b^{2}}. \label{Theta}
\end{equation}%
A straightforward consideration demonstrates that, for all values of $a,b,$
and $c$, expression (\ref{Theta}) is subject to constraint $\left\vert
\Theta \right\vert <1$, as it must be. When $b=0$, the theorem that the
spatially symmetric GS cannot have nodes, i.e., $\phi (x)\neq 0$ at $|x|<1/2$%
, if applied to ansatz (\ref{ans}), easily amounts to the following
constraint:%
\begin{equation}
-1<c/a<1/3. \label{or}
\end{equation}
Further, the substitution of ansatz (\ref{ans}) into Lagrangian (\ref{L})
yields
\begin{gather}
L=\allowbreak \left( \frac{1}{4}\pi ^{2}-\frac{1}{2}\mu +\varepsilon \right)
a^{2}+\left( \pi ^{2}-\frac{1}{2}\mu \right) b^{2}+\left( \frac{9}{4}\pi
^{2}-\frac{1}{2}\mu +\varepsilon \right) c^{2}\allowbreak +2\varepsilon ac
\notag \\
-\frac{g}{4}\left( \frac{3}{4}a^{4}+a^{3}c+3a^{2}b^{2}+3a^{2}c^{2}-3ab^{2}c+%
\frac{3}{4}\allowbreak b^{4}+3b^{2}c^{2}+\frac{3}{4}c^{4}\right) ,
\label{LL}
\end{gather}%
from which three variational equations follow:
\begin{equation}
\partial L/\partial (b^{2})=0, \label{d/db}
\end{equation}%
\begin{equation}
\partial L/\partial a=\partial L/\partial c=0. \label{dd/dadc}
\end{equation}%
In the general form, these equations are rather cumbersome. However, being
interested in the threshold at which the SSB sets in, one may set $b=0$ in
these equations (after performing the differentiation with respect to $b^{2}$
in Eq. (\ref{d/db})), which lead to the following system of three equations
for three unknowns $a$, $b$, and $\mu $:
\begin{equation}
2\pi ^{2}-\mu =\frac{3g}{2}\left( a^{2}-ac+c^{2}\right) , \label{bsimple}
\end{equation}%
\begin{gather}
\left( \frac{1}{2}\pi ^{2}-\mu +2\varepsilon \right) a+2\varepsilon c-\frac{g%
}{4}\left( 3a^{3}+3a^{2}c+6ac^{2}\right) =0, \label{asimple} \\
\left( \frac{9}{2}\pi ^{2}-\mu +2\varepsilon \right) c\allowbreak
+2\varepsilon a-\frac{g}{4}\left( a^{3}+6a^{2}c+3c^{3}\right) =0.
\label{csimple}
\end{gather}
In particular, Eqs. (\ref{asimple}) and (\ref{csimple}) with $g=0$, while
Eq. (\ref{bsimple}) is dropped, offer an additional application: they
predict the GS chemical potential of the linear system ($g=0$), as the value
at which the determinant of the linearized version of Eqs. (\ref{asimple})
and (\ref{csimple}) for $\ a$ and $c$ vanishes:
\begin{equation}
\mu _{0}=\frac{5}{2}\pi ^{2}+2\varepsilon -2\sqrt{\pi ^{4}+\varepsilon ^{2}}
\label{GS}
\end{equation}%
(recall that $\mu _{0}(\varepsilon )$ cannot be found above in an exact
form). The latter approximation is meaningful once it yields the GS chemical
potential smaller than the above-mentioned exact value $2\pi^2$ corresponding
to the lowest excited state. This condition holds at%
\begin{equation}
\varepsilon <\left( 15/8\right) \pi ^{2}\approx 18.5. \label{<}
\end{equation}%
Further, it is easy to check that Eqs. (\ref{bsimple})-(\ref{csimple}) yield
no physical solutions at $\varepsilon =0$, in agreement with the obvious
fact that the SSB does not occur when the central barrier is absent, i.e.,
the potential is not split into two wells.
Finally, a particular exact solution of Eqs. (\ref{bsimple})-(\ref{csimple})
(which includes a particular value of $\varepsilon $) can be found by
setting $c=0$, i.e., assuming that the third harmonic vanishes in ansatz (%
\ref{ans}):%
\begin{equation}
a^{2}=3\pi ^{2}/\left( 2g\right) ,~\mu =-\pi ^{2}/4,~\varepsilon =3\pi
^{2}/16\approx 1.85. \label{b=c=0}
\end{equation}%
A noteworthy feature of this particular solution is that it has $\mu <0$.
Indeed, Eq. (\ref{NLSE}) suggests that a sufficiently strong nonlinear term
should make the chemical potential negative, as corroborated by Eq. (\ref%
{mu-sol}).
A consistent analysis of the VA for the present model, and its comparison
with numerical results will be reported elsewhere.
\section{Conclusion}
The objective of this paper was two-fold. First, a short overview was given
of the general topic of the SSB (spontaneous symmetry breaking) in nonlinear
one-dimensional models featuring the competition of the self-focusing cubic
nonlinearity and DWP (double-well-potential) structure. Physically relevant
examples of such systems are offered by nonlinear optical waveguides with
the transverse DWP structure, and by BEC trapped in two symmetric potential
wells coupled by tunneling of atoms. The SSB occurs at a critical value of
the nonlinearity strength, i.e., of the field's norm (which is tantamount to
the total power, in the case of the trapped optical beam).
The second part of the paper displayed a particular model, which is the
simplest one capable to grasp the SSB phenomenology: an infinitely deep
potential box, split into two wells by a delta-functional barrier set at the
center. Approximate analytical results predicting the SSB point have been
presented for two limit cases, \textit{viz}., the strong or weak split of
the potential box by the central barrier. In both cases, critical values of
the norm at the SSB point have been found, being, respectively, small and
large. For the generic (intermediate) case, a coarser approach based on the
VA (variational approximation) has been developed. The detailed analysis of
the VA and comparison of the predictions with numerical results will be
reported elsewhere.
|
2012.06123
|
\subsection{Deterministic Recurrent Models}
Recurrent networks were first used for future frame prediction in \cite{ranzato} when Ranzato et al. learnt a model to predict a quantized space of image patches.
Srivastava et al. \cite{srivastava} proposed a model to predict the future as well as the input sequence in order to prevent the model from storing information only about the last few frames.
Shi et al. \cite{convlstm} proposed an extension of the LSTM by replacing the fully connected structure with one that is fully convolutional which saw popular use to date for learning sequential data with spatial information.
Finn et al. \cite{finn} used an LSTM framework to model motion via transformations of groups of pixels.
Patraucean et al. \cite{patraucean} and Villegas et al. \cite{villegas} explicitly injected short term motion information through the use of optical flow.
Xu et al. \cite{xu} proposed a two-stream recurrent network to deal with the high and low frequency content often present in natural videos.
Kalchbbrenner et al. \cite{kalchbrenner} introduced a model that learns the joint distribution of the raw pixels to generate them one at a time.
Wang et al. \cite{wang-predrnn} proposed to improve the stacked LSTM by having the memory and hidden states flow in a zig-zagged manner from the highest unit of the current timestep to the lowest unit of the subsequent timestep.
This was further improved in \cite{wang-e3d} by replacing the 2D convolutions with 3D and a memory attention in the LSTM itself.
We also use 3D convolutions throughout our entire architecture but in contrast, our model is stochastic.
Stochastic recurrent networks vary in way they propagate uncertainty across time as well as the way inference is computed. For instance, Bayer et al. \cite{storn} and Goyal et al. \cite{zforcing} conditioned the generation only on the hidden states of the recurrent network whilst Chung et al. \cite{vrnn} and Fraccaro et al. \cite{srnn} have the output be some function over both the hidden states and the latent vector. Next, the LSTM state transitions in \cite{storn,vrnn,zforcing} are additionally conditioned on the latent vectors whereas in \cite{srnn} is not. The work of \cite{vrnn} was later extended in \cite{hierarchyvrnn} through a hierarchy of latent variables for future frame prediction. We propagate stochastic information in the same way as \cite{vrnn} except that the latent tensors themselves now contain richer spatial-temporal information since they are the result of 3D convolutions.
For inference, both \cite{srnn} and \cite{zforcing} run a deterministic recurrent network backwards through the sequence to form the approximate posterior whereas the posterior in \cite{storn} and \cite{vrnn} is computed using only information up till the present.
Similarly, our method for inference follows that of \cite{vrnn} but in contrast to \cite{vrnn} and in fact all existing methods, we jointly optimize both the KL divergence and a novel log likelihood criterion.
Future frame prediction is useful for various application such as early action prediction\cite{Ryoo2011,shi2018action,sadegh2017} and action anticipation~\cite{Furnari2019,Miech2019,qi2017predicting}.
Some anticipation methods generate future RGB images and then classify them into human actions using convolution neural networks ~\cite{Zeng2017a,Wang2017}.
However, generation of RGB images for the future is a very challenging task specially for longer action sequences.
Similarly, some methods aim to generate future motion images and then try to predict action for the future~\cite{Rodriguez2018}.
\subsection{Variational Recurrent Neural Network}
\label{sec:variationalconvlstm}
Figure \ref{fig:vrnn} provides a graphical illustration of the VRNN.
The VRNN uses a latent variable $\textbf{z}_{t}$ at each timestep of a recurrent network to capture the variations in the observed data.
It contains a VAE at every timestep whose mean $\mu_{t}$ and variance $\sigma_{t}$ are conditioned on the hidden unit $h_t$ of a recurrent network. These parameters are then used to sample the latent variable $\textbf{z}_{t}$ at each timestep.
Concisely, the forward pass can be completely described by the following set of recurrence equations where the subscripts p and q denote the prior and posterior distributions respectively, and the components $\text{f}_{\text{p}}$, $\text{f}_{\text{q}}$, $\text{f}_{\text{enc}}$, $\text{f}_{\text{dec}}$ are functions implemented using neural networks.
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.6\columnwidth]{vrnn.png}
\end{tabular}
\caption{Graphical illustration of the Variational Recurrent Network. The dotted lines denote the posterior network $\text{f}_{\text{q}}$ that is only used during training and is discarded at test time.}
\label{fig:vrnn}
\end{center}
\end{figure}
\begin{align}
\mu_{\text{p},t},\sigma_{\text{p},t} &= \text{f}_{\text{p}}(h_{t-1})\\
\mu_{\text{q},t},\sigma_{\text{q},t} &= \text{f}_{\text{q}}(h_{t-1}, \text{f}_{\text{enc}}(x_t))\\
z_{\text{p},t} &\sim N(\mu_{\text{p},t},\sigma_{\text{p},t})\\
z_{\text{q},t} &\sim N(\mu_{\text{q},t},\sigma_{\text{q},t})\\
\hat{x}_t &= \text{f}_{\text{dec}}(z_{\text{p},t}, h_{t-1})\\
h_{t} &= \text{LSTM}(\text{f}_{\text{enc}}(x_t),h_{t-1},z_{\text{p},t})
\end{align}
Here $N(\mu,\sigma)$ is a multivariate Gaussian distribution with mean $\mu$ and co-variance \textbf{diag}$(\sigma^2)$.
Note that the posterior network $\text{f}_{\text{q}}$ is used only during training and is discarded at test time.
The entire model is then trained end-to-end for future frame prediction by minimizing a sum of the reconstruction loss ($L_{\text{rec}}$), and latent loss ($L_{\text{latent}}$) expressed as:
\begin{align}
L = \lambda_{\text{rec}} L_{\text{rec}} + \lambda_{\text{latent}} L_{\text{latent}}
\end{align}
where $\lambda_{\text{rec}}$ and $\lambda_{\text{latent}}$ are the trade off hyper-parameters and the latent loss is the timestep-wise KL divergence ($L_{\text{KL}}$) between the prior ($p$) and posterior ($q$) distributions and is expressed as:
\begin{align}
& L_{\text{KL}} = \sum_{t=1}^T \text{KL}(q(z_t | X_{\le t}, Z_{< t}) || p(z_t | X_{< t}, Z_{< t})))= \\
& \sum_{t=1}^T \log (\sigma_{\text{q,t}}) - \log (\sigma_{\text{p,t}}) + \frac{\sigma_{\text{p,t}}^2 + (\mu_{\text{p,t}}-\mu_{\text{q,t}})^2}{2\sigma_{\text{q,t}}^2} - 0.5
\label{eq:kldenominator}
\end{align}
\begin{figure*}[t!]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=1\textwidth]{architecture/architecture.png}
\end{tabular}
\caption{Our proposed architecture for future frame prediction. The architecture inputs and outputs at each timestep a sequence of M video frames with the prediction made H timesteps into the future. The entire architecture is fitted with 3D convolutions. The 3D-ENC and 3D-DEC are mirrored versions of the 2-block 3D-ResNet18 as shown in Figure \ref{fig:encoderdecoder}. We use 2 LSTM layers and a shallow 3D conv network to generate $\mu$ and $\sigma$ that are then sampled from to produce z. The block $[\circ,\circ]$ indicates a concatenation along the 4th axis. The values above each component (3D-ENC, CONVLSTM, Z, 3D-DEC) indicate the sizes of the output tensor.}
\label{fig:architecture}
\end{center}
\end{figure*}
\subsection{New log-likelihood regularized KL divergence}
\label{sec:loglikelihood}
Typically, the KL divergence-based latent loss is used to regularize the latent space, enforcing it to be a Gaussian distribution with known parameters.
We further enhance this regularization by appending the negative of the log-likelihood term to the latent loss.
The objective here is to maximize the likelihood of the prior mean distribution w.r.t. the posterior.
This is done by minimizing the negative likelihood as shown in Eq.~\ref{eq:lldenominator} by assuming that the prior, posterior and \emph{the conditional prior mean} given the posterior all follow a Gaussian distribution.
\begin{align}
- L_{\text{LL}} &= - \log \prod_{t=1}^T p(\mu_{\text{p},t} | \mu_{\text{q},t}, \sigma_{\text{q},t}) \\
&= \sum_{t=1}^T \log ({\sigma_{\text{q},t}}) + ( \frac{\mu_{\text{p},t} - \mu_{\text{q},t}}{\sigma_{\text{q},t}} )^2
\label{eq:lldenominator}
\end{align}
The proposed latent loss is thus expressed together as:
\begin{align}
& L_{\text{KL}} - L_{\text{LL}} \nonumber \\
&= \log (\sigma_{\text{q}}) - \log (\sigma_{\text{p}}) + \frac{\sigma_{\text{p}}^2 + (\mu_{\text{p}}-\mu_{\text{q}})^2}{2\sigma_{\text{q}}^2} - 0.5 \nonumber \\
&+ \log ({\sigma_{\text{q}}}) + ( \frac{\mu_{\text{p}} - \mu_{\text{q}}}{\sigma_{\text{q}}} )^2 \\
&= 2\log (\sigma_{\text{q}}) - \log (\sigma_{\text{p}}) + \frac{\sigma_{\text{p}}^2 + 3(\mu_{\text{p}}-\mu_{\text{q}})^2}{2\sigma_{\text{q}}^2} - 0.5
\end{align}
Interestingly, the above equation is similar to the KL divergence (eq \ref{eq:kldenominator}) but with some of its components weighted differently. In particular, the log posterior variance, $\log (\sigma_q)$, has been scaled by a factor of 2 and the squared difference of mean, $(\mu_p - \mu_q)^2$, by a factor of 3. This modification has 2 effects. First, it puts more emphasis on sample diversity since the log prior variance $\log(\sigma_p)$ put out by the network must now be higher in order to match the scaled log posterior variance $2\log(\sigma_q)$. Secondly, the scaled difference of mean $3(\mu_p - \mu_q)^2$ serves to balance out the additional weight assigned to the variance term and thus encourages the model to continue generating samples that are representative of the dataset. As such, the log-likelihood regularized KL divergence should have no adverse effects on the model since it is simply the KL divergence with a reweighting of its components and would argue it to be more forceful if one needs to have a greater emphasis on sample diversity.
Interestingly, the weights for each component can also be customized although their individual effects will not be investigated since it is not the purpose of this paper.
All-in-all, our new loss function for training the VRNN is expressed together as: $L = \lambda_{\text{rec}} L_{\text{rec}} + \lambda_{\text{latent}} ( L_{\text{KL}} - L_{\text{LL}} )$.
\subsection{Our 3D Convolutional VRNN}
\label{sec:convolutionalrecurrentnetworks}
The ConvLSTM was proposed in \cite{convlstm} to address the shortcomings of Fully-Connected LSTM, namely that latter always ends up decimating any spatial information contained in the input tensor.
Intuitively, if the states are viewed as the hidden representations of moving objects, then a ConvLSTM with a larger kernel should be able to capture faster motions while one with a smaller kernel can capture slower motions.
However, if the input at each timestep is a single image, then the hidden states are the only component that carry motion information in both Fully-Connected and ConvLSTMs.
In our work, we counteract this limitation by replacing all 2D convolutions (and de-convolutions) with 3D to enable every component to retain motion information instead of only the hidden states. The benefits of this are two-fold. First, the 3D ConvLSTM is no longer completely reliant on the hidden states for motion information since there is an additional source coming from the 3D image encoder. Specifically, the use of 3D convolutions on multiple frames result in an input tensor that carries short-term spatiotemporal information as opposed to a VRNN that run a 2D convolution on a single frame at every timestep. Second, we can now vary the window size and output horizon at each timestep without needing to redesign the architecture. For example, let us define \textbf{M} to be the window size, or the number of input frames to our model at each timestep and $\mathcal{H}$ the output horizon (output frames), or how far into the future should the model predict. Then, 3D convolutions (and de-convolutions) allow us to set a large \textbf{M} to efficiently capture large motions when dealing with datasets where the motion between frames is prevalently large and conversely, a large $\mathcal{H}$ to predict many frames into the future at once with minimal reconstruction errors if said motion between frames is small. All in all, this upgrade renders our 3D convolutional VRNN more effective and general than its 2D counterpart.
Our proposed architecture is shown in Figure \ref{fig:architecture}. The encoder (3D-ENC) takes at each timestep a clip of M video frames of shape [M,H,W,C] to produce a tensor of shape [M/2, H/4, W/4, C/4] where H,W,C denote the height, width and channels respectively. This tensor is then concatenated with the latent tensor Z (a zero tensor in the 1st timestep) along the 4'th channel indicated by $[\circ,\circ]$ then passed to the 3D ConvLSTM with 2 hidden layers for motion learning. The LSTM hidden states at the second level with a shape of [M/2, H/4, W/4, 128] are then fed through a shallow 3D CNN with two heads to produce the parameters of the prior distribution with shape [M/2, H/4, W/4, 16] that are later sampled to produce the latent tensor Z. This latent tensor is then concatenated with the hidden states and finally propagated through the decoder (3D-DEC) to predict the frames H timesteps into the future. The model is applied recursively by using the newly generated frame as input if the ground truth is not available. Specifically, if the frames are observed up to time C, then the model will use the ground truth as input up till time C-M then a combination of the ground truth and predicted frames between time C-M to C, and then finally, only the predicted frames as input from time C onwards. During training, a separate 3D encoder (not shown in Figure \ref{fig:architecture}) is used to generate the posterior distribution to optimize the KL divergence.
\begin{figure}
\begin{center}
\begin{tabular}{c} \includegraphics[width=1\columnwidth]{architecture/encoder.png}\\
(a) 2-block 3D ResNet-18 encoder\\ \includegraphics[width=1\columnwidth]{architecture/decoder.png}\\
(b) 2-block 3D ResNet-18 decoder
\end{tabular}
\caption{Architecture of our encoder and decoder. The encoder outputs a 4D tensor with a spatial resolution of (H/4,W/4). Exception the decoder, each filter output is followed by a 3D batch-norm and ReLU. An downsampling operation with stride 2 is indicated by "//2" and an upsampling with stride 2 by "x2".}
\label{fig:encoderdecoder}
\end{center}
\end{figure}
As shown in Figure \ref{fig:encoderdecoder}, we use a 2-block 3D ResNet-18 for both the encoder and decoder in contrast to \cite{hierarchyvrnn} that use a full ResNet. We find this to be sufficient especially since the 3D ConvLSTM itself serve as an extension of the 3D CNN for learning complex spatiotemporal features given a window of \textbf{M} frames.
Furthermore, by truncating the number of blocks to 2, we reduce the total number of parameters significantly which allow us to devote additional resources to our 3D ConvLSTM with 128 hidden units that contains 7m parameters per level. However, we also want to avoid the other end of not having a feature extractor at all \cite{wang-predrnn,wang-e3d} since they have been shown to extract useful features that tend to be task specific at the higher blocks and more general purpose at the lower blocks. In short, we propose to use a smaller feature extractor CNN and a larger LSTM. We show in our experiments that modelling the architecture in such a manner allows us to outperform the state-of-the-art while requiring fewer parameters.
\subsection{Comparison to state-of-the-art}
\label{sec.soa}
We compare our approach to several state-of-the-art methods using publicly available source code and model where available with default parameters and using standard metrics such as frame-wise Mean Squared Error (MSE), Structural Similarity Index (SSIM) and Peak Signal to Noise Ratio (PSNR). We sample 50 predictions from the stochastic models for each ground truth test sequence and average the metrics across the test set. Note that sampling is done only for the purpose of evaluation in order to get the average performance and is not required for deployment. \\
\noindent
\textbf{Training Details:}
We initialize the weights of our truncated 3D ResNet-18 encoder and decoder with weights pre-trained on the Kinetics-400 dataset \cite{kinetics} and all other components using PyTorch's default initializer.
We use the Adam optimizer with default hyperparameters, a learning rate of $10^{-3}$ with no weight decay, a batch size of 6 and the L1+L2 reconstruction loss that was also used in \cite{wang-predrnn,wang-e3d}. We train the model using beta warm-up \cite{betawarmup} and have it gradually predict into the future using its own predictions as input \cite{ss}. \\
\noindent
\textbf{The Moving MNIST dataset}
\cite{srivastava} consists of two digits (0 to 9) of size 28 x 28 moving inside a 64 x 64 patch.
The digits are chosen randomly from the MNIST training set and placed at random locations inside the patch.
Each digit is assigned a velocity whose direction is chosen uniformly at random on a unit circle and whose magnitude is also chosen uniformly at random over a fixed range.
The digits bounce off the edges of the 64 x 64 frame and overlap as they move past each other. The training set contains 10,000 sequences while the validation and test sets 1,000 sequences each. By default, the sequences are all 20 frames long and the models are trained to predict the next 10 frames given the first 5 or 10 as input.
Table \ref{table:quantitativemnist} shows the performance of the models when using 5 frames to predict 10 and 15 frames into the future and when using 10 frames to predict 10 and 20 frames into the future. Our method demonstrates its promise, outperforming both the state-of-the-art deterministic (E3D-LSTM \cite{wang-e3d}) and stochastic (VRNN \cite{vrnn}) models, with the latter by a large margin. We also improve over the E3D-LSTM \cite{wang-e3d} despite having fewer parameters.
We were only able to train the smallest variant of the models presented in \cite{hierarchyvrnn} which nevertheless contains 62 million parameters. Interestingly, we also outperform them. These results thus indicate the impact of our new design and the novel latent loss.
\begin{table*}[t]
\begin{center}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|l|l|cc|cc|cc|cc|r|}
\hlin
\multirow{2}{*}{Type} & \multirow{2}{*}{Model} &
\multicolumn{2}{c|}{$\text{x}_{1:5} \rightarrow \hat{\text{x}}_{6:15}$} & \multicolumn{2}{c|}{$\text{x}_{1:5} \rightarrow \hat{\text{x}}_{6:20}$} & \multicolumn{2}{c|}{$\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:20}$} & \multicolumn{2}{c|}{$\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:30}$} & \multirow{2}{*}{\# Params} \\
& & SSIM & MSE & SSIM & MSE & SSIM & MSE & SSIM & MSE & \\
\hline
\multirow{3}{*}{Deterministic} \ & 2D ConvLSTM \cite{convlstm} & 0.662 & 111.1 & 0.482 & 154.3 & 0.763 & 82.2 & 0.660 & 112.3 & \textbf{2.8M} \ \ \\
& PredRNN++ \cite{wang-predrnn} & 0.793 & 66.2 & 0.769 & 79.2 & 0.870 & 47.9 & 0.821 & 57.7 & 15.4M \ \ \\
& E3D-LSTM \cite{wang-e3d} & 0.853 & 53.4 & 0.801 & 64.1 & \textbf{0.910} & 41.3 & 0.872 & 47.6 & 38.7M \ \ \\
\hline
\multirow{2}{*}{Stochastic} & Variational 2D ConvLSTM \cite{vrnn} & 0.733 & 91.1 & 0.564 & 126.4 & 0.816 & 60.7 & 0.773 & 83.5 & 2.9M \ \ \\
& Improved VRNN \cite{hierarchyvrnn} & 0.772 & 123.1 & 0.728 & 162.2 & 0.776 & 129.2 & 0.699 & 194.3 & 62M \ \ \\
& Variational 3D ConvLSTM (Ours) & \textbf{0.864} & \textbf{51.4} & \textbf{0.805} & \textbf{63.2} & 0.896 & \textbf{39.4} & \textbf{0.874} & \textbf{47.54} & 12.9M \ \ \\
\hline
\end{tabular}
}
\end{center}
\caption{Results on the Moving MNIST dataset when using 5 frames to predict 10 ($\text{x}_{1:5} \rightarrow \hat{\text{x}}_{6:15}$) and 15 ($\text{x}_{1:5} \rightarrow \hat{\text{x}}_{6:20}$) frames into the future, and when using 10 frames to predict 10 ($\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:20}$) and 20 ($\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:30}$) frames into the future. The metrics are computed frame-wise. Higher SSIM or lower MSE scores indicate better results. Finally, the rightmost column indicate the number of parameters for the various models.}
\label{table:quantitativemnist}
\end{table*}
\begin{table*}
\centering
\setlength\tabcolsep{4pt}
\begin{tabular}{@{} lr @{}}
\small
\begin{tabular}{@{}|l|cc|cc|cc|@{}}
\hline
\multirow{2}{*}{Model} & \multicolumn{2}{c|}{$\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:30}$} & \multicolumn{2}{c|}{$\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:50}$} & \multicolumn{2}{c|}{$\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:70}$} \\
& SSIM & PSNR & SSIM & PSNR & SSIM & PSNR \\
\hline
2D ConvLSTM \cite{convlstm} & 0.712 & 23.58 & 0.639 & 22.85 & 0.551 & 20.13 \\
PredRNN++ \cite{wang-predrnn} & 0.865 & 28.47 & 0.741 & 25.21 & 0.702 & 23.51 \\
E3D-LSTM \cite{wang-e3d} & \textbf{0.879} & \textbf{29.31} & 0.810 & 27.24 & 0.798 & 26.82 \\
\hline
Variational 2D ConvLSTM \cite{vrnn} & 0.787 & 25.76 & 0.733 & 24.83 & 0.672 & 23.13 \\
Variational 3D ConvLSTM (Ours) & 0.866 & 28.31 & \textbf{0.852} & \textbf{27.89} & \textbf{0.846} & \textbf{27.66} \\
\hline
\end{tabular}
&
\includegraphics[width=0.34\linewidth,valign=c]{kth.png}
\end{tabular}
\vspace{0.05cm}
\caption{Results on the KTH action dataset when using 10 frames to predict 20 ($\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:30}$), 40 ($\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:50}$), and 60 ($\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:70}$) time steps into the future. The metrics are computed frame-wise. Higher SSIM and PSNR scores indicate better results. The bar chart on the right highlights the difference between our model and the E3D-LSTM. Our model performs much better for longer predictions.}
\label{table:quantitativekth}
\end{table*}
We present some visual results in Figure \ref{fig:qualitativemnist} where the first row illustrates the input sequence $\textbf{x}_{1:10}$, the second the ground truth for the predicted sequence $\textbf{x}_{11:20}$ and all subsequent rows the predictions made by the various models $\hat{\textbf{x}}_{11:20}$.
It can firstly be seen that the injection of stochasticity causes the Variational ConvLSTM to output predictions that are blurrier than its deterministic counterpart.
This could principally be due to the fact that information coming from the latent nodes act as noise and thus interferes with reconstruction.
Unlike the ConvLSTM however, the digits generated by the Variational ConvLSTM are closer to the ground truth resulting in a performance that is generally superior as evidenced by the quantitative scores in Table \ref{table:quantitativemnist}. We can then observe our model producing the best results.
This signals that the blurry reconstructions manifested by the Variational ConvLSTM are counteracted by replacing all 2D convolutions with 3D.
Intuitively, our novel loss also acts as a stronger regularizer for the reconstructions.\\
\noindent
\textbf{The KTH dataset}
\cite{kth} consists of humans performing 6 types of actions: boxing, clapping, waving, jogging, running, and walking under 4 scenarios: outdoors, outdoors with scale variation, outdoor with different clothes, and indoors with a homogeneous and static background. Each video is recorded at 25 fps and lasts an average of 4 seconds. We follow the experimental setup in \cite{villegas} using persons 1-16 for training and 17-25 for testing and resize each frame to 128x128 pixels. The models are trained to predict the next 10 frames given the first 10 as input.
Table \ref{table:quantitativekth} presents the performance of the various models when predicting the next 20, 40 and 60 frames. It can be observed that our model lags slightly behind the E3D-LSTM when predicting short term but performs much better when tasked to predict further into the future. This difference is highlighted on the bar chart beside Table \ref{table:quantitativekth} that shows the performance of our model degrading at a slower rate than the E3D-LSTM.
The results can be explained by Figure \ref{fig:qualitativekth} where each row represents the output sequence $\hat{\textbf{x}}_{11:50}$ spaced 2 frames apart.
It can be observed from the figures that the predictions coming from our model are blurrier than E3D, resulting in metrics that are inferior although we make up for it by being able to predict the individual re-entering the scene.
The quantitative scores also show that our variational method is much better than the Variational 2D ConvLSTM \cite{vrnn}.
These findings once again demonstrate the effectiveness of our architecture for modelling spatiotemporal data.
\begin{table*}
\begin{center}
\resizebox{1\textwidth}{!}{
\begin{tabular}{|l|cc|cc|cc|cc|r|}
\hline
\multirow{2}{*}{Model} &
\multicolumn{2}{c|}{$\text{x}_{1:5} \rightarrow \hat{\text{x}}_{6:15}$} &
\multicolumn{2}{c|}{$\text{x}_{1:5} \rightarrow \hat{\text{x}}_{6:20}$} &
\multicolumn{2}{c|}{$\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:20}$} &
\multicolumn{2}{c|}{$\text{x}_{1:10} \rightarrow \hat{\text{x}}_{11:30}$} &
\multirow{2}{*}{\# Params}\\
& SSIM & MSE
& SSIM & MSE
& SSIM & MSE
& SSIM & MSE &\\
\hline
2D ConvLSTM \cite{convlstm} & 0.662 & 111.1 & 0.482 & 154.3 & 0.763 & 82.2 & 0.660 & 112.3 & \textbf{2.8M} \ \ \ \\
Variational 2D ConvLSTM \cite{vrnn} & 0.733 & 91.1 & 0.564 & 126.4 & 0.816 & 60.7 & 0.773 & 83.5 & 2.9M \ \ \ \\
Variational 3D ConvLSTM & 0.857 & 52.1 & 0.797 & 63.8 & 0.887 & 41.8 & 0.868 & 49.7 & 12.9M \ \ \ \\
Variational 3D ConvLSTM + LL Criterion (ours) & \textbf{0.864} & \textbf{51.4} & \textbf{0.805} & \textbf{63.2} & \textbf{0.896} & \textbf{39.4} & \textbf{0.874} & \textbf{47.5} & 12.9M \ \ \ \\
\hline
\end{tabular}
}
\end{center}
\caption{Ablation study on the Moving MNIST dataset. The metrics are computed frame-wise. Higher SSIM or lower MSE scores indicate better results.}
\label{tab:ablationmnist-components}
\end{table*}
\begin{figure*}[t]
\scriptsize
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.41\textwidth]{ablation/mnistssim_final.png}& \includegraphics[width=0.38\textwidth]{ablation/mnistmse.png}\\
\includegraphics[width=0.41\textwidth]{ablation/kthssim.png}& \includegraphics[width=0.38\textwidth]{ablation/kthpsnr.png}\\
\end{tabular}
\caption{The effect of \textbf{M} window size and the output horizon $\mathcal{H}$ on the performance. The first row shows the SSIM and MSE scores on the moving MNIST action dataset, while the second row the SSIM and PSNR on the KTH action dataset.}
\label{fig:ablationkth-inputoutput}
\end{center}
\end{figure*}
\subsection{Ablation Study}\label{sec:ablationstudies}
\textbf{Effectiveness of each component:}
We quantize the effect of each contribution in Table \ref{tab:ablationmnist-components} on the moving MNIST dataset.
It can be seen that the introduction of stochasticity into the recurrent network allows it to better tackle uncertainties in the recurrent dynamics which results in better predictions, significantly lowering the MSE from 82.2 to 60.7.
Additionally, swapping out the 2D convolutions in place for 3D brings about significant improvements to the model, lowering the MSE by approximately another 20 points.
This makes sense since 3D convolutions operate on both the spatial and temporal axis, letting the architecture capture relationships in said dimensions.
Finally, it is quite apparent that the introduction of the log-likelihood criterion has a noticeable effect, further bringing down the MSE by approximately 2 points.
This can be interpreted in various ways: (1) that the resulting loss function empirically helps the network traverse towards a better local minima and (2) that the added regularizer helps the recurrent model strengthen its ability at expressing complex distributions.
Intuitively, appending the log-likelihood criterion to the KL divergence has some conceptual similarity to the use of the L1+L2 loss functions that has been empirically shown in \cite{wang-predrnn,wang-e3d} to be better than the individual counterparts.
In conclusion, each component brings a definitive upgrade to the model and together, lend it the advantage it needs to outperform the deterministic and stochastic state-of-the-art models~\cite{vrnn,wang-e3d,wang-predrnn,hierarchyvrnn}.
\noindent
\textbf{Window size and output horizon:}
Recall from section \ref{sec:convolutionalrecurrentnetworks} that the advantages 3D convolutions have over its 2D counterpart when paired with an LSTM is that (1) the LSTM state transitions are no longer completely reliant on its hidden states for motion information and (2) that one could vary the window size (\textbf{M}) and the output horizon ($\mathcal{H}$) at each timestep without having to change the architecture. In this experiment, we conduct additional studies on our model where we varied \textbf{M} and $\mathcal{H}$, the number of input frames and the output horizon respectively at each timestep to study the effects different design choices have on our model.
As expected, the plots in Figure \ref{fig:ablationkth-inputoutput} show a degradation in the metrics over time regardless of the configuration in use. The KTH plots exhibits an exponential decay whereas the moving MNIST is more linear. The plots also show that a smaller window size is better for the MNIST dataset but has no clear difference for the KTH action dataset.
Interestingly, there are cases that favour a large window size and output horizon and some other cases, that do not.
For longer predictions however, the plots show that the various configurations are quite similar in terms of performance.
\section{Introduction}
\input{2introduction}
\section{Related Work}
\input{3relatedwork}
\section{Our model}
\input{4method}
\section{Experiments}
\label{sec:experiments}
\input{5experiments}
\section{Conclusion}
\label{sec:conclusion}
\input{6conclusion}
{\small
\bibliographystyle{ieee_fullname}
|
2103.11298
|
\section{Introduction}
\label{introduction}
\begin{figure*}[t]
\centering
{\includegraphics[width=0.99\linewidth]{Figures/performance_introduction_v2.pdf}}
\caption{{\bf Exemplar visual results of snow removal.} The goal of image desnowing is to remove snow and restore high-quality snow-free images. The various scene, occlusion and illumination variations of objects make it difficult to accomplish this task. Our proposed \textit{DDMSNet} is able to remove the snow and recover clean images with the aid of semantic and geometric priors.}
\label{fig:performance_introduction}
\end{figure*}
As a common weather condition, snow can greatly affect the visibility of scene and objects in captured images. Its presence not only leads to poor visual quality but also degenerates the performance of subsequent image processing tasks, like object detection \cite{dalal2005histograms}, object tracking \cite{kaempfer2007three} and scene analysis \cite{itti1998model}. Therefore, snow removal from images is an important task and has attracted increasing attention in the computer vision community.
Compared with its counterpart task of image deraining, the snow removal is more challenging due to two reasons. Firstly, although the rain can also affect the visibility of objects in the scene, it is transparent which provides more scene information to remove rain and recover the high-quality clean images. On the contrary, the snow is opaque which makes it difficult for desnowing models to recover the occluded regions.
Secondly, based on the definition of \cite{liu2018desnownet}, we consider only removing the snowflake in the air, rather than the snow falling on the ground or buildings. Because only the snowflake in the air can cover objects and has obviously negative impact on the vision-based intelligent systems. Therefore, it is important for snow removal models to understand semantic \cite{fourure2017residual} and geometry information.
Existing single image desnowing methods adopt image priors \cite{wang2017hierarchical} or develop deep convolutional neural network (CNN) \cite{liu2018desnownet} to remove snow via learning a snow map or directly transferring the snowy images to their corresponding clean versions. Though the current state-of-the-art methods have achieved great success in snow removal, they ignore the geometric information which can affect the visibility of the same between the camera and objects, and the semantic information which can help to remove the snow in the air through understanding scenes. In addition, most of them focus on removing snow, but ignore to fill in the regions which are occluded by snow. Therefore, the recovered images fail to remove heavy snow and exhibit bad details.
To this end, we design a new deep CNN for snow removal via learning semantic-aware and geometry-aware representation. We leverage the semantic and geometric information as priors to train a deep learning based framework. The proposed framework consists of four sub-networks: a coarse snow removal network, a semantic segmentation network, a depth estimation network, and \textit{DDMSNet}. The snowy images are first fed into the coarse snow removal network to generate the
coarsely desnowed images, which are then utilized to predict semantic segmentation and depth maps via a pre-trained semantic segmentation network and a depth estimation network. In order to generate finer results, a \textit{DDMSNet} is proposed. Different from the traditional multi-scale networks, the \textit{DDMSNet} applies dense connections between different scales in different dimensions, thus is able to achieve better performance to recover clean images with better details. In addition, \textit{DDMSNet} takes the coarsely desnowed images, maps of semantic segmentation and depth as input to calculate the semantic-aware and geometry-aware representation in an attention guided manner to restore clean images.
Moreover, considering that the snow always degrades the quality of images captured in the outdoor scenarios, which significantly deteriorates the performance of various core applications in autonomous driving, we create two new datasets specialized for street scenarios. These two datasets can be conveniently employed to evaluate algorithms of autonomous driving in the condition of snow. Along with another public dataset, the proposed method achieves the state-of-the-art performance for the task of snow removal.
Overall, the contributions of our method are summarized as follows.
\begin{itemize}
\item
Firstly, we propose a deep dense multi-scale network, named \textit{DDMSNet}, for single image desnowing. Different from previous multi-scale networks, which mainly capture features from pixel-level input, the proposed \textit{DDMSNet} can extract multi-scale representation from pixel-level and feature-level input.
\item
Secondly, we exploit to use semantic and geometric information as priors for snow removal. Under a map-guided scheme, semantic and geometric features are obtained in different stages to help remove snow and recover clean images.
\item
Thirdly, the two datasets we created will benefit research in the community. Experiments on three datasets show that the proposed method achieves the state-of-the-art performance on snow removal. Meanwhile, the desnowed images can improve the performance of many core applications, like semantic segmentation and depth estimation.
\end{itemize}
\section{Related Work}
Our work in this paper is closely related to snow removal, rain removal and haze removal, which are briefly introduced respectively in the following.
\subsection{Snow Removal}
Removing snow from a single image is a highly ill-posed problem. Its formulation can be described as,
\begin{equation}
O = A \odot M + B \odot (1 - M) \,,
\end{equation}
where $O$, $A$, $B$ and $M$ are the observed snowy image, the chromatic aberration map, the latent clean image and the snow mask, respectively.
Traditional methods utilize priors of snow-driven features to recover the clean images from the snowy versions. Bossu \textit{et al.} \cite{bossu2011rain} use a classical MoG model to separate the foreground from background. Then the snow can be detected from the foreground and removed to recover clean images under the help of HOG features. Similarly, Pei \textit{et al.} \cite{pei2014removing} extract features in color space and shape to detect the position of snow to help remove snow. In addition, Rajderkar \textit{et al.} \cite{rajderkar2013removing} and Xu \textit{et al.} \cite{xu2012improved,xu2012removing} employ frequency space separation and color assumptions to model the characteristics of snow for the snow removal task.
Recently, deep learning witnesses great success in low-level image enhancing tasks such as super super-resolution \cite{ledig2017photo,johnson2016perceptual}, image deblurring \cite{zhang2018adversarial,zhang2020deblurring,li2021arvo,zhang2020every}, image deraining \cite{liu2018desnownet,zhang2021dual,zhang2020beyond}, which also include snow removal. Specially, Liu \textit{et al.} \cite{liu2018desnownet} propose a DesnowNet, which is the first deep learning based method to remove snow from a single image. The DesnowNet adopts translucency and residual generation modules to recover image details obscured by snow. In order to generate realistic desnowed images, Li \textit{et al.} \cite{li2019single} adopt the GAN framework to restore better details. Li \textit{et al.} \cite{li2019stacked} introduce a multi-scale network for snow removal. However, they only consider the different scales in the pixel-level space, ignoring the feature space. More recently, Li \textit{et al.} \cite{li2020all} use the Network Architecture Search (NAS) framework to obtain a network, which achieves the state-of-the-art performance on snow removal. However, they ignore the semantic and geometric information, which are important priors for image restoration.
\subsection{Rain Removal}
Image deraining aims to remove the rain from rainy images and recover clean versions. The main difference from snow removal is that the snow is opaque and thus it is more difficult to restore the occluded details. In the recent decades, a set of methods are proposed to successfully remove rain via modeling the physical characteristics of rain \cite{kang2011automatic,luo2015removing,li2016rain,chang2017transformed,zhu2017joint,du2018single}. Kang \textit{et al.} \cite{kang2011automatic} propose a framework to first decompose the rainy images into low-frequency and high-frequency layers, and then use dictionary learning to remove rain in the high frequency layer. Chen \textit{et al.} \cite{chen2014visual} and Luo \cite{luo2015removing} use classified dictionary atoms and discriminative sparse coding to separate the rain and background. In \cite{li2016rain}, Li \textit{et al.} introduce a method to remove rain streaks via Gaussian mixture models.
It also has witnessed further promising achievement by the deep learning based deraining methods \cite{li2019single,zhang2019image,fu2017clearing,fu2017removing,yang2017deep,zhang2018density,li2018recurrent,eigen2013restoring,qian2018attentive,zheng2019residual}. Yang \textit{et al.} propose a deep CNN model for joint rain detection and removal. Fu \textit{et al.} \cite{fu2017removing} develop a deep detail network to remove rain and maintain texture details. However, it is difficult to remove heavy rain. Li \textit{et al.} \cite{li2019heavy} introduce a two-stage network to remove heavy rain. The first physics based stage decomposes the entangled rain streaks and rain accumulation, while the second model-free stage includes a conditional GAN to produce the final clean images.
\subsection{Haze Removal}
Image dehazing \cite{schechner2001instant,shwartz2006blind,narasimhan2000chromatic,narasimhan2003contrast,nayar1999vision} aims to remove haze with other additional information such as atmospheric cues \cite{cozman1997depth,narasimhan2002vision} and depth information \cite{kopf2008deep,tarel2009fast}.
Early image dehazing methods rely on prior information to estimate the transmission maps and atmospheric light intensity. He \textit{et al.} \cite{he2010single} calculate a dark channel prior (DCP) based on the statistics of the outdoor images to help estimate the transmission map. Apart from the DCP-based methods \cite{tarel2009fast,meng2013efficient,li2015nighttime,nishino2012bayesian}, the attenuation prior is also utilized to recover clean images. Fattal \cite{fattal2014dehazing} derives a local formation model to explain the color-line in the context of hazy images and recover clean images. Berman \textit{et al.} \cite{berman2016non} introduce a novel non-local method based on the assumption that images can be present with a few hundreds of colors. Though these methods have shown their effectiveness in image dehazing, their performance in the real-world scene is not satisfied.
With the advance in deep learning methods, recent years have witnessed significant success in image dehazing \cite{cai2016dehazenet,ren2016single,zhang2018densely,li2017aod}. Many approaches develop different kinds of CNN models to recover clean images via estimating the transmissions and atmospheric light. Specially, Ren \textit{et al.} introduce a multi-scale dehazing network to remove haze with the coarse-to-fine scheme. Zhang and Patel \cite{zhang2018densely} and Li \textit{et al.} \cite{li2017aod} build a pyramid network and AOD-Net to estimate the transmission and atmospheric light for image restoration. Cheng \textit{et al.} \cite{cheng2018semantic} propose a deep semantic dehazing network which verifies the effectiveness of semantic information for image dehazing.
\begin{figure*}[t]
\centering
\subfigure[The coarse snow removal network. D and U represent the down-sampling and up-sampling modules, respectively.]{
\label{fig:overall_arc:a}
\includegraphics[width= 0.99\linewidth]{Figures/coarse_net_2.pdf}}
\subfigure[Semantic and depth estimation]{
\label{fig:overall_arc:b}
\includegraphics[width=0.85\linewidth]{Figures/semantic_depth_1.pdf}}
\caption{{\bf The architecture of the proposed framework.} The coarse snow removal network consists of several Denseblocks \cite{huang2017densely}. The semantic segmentation and depth estimation networks are from \cite{tao2020hierarchical} and \cite{yin2019enforcing}, respectively.}
\label{fig:overall_arc}
\end{figure*}
\begin{figure*}[t]
\centering
{\includegraphics[width=0.99\linewidth]{Figures/finer_net_3.pdf}}
\caption{{\bf The fine snow removal network.} The fine snow removal network is a dense multi-scale network, which extracts multi-scale features from both the RGB space and latent feature spaces. ``X" represents the scale of down-sampling. ``Up" means the up-sampling module.}
\label{fig:overall_arc_2}
\end{figure*}
\section{Our method}
\label{sec:method}
In this section, we first give an overview of the proposed snow removal framework. Then the details of the proposed methodology are introduced.
\subsection{Overall}
\label{sec:overall}
The goal of our work is to remove snow and recover clean images from their corresponding snowy versions. In order to improve the capability of restoration, we introduce the proposed \textit{DDMSNet} (Sec. \ref{sec:network}), which is able to extract multi-scale features from multiple dimensional representations. Considering that the semantic and geometric information can provide useful priors, we discuss in Sec. \ref{sec:semantic-aware} and \ref{sec:geometry-aware} how to obtain semantic-aware and geometry-aware representation using a map-guided manner to improve the performance. Finally, the loss functions are represent in Sec. \ref{sec:loss} to illustrate how to train the proposed framework. Specially, the overall of the framework is shown in Fig. \ref{fig:overall_arc}. The steps of the procedure are listed as follows.
\begin{itemize}
\item
A coarse image removal network is built to obtain coarsely desnowed images via reducing snow in the input images.
\item
A semantic segmentation and a depth estimation network are utilized to extract semantic labels and depth knowledge from the coarsely desnowed images.
\item
The coarsely desnowed images, semantic labels and depth maps are fed into the \textit{DDMSNet}. Based on a map-guided manner, semantic-aware and geometry-aware representation are obtained to help remove snow and obtain finer results.
\end{itemize}
\subsection{Network Architecture}
\label{sec:network}
\textit{Coarse snow removal network.} We adopt a coarse-to-fine strategy to handle the task of snow removal. An input snowy image is addressed in a coarse stage and then processed by a fine stage with the semantic and geometry cues. To eliminate the possible negative effects of snow on the subsequent semantic understanding and depth estimation, we firstly in the coarse stage develop a coarse snow removal network to obtain pre-desnowed results, which can be represented as:
\begin{equation}
Y_{c} = G_{c}(O) \,,
\end{equation}
where $O$, $Y_{c}$ and $G_{c}$ are the observed snowy image, coarsely desnowed images and the coarse snow removal network, respectively.
To be specific, the coarse snow removal network consists of three modules, \textit{i.e.,} a pre-processing module, a core module and a post-processing module. The pre-processing module includes one convolutional layer and a residual dense block. The channel number of the output feature map is 16. The core module consists of 5 dense blocks \cite{huang2017densely}. Each dense block contains 4 convolutional layers. Within each dense block, there are densely connected shortcuts among the convolutional layers. The number of feature maps is 16 in the sequential dense blocks. The feature maps in the same row have the same scale. \textit{D} represents the down-sampling module. Therefore, the scale of feature maps in the top row is double to the second row, whose scale is also double to the third row. Following the core module there is a post-processing module, which contains another dense block. This dense block includes 4 convolutional layers with ReLU as the activation function. In addition, there are another two convolutional layers in the post-processing module. The output is an image which is the coarse result of the snow removal.
\textit{Deep dense multi-scale network (DDMSNet).} In order to obtain finer results, this paper introduces a new multi-scale network, which is shown in the Fig. \ref{fig:overall_arc_2}. The backbone in the Fig. \ref{fig:overall_arc_2} can refer to the Fig. \ref{fig:overall_arc}. We name it as deep dense multi-scale network because it not only extracts multi-scale features from multi-scale RGB images, but also extracts multi-scale features from single-scale RGB images. There are a set of sub-networks in the proposed DDMSNet, each sub-network corresponding to a scale. Specially, each sub-network consists of a semantic-guided attention module, a transfer module (which is the same as that in Fig. \ref{fig:overall_arc:a}) and a depth-guided attention module.
Roughly, in each sub-network of Fig. \ref{fig:overall_arc_2}, the semantic-aware module takes a RGB image and its corresponding semantic labels as input. With several convolutional layers and residual dense blocks (RDB) \cite{wang2018esrgan} in the semantic-aware module, feature maps are extracted, which are forwarded to the following transfer module to learn how to remove snow.
The transfer module is an attention-based multi-scale structure. It takes as input the feature maps generated by the semantic-attention module and extracts three-scale features. As the ``Backbone" in Fig. \ref{fig:overall_arc} shows, the rows represent different scales and the \textit{D} in each column are the connections between different scales. Each row consists of five RDB structures and the number of feature maps is fixed during the processing. The columns consist of down-sample and up-sample modules. The down-sample module consists of two convolutional layers to reduce the feature map to its half, while the up-sample module also includes two convolutional layers to increase the number of the feature maps by a factor of $2$, whose details can refer to the Fig. \ref{fig:overall_arc}. In order to concatenate the features from different scales, we adopt a channel attention manner,
\begin{equation}
F_a = \alpha_r F_r + \beta_c F_c \,,
\end{equation}
where $F_r$ and $F_c$ are the features from the row and column, respectively. $\alpha_r$ and $\beta_c$ are weights to balance different features.
Finally, the geometry-aware module takes the features from the transfer module and geometry labels as input to recover final snow-free images. To be specific, the geometry module consists of 3 layers of convolutional operation, and 3 residual dense blocks. The number of output feature channel is three, corresponding to an RGB image.
\subsection{Semantic-aware Representation}
\label{sec:semantic-aware}
In this section, we propose a method to explore the semantic information to help remove snow. We first use a pretrained semantic segmentation network \cite{tao2020hierarchical} to predict semantic labels based on the coarsely desnowed images, and then learn a semantic-aware representation under the guidance of semantic labels. Fig. \ref{fig:guid} shows the architecture of the mechanism. It processes the input features and semantic labels. The output is input into a Softmax function to transfer a set of attention weights ${A_1, A_2, ..., A_n}$ to ${W_1, W_2, ..., W_n}$. Each of them corresponds to a type of objects,
\begin{equation}
\label{semantic}
W_i = \frac{e^{A_i}}{\sum_{c=1}^{n} e^{A_c}} \,,
\end{equation}
where $c$ is the channel of features.
The feature map before the semantic-aware representation has $30$ channels. We set $n$ as 30 because the semantic label set has about $30$ types of different objects. Therefore, feature maps are divided into $30$ groups. The semantic-aware representation is obtained based on $W$ and feature channels in an element-wise manner. After that, we use group convolution in the $30$ groups to process the semantic-aware representation and merge all the features from different groups via a $1 \times 1$ CNN layer.
\subsection{Geometry-aware Representation}
\label{sec:geometry-aware}
Similar to the method of generating semantic-aware representation, we first use a depth estimation network \cite{yin2019enforcing} to obtain depth information based on the coarsely desnowed images. Then the depth information is combined with features extracted from input images to obtain the geometry-aware representation. However, it is different from the above part of semantic-aware representation in terms of two aspects. Firstly, we concatenate the geometry information in the last layer of \textit{DDMSNet}. The input features are the output of the transfer module, rather than the low-level features. Secondly, the group number in Eq. \ref{semantic} is set to $8$, rather than $30$. Finally, the geometry-aware representation is input into two CNN layers to recover clean images.
\subsection{Loss Function}
\label{sec:loss}
We train the coarse snow removal network and the \textit{DDMSNet} with a smooth $L_1$ and the perceptual loss function. For the coarse desnowing network, the smooth $L_1$ loss function can be represented as:
\begin{equation}
\mathcal{L}_1 = \frac{1}{N} \sum_{x=1}^{N} \sum_{i=1}^{3} Q (I'_i(x) - I_i(x)) \,,
\end{equation}
where
\begin{equation}
\left\{
\begin{array}{lr}
Q(e)=0.5e^2, \ \ \ \ \ \ \ if \ |e|<1, \\
Q(e)=|e|-0.5, \ \ otherwise. &
\end{array}
\right.
\end{equation}
$I'_i(x)$ and $I_i(x)$ are the intensity of the $i$-th color channel of pixel $x$ in the desnowed image and the ground-truth image, respectively. $N$ is the batch size.
\begin{figure*}[t]
\centering
{\includegraphics[width=0.8\linewidth]{Figures/attention_net_1.pdf}}
\caption{{\bf The Architecture of network to learn the semantic and geometry attention weights}.}
\label{fig:guid}
\end{figure*}
\begin{figure*}[!tb]
\centering
\subfigure[Input]{
\label{ablation:a}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/input.png}}
\subfigure[SnowCNN]{
\label{ablation:b}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/desnowNet.png}}
\subfigure[MSNet]{
\label{ablation:c}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/ordinary.png}}
\subfigure[DDMSNet]{
\label{ablation:d}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/nocf.png}}
\subfigure[DDMSNet (+)]{
\label{ablation:d}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/imgmult.png}}
\subfigure[DDMSNet (S)]{
\label{ablation:e}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/semantic.png}}
\subfigure[DDMSNet(G)]{
\label{ablation:e}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/depth.png}}
\subfigure[DDMSNet(S+G)]{
\label{ablation:f}
\includegraphics[width=0.24\linewidth ]{Figures/Example_ablation/final.png}}
\caption{ {\bf Exemplar results on the SnowKITTI2012 dataset.} From top to bottom: input, SnowCNN, MSNet, DDMSNet, DDMSNet(+), DDMSNet(S), DDMSNet(G) and DDMSNet(S+G). Best viewed in color.}
\label{fig:ablation}
\end{figure*}
The perceptual loss function is defined as:
\begin{equation}
{\mathcal{L}_{p}} = \frac{1}{{CWH}}\sum\limits_{c = 1}^C \sum\limits_{x = 1}^W {\sum\limits_{y = 1}^H {{{(G(I^{clean})_{x,y,c} - G(I^{de})_{x,y,c})}^2}} }\, ,
\end{equation}
where $C$, $W$ and $H$ are the channel, width and height of feature maps extracted from a pre-trained VGG16 \cite{simonyan2014very} model. $I_{x,y,c}^{clean}$ is the pixel value of clean images at location $\left(x, y, c\right)$, and $G(I^{de})_{x,y,c}$ corresponds to the value of desnowed images.
The loss functions of the \textit{DDSMNet} are similar to those of the coarse desnowing network. The main differences come from the multi-scale scheme. Therefore, the smooth $L_1$ and perceptual loss functions for \textit{DDSMNet} are,
\begin{equation}
\mathcal{L}_1^{f} = \frac{1}{M} \sum_{m=1}^{M} \mathcal{L}_1^{m} \,,
\end{equation}
\begin{equation}
\mathcal{L}_p^{f} = \frac{1}{M} \sum_{m=1}^{M} \mathcal{L}_p^{m} \,,
\end{equation}
where $M$ is the number of different scales we adopt. We set it as $3$ in our paper. Here the superscript $f$ indicates the loss functions are for the fine snow removal network.
The total loss function is defined by combining two different loss functions as follows,
\begin{equation}
\mathcal{L} = \mathcal{L}_1 + \beta * \mathcal{L}_p \,,
\end{equation}
where $\beta$ is set to $0.05$ in our paper. Note that, the formulation of loss function applies to both the coarse and the fine snow removal networks.
\begin{table*}
\centering
\caption{Performance comparison of different architectures on the SnowKITTI2012 dataset, in terms of PSNR and SSIM. Here ``Small", ``Medium" and ``Large" indicate the particle size of the snow.}
\begin{tabular}{l | c c c }
\toprule
Methods & Small & Medium & Large \\
\hline
\textit{SnowCNN} & 34.94/0.9724 & 29.83/0.9396 & 31.14/0.9323 \\
\textit{MSNet} & 36.17/0.9792 & 32.61/0.9586 & 32.70/0.9473 \\
\hline
\textit{DDMSNet} & 37.99/0.9840 & 34.24/0.9677 & 34.12/0.9572 \\
\textit{DDMSNet (+)} & 37.23/0.9841 & 34.66/0.9720 & 34.16/0.9642 \\
\textit{DDMSNet(G)} & 38.15/0.9871 & 34.54/0.9736 & 34.94/0.9691 \\
\textit{DDMSNet(S)} & 38.89/0.9864 & 35.41/0.9740 & 35.22/0.9678 \\
\hline
\textit{DDMSNet(G+S)} & \textbf{39.53/0.9877} & \textbf{35.50/0.9745} & \textbf{35.55/0.9700} \\
\bottomrule
\end{tabular}%
\label{table:ablation}
\end{table*}%
\begin{figure*}[t]
\centering
{\includegraphics[width=0.99\linewidth]{Figures/kitti2012.pdf}}
\caption{{\bf Exemplar results on the SnowKITTI2012 dataset}. From top to bottom: input, DesnowNet, RESCAN, SPANet, and Ours. Best viewed in color.}
\label{fig:kitti}
\end{figure*}
\section{Experiments}
In this section, we evaluate our proposed method on three datasets of snowy images. Considering that the existing datasets such as the Snow100K dataset \cite{liu2018desnownet} lack snowy images of street scenes, we firstly create two new datasets of snowy images of street scenes, which are introduced in Sec. \ref{dataset}. Then the implementation details of our framework are introduced in Sec. \ref{implementation}. We conduct an ablation study to show the effectiveness of different modules in Sec. \ref{ablation}. Finally, we compare the proposed method with the state-of-the-art methods on both synthesized snowy images and real-world images to demonstrate its superiority in Sec. \ref{comparison}.
\subsection{Datasets}
\label{dataset}
\textbf{SnowKITTI2012 dataset.} We first use Photoshop to create a synthetic SnowKITTI2012 dataset based on the public KITTI 2012 dataset \cite{geiger2013vision}. The training and testing sets of the proposed dataset include $1,500$ and $1,000$ pairs of images, respectively. In order to model different types of snow, each set contains three kinds of snow including light, medium and heavy snow. The size of images in both the training and the testing sets is $884 \times 256$.
\textbf{SnowCityScapes dataset.} The SnowCityScapes dataset is created based on the Cityscapes dataset \cite{cordts2016cityscapes}. The training and testing sets consist of $2,000$ and $2,000$ pairs of images, respectively. The size of images in both the training and the testing sets is $512 \times 256$. Similar to the SnowKITTI2012 dataset, we also provides three kinds of snowy images.
\textbf{Snow100K dataset.} This dataset is created by Liu \textit{et al.} \cite{liu2018desnownet}. In order to model snowy images, they first produce $5,800$ snowy masks and download $100K$ clean images. Then snowy images are synthesized based on the clean images and snowy masks. This dataset provides three kinds of snow, \textit{i.e.}, small, medium, and large particle sizes. They also provide $1,329$ realistic snowy images to evaluate models in terms of generalization in the real world.
\subsection{Implementation Details}
\label{implementation}
In this paper, we use a Gaussian distribution with zero mean and a standard deviation of $0.01$ to initialize the parameters of our proposed networks. The size of mini-batch during the training stage is set to $8$ for updating the models. In order to boost the variance of the data, we augment data by cropping $224 \times 224$ patches from images at random locations, and randomly flipping them along the horizontal direction. The learning rate is set as $10^{-4}$ and then we decrease it to $10^{-6}$ after the training loss achieves convergence.
\subsection{Ablation Study}
\label{ablation}
\begin{table*}[t]
\centering
\caption{Performance comparison with state-of-the-art methods on the SnowKITTI2012, SnowCityScapes and Snow100K datasets.}
\begin{tabular}{l | c | c c c c}
\toprule
Dataset & Snow & RESCAN & SPANet & DesnowNet & Ours \\
\hline
& Small & 35.68/0.9735 & 35.90/0.9781 & 32.11/0.9423 & \textbf{39.53/0.9877} \\
\textit{SnowKITTI2012} & Medium & 31.81/0.9489 & 32.08/0.9559 & 29.11/0.8903 & \textbf{35.50/0.9740} \\
& Large & 32.33/0.9360 & 32.49/0.9468 & 29.14/0.8663 & \textbf{35.55/0.9700} \\
\hline
& Small & 38.59/0.9815 & 39.66/0.9872 & 35.39/0.9603 & \textbf{42.24/0.9913} \\
\textit{SnowCityScapes} & Medium & 33.63/0.9627 & 35.73/0.9741 & 33.58/0.9382& \textbf{38.30/0.9826} \\
& Large & 34.24/0.9624 & 35.50/0.9669 & 33.39/0.9148 & \textbf{38.60/0.9822} \\
\hline
& Small &31.51/0.9032 & 29.92/0.8260 & 32.33/0.9500 & \textbf{34.34/0.9445} \\
\textit{Snow100K} & Medium & 29.95/0.8860 & 28.06/0.8680 & 30.87/0.9409 & \textbf{32.89/0.9330}\\
& Large & 26.08/0.8108 & 23.70/0.7930 & 27.17/0.8983 & \textbf{28.85/0.8772} \\
\bottomrule
\end{tabular}%
\label{table:sota}
\end{table*}%
The proposed \textit{DDMSNet} has the advantage of extracting dense multi-scale features from the input images. Meanwhile, the semantic-aware and geometry-aware representations provide semantic and geometric priors to help update the \textit{DDMSNet} to learn how to remove snow and restore clean images. In order to verify their effectiveness, we perform an ablation study by evaluating seven variant networks: \textit{SnowCNN}, \textit{MSNet}, \textit{DDMSNet}, \textit{DDMSNet (+)}, \textit{DDMSNet (S)}, \textit{DDMSNet(G)} and \textit{DDMSNet(S+G)}.
\begin{itemize}
\item
\textbf{SnowCNN} is a plain CNN network consisting of one convolutional layer, $7$ RRDB and another convolutional layer. The input of this model is a pair of (clean and snowy) images of original size without scaling.
\begin{figure*}[t]
\centering
{\includegraphics[width=0.99\linewidth]{Figures/cityscapes.pdf}}
\caption{{\bf Exemplar results on the SnowCityScapes dataset}. From top to bottom: input, DesnowNet, RESCAN, SPANet and Ours. Best viewed in color.}
\label{fig:cityscapes}
\end{figure*}
\item
\textbf{MSNet} is a multi-scale version of the above SnowCNN architecture. The main difference is that MSNet uses a multi-scale scheme to conduct the snow removal task like \cite{nah2017deep}. The input images are resized to different scales to help achieve finer results. The sub-networks in different scales share weights in our experiments.
\begin{figure*}[t]
\centering
{\includegraphics[width=0.99\linewidth]{Figures/snow100k_v2.pdf}}
\caption{{\bf Exemplar results on the Snow100K dataset}. From top to bottom: input, DesnowNet, RESCAN, SPANet and Ours. Best viewed in color.}
\label{fig:snow100k}
\end{figure*}
\item
\textbf{DDMSNet} is a dense multi-scale version of the above MSNet architecture. The main difference is that this model not only extracts features from images in different scales, but also extracts multi-scale features from images in a fixed scale.
\item
\textbf{DDMSNet(+)} is a coarse-to-fine version of the above DDMSNet architecture. We first use the SnowCNN to generate coarsely desnowed images and then feed them into the DDMSNet model to obtain finer results.
\item
\textbf{DDMSNet(S)} and \textbf{DDMSNet(G)} are two variants of the above DDMSNet(+). They majorly differ from the DDMSNet(+) due to that we use semantic-aware and geometry-aware representations to help update the DDMSNet, respectively.
\begin{figure*}[t]
\centering
{\includegraphics[width=0.99\linewidth]{Figures/real_v2.pdf}}
\caption{{\bf Exemplar results on the real-world snowy frames}. From left to right: input, DesnowNet, RESCAN, SPANet and Ours. Best viewed in color.}
\label{fig:real}
\end{figure*}
\item
\textbf{DDMSNet(D+S)} is our final model. The snowy images are fed into a SnowCNN to obtain the coarse results, which are fed into the DDMSNet to learn the semantic-aware and geometry-aware representations to help restore the final clean images.
\end{itemize}
Fig. \ref{fig:ablation} and Table \ref{table:ablation} show the ablation study results of different variants on the SnowKITTI2012 dataset. In general, both the plain SnowCNN and MSNet achieve reasonable performance, but the performance is inferior when compared with the variants of our DDMSNet. The DDMSNet(+) performs better than the original DDMSNet, suggesting the effectiveness of the ``coarse-to-fine" strategy. With additional semantic-aware and the geometry-aware representation, the DDMSNet(S) and DDMSNet(G) respectively obtain better results compared with the variant DDMSNet(+). This verifies the usefulness of the introduced semantic-aware and geometry-aware attentions in the fine snow removal network. Finally, the DDMSNet(S+G) achieves the best performance without doubt, indicating the effect of the combination of semantic-aware and geometry-aware representations.
It is also notable that these variants perform best in the case of small particle size, i.e., light snow. And in general they perform worse when the snow becomes heavier, though with a few inconsistent cases. This is reasonable and consistent with human perception.
\subsection{Comparison with Existing Methods}
\label{comparison}
To verify our model, we compare it with the existing state-of-the-art methods on the three datasets described above. To the best of our knowledge, there seems difficult to find existing methods specifically for snow removal, except the \textit{DesnowNet} \cite{liu2018desnownet}. To make the comparison more convincing, we additionally employ two learning-based rain removal approaches, \textit{RESCAN} \cite{li2018recurrent} and \textit{SPANet} \cite{wang2019spatial}, for the comparison. To adopt to the snow scenery, we retrain their networks with the snow dataset in the comparison.
Table \ref{table:sota} presents the quantitative results of different methods on the three datasets of SnowKITTI2012, SnowCityScapes and Snow100K. It shows that, the RESCAN and SPANet perform better than DesnowNet on the datasets of SnowKITTI2012 and SnowCityScapes, while worse on the Snow100K dataset. On all the three datasets, the performance achieved by our proposed method is significantly better than that of the counterparts.
Fig. \ref{fig:kitti}, Fig. \ref{fig:cityscapes} and Fig. \ref{fig:snow100k} represent the visual comparison results on the three datasets of SnowKITTI2012, SnowCityScapes and Snow100K. Compared with RESCAN, SPANet and DesnowNet, the results of our method exhibit fewer artifacts. And we achieve the best visually appealing results for snow removal.
\subsection{Performance in Real-World Scenarios}
\label{real_world}
In order to further verify the effectiveness of the proposed model in the real-world scenery, we compare the performance of our method with current state-of-the-art methods on real-world snowy images from the dataset of Snow100K. The qualitative results are shown in Fig. \ref{fig:real}, which validate that our method outperforms the current methods on real-world snowy images.
\section{Conclusion}
We propose a new multi-scale network, named as Deep Dense Multi-scale Network (\textit{DDMSNet}), and demonstrate its superior performance for snow removal. We exploit the semantic and geometric information as global priors to better remove snow and restore the clean images. Furthermore, based on the public KITTI and Cityscapes datasets, we synthesize two large-scale snowy datasets for snow removal. Experimental results demonstrate that the proposed method performs better than previous methods and achieves state-of-the-art performance.
\section*{Acknowledgment}
This work is funded in part by the ARC Centre of Excellence for Robotics Vision (CE140100016), ARC-Discovery (DP 190102261) and ARC-LIEF (190100080) grants, as well as a research grant from Baidu on autonomous driving. The authors gratefully acknowledge the GPUs donated by NVIDIA Corporation. We thank all anonymous reviewers and editors for their constructive comments.
\input{egbib.bbl}
\bibliographystyle{IEEEtran}
|
2106.02004
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\section{Introduction}
A theory to explain why the nucleus of an atom does not fall apart, in spite of the powerful repelling
electric force between protons, was proposed by C.N. Yang and R.L. Mills in 1954 \cite{YM54}. After quantization and systematic elimination of the divergences in the approximation schemes needed for the computation of experimental predictions, the theory is now widely regarded as
one of the most successful theories of fundamental physics. But the internal consistency of
the theory remains in question: It is not clear what the approximations are approximating.
After 70 years of intense efforts by many mathematicians and physicists to find the
mathematical structures into which the computations fit, a solution does not yet seem to be in sight, in spite of the many
different approaches that have been explored.
The aim of our work is to understand the configuration space for a Yang-Mills field and,
by completion, to find an infinite dimensional support space for the presumed ground state measure,
which would help to give meaning to the approximations. Informally, the configuration space
for a Yang-Mills field is a quotient space $\mathcal{A}/\mathcal{G}$ where $\mathcal{A}$ is some (yet to be determined)
space of 1-forms on $\mathbb{R}^3$ with values in the Lie algebra of some compact
Lie group $K$. The Lie group $K$ is determined experimentally, and for example $K = SU(3)$ is now regarded as the correct group for strong interactions in the nucleus of a particle. $\mathcal{G}$ is some (yet to be determined) set of functions $g:\mathbb{R}^3 \to K$, which is a group under pointwise multiplication and which acts naturally on $\mathcal{A}$. The case $K = U(1)$ has provided further understanding of what the configuration space should be, and it is now clear that the very large completion needed to support the ground state measure must contain distributions over $\mathbb{R}^3$ of large negative Sobolev index.
Various classes of distributions over $\mathbb{R}^3$ can be characterized by the behavior of the solution to the standard heat equation on $\R^3$ whose initial value is a distribution. See for example \cite{Lio} and \cite{Tai1} for some classical developments of this identification. For our purposes the classical heat equation must be replaced by the Yang-Mills heat equation in order for the gauge group to commute with the flow of the solution.
The Yang-Mills heat equation was first used by D. Zwanziger \cite{Z} over $\mathbb{R}^4$
as a zeroth order approximation
of stochastic quantization. Independently, S.K. Donaldson \cite{Do1} introduced
the Yang-Mills heat equation over a complex surface (4 real dimensions) as a tool to study the existence of irreducible Yang-Mills connections on the projective plane. He demonstrated that a vector bundle on a complex surface is stable if and only if it admits a Hermitian Yang-Mills connection.
In our work we have been considering the Yang-Mills heat equation in a product bundle over a compact 3-dimensional Riemannian manifold $M$ with possibly nonempty smooth boundary. We consider a $K$-product-bundle over $M$, where $K$ is a compact, possibly nonabelian, connected Lie group with Lie algebra $\mathfrak{k}$. A connection over a product bundle allows us to take derivatives of its various differential structures, and in general there exist multiple options for a connection depending on the properties of the space that we would like to preserve. In the case of the tangent bundle for example, the Levi-Civita connection is the unique metric-preserving connection with zero torsion on tangent vectors. Over a $K$-product-bundle, we would also like to differentiate the structures that are generated by the Lie group and the Lie algebra, and differentiation `should' be invariant under a change of variables on the manifold, but also under the action of the Lie group on the bundle fibers. Such derivatives are given by the space of connections, which is an affine space over $\Lambda^1(\mathfrak{k})$, the set of 1-forms with values in the Lie algebra $\mathfrak{k}$. The sections of the Lie group $K$ form the group of symmetries of the bundle, known as the gauge transformations.
The energy of a connection, also referred to as its magnetic energy in dimension 3, is given by the Yang-Mills functional
\begin{equation}
\mathcal{YM}(A)= \int_M |B|^2\, dx
\end{equation}
where $\;B:= dA + A \wedge A \; $ is the curvature of $A$ and $dx$ denotes the Riemannian volume element of the manifold. The variational equation for extrema of the Yang-Mills functional is the (degenerate elliptic) Yang-Mills equation $d^*_A B=0$, where $d_{A}^*$ is the gauge covariant coderivative. On compact manifolds of dimension up to 4 existence and regularity of solutions to the Yang-Mills equation has been extensively studied by C. H. Taubes and K. Uhlenbeck whereas T. Isobe and A. Marini studied the case of nonempty boundary (see \cite{Ma7,Ma5,Ma3,Uh2,Uh1,Tau1,Tau2}). Taking the negative gradient flow corresponding to the Yang-Mills functional gives rise to the (degenerate parabolic) Yang-Mills heat equation.
The Yang-Mills heat equation is given by
\begin{equation} \label{ymh}
\p A(t)/ \p t = -d_{A(t)}^* B(t), \ \ \text{for} \ \ t >0\ \ \text{with}\ \ A(0) = A_0,
\end{equation}
for some adequate initial condition $A_0$, and where $B(t)$ denotes the curvature of $A(t)$. The major difficulty when studying its solutions is the fact that the second order operator on the right side is not fully elliptic. In fact, its solutions are invariant under the infinite group of gauge transformations and as a result the heat operator does not smooth out all initial data. This can be illustrated by the following simple example. Consider the connection $A(t)= g^{-1} dg$ where $g$ is a time-independent gauge. $A(t)$ has curvature $B(t)=0$, therefore $A(t)= g^{-1} dg$ is a solution to \eqref{ymh} with initial condition $A(0)= g^{-1} dg$. In other words, $A(t)$ can be as irregular as $A(0)$, even if the curvature $B(t)$ is smooth. In addition, \eqref{ymh} is nonlinear in the nonabelian case, as a cubic term in $A$ appears in the right side. The above obstructions have limited results on the regularity and uniqueness of strong solutions to manifolds of dimension less than or equal to 3, due to the dimension restrictions of Sobolev embedding theorems.
Over compact manifolds with empty boundary in dimensions 2 and 3, J. Rade \cite{Rade} proved existence and uniqueness of strong solutions for initial data in $H_1$. Rade also demonstrated that the solution converges in $H_1$ to a Yang-Mills connection as $t\to \infty$, and that a Yang-Mills connection for any energy $\lambda$ may be realized as a limit. His method of proof constituted in adding a parabolic equation that the curvature $B(t)$ satisfies to \eqref{ymh} and then solving the system. This technique is known as DeTurck's trick who first used it in the context of the parabolic Ricci flow \cite{DT}, and the method was also used by J. Ginibre and G. Velo in the context of the hyperbolic Yang-Mills equation \cite{GV1,GV2}.
On the other hand, it is well known that singularities may develop at finite time for dimensions 4 or higher, unless one assumes some strong symmetric properties for the solutions \cites{Do1,HT1,Stru2,SST}. Over Euclidean space $\mathbb{R}^n$ for $n\geq 5$ J. Grotowski showed that solutions can blow up at finite time even for smooth initial data \cite{Grot}. For the nature of the singularity formation see for example \cites{Ga,Na,We}. In the context of weak solutions M. Struwe, A. Schlatter and A. Tahvildar \cites{Stru,SST} prove the existence of weak solutions (with finitely many isolated singularities) in dimension 4, whereas over compact 4-manifolds A. Waldron recently showed that smooth solutions can be extended for all time \cite{Wal}.
In the case where the underlying space is a noncompact Riemannian manifold, and for initial data in $H_1$, L. Sadun \cite{Sa} proved the existence of solutions in dimension 3, although he did not provide any uniqueness nor regularity results (see also \cite{Ha} for the Yang-Mills Higgs flow). M.-C. Hong and G. Tian proved the existence of a smooth solution $A(t)\in C(\,[\tau,T]; H_1)$, but did not study its uniqueness properties. They then used this solution to find certain symmetric solutions to the Yang-Mills equation over $\mathbb{R}^4$ and then construct non-self-dual Yang-Mills connections over $S^4$ \cites{Ho,HT2,HT1}.
We began our study of the Yang-Mills heat equation motivated by its potential application to the regularization of quantized Yang-Mills fields. It appears that a gauge invariant regularization method for Wilson loop variables will be necessary for the construction of quantized
Yang-Mills fields, as the standard methods for regularizing a quantum field are inapplicable to gauge fields, even though they have been successful in studying scalar field theories. At the same time, lattice regularization of Wilson loop functions of the gauge fields has been the only useful gauge invariant regularization procedure so far, but has not produced a continuum limit.
In this context, the Yang-Mills heat equation offers a way to regularize a large class of irregular connection forms $A_0$, by providing a gauge invariant and essentially smooth connection $A(t)$ corresponding to $A_0$, along which the Wilson loop functions will be well defined. In our work with L. Gross, we were interested in the existence and uniqueness of strong solutions to \eqref{ymh} over manifolds with boundary. Our focus on manifolds with boundary was due to the local nature of the Wilson loop variables, and the fact that local quantum field theory requires the use of locally defined observables such as $A_0$ and functions of $A_0$. The existence problem requires assuming boundary conditions for the solution and the initial condition. Our Neumann and Dirichlet boundary conditions were the natural ones from the analytical point of view, but for the intended applications in quantum field theory the gauge invariant Marini boundary conditions will most likely prove to be the important ones.
The presence of boundary did not allow for the use of DeTurck's trick as in \cite{Rade} due to complications in the boundary conditions for the curvature. Instead our method was to use a symmetry breaking technique on the right side of the equation, in order to make it parabolic (see \eqref{ymp}) and was similar to the technique employed in \cite{Do1,Sa,Z}. Our long-time existence followed a more classical path, compared to the semiprobabilistic methods used by A. Pulemotov in \cite{Pu}. A key element in our approach was the fact that we avoided the use of negative Sobolev spaces, which do not behave well under the heat kernel when the manifold has boundary. At the same time, our strict adherence to gauge invariant estimates was responsible for much of the novelty in our approach.
The parabolic equation \eqref{ymp} used in our approach was also recently used by S.-J.Oh and S.-J. Oh and D. Tataru on $\mathbb{R}^3$ ($\mathbb{R}^4$ resp.) together with the hyperbolic one to give a novel way to study the Cauchy problem for the hyperbolic equation in 3+1 (4+1 resp.) space-time \cites{Oh,OT}. They considered $H_1$ initial data and showed that over $\mathbb{R}^4$ the solution is either global, or it will blow up to a soliton. Under an additional $L^3$ assumption for the curvature over time, they proved that the solution will converge to a zero-curvature connection. They also used solutions to \eqref{ymh} to study the hyperbolic equation in 4+1 space-time.
We studied the Yang-Mills heat equation in two main cases. In Section \ref{S2} we will provide a summary of the existence and uniqueness, as well as regularity results that the author obtained together with L. Gross for initial data in $H_1$, referred to as the finite energy case. In dimension 3 however, the critical Sobolev exponent in the sense of scaling for the Yang-Mills heat equation is one half. This is due to the fact that the Sobolev $H_a(\mathbb{R}^3)$ norm of a 1-form is invariant under the scaling $\vec{x}\to c \vec{x}$ for $\vec{x}\in \mathbb{R}^3$ if and only if $a=1/2$. As a result, one anticipates that the most general case in which the Cauchy problem would be solvable is for initial data $A_0 \in H_{1/2}(\mathbb{R}^3)$. Gross studied this case in the recent article \cite{Gr2}. We will give an overview of the results of Gross and the author for initial in data in $H_a$, with $1/2\leq a<1$, under the assumption of finite action, in Section \ref{FA}. We will also provide an account of some recent results of L. Gross which are aimed at producing a Hilbert space structure to an appropriate configuration space for the set of solutions to the Yang-Mills heat equation in Section \ref{S4}.
{\bf Acknowledgment} {The author would like to thank Leonard Gross for introducing her to the Yang-Mills heat equation, for his support and teachings throughout the years, and also for a detailed historic account of the problem.}
\section{Technical Description}
We let $M$ be a smooth Riemannian manifold of dimension 3 with possibly nonempty smooth boundary $\p M$. We study the Yang-Mills heat equation on a product bundle $M\times \V \rightarrow M$, where $\V$ is a finite dimensional real (resp. complex) vector space with an inner product. $K$ will denote a compact, possibly nonabelian, connected and orthogonal (resp. unitary) group of the space $End\ \V$, of operators on $\V$ to $\V$. We denote the Lie algebra of $K$ by $\mathfrak{k}$, which may be identified with a real subspace of $End\ \V$. We assume that we have an $Ad\ K$ invariant inner product $\langle \cdot, \cdot \rangle$ on $\mathfrak{k}$, with norm denoted by $|\cdot|_{\frak k}$. We will not distinguish between $|\xi|_{\frak k}$ and $|\xi|_{End \V}$, since they are equivalent norms and will usually denote this norm $|\cdot|$ for simplicity. For $\mathfrak{k}$-valued $p$-forms $\omega, \phi$ we define $(\omega,\phi)=\int_M \langle \omega, \phi \rangle \, dx$ and the $L^2$ norm of a form by $\|\omega\|_2= (\omega,\omega)^{1/2}$. The $L^p$ norm is defined similarly for $1\leq p\leq \infty$ and denoted as $\|\omega\|_p$.
Over a product bundle, a connection can be identified with a $\mathfrak{k}$-valued 1-form, and in local coordinates a connection may be written as $A=\sum_i A_i \, dx^i\ $ with $\ A_i \in \mathfrak{k}$. The curvature of $A$ is defined as
\[
B:= dA + (1/2)[A\wedge A]
\]
where $[A\wedge A]=\sum_{i,j}[A_i , A_j] \, dx^i\wedge dx^j$ and $[A_i , A_j]$ is the commutator in the Lie algebra. Each connection induces an exterior derivative $d_A: \Lambda^p(\mathfrak{k})\to \Lambda^{p+1}(\mathfrak{k})$,
such that for any $p$-form $\omega$
\[
d_A \omega = d\omega + [A \wedge \omega],
\]
and its adjoint $d_A^*: \Lambda^{p+1}(\mathfrak{k})\to \Lambda^{p}(\mathfrak{k})$
\[
d_A^* \omega = d^*\omega + [A \lrc \omega].
\]
A gauge transformation $g$ acts on a connection $A$ via
\[
A^{g} = g^{-1} dg + g^{-1}A\,g.
\]
$A, \;A^{g}$ are distinct matrix connections representing the same connection, and $g$ corresponds to a change in trivialization. Two connections are called gauge equivalent, if they lie in the same orbit of this action. In this context, both the space of connections and the group of gauge transformations are infinite dimensional sets.
We denote the gauge-covariant $W_1$ norm by
\[
\|\omega\|_{W_1}= \|\nabla \omega\|_2 + \|\omega\|_2
\]
which is defined independently of boundary conditions, and where $\nabla$ is the Riemannian covariant derivative on forms. Since our work is carried out over manifolds with boundary, we will distinguish the Sobolev space $W_1$, which does not assume any boundary conditions on the form $\omega$, from the Sobolev space $H_1$, and in general $H_a$, which does include boundary conditions for the form.
In our work we have been primarily interested in the existence and uniqueness of strong solutions to the Yang-Mills heat equation, which we define below.
\begin{definition} \label{SS1} Let $0 < T \leq \infty$. A {\bf \it strong solution} to the Yang-Mills heat equation over $[0, T)$
is a continuous function
\begin{equation*}
A(\cdot): [0,T) \rightarrow W_1(M; \Lambda^1 \otimes \frak k )
\end{equation*}
such that
\begin{align}
a)& \ B(t):= dA(t)+ (1/2) [A(t)\wedge A(t)] \in W_1 \ \text{for each}\ \ t\in (0,T),\ \notag \\
b)& \ \text{the strong $L^2(M)$ derivative $A'(t) \equiv dA(t)/dt $}\
\text{exists on}\ (0,T), \text{and} \notag \\
&\ \ \ \ \ \ A'(\cdot): (0, T) \rightarrow L^2(M) \ \text{is continuous}, \notag \\
c)& \ A'(t) = - d_{A(t)}^* B(t)\ \ \text{for each}\ t \in(0, T). \notag
\end{align}
A solution $A(\cdot)$ that satisfies all of the above conditions
except for $a)$ will be called an {\it almost strong solution}. In this
case the spatial exterior derivative $dA(t)$, which appears in the
definition of the curvature, must be interpreted in the weak sense.
A strong solution will be called locally bounded if
\begin{align*}
&d) \ \| B(t)\|_\infty\ \text{is bounded on each
bounded interval $ [a,b) \subset (0, T)$ and} \\
&e) \ t^{3/4} \| B(t)\|_\infty\
\text{is bounded on some interval $(0, b)$ with $0<b <T$.}
\end{align*}
\end{definition}
The above definition of a strong solution was the one used for problems where the initial condition has finite energy, in other words it belongs to $W_1$, with boundary conditions which we will discuss below. As will see in Section \ref{FA}, Gross was able to generalize the existence and uniqueness properties for less regular initial conditions, namely for $A_0$ in a Sobolev space $H_a(M)$ for $1/2\leq a < 1$. In this latter case since the initial condition does not belong to $W_1$, the solution itself cannot be continuous in $W_1$ at $t=0$, although the flow regularizes the initial condition for $t>0$. To clarify this distinction we will call these strong solutions of the second type, even though they are referred to as simply strong solutions in all of the literature.
\begin{definition} \label{SS2} Let $0 < T \leq \infty$. A {\bf \it strong solution of the second type} to the Yang-Mills heat equation over $[0, T)$
is a continuous function
\begin{equation*}
A(\cdot): [0,T) \rightarrow L^2(M; \Lambda^1 \otimes \frak k )
\end{equation*}
such that
\begin{align}
a)& \ A(t) \in W_1 \ \text{for all} \ t \in (0,T) \ \text{and} \ A(\cdot): (0,T) \rightarrow W_1 \ \text{is continuous}, \notag \\
b)& \ B(t) \in W_1 \ \text{for each}\ \ t\in (0,T),\ \notag \\
c)& \ \text{the strong $L^2(M)$ derivative $A'(t) \equiv dA(t)/dt $}\
\text{exists on}\ (0,T), \text{and} \notag \\
&\ \ \ \ \ \ A'(\cdot): (0, T) \rightarrow L^2(M) \ \text{is continuous}, \notag \\
d)& \ A'(t) = - d_{A(t)}^* B(t)\ \ \text{for each}\ t \in(0, T). \notag
\end{align}
A solution $A(\cdot)$ that satisfies all of the above conditions
except for $a)$ will be called an {\it almost strong solution of the second type}. In this
case the spatial exterior derivative $dA(t)$, which appears in the
definition of the curvature, must again be interpreted in the weak sense.
\end{definition}
If the boundary of the manifold is nonempty, then we must impose boundary conditions on the solutions.
\begin{definition}\label{defbdyconds}
For a strong solution to the Yang-Mills heat equation we will consider three types of boundary conditions:
\noindent
{\it Neumann boundary conditions:}
\begin{align}
&i)\ \ \ A(t)_{norm} =0\ \ \text{and} \label{N1}\\
&ii)\ \ B(t)_{norm}=0\ \ \label{N2}
\end{align}
\noindent
{\it Dirichlet boundary conditions:}
\begin{align}
&i)\ \ \ A(t)_{tan} =0\ \ \ \text{and} \label{D1}\\
&ii)\ \ B(t)_{tan} =0. \ \ \ \label{D2}
\end{align}
{\it Marini boundary conditions:}
\begin{align}
&i)\ \ B(t)_{norm} =0. \ \ \ \label{M}
\end{align}
\end{definition}
The above Neumann and Dirichlet boundary conditions are related to the respective conditions for the Hodge Laplacian on differential forms but they are weaker, reflecting the weak parabolicity of the problem. The definition of tangential and normal components of a form on the boundary is the one generalized from the classical case of forms on manifolds with smooth boundary. For their precise definition as well as a further discussion of the Marini condition and our weaker boundary conditions see Section 2 in \cite{ChG1}. Here we would simply like to remark that the Marini boundary condition is a nonlinear condition which is gauge invariant.
As we have mentioned, the method of proof of our main results includes a symmetry breaking technique that replaces the original equation with a parabolic one where the Hodge Laplacian on 1-forms appears. We recall that
\begin{equation}
-\Delta = d^* d + d d^*, \label{Lap1}
\end{equation}
where $d$ denotes the closed version of the exterior derivative operator
with $C_c^\infty(\R^3, \Lambda^1\otimes \kf)$ as a core. When the boundary of the manifold is nonempty there are many ways to define an adequate Sobolev space for the domain of this operator. The Sobolev spaces for $\kf$ valued 1-forms that are associated to the boundary conditions that we considered can be obtained from the corresponding Laplacian. The classical Neumann and Dirichlet boundary conditions are given by
\begin{align*}
&\w_{norm} =0 \ \ \text{and} \ \ (d\w)_{norm}=0, \ \ \ \text{\it Neumann conditions} \\
&\w_{tan} =0\ \ \ \ \text{and} \ \ (d^*\w)_{\p M} =0, \ \ \ \text{\it Dirichlet conditions}.
\end{align*}
Alternatively, the Neumann (resp. Dirichlet) Laplacian can be defined by \eqref{Lap1},
wherein $d$ is taken to be the maximal (resp. minimal) exterior derivative operator over $M$.
See \cite{ChG1} for further discussion of these domains. In both cases the Laplacian is
a nonnegative, self-adjoint operator on the appropriate domain.
For $0 \le a \le 1$ we define the Sobolev spaces
\begin{equation}
H_a = \text{Domain of} \ (-\Delta)^{a/2} \ \text{on} \ L^2(M; \Lambda^1\otimes \mathfrak{k}) \notag
\end{equation}
with norm
\begin{equation}
\|\w\|_{H_a} = \|(1-\Delta)^{a/2} \w\|_{ L^2(M; \Lambda^1\otimes\mathfrak{k})}. \label{Lap5}
\end{equation}
The following embedding property holds
\[
\|\w\|_{H_a} \leq c_{a,b} \|\w\|_{H_b} \ \text{whenever} \ 0\leq a \leq b,
\]
for some constant $c_{a,b}$ independent of $M$.
For solutions corresponding to $A_0\in H_a(M)$, one must define an appropriate group of gauge transformations that would work well when applying the symmetry breaking method for existence of solutions. For $a\in (1/2, 1]$ Gross defines the gauge group $\mathcal{G}_{1+a}$ which is in fact a Hilbert manifold \cite{Gr2}. For the case $a=1/2$ however, the corresponding group does not have a tangent space at the identity. This makes the analysis of the critical case $A_0\in H_{1/2}(M)$ all the more interesting. Below we give the full definition of $\mathcal{G}_{1+a}$ following \cite{Gr2}. In this case the underlying manifold $M$ is either $\mathbb{R}^3$ or the closure of a bounded open set in $\mathbb{R}^3$ with smooth boundary.
\begin{definition}[The gauge group $\mathcal{G}_{1+a}$.] \label{defgg} A measurable function
$g:M\rightarrow K \subset End\ \V$ is a bounded function into the linear space
$End\ \V$, therefore its weak derivatives are well defined. Following \cite{Gr2} we will write $g \in W_1(M;K)$
if $\|g - I_\V\|_2 <\infty$ and the derivatives $\p_j g \in L^2(M; End\ \V)$.
The 1-form $g^{-1} dg := \sum_{j=1}^3 g^{-1}(\p_j g)dx^j$ is then an a.e. defined $\kf$ valued 1-form.
The Sobolev norm $\|g^{-1}dg \|_{H_a}$ is defined as in \eqref{Lap5}.
For an element $g \in W_1(M;K)$
the restriction $g|_{\p M}$ is well defined almost everywhere on $\p M$ by a Sobolev trace theorem.
The three versions of $\G_{1+a}$ that we will need are given in the following definitions.
\begin{align*}
\G_{1+a}(\R^3)
= \Big\{g \in W_1(\R^3; K): g^{-1}dg \in H_{a}(\R^3;\Lambda^1\otimes \kf) \Big\}, \qquad \qquad \ \ \
\end{align*}
If $M \ne \R^3$ define
\begin{align*}
\G_{1+a}^N(M) &= \Big\{g \in W_1(M; K): g^{-1}dg \in H_{a}(M;\Lambda^1\otimes \kf) \Big\}, \\
\G_{1+a}^D(M) & = \Big\{g \in W_1(M; K): g^{-1}dg \in H_{a}(M;\Lambda^1\otimes \kf),\ g = I_\V\ \text{on}\ \p M \Big\}.
\end{align*}
It should be understood that the two spaces denoted $H_{a}(M;\Lambda^1\otimes \kf)$ are those
determined by Neumann, respectively Dirichlet, boundary conditions. It was proved in \cite[Theorem 5.3]{Gr2}
that all three versions of $\G_{1+a}$ are complete topological groups in the metric
$\rho_a(g,h) = \| g^{-1} dg - h^{-1} dh\|_{H_{a}} +\| g-h\|_{L^2(M; End\, \V)}$.
\end{definition}
The apriori energy estimates needed in our proofs must be in terms of gauge covariant derivatives, which reflect the many symmetries of solutions to the Yang-Mills heat equation, because neither the connection form nor its curvature is smoothed by the flow. As a result, it was necessary to express Sobolev inequalities in terms of the gauge covariant exterior derivative $d$ and its adjoint $d^*$. These will be elaborated on in Subsection \ref{Apr}.
\section{The Yang-Mills heat equation under finite energy} \label{S2}
In \cite{ChG1} we considered the Yang-Mills heat equation on 3-manifolds with smooth boundary when the initial condition $A_0$ is a connection lying in the first order Sobolev space $W_1(M)$, with an appropriate `half' boundary condition. We showed that there exists a unique solution to \eqref{ymh} satisfying Dirichlet or Neumann type boundary conditions. We also considered Marini boundary conditions where we proved existence and uniqueness for the flow whenever the initial data $A_0$ is $C^2$. Our main existence and uniqueness results in \cite{ChG1} are summarized below.
\begin{theorem} \label{thm1}
Suppose that $A_0 \in W_1$ and $(A_0)_{norm} =0.$ Then there exists a locally bounded strong solution $A(\cdot)$ over $[0, \infty)$ to \eqref{ymh} such that $A(0) = A_0$, which satisfies
the Neumann boundary conditions \eqref{N1}, \eqref{N2} as follows
\begin{equation} \label{N4}
A(t)_{norm} =0\ \ \text{for all}\ t \ge 0 \ \ \text{and} \ \ B(t)_{norm}=0\ \ \text{for all}\ t >0.
\end{equation}
Moreover, if $A_1$ and $A_2$ are two locally bounded strong solutions which agree at $t=0$ and satisfy \eqref{N2} for $t>0$, then $A_1=A_2$ for all $t\in [0,\infty)$.
In the Dirichlet case, whenever $A_0 \in W_1$ and $(A_0)_{tan} =0,$ then there exists a locally bounded strong solution $A(\cdot)$ over $[0, \infty)$ to \eqref{ymh}, such that $A(0) = A_0$ which satisfies
the Dirichlet boundary conditions \eqref{D1}, \eqref{D2}
\begin{equation} \label{D4}
A(t)_{tan} =0\ \ \ \text{for all}\ t \ge 0 \ \ \text{and} \ \ B(t)_{tan} =0 \ \ \ \text{for all}\ t >0.
\end{equation}
If $A_1$ and $A_2$ are two locally bounded strong solutions which agree at $t=0$ and satisfy \eqref{D1} for all $t\geq0$, then $A_1=A_2$ for all $t\in [0,\infty)$.
For the case of Marini boundary conditions, whenever $A_0 \in C^2$ then there exists a unique locally bounded strong solution $A(\cdot)$ over $[0, \infty)$ to \eqref{ymh}, such that $A(0) = A_0$ which satisfies the Marini boundary condition
\begin{equation} \label{M2}
B(t)_{norm} =0 \ \ \ \text{for all}\ t > 0.
\end{equation}
\end{theorem}
Observe that the boundary condition for the uniqueness property is not symmetric; the Neumann problem only requires the boundary condition for the curvature, whereas the Dirichlet problem requires both, since $ A(t)_{tan} =0$ implies $B(t)_{tan} =0$ when $B(t) \in W_1$. In the case of Marini boundary condition, the boundary condition itself suffices for the uniqueness result.
As Gross later observed in \cite{Gr2}*{Theorem 2.24} the above theorem will also hold in the case $M=\mathbb{R}^3$, as all the steps in our proof will go through without any significant modifications since we never use that the volume of $M$ is finite.
Many of the apriori energy estimates that we required for the proof of the above theorems are usually formulated in terms of gauge covariant derivatives. In our case however, neither the connection form nor its curvature are smoothed by the flow. It was therefore necessary to express Sobolev inequalities in terms of the gauge covariant exterior derivative $d_A$ and its adjoint. We achieved this by proving a gauge invariant version of the Gaffney-Friedrichs inequality. The curvature of the connection form $A$ that appeared in these inequalities contributed to some of the technical difficulties that needed to be resolved. However, our necessity to adhere to gauge invariant estimates was one of the innovative elements of our approach. It is also noteworthy that our method allowed us to obtain estimates for $\|A(t)\|_{W_1}$ although it is not a gauge invariant quantity.
\subsection{Existence by symmetry breaking}
The proof of the existence of solutions to the Yang-Mills heat equation relied on a symmetry breaking technique which consisted in adding a Zwanziger gauge fixing term $-d_A d^*A$ to the right side of \eqref{ymh}. To distinguish the solution to \eqref{ymh} from the solution to the modified equation we will denote the latter by $C(t)$. The modified equation then becomes
\begin{equation} \label{ymp}
\p C(t)/ \p t = -d_{C(t)}^* B_C(t) - d_{C(t)}\, d^{*}C(t)
\end{equation}
where $B_C$ is the curvature of $C$. The Zwanziger term turns the second order operator on the right side into an elliptic one equal to $\Delta C + V(C)$ where $V$ is a nonlinear term of the type $V(C)= C^3 + C \cdot \partial C$. Although the solution to this modified parabolic equation is no longer gauge invariant, it be transformed to a solution of the original equation using a time-dependent gauge transformation. This method was first proposed by D. Zwanziger \cite{Z} in the context of stochastic quantum field theory, and a similar approach was used by S. K. Donaldson and separately L. Sadun \cites{Do1,Sa} in the context of the classical Yang-Mils heat equation. We refer to this method as the Zwanziger-Donaldson-Sadun (ZDS) procedure. In our work we had to be slightly more careful with our gauge fixing term due to the boundary conditions.
The parabolic equation does have a smooth unique solution for initial conditions in $W_1$, with boundary conditions given by
\begin{align}
&(N) \ \ C(t)_{norm}=0 \ \ \text{for} \ t\ge 0, \ \ \left(B_C(t)\right)_{norm}=0 \ \ \text{for} \ t>0 \label{N3}\\
&(D) \ \ C(t)_{tan}=0 \ \ \text{for} \ t\ge 0, \ \ d^*C (t)_{tan}= d^*C (t)\big|_{\p M}=0 \ \ \text{for} \ t>0. \label{D3}
\end{align}
These boundary conditions correspond the classical absolute (Neumann) and relative (Dirichlet) boundary conditions respectively, for real valued forms. We proved the following existence and uniqueness theorem for the parabolic equation \cite{ChG1}.
\begin{theorem}
Let $A_0 \in W_1$ satisfying either $(A_0)_{norm} =0$ or respectively $ \,(A_0)_{tan} =0$. Then, there exists $T>0$ and a continuous function $C:[0,\infty) \to W_1$ such that $C(0)=A_0$ and
\begin{align}
1)& \ B_C(t) \in W_1 \ \text{and} \ d^*C(t) \in W_1 \ \text{for each} \ t \in (0,T), \notag \\
2)& \ \text{the strong $L^2(M)$ derivative $dC(t)/dt $}\
\text{exists on for each}\ t\in (0,T), \notag \\
3) &\ C(t) \ \text{satisfies} \ \eqref{ymp} \ \text{together with the boundary conditions} \ \eqref{N3}, \ \text{respectively}\ \eqref{D3}, \notag \\
&\ \ \ \text{for each} \ t \in (0,T), \notag \\
4)& \ t^{3/4}\|B_C(t)\|_{\infty} \ \text{is bounded on}\ \in(0, T). \notag
\end{align}
The solution is unique under the above conditions. Moreover, $C(\cdot)$ belongs to $C^{\infty}(\,(0,\infty)\times M;\Lambda^1 \otimes \frak k)$.
\end{theorem}
The proof of the above theorem was based on a classical conversion of the differential equation into an integral one, together with a contruction mapping argument into an appropriate Banach space that involved the $W_1$ norm of $C$ and the $L^\infty$ norms of $C, \; dC$ and $d^*C$. A regularity argument then allowed us to prove the higher order estimates. Uniqueness and the full boundary conditions for $C(t)$ follow in a similar way as in \cite{Tay3}, and the specific boundary conditions for the Yang-Mills problem follow from the symmetry properties of the operators involved.
For the smooth case, one can obtain the solution to the Yang-Mills heat equation from the parabolic equation \eqref{ymp} using the following gauging procedure. For a smooth solution $C(t)$ to \eqref{ymp} with $C(0)=A_0$ that satisfies the boundary conditions \eqref{N3} (resp. \eqref{D3}), we define the flow of gauge transformations $g(t):[0,\infty)\to C^{\infty}(M;K)$ as the solution to the initial value problem
\[
(\p g(t)/ \p t) \; g(t)^{-1}= d^{*} C(t), \ \ \ g(0) = I_K
\]
where $I_K$ is the identity element of the gauge group. Then,
\[
{A}(t)= C(t)^{g(t)}=g(t)^{-1} dg(t) + g(t)^{-1} C(t) g(t)
\]
is a solution to \eqref{ymh} with $A(0)=A_0$ that satisfies the Neumann boundary conditions \eqref{N1} and \eqref{N2} (resp. \eqref{D1} and \eqref{D2}) as in Theorem \ref{thm1}.
However, for initial data in $W_1$ singularity issues arise for $d^*C$ as $t\downarrow 0$ which make it difficult to obtain sufficient regularity for $g(t)$ so that $A(t) \in W_1(M)$ for $t\geq 0$. As we have mentioned, this is due to the fact that $\|A(t)\|_{W_1}$ is not a gauge invariant quantity. In \cite{ChG1} we addressed this difficulty by trying to avoid the singular point at $t=0$ for the gauge flow. In particular, we considered a solution $g_\epsilon (t)$ to the same equation as above, but only for $t\geq \epsilon$ and with initial condition $g_\epsilon (\epsilon)=I_K$. Then for $t\geq \epsilon,$ $\,A_\epsilon(t) =C(t)^{g_\epsilon (t)}\,$ is a sequence of smooth solutions which strongly converges to a $W_1$ solution to the Yang-Mills heat equation solution as $\epsilon \downarrow 0$. The proof of this convergence was the most novel part of our work, and it relied on the gauge invariant Gaffney-Friedrichs inequality described in the following subsection.
\subsection{A priori estimates and a new Gaffney-Friedrichs inequality} \label{Apr}
For any $\frak k$ valued $p$-form $\w$ on $M$ the covariant $W_1$ norm with respect to (a sufficiently smooth) connection $A$ is defined as
\[
\|\w\|_{W_1^A(M)}^2 = \|\n^A \w \|_2^2 + \| \w \|_2^2.
\]
where $\n^A$ is the covariant derivative induced by $A$. This gauge covariant norm is the one that can be used to control $L^p$ norms via Sobolev inequalities. For example, using the Sobolev and Kato inequalities one can show that
\begin{equation} \label{eqSob}
\|\w \|_6^2 \le C(M) \|\w\|_{W_1^A(M)}^2 \
\ \text{for any}\ \w\ \text{and}\ A \in W_1(M)
\end{equation}
where $C(M)$ is a constant that depends only on the geometry of $M$, but not on $A$. However, it is the Hodge version of the energy that relates well with the Yang-Mills equation
\[
\|d_A \w \|_2^2 + \|d_A^* \w \|_2^2 + \lambda \| \w\|_2^2
\]
due to the various symmetries that the solution and its curvature exhibit.
The Gaffney-Friedrichs inequality that we proved in \cite{ChG1} is an important tool that allows us to relate the two and thus prove critical apriori estimates for solutions.
\begin{theorem}[Gaffney-Friedrichs inequality] \label{thmGF}
Let $M$ be a compact smooth 3-manifold with smooth boundary. Suppose that $A\in W_1(M)$ and its curvature $B$ satisfies $\|B\|_2<\infty$. Then for any $p$-form $\w$ in $W_1(M)$ which satisfies either
\[
\w_{norm} =0 \ \ \ \text{or}\ \ \ \w_{tan} =0
\]
the following inequality holds
\[
(1/2) \|\w\|_{W_1^A(M)}^2\le \|d_A \w \|_2^2 + \|d_A^* \w \|_2^2 + \lambda(B) \| \w\|_2^2
\]
where
\[
\lambda(B) := \lambda_M + \gamma_2 \|B\|_2^{4},
\]
and $\lambda_M$, $\gamma_2 $ depend only on the geometry of $M$ and its boundary, but not on the size of $M$, neither on $A$.
\end{theorem}
Note that in \cite{ChG1} we also have a version of Theorem \ref{thmGF} for $B \in L^p$, for any $p \in [2,\infty]$. Also, the constant $\lambda_M$ is zero in case the boundary of $M$ is convex.
Smooth solutions to \eqref{ymh} have a lot of symmetries. For example since $B$ satisfies the Bianchi identity $d_A B=0$, it follows that $B'= d_A A'$ and as a result the following differential inequality holds,
\begin{equation} \label{eqFE}
d/dt(\|B\|_2^2) = 2(B', B) = 2(d_A A', B)= -2 \|A'\|_2^2 \leq 0
\end{equation}
which in turn implies that the energy of a solution, $\|B(t)\|_2^2$, is nonincreasing with respect to $t$, and therefore uniformly bounded by $\|B_0\|_2^2$. At the same time, by combining equation \eqref{eqSob} and Theorem \ref{thmGF} (and under the appropriate boundary conditions) we can obtain an upper bound for the $L^6$, as well as the $W_1$ norm of the gauge invariant quantities $B$ and $A'$ since
\begin{equation*}
\begin{split}
\|B\|_6^2 &\leq c (\|d_A B\|_2^2 + \|d_A^* B\|_2^2 + \|B\|_2^2) \\
& = c (\|A'\|_2^2 + \|B\|_2^2)
\end{split}
\end{equation*} and
\begin{equation*} \begin{split}
\|A'\|_6^2 &\leq c (\|d_A A'\|_2^2 + \|d_A^* A'\|_2^2 + \|A'\|_2^2) \\
& = c (\|B'\|_2^2 + \|A'\|_2^2)
\end{split}\end{equation*}
with respect to the $L^2$ norms $A, A'$ and $B$. Moreover we have the differential inequality
\begin{equation*}
d/dt(\|A'\|_2^2) \leq - \|B'\|_2^2 + c (\|B_0\|_2) \left[ \|A'\|_2^2 + \|A'\|_6^2 \right]
\end{equation*}
The above pointwise and integral identities can be used in combination with the Gaffney Friedrichs inequality to obtain $L^6$ and $L^2$ bounds for $A'$ and $B$ with respect to the energy of the initial condition, $\|B_0\|_2$. After some careful work and via interpolation arguments and the use of H\"older's inequality we can also use such estimates to prove that a smooth solution with $A_0\in W_1$ will remain in $W_1$ for all $t>0$ \cite{ChG1}*{Sections 5, 6}.
\subsection{Existence and Uniqueness} The various apriori estimates obtained in the process outlined above, can be used to prove integrability estimates for the sequence of gauge transformations $g_\epsilon$. Ultimately they allowed us show that $A_\epsilon$ and $B_\epsilon$ are uniformly Cauchy in the $H_1$ norm as $\epsilon \downarrow 0$, and also to prove all the regularity properties of a strong solution. Uniqueness over $[0,T)$ is a consequence of Gronwall's Lemma, because given our boundary conditions we can prove the inequality
\[
d/dt\|A_1(t) -A_2(t)\|_2^2\leq c(\|B_1(t)\|_{\infty}+\|B_2(t)\|_{\infty}) \, \|A_1(t) -A_2(t)\|_2^2
\]
where $B_1$ and $B_2$ are the curvatures of $A_1$ and $A_2$ respectively.
For long time existence, the following regularization result was necessary.
\begin{lemma} \label{lemReg} Suppose that $A$ is a locally bounded strong solution
over $[0, T)$ for some $T \leq \infty$. Let $ 0 < t <T$ and define
$\beta = \sup_{0\le s \le t} \|A(s)\|_{W_1}$. Then there exists $\tau(\beta) >0$, such that, for any $[a, b] \subset (0, t]$
with $b-a <\tau$, there exists a sequence $A_n$ of smooth solutions over $[a,b]$ such that
\begin{align*}
\sup_{ a \le s \le b} \Big\{ \|A_n(s)& - A(s)\|_{W_1}
+ \|A_n'(s) - A'(s)\|_{L^2} \notag \\
&+ \|B_n(s) - B(s)\|_{W_1}
+\| B_n(s) - B(s)\|_\infty \Big\} \rightarrow 0
\end{align*}
as $n\rightarrow \infty$.
\end{lemma}
Apart from long-time existence, this Lemma also allowed us to prove further regularity properties for solutions. For example, given that the norms of $B$ and $A'$ are gauge invariant, we can prove the same $L^\infty$ bounds for the rough solution from the nearby smooth solutions, and therefore show that our strong solution to \eqref{ymh} is a locally bounded one.
\subsection{Neumann heat kernel domination} In \cite{ChG2} we continued our regularization program, started in \cite{ChG1}. First, we improved our previous pointwise estimates for the gauge invariant quantities $|B(t)|$ and $|A'(t)|$ as $t\downarrow 0$ whenever $A(t)$ is a strong solution to \eqref{ymh}. Our method required that the boundary of $M$ be smooth and convex in the sense that its second fundamental form is non-negative, so that the heat operator of the Neumann Laplacian on functions is bounded. Our estimates depended on the initial energy of the flow, $\|B_0\|_{L^2}$.
\begin{theorem} \label{thmNeu} Let $M$ be a compact 3-manifold with smooth convex boundary. Suppose that $A(t)$ is a locally bounded strong solution on $[0,T)$
that satisfies either the Neumann \eqref{N4}, Dirichlet \eqref{D4} or Marini \eqref{M2} boundary conditions. Then there exists $\tau>0$, depending only on $\|B_0\|_2$,
such that
\begin{align*}
\|B(t)\|_\infty &\le 2c_N \|B_0\|_2 \; t^{-3/4},\ \ \text{for} \ \ 0 <t \le 2\tau \ \ \text{and} \\
\|B(t)\|_\infty &\le 2c_N \|B_0\|_2 \; \tau^{-3/4}, \ \ \text{for} \ \ \tau \le t <\infty.
\end{align*}
Moreover, if $\|A'(0)\|_2 <\infty$ then there exists $\gamma >0$ such that
\[
\|A'(t)\|_\infty \le \gamma \|A'(0)\|_2 \; t^{-3/4}, \ \ \text{for} \ \ 0 < t \le 2\tau .
\]
\end{theorem}
For the proof of this theorem we used the fact that whenever $A(t)$ is a smooth strong solution to the Yang-Mills heat equation \eqref{ymh} that satisfies Neumann, Dirichlet or Marini boundary conditions, as in the Theorem above, then both its curvature $B(t)$ and $A'(t)$ satisfy a parabolic equation with reasonable potential terms (and with respective classical boundary conditions). Moreover, whenever the boundary is convex, both functions $|B(t)|^2$ and $|A'(t)|^2$ satisfy classical sub-Neumann boundary conditions for any one of the boundary conditions on $A(t)$.
The key element of the proof is the use of a Neumann domination technique. Namely, over a manifold with convex boundary the heat kernel on forms is dominated by the Neumann heat kernel on functions. For this domination technique, the boundary conditions for the connection and the convexity of the boundary are key. Finally, we can use the ultracontractivity property of the Neumann Laplacian on functions over these manifolds to control $\|B(t)\|_\infty$, $\|A'(t)\|_\infty$ even near $t=0$. Note that the constant $c_N$ that appears in the theorem is
\[
c_N = \sup_{0<t \le 1} t^{3/4} \|e^{t\Delta_N}\|_{2\rightarrow \infty}
\]
where $\|e^{t\Delta_N}\|_{2\rightarrow \infty}$ is the ultracontractivity norm of the Neumann Laplacian on functions. The constant $\gamma$ depends only on $c_N$ and $\|B_0\|_2.$ After proving that the estimates of Theorem \ref{thmNeu} hold for a smooth solution, we then used the regularization Lemma \ref{lemReg} to prove them for a locally bounded strong solution to \eqref{ymh} with bounded initial energy.
In addition to the small time estimates, we were also interested in the long-time convergence properties of our solutions. Motivated from the general realization that a gauge invariant regularization method for Wilson loop variables might be necessary for the construction of quantized Yang-Mills fields \cites{Ba,Sei}, we considered the Wilson loop functions in our setting. To define the Wilson loop function, we first recall that each connection has a parallel transport operator along curves, which in turn determines the connection. Instead of proving the convergence of the connection itself, we were able to show the convergence of these parallel transport operators. In particular, in \cite{ChG2} we proved that the Wilson loop functions, gauge invariantly regularized, will converge as time goes to infinity for any initial gauge potential $A_0\in H_1$.
\begin{definition} For a smooth End $\V$ valued
connection form $A$ on the interior of $M$ and a piecewise
$C^1$ path $\gamma: [0,1] \to M$, the parallel transport operator along $\gamma$
is defined by the solution to the ordinary differential equation
\begin{equation*}
g(t)^{-1} dg(t)/dt = A\< d \gamma(t)/dt \>,\ \ \ g(0) = I_{\V}.
\end{equation*}
We set $//_\gamma^A = g(1)$ and note that this map satisfies the classical properties of a parallel transport (see Notation 3.4 in \cite{ChG2}).
The Wilson loop function is defined as $W_\gamma (A) \equiv trace\ //_\gamma^A$ where
the trace is computed in some finite dimensional unitary representation of $K$.
\end{definition}
We will denote the set of closed loops at a fixed point $x_0$ by
\[
\Gamma_0 = \{ \gamma \;\big| \ \gamma \text{ is a piecewise} \ C^1 \ \ \text{function}
\gamma: [0,1] \to M, \ \text{satisfying} \ \gamma(0) = \gamma(1) = x_0 \}
\]
\begin{theorem} \label{LTB}
Suppose that M is a compact convex subset of $\mathbb{R}^3$ with smooth boundary, and let
$A(\cdot)$ be a locally bounded strong solution of the Yang-Mills heat equation \eqref{ymh} over $[0,\infty)$, satisfying
Dirichlet or Neumann boundary conditions. Choose a point $x_o$ in the interior of $M$. Suppose that $\{t_i\}$ is a sequence of times going to $\infty$. Then, there exists a subsequence $t_j$ and gauge functions $k_j\in W_1(M; K)$ such that
\begin{enumerate}[$1)$]
\item $k_j^{-1} dk_j \in W_1(M; \mathfrak{k})$ for all $j$
\item $\alpha_j= A(t_j)^{k_j}$ is in $C^\infty(M;\Lambda^1\otimes\mathfrak{k})$, and
\item for each $\gamma \in \Gamma_0$, the operators $//_\gamma^{\alpha_j}$
converge, as operators from $\V$ to $\V$, to a map $P(\gamma)$ as $j\rightarrow \infty$. The map $P$ can be extended to a parallel transport system on the set of loops.
\end{enumerate}
\end{theorem}
For more detailed properties of $P$ and parallel transport systems we refer the interested reader to \cite{ChG2}*{Section III}.
In general $W_\gamma(A)$ is highly singular as a function of the connection $A$ when $A$ varies over the very large space of typical gauge fields required in quantized theory. As we illustrate in \cite{ChG2} the function $A \mapsto trace\ //_\gamma^A$ is fully gauge invariant, in the sense that $trace\ //_\gamma^{A^k} = trace //_\gamma^A$ whenever $\gamma$ is a closed curve in $M^{int}$, $A$ is a smooth connection form and $k$ is a smooth gauge transformation. At the same time, the Yang-Mills heat equation is itself fully gauge invariant: if we transform the initial data $A_0$ by a gauge transformation and then propagate, we arrive at the same gauge field as if we first propagate $A_0$ and then gauge transform. Moreover, the flow regularizes the initial data well enough so that the Wilson loop function $W_\gamma(A(t))$ is well-defined for any fixed time $t>0$, even when $W_\gamma(A_0)$ fails to be so, since $W_\gamma(A(t))$ is gauge invariant. As a result, the Yang-Mills heat equation offers a gauge invariant regularization procedure for the Wilson loop function for some class of irregular connection forms.
Theorem \ref{LTB} implies that for any initial gauge potential $A_0\in H_1$ there exists a sequence of times going to infinity for which the functions $\;trace //_\gamma^{A(t_j)}$, in other words $W_\gamma(A(t_j))$, converge for all piecewise $C^1$ loops $\gamma$ starting at $x_0$. The proof relies on the fact that the norm of the Wilson loop functions is controlled by the $L^\infty$ norm of the curvature of the connection, which is bounded in this case for $t\geq 1$. The underlying space was a compact convex subset of $\mathbb{R}^3$ with smooth boundary, since the case of interest for quantum field theory is that in which $M$ is the closure of a bounded open set $O$ in $\mathbb{R}^3$ with smooth boundary.
\section{The Yang-Mills heat equation under finite action} \label{FA}
More recently, Gross has considered the existence and uniqueness of solutions to the Yang-Mills heat equation for less regular initial data $A_0\in H_{a}(M)$ for $1/2\leq a < 1$ \cite{Gr2}. He considered the case when $M$ is either all of $\mathbb{R}^3$ or the closure of a bounded domain in $\mathbb{R}^3$ with smooth convex boundary.
As we have already mentioned, the critical case in dimension 3 is when $a=1/2$, which is the most general case in which one anticipates existence and uniqueness of solutions. In fact, the techniques used in the proof of the existence and uniqueness theorems for $a>1/2$ break down as $a\downarrow 1/2$ and further illustrate the way in which $a=1/2$ is critical.
Recall that as defined in \eqref{Lap5} the $H_{a}$ norms are not in themselves gauge invariant for $1/2\leq a <1$. One of the central ideas of Gross, was that the functional that does capture in a gauge invariant way the $H_a$ norm of $A_0$ is the following.
\begin{definition}[Finite $a$-action] An almost strong solution of the second type $A(\cdot)$ to the Yang-
Mills heat equation has finite $a$-action if
\begin{equation}\label{fa1a}
\rho_{a}(t)= \int_0^t s^{-a} \|B(s) \|_2^2\, ds < \infty \ \ \ \text{for some} \ \ t>0,
\end{equation}
where $B(s)$ is the curvature of $A(s)$. This definition is of interest for $1/2\leq a<1$ .
\end{definition}
The finite action property for a solution, does control many of the estimates needed in this more general setting. Gross' use of this term was motivated by the observation that when this functional is finite, then the initial condition $A_0$ has an extension to a time interval in Minkowski space with a finite magnetic contribution to the Lagrangian. In the setting of the Yang-Mills heat equation, it allowed Gross to prove the existence and uniqueness of strong solutions of the second type $A(t)$ which belong to $W_1(M)$ for $t>0$, and whose curvature also belongs to $W_1(M)$ for $t>0$ (see Definition \ref{SS2}). The lack of stronger regularity at $t=0$ is the main difference between strong solutions in the finite action case and strong solutions in the sense of Definition \ref{SS1}. The latter, as anticipated, are no longer possible given the less regular initial value $A_0$. We state below the two main theorems in \cite{Gr2}.
\begin{theorem}[Gross \cite{Gr2}] \label{thmFAa} Let $1/2<a<1$ and assume that $M$ is either all of $\mathbb{R}^3$ or is the closure of a bounded domain in $\mathbb{R}^3$ with smooth convex boundary. Suppose that $ A_0 \in H_{a}(M)$. Then
\begin{enumerate}[$1)$]
\item there exists an almost strong solution of the second type $A(t)$ to \eqref{ymh} over $[0,\infty)$ with $A(0)=A_0$
which satisfies the following properties.
\item There exists a gauge function $g_0\in \G_{1+a}$ such that $A(t)^{g_0}$ is a strong solution of the second type.
\item $A(\cdot)$ and $A(\cdot)^{g_0}$ are continuous functions on $[0,\infty)$ into $H_a$.
\item Both $A(\cdot)$ and $A(\cdot)^{g_0}$ have finite $a$-action.
\item If $M\neq \mathbb{R}^3$ then the curvature of both $A(\cdot)$ and $A(\cdot)^{g_0}$ satisfies the boundary condition \eqref{N2}, or respectively \eqref{D2}, for all $t>0$, depending on the boundary condition of $A_0$. Moreover $A(\cdot)^{g_0}$ satisfies the Neumann boundary condition \eqref{N1}, or respectively the Dirichlet boundary condition \eqref{D1}, for all $t>0$.
\item Strong solutions of the second type are unique among solutions with finite $a$-action under the boundary condition \eqref{N2}, respectively \eqref{D1}, for all $t>0$ when $M\neq \mathbb{R}^3$.
\end{enumerate}
\end{theorem}
In other words, any connection form $A_0\in H_{a}$ is, after gauge transformation, the initial value of a strong solution of the second type. Uniqueness holds when properly formulated, and note that since $A(t)$ need not be in $W_1$ in this case, the boundary conditions \eqref{N1} and \eqref{D1} are only meaningful for the gauge transformed solution $A(t)^{g_0}$. A similar result holds true for the case $a=1/2$ however, to gain continuity at $t=0$ into $H_{1/2}$ and to prove that the solution has finite $(1/2)$-action one must assume that the $H_{1/2}$ norm of $A_0$ is sufficiently small.
\begin{theorem}[Gross \cite{Gr2}] \label{thmFA12} Assume $M$ is either all of $\mathbb{R}^3$ or is the closure of a bounded domain in $\mathbb{R}^3$ with smooth convex boundary. Suppose that $ A_0 \in H_{1/2}(M)$. Then,
\begin{enumerate}[$1)$]
\item there exists an almost strong solution of the second type $A(t)$ to \eqref{ymh} over $[0,\infty)$ with $A(0)=A_0$. The curvature of $A(t)$ satisfies the boundary condition \eqref{N2}, or respectively \eqref{D2}, for all $t>0$ when $M\neq \mathbb{R}^3$, depending on the boundary condition of $A_0$.
\item There exists a gauge function $g_0$ such that $A(t)^{g_0}$ is a strong solution of the second type, and $A(t)^{g_0}$ satisfies the boundary conditions \eqref{N1} and \eqref{N2}, or respectively \eqref{D1} and \eqref{D2}, for all $t>0$ when $M\neq \mathbb{R}^3$.
\item If $\; \|A_0\|_{H_{1/2}}$ is sufficiently small, then $A(\cdot)$ and $A(\cdot)^{g_0}$ have finite $(1/2)$-action. In this case one may choose $g_0 \in \mathcal{G}_{3/2}$.
\item If $\; \|A_0\|_{H_{1/2}}$ is sufficiently small, then $A(\cdot)$ is a continuous function from $[0,\infty)$ into $H_{1/2}$. If in addition $g_0$ is chosen to lie in $\mathcal{G}_{3/2}$ then $A(\cdot)^{g_0}:[0,\infty) \to H_{1/2}$ is also continuous.
\item Strong solutions are unique among solutions with finite $(1/2)$-action under the boundary condition \eqref{N2}, respectively \eqref{D1}, for all $t>0$ when $M\neq \mathbb{R}^3$.
\end{enumerate}
\end{theorem}
We note that the gauge transformation $g_0$ in Theorem \ref{thmFAa} that converts an almost strong
solution $A$ to a strong, $A^{g_0}$ one is not unique. In particular, if $g_1$ is an element of ${\mathcal G}_2$, then $A^{g_0 g_1}$ is also a strong solution. It would be interesting to know whether this is the full-extent of non-uniqueness.
As Gross illustrates in \cite{Gr2}*{Theorem 7.1} the solution $A(t)^{g_0}$ produced by the two theorems above is actually in $C^\infty\left( (0,T]\times M; \Lambda^1\otimes \mathfrak{k} \right)$ for some $T<\infty$. The importance of this property will be illustrated in Theorem \ref{thmEst} and Corollary \ref{corlEst}, where it allowed us to obtain gauge covariant derivatives of all orders and prove improved $L^p$ and $W_1$ estimates for them for small time \cite{ChG3}.
The proofs of Theorems \ref{thmFAa} and \ref{thmFA12} use similar techniques to the finite energy case, with a lot of technical subtlety and an augmented parabolic equation. The gauge transformation $g_0$ that converts an almost strong solution to a strong one is unavoidable since, as we have mentioned in the introduction, the solution $A(t)$ can be as irregular as $A(0)$ even if its curvature is smooth. As a result, we cannot expect that any $A_0\in H_{a}$ will be the initial value to a strong solution for $a<1$. This is also reflected in the different way that the ZDS procedure is used in the proof of Theorem \ref{thmFA12}. If the initial data is in $H_1$, then the gauge transformation flow $g(t)$ produced by the augmented parabolic equation is only used to produce the strong solution $A$ from $C$. But for $A_0\in H_a$ the ZDS procedure produces a gauge transformation $g_0$ such that $A_0^{g_0}$ is the initial value to a strong solution of the second type.
As in the finite energy case, the difficulty in the ZDS procedure arises from the singular behavior of $d^*C(t)$ as $t\downarrow 0$, since $d^*C(0)$ need only belong to $H_{-a}$ in this case. A significant part of \cite{Gr2} was dedicated to proving that $t\mapsto g(t)$ is a continuous function into $\G_{1+a}$ for $A_0\in H_a$. Gaffney-Friedrichs inequalities in combination with Neumann domination results were the ones that enabled the use of Sobolev inequalities that led to $L^p$ estimates for all $p\leq \infty$. The finite $a$-action condition was key in obtaining many of the $H_a$ estimates. Uniqueness for $a>1/2$ also relied on a Gronwall type argument, but for $a=1/2$ it was necessary to follow a more specialized proof.
Finally, we remark that the notion of solution to the Yang-Mills heat equation in the finite action case with $A_0\in H_{1/2}$ allowed for the first spatial derivatives of $A(t)$ to exist in some generalized sense. On the other hand, the weak curvature of $A(t)$ is actually in $H_1$ for all $t>0$, and in consequence certain second order derivatives of $A(t)$ can be defined in the classical sense. This is unusual for typical weak solutions in heat equations, but reflects the many symmetries that are satisfied by higher order derivatives of solutions in the Yang-Mills setting.
In \cite{ChG3} together with Gross we more carefully considered the small-time behavior of solutions to \eqref{ymh} for initial data $A_0\in H_{1/2}$. We were interested in the smoothness properties of the solution for $t >0$, and given the fact that in general the higher order covariant derivatives of $A$ itself need not belong to $W_1(M)$, we concentrated only on gauge covariant derivatives. Our main result was the following.
\begin{theorem} \label{thmEst} Assume that $A_0\in H_{1/2}(M; \Lambda^1 \otimes \frak k )$.
Suppose that $A(\cdot)$ is a strong solution of the second type to \eqref{ymh}
over $[0,\infty)$
with initial value $A_0$ and having finite action. If $\|A_0\|_{H_{1/2}}$ is sufficiently small
then there exists $T>0$ and
standard dominating functions $C_{nj}$ for $\ j=1,\ldots 4$
and $n=1,2,\dots$, such that, for $0<t < T$, the following estimates hold.
\begin{align*}
t^{2n-\frac 12}\|A^{(n)}(t)\|_2^2 \ +
&\int_0^t s^{2n- \frac 12} \|B^{(n)}(s) \|_2^2\, ds \le C_{n1} (t) \ \ \ \ \ \\
t^{(2n- \frac 12)}\|B^{(n-1)}(t)\|_6^2 \ +
& \int_0^t s^{2n- \frac 12} \|A^{(n)}(s) \|_6^2 \,ds \le C_{n2}(t) \\
t^{2n+\frac 12} \| B^{(n)} ( t)\|_2^2 \ +
& \int_0^t s^{2n+\frac 12} \| A^{(n+1)} (s) \|_2^2 \,ds \le C_{n3} (t) \\
t^{2n+\frac 12} \| A^{(n)} (t)\|_6^2 \ +
& \int_0^t s^{2n+\frac 12} \| B^{(n)}(s)\|_6^2 \,ds \le C_{n4}(t).
\end{align*}
Moreover the third estimate also holds for $n =0$.
\end{theorem}
In the above theorem $A^{(n)}$ and $B^{(n)}$ denote the $n$th order time-derivatives of $A$ and $B$ respectively. A standard dominating function is a function $C:[0, \infty) \rightarrow [0,\infty)$ of the form $C(t) =\hat C(t, \rho_{1/2}(t))$ , where $\rho_{1/2}(t)$ is the finite $1/2$-action functional at time $t$, such that $\hat C:[0,\infty)^2 \rightarrow [0, \infty)$ is continuous and non-decreasing in each variable, $\hat C(0,0) =0$ and $\hat C$ is independent of the solution $A(\cdot)$.
The estimates of Theorem \ref{thmEst} provided information about the order of the singularity as time $t\downarrow 0$ which are consistent with what is expected in parabolic equations for the respective order of the derivative and for initial data in the Sobolev space $H_{1/2}$. At the same time, estimates for $A^{(n)}$ and $B^{(n)}$ are essentially estimates for higher order covariant exterior derivatives and coderivatives of $A$ and $B$. For example, we know that $A'(t)=-d^*_{A(t)} B(t)$ and $B'(t)= d_{A(t)} A'(t)$ therefore our small-time estimates are in fact $L^2$ and $L^6$ estimates for first order spacial derivatives of $A$ and $B$. In \cite{ChG3} the identities we proved for $A^{(n)}$ and $B^{(n)}$ would also imply $L^p$ estimates for higher order spacial derivatives of $A$ and $B$.
A central idea behind the proof of the theorem was to use Gross' result in \cite{Gr2} which states that a strong solution of the second type $A$ to \eqref{ymh} with $A_0\in H_{1/2}$ and $\|A_0\|_{H_{1/2}}$ small enough is gauge equivalent to a smooth solution $\hat{A}=A^{g_0}$. The smooth solution $\hat{A}$ exists for small time, satisfies the same (respective) boundary conditions and also has finite $(1/2)$-action. At the same time, all $n$th order time derivatives $\hat{A}^{(n)}(t)$ for $n\geq 1$, and $\hat{B}^{(n)}(t)$ for $n\geq 0$ are well defined for the smooth solution, and in addition, their norms are gauge invariant quantities. As a result, all quantities on the left side of the inequalities of Theorem \ref{thmEst} are gauge invariant and since the estimates hold for $\hat{A}$, they will automatically hold for the original solution $A$.
The remaining proof consisted in showing that the gauge covariant exterior derivatives
and coderivatives of $A^{(n)}(t)$ and $B^{(n)}(t)$ (for the smooth solution) can be expressed in terms of lower order time derivatives. These differential identities then led to integral identities for the $L^p$ norms of these quantities which in turn were used to establish bounds on the initial behavior by induction on $n$. This was similar to the process described in Subsection \ref{Apr}, and took advantage of the many symmetries that characterize higher order derivatives of solutions to \eqref{ymh}. The proof made an extensive use of the Gaffney-Friedrichs inequality of Theorem \ref{thmGF} and the Sobolev embedding \eqref{eqSob}. Particular care had to be taken in the case that the boundary of the manifold was nonempty, so that the correct boundary conditions would hold for all quantities.
The proof of the theorem also led to short-time estimates for the $H_1$ norm of the higher order time-derivatives of $A$ and $B$.
\begin{corollary}\label{corlEst} Under the hypotheses of Theorem \ref{thmEst} there exists $T>0$
and standard dominating functions $C_{nj}$ for $j=5,6$ and $n=1,2,...$ such that, for $0 < t <T$, the following estimates hold.
\begin{align*}
\qquad t^{(2n- \frac 12)}\|B^{(n-1)}(t)\|_{H_1^A}^2 + & \int_0^t
s^{2n- \frac 12} \|A^{(n)}(s) \|_{H_1^A}^2 \,ds \le C_{n5}(t)
\\
\qquad\ \ \ \ \ t^{2n+\frac 12} \| A^{(n)} (t)\|_{H_1^A}^2 + & \int_0^t
s^{2n+\frac 12} \| B^{(n)}(s)\|_{H_1^A}^2 \,ds \le C_{n6}(t).
\end{align*}
\end{corollary}
In this setting we make the following interesting observation. Let $\mathcal{Y}$ denote the set of almost strong solutions to the Yang-Mills heat equation over $M$ with initial value $A_0\in H_{1/2}$ and having finite action. Theorem \ref{thmFA12} tells us that the group $\G_{3/2}$ acts on $\mathcal{Y}$ through its action on $A(0)$ for each $A \in \mathcal{Y}$. Since all functionals that appear on the both sides of the estimates in Theorem \ref{thmEst} and Corollary \ref{corlEst} are gauge invariant, then they all descend to functions of the initial values on the quotient space ${\mathcal{C}} \equiv \mathcal{Y}/\G_{3/2}$. In other words, our estimates are in fact estimates on $\mathcal{Y}/\G_{3/2}$.
\section{The configuration space for the Yang-Mills heat equation} \label{S4}
Relating these results to the main questions from Quantum Field theory, it is natural to ask what would be a well-defined configuration space for the Yang-Mills heat equation. A configuration space for classical Yang-Mills fields is generally defined as a quotient space $\mathcal{C} = \mathcal{Y}/\mathcal{G}$ where $\mathcal{Y}$ is an appropriately chosen space of connections and $\mathcal{G}$ is an appropriate group of gauge transformations. The structure of the configuration space for classical Yang-Mills Fields. remains a central but still elusive problem in Mathematical Physics. At the same time, in order to carry out quantization for the classical Yang-Mills field, in other words assign a metric or measure structure to the configuration space, it is important to choose $\mathcal{Y}$ and $\mathcal{G}$ such that the quotient space is a complete metric space in a natural metric and in particular a Hilbert manifold. For these structures it is appropriate to start this process over a compact subset of $\mathbb{R}^3$ and then extend to the whole space, which also motivated our study of compact 3-manifolds with boundary.
Gross anticipates that in this setting if $\mathcal{Y}_a$ is the space of strong solutions to the Yang-Mills heat equation over $\mathbb{R}^3$ with finite $a$-action and initial value $A_0\in H_{a}(\mathbb{R}^3)$, then $\mathcal{Y}_a/\G_{1+a}$ is complete metric space, and in fact it is a Hilbert manifold for $1/2< a<1$ \cite{Gr4}. A similar result should also hold for $a=1/2$. See also \cite{Gr1} for a notion of configuration space for Yang-Mills fields in the context of the Maxwell-Poisson equation.
To this end, he has recently worked on defining an appropriate `tangent space' to each solution to the Yang-Mills heat equation. The notion of a tangent space to a solution is similar to the one from differential geometry. Here one considers paths $[r_1,r_2] \ni r \mapsto A_r(\cdot)$ where $ A_r(\cdot)$ is a solution to \eqref{ymh} with $A_r(0)\in H_{a}$. If $v_r(s) = \partial_r A_r(s)$ in the $L^2$ sense for each $r$, then $v_r$ gives the analogue of a tangent vector to $A_r$. Moreover, these tangent vectors must be solutions to the variational equation
\begin{equation} \label{veq}
-v'(t)= d^*_{A(t)} d_{A(t)} v(t) + [v(t) \lrc B(t)].
\end{equation}
In a recent preprint Gross proved the existence of solutions to the variational equation for initial conditions $v_o \in H_{a}(M)$ when $A(t)$ is a strong solution to \eqref{ymh} with finite action \cite{Gr3}. He considered only the case where $M$ is either all of $\mathbb{R}^3$ or a bounded subset of it with smooth convex boundary. In this context, the definition of strong solution is slightly more general than Definition \ref{SS2}.
\begin{definition} \label{SS3} A {\bf \it strong solution of the third type} to the Yang-Mills heat equation over $(0, \infty)$
is a continuous function
\begin{equation*}
A(\cdot): (0,\infty) \rightarrow L^2(M; \Lambda^1 \otimes \frak k )
\end{equation*}
which satisfies the conditions $a)-d)$ of Definition \ref{SS2} and in the case the boundary of the manifold $M$ is nonempty $A$ is assumed to satisfy the boundary conditions $A(t)_{norm}=0$ in the Neumann case and $A(t)_{tan}=0$ in the Dirichlet case.
\end{definition}
In particular, the strong solutions $A(t)$ for which the variational equation is defined, need not have initial condition in $H_a$, nor any type of continuity at $t=0$.
\begin{definition} \label{SSV} A {\bf \it strong solution} to the variational equation \eqref{veq} over $[0, \infty)$
is a continuous function
\begin{equation*}
v : [0,\infty) \rightarrow L^2(M; \Lambda^1 \otimes \frak k )
\end{equation*}
such that
\begin{align}
a)& \ v(t) \in H_1^{\textup{A}} \ \text{for all}\ \ t\in (0,\infty),\ \text{and} \ \ v : (0,\infty) \rightarrow H_1^{\textup{A}} \ \text{is continuous} \notag \\
b)& \ d_{A(t)}v(t) \in H_1^{\textup{A}} \ \text{for each}\ \ t\in (0,\infty),\ \notag \\
c)& \ \text{the strong $L^2(M)$ derivative $v'(t) \equiv dv(t)/dt $}\
\text{exists on}\ (0,\infty), \text{and} \notag \\
d)& \ \text{the variational equation \eqref{veq} holds on}\ (0, \infty). \notag
\end{align}
A solution $v(\cdot)$ that satisfies all of the above conditions
except for $a)$ will be called an {\it almost strong solution}. In this
case the spatial exterior derivative $dv(t)$ must be interpreted in the weak sense.
\end{definition}
In the above definition, the Sobolev norm $H_1^{\textup{A}}$ is defined as
\[
\| \w \|_{H_1^{\textup{A}}(M)}^2
= \int_M |\p_j^{\textup{A}} \w(x)|_{ \Lambda^1\otimes\frak k}^2 + | \w(x)|_{ \Lambda^1\otimes\frak k}^2 d\, x\ \
+ \| \w \|_2^2,
\]
since we are over Euclidean space, and $\textup{\rm A}=A(T)$ for some $0<T<\infty$. The $H_a^\textup{\rm A}$ norm is defined similarly to \eqref{Lap5}, with $\Delta$ replaced by $\Delta_\textup{\rm A}$.
\begin{theorem}[Gross \cite{Gr3}] \label{thmVE} Assume $M$ is either all of $\mathbb{R}^3$ or is the closure of a bounded domain in $\mathbb{R}^3$ with smooth convex boundary. Assume that $1/2\leq a<1$ and $1/2\leq b<1$. Let $A(\cdot)$ be a strong solution to the Yang-Mills heat equation over $(0,\infty)$ with finite
$a$-action and such that for each $s\in [0,\infty)$ the function
\[
[0,\infty) \ni t \mapsto A(t)- A(s) \ \ \text{is continuous into} \ L^3(M; \Lambda^1\otimes\frak k).
\]
Let $v_0 \in H_b^\textup{\rm A}(M;\Lambda^1\otimes\frak k)$. Then
\begin{enumerate}[$1)$]
\item There exists an almost strong solution $v(\cdot)$ to the variational equation
\eqref{veq} over $[0,\infty)$ with initial value $v_0$.
\item For each real number $\tau > 0$ there exists a vertical almost strong solution
$d_{A(t)}\alpha_\tau$ for some $\alpha_\tau \in H_1^{\textup{A}}(M;\frak k)$ such that the function
\[
v_\tau (t) \equiv v(t) - d_{A(t)}\alpha_\tau, \ t \geq 0
\]
is a strong solution to the variational equation with initial value $v_0 - d_{A(0)}\alpha_\tau$.
Moreover
\[
\sup_{0\leq t\leq 1} \| v(t) - v_\tau (t)\|_2 \to 0 \ \ \text{as} \ \ \tau \downarrow 0.
\]
\item If $\|A(t)\|_{L^3(M)} < \infty $ for some $t > 0$ then
\[
v : [0,\infty) \to H_b^\textup{\rm A}
\]
is continuous.
\item Strong solutions are unique when they exist.
\end{enumerate}
\end{theorem}
Theorem \ref{thmVE} implies that the solution satisfies $d_{A(t)} v(t) \in H_1,$ but fails to be in $H_1$ up to a vertical solution. In other words, as Gross mentions in \cite{Gr3}, the above result is the infinitesimal analogue of the existence theorem for the Yang-Mills heat equation, where now the infinitesimal analogue of a gauge transformation is played by the vertical vectors. To achieve $v\in H_b^\textup{\rm A}$, one must make the additional assumption that $A$ is in $L^3$. This is known for initial data $A_0\in H_{1/2}$, but need not hold in general
Recently, Gross in \cite{Gr4}, and in a separate work with the author, have been considering the topological properties of the natural configuration space that arises in the context of the Yang-Mills heat equation over compact subsets of 3-dimensional Euclidean space with smooth boundary and $\mathbb{R}^3$ itself. They are interested in providing a space of solutions $\mathcal{Y}$ to \eqref{ymh} as well as an appropriate space of gauge transformations $\G$ such that $\mathcal{Y}/\mathcal{G}$ is an infinite dimensional complete manifold, with a metric structure. The spaces $\mathcal{Y}$ and $\mathcal{G}$ should correspond to a general class of initial conditions, and in particular one that would be relevant for quantum field theory applications. They anticipate that the space of solutions with initial value in $A_0 \in H_{1/2}(M)$, or alternatively $A_0 \in H_{a}(M)$, and an appropriate gauge group will provide such a space. The previous work of Gross regarding the solutions to the variational equation will be critical, as the metric will correspond to a norm on the relevant tangent vectors, which will eventually allow us to find a homeomorphism between small neighborhoods of the space and the relevant tangent space.
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1910.09999
|
\section{Introduction}
Let $G$ be a graph. A {\sl signed graph} is a pair $(G,\Sigma)$ with $\Sigma\subseteq E(G)$, each edge in $\Sigma$ is labelled by $-1$ and other edges are labelled by 1. The graph $G$ can be viewed as the signed graph $(G,\emptyset)$.
A circuit $C$ of $G$ is \emph{balanced} if $|C\cap\Sigma|$ is even, otherwise it is \emph{unbalanced}. We say that a subgraph of $(G,\Sigma)$ is \emph{unbalanced} if it contains an unbalanced circuit, otherwise it is \emph{balanced}.
Signed graphs is a special class of ``biased graphs", which was defined by Zaslavsky in \cite{Zas89, Zas91}. Just as biased graphs, there are two interesting classes of matroids, the class of signed-graphic matroids and the class of even-cycle matroids, associated with signed graphs, which in fact are special classes of ``frame matroids" and ``lifted-graphic matroids" associated with biased graphs, respectively.
A {\sl barbell} is a union of two unbalanced circuits sharing exactly one vertex or a union of two vertex-disjoint unbalanced circuits together with a minimal path joining them.
A \emph{signed circuit} of $(G,\Sigma)$ is a balanced circuit or a barbell.
We say the matroid with $E(G)$ as its ground set and with the set of all signed circuits as its circuit set is the {\sl signed-graphic} matroid defined on $(G,\Sigma)$. We say that $(G,\Sigma)$ is \emph{flow-admissible} if each element of $E(G)$ is in a circuit of its signed-graphic matroid, that is, each edge of $G$ is in a signed circuit of $(G,\Sigma)$.
For a positive integer $k$, we say that a signed graph $(G,\Sigma)$ has a {\it $k$-cover} if there is a family $\mathcal{C}$ of signed circuits of $(G,\Sigma)$ such that each edge of $G$ belongs to exactly $k$ members of $\mathcal{C}$. For ordinary graphs $G$ (signed graph $(G,\Sigma)$ with $\Sigma =\emptyset$), a $k$-cover of $G$ is just a family of circuits which together covers each edge of $G$ exactly $k$ times. In \cite{BJJ}, Bermond, Jackson and Jaeger proved that every bridgeless graph $G$ has a 4-cover. Fan proved that every bridgeless graph $G$ has a 6-cover in \cite{F}. Together it follows that every bridgeless graph $G$ has a k-cover, for every even integer $k$ greater than 2. The only left case that $k=2$ is the famous Circuit Double Cover Conjecture: every bridgeless graph $G$ has a 2-cover, which is still open and believed to be very hard. It is somehow a surprise that it is even unknown whether there is an integer $k$ such that every signed graph $(G,\Sigma)$ has a $k$-cover. In \cite{F-2}, Fan showed that for each positive integer $k\leq 5$, there are infinitely many flow-admissible signed graphs that have no $k$-cover, and proposed the following conjecture.
\begin{conjecture}\label{Fan}
Every flow-admissible signed graph has a $6$-cover.
\end{conjecture}
In this paper, we prove
\begin{theorem}\label{main thm}
Conjecture \ref{Fan} holds for signed Eulerian graphs.
\end{theorem}
In \cite{CLLZ}, Cheng, Lu, Luo, and Zhang proved that each signed Eulerian graph with even number of negative edges has $2$-covers. We will prove Theorem \ref{main thm} from a different aspect, and our proof does not rely on their result.
This paper is organised as follows. Definitions and results needed in the proof of Theorem \ref{main thm} are given in Section 2. Theorem \ref{main thm} will be proved in Section 4 by contradiction. All ``small" signed Eulerian graphs occurring in Section 4 in the proof of contradiction are dealt with in Section 3.
\section{Preliminaries}
Let $G$ be a finite graph. Let $loops(G)$ denote the set of loops in $G$. Let $\Delta(G)$ and $\delta(G)$ be the maximal and minimal degree of $G$, respectively. For a positive integer $k$, let $V_k(G)$ be the subset of $V(G)$ consisting of degree-$k$ vertices of $G$, and let $kG$ be the graph obtained from $G$ by replacing each edge in $G$ with a parallel class with exactly $k$ edges. For an $(u,v)$-path $P$ of $G$, we say that $P$ is {\sl pendant} if $u\in V_1(G)$, $v$ is of degree at least three and all internal vertices of $P$ are in $V_2(G)$.
A subgraph $H$ of $G$ is \emph{spanning} if $V(H)=V(G)$. In this paper, we will also use $H$ to denote its edge-set. For example, we will let $G\backslash H$ denote $G\backslash E(H)$. If exactly one component of $G$ has edges, then we say that $G$ is {\sl connected up to isolated vertices}. Evidently, a connected graph is also connected up to isolated vertices, but the converse maybe not true.
We say that $G$ is {\sl even} if every vertex of $G$ is of even degree. If an even graph is connected, we say that it is {\sl Eulerian}.
A {\sl circuit} is a connected 2-regular graph. A circuit $C$ of $G$ is {\sl non-separating} if $G\backslash C$ is connected, otherwise, it is {\sl separating}. A \emph{theta graph} is a graph that consists of a pair of vertices joined by three internally vertex-disjoint paths.
Let $\mathcal{C}$ be a circuit-decomposition of an Eulerian graph $G$. Let $H$ be a graph with $\mathcal{C}$ as its vertex set, where two vertices in $H$ are adjacent if and only if their corresponding circuits in $G$ have common vertices. We say that $H$ is {\sl determined} by $\mathcal{C}$.
\begin{lemma}\label{remove C}
Let $G$ be an Eulerian graph with $\Delta(G)\geq4$. Let $C$ be a circuit of $G$. Then there is a circuit $C^{'}$ of $G$ with $C\cap C'=\emptyset$ such that $G\backslash C^{'}$ is connected up to isolated vertices.
\end{lemma}
\begin{proof}
Since $G$ is Eulerian, $G$ has a circuit-decomposition $\mathcal{C}$ containing $C$. Let $H$ be the graph determined by $\mathcal{C}$. Since $G$ is connected with $\Delta(G)\geq4$, the graph $H$ is connected with at least two vertices. Let $T$ be a spanning tree of $H$. Since $T$ has at least two degree-1 vertex, $T$ has a degree-1 vertex, say $C'$, which is not $C$. Then $C'$ is the circuit as required by the lemma.
\end{proof}
\begin{lemma}\label{2-conn}
Let $G$ be a $2$-connected graph with $\vert V(G)\vert\geq 3$. For any vertex $v$ of $G$, there is an edge $e$ of $G-v$ such that $G-V(e)$ is connected.
\end{lemma}
\begin{proof}
Let $C$ be a circuit of $G$ passing through $v$ with $|C|$ as large as possible. Evidently, $|C|\geq3$ as $\vert V(G)\vert\geq 3$ and $G$ is 2-connected. Let $e$ be an edge of $C$ that is not incident with $v$. Then $G-V(e)$ is connected, otherwise we can find a longer circuit going through $v$.
\end{proof}
A set $\Sigma'\subseteq E(G)$ is a {\sl signature} of $(G,\Sigma)$ if $(G,\Sigma)$ and $(G,\Sigma')$ have the same balanced circuits and the same unbalanced circuits. Evidently, for any edge-cut $C^*$ of $G$, the set $\Sigma\triangle C^*$ is a signature of $(G,\Sigma)$. We say that $(G,\Sigma')$ is obtained from $(G,\Sigma)$ by \emph{switching}. In (\cite{CDFP}, Lemma 3.5.), Chen, DeVos, Funk, and Pivotto proved that all edges of a balanced signed graph can be labelled by 1 by switching. Since each edge-cut of a subgraph of $(G,\Sigma)$ is contained in an edge-cut of $G$, by (\cite{CDFP}, Lemma 3.5.), we have
\begin{lemma}\label{switch}
All edges of a balanced signed subgraph of $(G,\Sigma)$ can be labelled by $1$ by switching.
\end{lemma}
The following two results are obvious, which will be frequently used in Section 3 without reference.
\begin{lemma
Each signed theta-graph has a balanced circuit and can not have exactly two balanced circuits.
\end{lemma}
\begin{lemma
Every $2$-edge-connected signed graph containing two edge-disjoint unbalanced circuits is flow-admissible.
\end{lemma}
In (\cite{Maca-1}, Theorem 4.2.), M\'a\v cjov\'a and \v Skoviera proved that a flow-admissible signed Eulerian graph with odd number of negative edges contains three edge-disjoint unbalanced circuits. On the other hand, since each unbalanced Eulerian signed graph with even number of negative edges contains two edge-disjoint unbalanced circuits, we have
\begin{lemma}\label{decomposition}
A flow-admissible unbalanced signed Eulerian graph contains two edge-disjoint unbalanced circuits.
\end{lemma}
For simplicity, we will also use $G$ to denote a signed graph defined on $G$.
\section{Signed Eulerian graphs with special circuit decompositions}
Recall that $kG$ is the graph obtained from $G$ by replacing each edge in $G$ with a parallel class having exactly $k$ edges. For any integer $k\geq3$, let $N_k$ be a circuit of length $k$. Let $N$ be a subdivision of $2N_k$, and $C$ a circuit of $N$. We say that $C$ is {\sl small} if $|V(C)\cap V_4(N)|=2$, otherwise, $C$ is {\sl long}. When $C$ is small, we also say that a vertex in $V(C)\cap V_4(N)$ is an {\sl end} of $C$. For any edges $e_1, e_2$ of $N$, which are in a small circuit of $N$ and such that $N\backslash\{e_1, e_2\}$ is connected, if we label $\{e_1, e_2\}$ by $-1$ and all other edges by 1, all small circuits are balanced and all long circuits are unbalanced. We say that such signed Eulerian graph is a {\sl necklace} of {\sl length} $k$. Evidently, necklaces have a 1-cover.
{\bf In the rest of this section, we will always let $G$ denote a flow-admissible signed Eulerian graph with $\delta(G)\geq4$ such that $G\backslash loops(G)$ is $2$-connected, and $\mathcal{C}$ a circuit-decomposition of $G$, and let $H$ be the graph determined by $\mathcal{C}$}. We say that $\mathcal{C}$ is {\sl optimal} if it satisfies the following properties:
\begin{itemize}
\item[(CD1)] $\mathcal{C}$ is chosen with the number of unbalanced circuits as large as possible.
\item[(CD2)] subject to (CD1), $\mathcal{C}$ is chosen with $|\mathcal{C}|$ as large as possible.
\end{itemize}
{\bf In the rest of this section, we will always assume that $\mathcal{C}$ is optimal}.
For any $C\in \mathcal{C}$, we say that $C$ is a {\sl balanced} vertex of $H$ if $C$ is a balanced circuit of $G$, otherwise it is {\sl unbalanced}.
\begin{lemma}\label{adjacent-circuits}
For every pair of adjacent vertices $C_i$ and $C_j$ in $H$, if $C_i$ is balanced, we have
\begin{enumerate}
\item $1\leq |V_G(C_i)\cap V_G(C_j)|\leq2$,
\item $C_i\cup C_j$ is balanced when $C_j$ is balanced, and
\item $C_i\cup C_j$ is not flow-admissible when $C_j$ is unbalanced.
\end{enumerate}
\end{lemma}
\begin{lemma}\label{adjacent-ub-C}
For every pair of adjacent unbalanced vertices $C_i$ and $C_j$ in $H$, if $|V_G(C_{i})\cap V_G(C_{j})|\geq 3$, then $C_{i}\cup C_{j}$ is a necklace.
\end{lemma}
\begin{proof}
Since $C_i$ and $C_j$ are unbalanced, for any circuit decomposition $\mathcal{C}'$ of $C_{i}\cup C_{j}$, either all circuits in $\mathcal{C}'$ are balanced or at least two of them are unbalanced. If $C_{i}\cup C_{j}$ has an unbalanced circuit avoiding some vertex in $V_4(C_{i}\cup C_{j})$, then $C_{i}\cup C_{j}$ can be decomposed into at least three circuits and two of which are unbalanced, which is not possible as $\mathcal{C}$ is optimal. So each circuit in $C_{i}\cup C_{j}$ avoiding a vertex in $V_4(C_{i}\cup C_{j})$ is balanced. Hence, $C_{i}\cup C_{j}$ is a necklace.
\end{proof}
We say that $G$ is {\sl cover-decomposable} if $G$ can be decomposed into two proper edge-disjoint flow-admissible signed Eulerian subgraphs.
\begin{lemma}\label{final-structure}
If $H$ is isomorphic to a graph pictured as Figure \ref{Figure special graph} and $G$ has no balanced loops, then $G$ is cover-decomposable or has a $6$-cover.
\end{lemma}
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=3.5 cm]{Figure.pdf}
\caption{All degree-3 vertices are balanced, and others are unbalanced. All $f_i$ are loops of $G$. }
\label{Figure special graph}
\end{center}
\end{figure}
\begin{proof}
Assume otherwise. When $H$ is isomorphic to the graph pictured as Figure \ref{Figure special graph} (d), since $C_i\cup C_j$ has a 1-cover for all $1\leq i<j\leq 3$ by Lemma \ref{adjacent-ub-C}, $G$ has a 2-cover. So $H$ is isomorphic to a graph pictured as Figure \ref{Figure special graph} (a)-(c). Note that, $1\leq|V_G(C_i)\cap V_G(C_j)|\leq 2$ when $C_i$ is balanced by Lemma \ref{adjacent-circuits}. Since $G$ has a 6-cover when some $C_i$ is a loop, no $C_i$ is a loop.
When $|V_G(C_i)\cap V_G(C_j)|=1$ for all $1\leq i<j\leq 3$, since $C_1\cup C_2\cup C_3$ is isomorphic to a $2K_3$-subdivision, the lemma holds. Hence, $|V_G(C_i)\cap V_G(C_j)|\geq2$ for some $1\leq i<j\leq 3$.
Assume that $|V_G(C_i)\cap V_G(C_3)|=2$ for some $1\leq i\leq 2$. Either there is a balanced circuit $C$ of $C_i\cup C_3$ such that $G\backslash C$ is connected or $G$ has a 2-cover. For the first case, since $G\backslash C$ contains two edge-disjoint unbalanced circuits, it is flow-admissible, so $G$ is cover-decomposable. Hence, $H$ is isomorphic to a graph pictured as Figure \ref{Figure special graph} (b) or (c), $m=|V_G(C_1)\cap V_G(C_2)|\geq2$ and $|V_G(C_i)\cap V_G(C_3)|=1$ for each $1\leq i\leq 2$.
When $m=2$, by simple computation, the lemma holds. Hence, $m\geq3$, so $H$ is isomorphic to the graph pictured as Figure \ref{Figure special graph} (c) and $C_1\cup C_2$ is a necklace of length $m$ by Lemmas \ref{adjacent-circuits} and \ref{adjacent-ub-C}.
Assume that $G$ is a counterexample to the lemma with $|V(G)|$ as small as possible. When $C_3$ does not share a vertex with a small circuit $C$ of $C_1\cup C_2$, delete $C$ and identify its two ends as a new vertex. Let $G'$ be the new graph. Then $G'$ is cover-decomposable or has a 6-cover by the choice of $G$, so is $G$ since $C$ is balanced. Hence, $C_3$ intersects all small circuits of $C_1\cup C_2$. Moreover, since $m\geq3$ and $|V_G(C_i)\cap V_G(C_3)|=1$ for each $1\leq i\leq 2$, there are edge-disjoint long circuits $C'_1, C'_2$ of $C_1\cup C_2$ with $|V_G(C'_1)\cap V_G(C_3)|=2$ and $|V_G(C'_2)\cap V_G(C_3)|\geq1$. Since $C'_1, C'_2$ are unbalanced and $C'_1\cup C'_2=C_1\cup C_2$, the graph determined by $\{C'_1, C'_2, C_3, \{f_3\}\}$ isomorphic to the graph pictured as Figure \ref{Figure special graph} (c). Since $|V_G(C'_1)\cap V_G(C_3)|=2$, the lemma holds by similar analysis in the second paragraph of the proof.
\end{proof}
Let $C$ be a separating circuit of a graph $G$ with $u,v\in V(C)$. Let $P$ be an $(u,v)$-path of $C$. For a component $G'$ of $G\backslash C$, if $V(G')\cap V(P)\neq\emptyset$ we say that $G'$ {\sl intersects} $P$; if $V(G')\cap V(C)\subseteq V(P)-\{u,v\}$ we say that $G'$ {\sl properly intersects} in $P$.
\begin{lemma}\label{cover-decomposable}
Let $C$ be a separating circuit of $G$ such that all components of $G\backslash C$ are unbalanced. Let $C'$ be a circuit-component of $G\backslash C$ with $\{u,v\}=V(C)\cap V(C')$. Let $P_1$ and $P_2$ be the $(u,v)$-paths of $C$. When $C$ is balanced or $G\backslash C$ has three components, one of the following holds.
\begin{itemize}
\item[(1)] $G$ is cover-decomposable, or
\item[(2)] $G\backslash C$ has exactly three components, none of which is flow-admissible and one of which properly intersects $P_i$ for each $1\leq i\leq2$.
\end{itemize}
\end{lemma}
\begin{proof}
Assume that (1) is not true. Without loss of generality we may assume that $C'=\{e,f\}$.
Since $C'$ is unbalanced, we may assume that $P_1\cup\{e\}$ and $P_2\cup\{x\}$ are balanced for some $x\in\{e,f\}$. Since $G\backslash C$ has two components, besides $C'$, some component of $G\backslash C$ intersects in some $P_i$, say $P_2$. Since $C$ is balanced or $G\backslash C$ has three components, $G\backslash (P_1\cup\{e\})$ has two edge-disjoint unbalanced circuits. Since (1) does not hold, $G\backslash (P_1\cup\{e\})$ is disconnected, so some component of $G\backslash C$ properly intersects in $P_1$ and is not flow-admissible. Repeated the analysis, a component of $G\backslash C$ properly intersects in $P_2$ and is not flow-admissible. So $G\backslash C$ has three components.
Let $G_i$ be the union of components of $G\backslash C$ that properly intersect $P_i$ for each $1\leq i\leq2$. Then $G_1$ and $G_2$ are unbalanced and not flow-admissible. Assume that $G_1$ is disconnected. Since $G_1$ contains two edge-disjoint unbalanced circuits, $G_1\cup P_1\cup\{x\}$ and its complement are flow-admissible, implying that (1) holds. Hence, $G_1$ is connected, so is $G_2$ by symmetry. Besides $C', G_1$ and $G_2$, assume that $G\backslash C$ has another component $G_3$. Since $G_3$ is unbalanced and intersects $V(P_1)$ and $V(P_2)$ by the definition of $G_1$ and $G_2$, both $G_1\cup P_1\cup \{f\}$ and its complement are flow-admissible, a contradiction. So $G\backslash C$ has exactly three components $C', G_1$ and $G_2$, that is, (2) holds.
\end{proof}
\begin{lemma}\label{star+1}
Let $H$ be a tree with a unique vertex $C$ of degree at least three with all leaf vertices unbalanced and all pedant paths having at most two edges. When $C$ is balanced, $V_2(H)=\emptyset$. When $C$ is unbalanced, all degree-$2$ vertices of $H$ are balanced triangles and leaf vertices that are adjacent with degree-$2$ vertices are loops. Then $G$ is cover-decomposable or has a $6$-cover.
\end{lemma}
\begin{proof}
Assume that the lemma is not true. Since $G$ has a 6-cover when each component of $G\backslash C$ is a loop, there is a vertex $C'$ in $H$ adjacent with $C$ with $|C'|\geq2$. Set $m=|V_G(C)\cap V_G(C')|$.
Since $G\backslash loops(G)$ is 2-connected and $\delta(G)\geq4$, we have $m\geq2$.
We claim that $C'$ is balanced or $|C'|\neq2$. Assume otherwise. Then $C'$ is a component of $G\backslash C$ as all degree-2 vertices of $H$ are balanced. Let $\{u, v\}=V_G(C')\cap V_G(C)$, $P_1$ and $P_2$ be the $(u,v)$-paths of $C$.
By Lemma \ref{cover-decomposable}, $G\backslash C$ has exactly three components $C', G_1$ and $G_2$, where $G_1$ and $G_2$ properly intersect $P_1$ and $P_2$, respectively. When $C\cup G_1$ is a necklace, there is a small circuit $D$ of $C\cup G_1$ such that $G\backslash D$ is connected. Since $C'$ and $G_2$ are unbalanced, $G\backslash D$ is flow-admissible, so $G$ is cover-decomposable. Hence, $G_1$ is an unbalanced circuit of size at most 2 or $G_1$ consists of a balanced triangle and a loop, so is $G_2$ by symmetry. By simple computation, $G$ is cover-decomposable or has a 6-cover.
Assume that $C'$ is balanced. Then $C'\in V_2(H)$ is a triangle. So $C$ is unbalanced and $|V_G(C')\cap V_G(C)|=2$ by Lemma \ref{adjacent-circuits}. Let $u,v, P_1, P_2$ be defined as above. Let $e$ be the loop incident with $C'$ and $f$ the edge in $C'$ whose ends are $u,v$. Since $C$ is unbalanced, $P_1\cup\{f\}$ is balanced and $P_2\cup\{f\}$ is unbalanced. Evidently, (a) a component of $G\backslash C$ properly intersects $P_1$, otherwise $P_1\cup\{f\}$ and its component are flow-admissible; and (b) no component of $G\backslash C$ intersects $P_2-\{u,v\}$, otherwise the union $G'$ of $P_2\cup\{f\}$ and all components of $G\backslash C$ intersects $P_2-\{u,v\}$ and $G\backslash G'$ are flow-admissible. Then $P_2\cup (C'-\{f\})\cup\{e\}$ and its component are flow-admissible, a contradiction.
We may therefore assume that $C'$ is unbalanced with $|C'|\geq3$, implying that $C$ is unbalanced by Lemma \ref{adjacent-circuits}. By the choice of $C'$, for each component $G'$ of $G\backslash C$, either $G'$ is a loop or $|G'|\geq3$. When $|G'|\geq3$, $C\cup G'$ is a necklace by Lemma \ref{adjacent-ub-C}.
Let $D$ be a small circuit of $C\cup C'$. Since $G\backslash D$ has two edge-disjoint unbalanced circuits, $G\backslash D$ is disconnected, so a component $G_D$ of $G\backslash C$ properly intersects in $C\cap D$. Since $C\cup C'$ has three small circuits, $G_D$ is the unique component of $G\backslash C$ properly intersecting in $C\cap D$ and $C\cup C'$ has exactly three small circuits, implying $|C'|=3$, otherwise $G$ is cover-decomposable.
When $G_D$ is not a loop, there is a small circuit $D'$ of $C\cup G_D$ such that $G\backslash D'$ is connected, so $G$ is cover-decomposable. Hence, $G_D$ is a loop. By the choice of $C'$, each component $G'$ of $G\backslash C$ that is not a loop is an unbalanced triangle. When $C'$ is the unique component of $G\backslash C$ that is not a loop, $G$ has a 3-cover. When there is another component $G_1$ of $G\backslash C$ that is not a loop, let $D$ be a small circuit of $C\cup C'$ intersecting $G_1$. Let $G'$ be a union of $D\cup G_1$ and the loop incident with $D$. Then $G'$ and $G\backslash G'$ are flow-admissible, so $G$ is cover-decomposable.
\end{proof}
\section{Proof of Theorem \ref{main thm}.}
In this section, we prove Theorem \ref{main thm}, which is restated here in a slightly different way.
\begin{theorem}
Every flow-admissible signed Eulerian graph has a $6$-cover.
\end{theorem}
\begin{proof}
Assume that the result is not true. Let $G$ be a counterexample with $|V(G)|$ as small as possible. Evidently,
\begin{claim}
\begin{itemize}
\item $G$ is unbalanced with $\delta (G)\geq 4$;
\item $G$ has no balanced loops; and
\item $G$ is not cover-decomposable, in particular, if $C$ is a non-separating balanced circuit of $G$, then $G\backslash C$ is not flow-admissible.
\end{itemize}
\end{claim}
\begin{claim}\label{delete loop}
$G\backslash loops(G)$ is $2$-connected.
\end{claim}
\begin{proof}[Subproof.]
Assume otherwise. There are edge-disjoint Eulerian subgraphs $G_1, G_2$ of $G$ with $|E(G_1)|, |E(G_2)|\geq2$, with $\{v\}=V(G_1)\cap V(G_2)$, and with $E(G)=E(G_1)\cup E(G_2)$. Since $G$ is not cover-decomposable, $G_{1}$ and $G_2$ are unbalanced. Let $G_i^+$ be a signed graph obtained from $G_i$ by adding an unbalanced loop $e_i$ incident with $v$ for each integer $1\leq i\leq 2$. Since $G_1^+$ and $G_2^+$ are flow-admissible, both of them have $6$-covers by the choice of $G$. Since $|V(G_{1})\cap V(G_{2})|=1$, we can obtain a 6-cover of $G$ by combining 6-covers of $G_1^+$ and $G_2^+$, a contradiction.
\end{proof}
Let $\mathcal{C}$ be an optimal circuit decomposition of $G$ and $H$ the graph determined by $\mathcal{C}$. Since $G$ is connected, so is $H$. By Lemma \ref{decomposition}, at least two members of $\mathcal{C}$ are unbalanced. Hence, by Lemma \ref{adjacent-ub-C}, $|V(H)|\geq3$ and the following holds.
\begin{claim}\label{b-circuit in H}
Each balanced vertex of $H$ is a cut-vertex, in particular, each vertex in a leaf block of $H$ that is not a cut-vertex is unbalanced.
\end{claim}
By \ref{b-circuit in H}, for any vertex $C$ of $H$, all components of $G\backslash C$ are unbalanced.
For a subgraph $H'$ of $H$, we say that the subgraph of $G$ without isolated vertices whose edge set is a union of all circuits in $\mathcal{C}$ that label some vertex of $H'$ {\sl corresponds to } $H'$.
\begin{claim}\label{adjacent-circuits+1}
Let $e$ be a cut-edge of $H$ whose ends are $C_i$ and $C_j$. If $e$ is not a leaf edge and $H-\{C_i, C_j\}$ has exactly two components, then $C_i$ or $C_j$ is unbalanced.
\end{claim}
\begin{proof}[Subproof.]
Assume to the contrary that $C_i$ and $C_j$ are balanced. Let $G_1$ and $G_2$ be the subgraphs of $G$ corresponding to the two components of $H-\{C_i,C_j\}$ with $V(G_1)\cap V_G(C_i)\neq\emptyset$. It follows from \ref{b-circuit in H} that $G_1, G_2$ are unbalanced. Moreover, since $G\backslash loops(G)$ is $2$-connected, by Lemma \ref{adjacent-circuits}, we have $|V_G(C_i)\cap V_G(C_j)|=2$. Let $u\in V_G(G_1)\cap V_G(C_i)$ and $v\in V(G_2)\cap V_G(C_j)$. Since $ |V_G(C_i)\cap V_G(C_j)|=2$, the graph $C_i\cup C_j$ has a circuit $C$ avoiding $u$ and $v$ such that $(C_i\cup C_j)\backslash C$ is connected up to isolated vertices. Since $H-\{C_i, C_j\}$ has exactly two components, $G\backslash C$ is connected, so $G\backslash C$ is flow-admissible. Moreover, since $C_i\cup C_j$ is balanced by Lemma \ref{adjacent-circuits}, $C$ is balanced, so $G$ is cover-decomposable, a contradiction.
\end{proof}
\begin{claim}\label{remove balanced C}
For any separating circuit $C\in\mathcal{C}$, if $G'$ is a component of $G\backslash C$ that is not flow-admissible, then one of the following holds.
\begin{itemize}
\item[(1)] $G'$ is an unbalanced circuit such that $|G'|\leq2$ or $C\cup G'$ is a necklace. In particular, when $C$ is balanced, $|G'|\leq2$.
\item[(2)] $G'$ consists of a loop and a balanced triangle.
\end{itemize}
\end{claim}
\begin{proof}[Subproof.]
When $G'$ is a circuit, since $\delta(G)\geq4$, by Lemmas \ref{adjacent-circuits} and \ref{adjacent-ub-C}, (1) holds. Assume that $G'$ is not a circuit. When $G'$ consists of exactly two edge-disjoint circuits that share exactly one vertex, since $C$ only shares vertices with the balanced circuit of $G'$, Lemma \ref{adjacent-circuits} and \ref{delete loop} imply that (2) holds. So we may assume that $\Delta(G')\geq6$ or $|V_4(G')|\geq2$.
Since $G'$ is not flow-admissible, by switching we may assume that there is a unique edge $e$ of $G'$ labelled by $-1$ and all other edges in $G'$ are labelled by 1.
When $e$ is a loop, let $v$ be the end of $e$, and $B$ a block of $G'\backslash\{e\}$ containing $v$, and let $C'$ be a circuit of $B$ containing $v$; otherwise, let $\{v\}=\emptyset$, and $B$ the block containing $e$, and let $C'$ be a circuit of $B$ with $e\in C'$. If possible, we may further assume that $C'$ is chosen with $V_G(C')\cap V_2(G')\neq\emptyset$. By Lemma \ref{remove C}, there is a circuit $C_1$ of $G'\backslash loops(G')$ with $C'\cap C_1 =\emptyset$ such that $G'\backslash C_1$ is connected up to isolated vertices. Since $C_1$ is balanced and $G\backslash C_1$ has two edge-disjoint unbalanced circuits, $G\backslash C_1$ is not connected. Hence, $V_G(C)\cap V(G')\subseteq V_G(C_1)$ and $\emptyset\neq V_2(G')\subseteq V_G(C_1)$ as $e$ is the only edge in $G'$ which has a chance to be a loop. By the choice of $C'$,
the set $V_2(G')$ is contained in another block $B'$ of $G'$ with $B\neq B'$ as $C'$ contains no vertex in $V_2(G')$. Since $V_G(C)\cap V(G')\subseteq V(B')$ and $G\backslash loops(G)$ is $2$-connected, $|B|=1$, a contradiction to the choice of $B$.
\end{proof}
\begin{claim}\label{cn after delete C}
For any $C\in\mathcal{C}$, the graph $G\backslash C$ has at most two components.
\end{claim}
\begin{proof}[Subproof.]
Assume that $G\backslash C$ has three components. Since each component $G'$ of $G\backslash C$ is unbalanced, $G'$ is not flow-admissible. By \ref{remove balanced C}, $H$ is a tree with $C$ as a unique vertex of degree at least three whose pedant paths have at most two edges
When $C$ is balanced, \ref{adjacent-circuits+1} implies that $V_2(H)=\emptyset$. Hence, by \ref{remove balanced C} and Lemma \ref{star+1}, $G$ is cover-decomposable or has a 6-cover, a contradiction.
\end{proof}
\begin{claim}\label{C&C'+}
For any balanced vertex $C$ of $H$, each degree-$1$ vertex of $H$ adjacent with $C$ is a loop of $G$.
\end{claim}
\begin{proof}[Subproof.]
Let $C'$ be a degree-$1$ vertex of $H$ adjacent with $C$. Assume that $C'$ is not a loop of $G$. Then $|C'|=|V_G(C)\cap V_G(C')|=2$ by \ref{remove balanced C}. It follows from Lemma \ref{cover-decomposable} and \ref{cn after delete C} that $G$ is cover-decomposable, a contradiction.
\end{proof}
\begin{claim}\label{tree-case}
$H$ is not a tree.
\end{claim}
\begin{proof}[Subproof.]
Assume otherwise. By \ref{cn after delete C}, $H$ is a path. Evidently, at most one vertex in $V_2(H)$ is unbalanced, otherwise, $G$ is cover-decomposable. Since no balanced vertices of $H$ are adjacent by \ref{adjacent-circuits+1}, we have $|V(H)|\leq 5$, and when $|V(H)|\geq4$ exactly one vertex in $V_2(H)$ is unbalanced.
Assume that $H$ has two adjacent vertices $C_1, C_2$ with $|V_G(C_1)\cap V_G(C_2)|\geq3$. Then $C_1\in V_1(H)$, $|V(H)|\leq4$ and $C_1\cup C_2$ is a necklace by Lemma \ref{adjacent-ub-C}. Let $C_3$ be the other vertex adjacent with $C_2$ in $H$. When $V_G(C_2)\cap V_G(C_3)$ is in a small circuit of $C_1\cup C_2$, the graph $G$ has a 6-cover.
When $V_G(C_2)\cap V_G(C_3)$ is not in a small circuit of $C_1\cup C_2$, implying $|V_G(C_2)\cap V_G(C_3)|=2$, since $V_G(C_1)\cap V_G(C_3)=\emptyset$, the graph $C_1\cup C_2$ can be decomposed to two long circuits $C'_1, C'_2$ both of which share exactly one vertex with $C_3$. Hence, the graph determined by $\mathcal{C}-\{C_1, C_2\}+\{C'_1, C'_2\}$ is isomorphic to a graph pictured as Figure \ref{Figure special graph} (c) or (d). Lemma \ref{final-structure} implies that $G$ is cover-decomposable or has a 6-cover. Therefore, combined with Lemma \ref{adjacent-circuits} we can assume that every pair of adjacent vertices in $H$ share at most two vertices in $G$. Note that each degree-1 vertex of $H$ adjacent with a balanced vertex is a loop by \ref{C&C'+}. Hence, by simple computation, $G$ has a 6-cover, a contradiction.
\end{proof}
\begin{claim}\label{leaf-block}
$H$ is not $2$-connected whose leaf blocks are isomorphic to $K_{2}$.
\end{claim}
\begin{proof}[Subproof.]
Assume otherwise. When $H$ is not 2-connected, let $B$ be a leaf block of $H$ that is not isomorphic to $K_2$, and $v$ be the unique cut-vertex of $H$ in $V(B)$. When $H$ is 2-connected, let $B=H$ and $v$ any vertex of $B$.
By Lemma \ref{2-conn}, there is an edge $e$ in $B-v$ such that $B-V_H(e)$ is connected, so $H-V_H(e)$ is also connected. Without loss of generality assume that $C_{1}$ and $C_{2}$ are the ends of $e$. Then $C_{1}\cup C_{2}$ and $G\backslash C_{1}\cup C_{2}$ are connected.
Since $C_{1}\cup C_{2}$ is flow-admissible by \ref{b-circuit in H}, the graph $G\backslash C_{1}\cup C_{2}$ is not flow-admissible. Since $H$ is not isomorphic to the graph pictured as Figure \ref{Figure special graph} (d) by Lemma \ref{final-structure}, $H$ has exactly three unbalanced vertices and exactly two leaf blocks, one of which is $B$ that is isomorphic to $K_3$ and the other is isomorphic to $K_2$. Let $C_{1}C_{2}C_{3}\ldots C_{n}$ be a longest path in $H$. It follows from \ref{adjacent-circuits+1} that $n=4$. By \ref{C&C'+}, the circuit $C_4$ is a loop of $G$. That is, $H$ is isomorphic to the graph pictured as Figure \ref{Figure special graph} (c). Hence, $G$ is cover-decomposable or has a 6-cover by Lemma \ref{final-structure}, a contradiction.
\end{proof}
Let $B$ be a block of $H$ with $|V(B)|\geq3$. By \ref{tree-case} and \ref{leaf-block}, such $B$ exists and $B$ is not a leaf block. When $H$ has two blocks that are not isomorphic to $K_2$, it follows from \ref{b-circuit in H} and \ref{leaf-block} that $G$ is cover-decomposable.
Hence, $B$ is the unique block of $H$ that is not isomorphic to $K_2$. By \ref{b-circuit in H}, each vertex in $B$ that is not a cut-vertex of $H$ is unbalanced.
Let $u\in V(B)$ be a cut vertex of $H$. When $u$ is unbalanced or $H$ has two pendant paths using $u$, let $H_1$ be a union of all pendant paths using $u$, and $G_1$ the subgraph of $G$ corresponding to $H_1$. Since $|V(B)|\geq3$, by \ref{b-circuit in H} and \ref{leaf-block}, both $G_1$ and $G\backslash G_1$ are flow-admissible, a contradiction. Hence, $u$ is balanced and $H$ has exactly one pendant path using $u$. By the arbitrary choice of $u$, all cut-vertices of $H$ in $B$ are balanced. Using a similar strategy, all vertices in $V_2(H)-V(B)$ are balanced. Combined with \ref{adjacent-circuits+1}, we have $V_2(H)-V(B)=\emptyset$. That is, each pendant path of $H$ has exactly one edge. By \ref{C&C'+}, each vertex in $V_1(H)$ is a loop of $G$.
When there is a vertex in $V(B)$ that is not a cut-vertex of $H$, let $v$ denote the vertex. Otherwise, let $v$ be any vertex of $B$. By Lemma \ref{2-conn}, there is an edge $e\in B-v$ such that $B-V(e)$ is connected. Let $H_1$ be a union of $e$ and all pendant paths of $H$ using an end of $e$, and $G_1$ be the subgraph of $G$ corresponding to $H_1$. Since each vertex in $B$ that is not a cut-vertex of $H$ is unbalanced, $H_1$ contains two unbalanced vertices, so $G_1$ is flow-admissible. Since $H-V(H_1)$ is connected and has an unbalanced vertex, $H$ is isomorphic to a graph pictured as Figure \ref{Figure special graph} (a) or (b). Lemma \ref{final-structure} implies that $G$ is cover-decomposable or has a 6-cover, a contradiction.
\end{proof}
|
1506.01321
|
\section{Derivation of the excited states of an aggregate}
The eigenvalues of the matrix Hamiltonian in our main text (Equation 5) are solutions to the equation,
\begin{equation}
\left |
\begin{array}{cccccc}
\hbar\omega_0-\lambda & 0 & 0 & 0 & \cdots & 0\\
0 & \hbar\omega_1^{(1)}-\lambda & J & 0 & \cdots & 0\\
0 & J & \hbar\omega_1^{(1)}-\lambda & J & \cdots & 0\\
0 & 0 & J & \hbar\omega_1^{(1)}-\lambda & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & 0 & \cdots & \hbar\omega_1^{(1)}-\lambda
\end{array}\right |=0,
\label{eq:Hamiltonian_matrix}
\end{equation}
\noindent where $J$ represents the nearest-neighbour coupling. \Eref{eq:Hamiltonian_matrix} can be expressed in the form,
\begin{equation}
\left |
\begin{array}{cc}
\hbar\omega_0-\lambda & 0\\
0 & H_n\\
\end{array}\right |=0,
\end{equation}
\noindent where $H_n$ is the sub-matrix written explicitly in \Eref{eq:Hamiltonian_matrix}, and is representative of an aggregate with $n$ molecular units. The eigenvalues of \Eref{eq:Hamiltonian_matrix} can be determined by solving,
\begin{equation}
(\hbar\omega_0-\lambda)|H_n|=0.
\end{equation}
\noindent The first of these is the ground state energy $\lambda_0=\hbar\omega_0$, which readily yields the eigenvector $|0\rangle$. The other eigenvalues and eigenvectors require more consideration. The first step is to find an expression for $|H_n|$. Writing this determinant in terms of further sub-matrices gives the following recursive relationship,
\begin{equation}
|H_n| = (\hbar\omega_1^{(1)}-\lambda_m)|H_{n-1}|-J^2|H_{n-2}|.
\label{eq:recursive1}
\end{equation}
\noindent The next step is to identify that this recursive relationship can be put into the same form as the recurrence relation for Chebyshev polynomials~\cite{Chebyshev_1854} \textit{i.e.}
\begin{equation}
U_n(x)=2xU_{n-1}(x)-U_{n-2}(x).
\label{eq:Chebychev_rec}
\end{equation}
\noindent In doing this, and after a little algebra, an expression for the eigenvalues $\lambda_m$ is determined,
\begin{equation}
\lambda_m = \hbar\omega_1^{(1)}-2J\cos\left (\frac{m\pi}{n+1}\right ),
\end{equation}
\noindent where $1<m<n$. The first of these ($\lambda_1$) is the excited state of the aggregate ($\hbar\omega_1$) taken in our main text.\\
The eigenvectors $|m\rangle$ are determined by analysis of $H_n|m\rangle$. The $j^{th}$ element of $H_n|m\rangle$ is,
\begin{equation}
Jm_{j-1}+\hbar\omega_1^{(1)}m_j+Jm_{j+1}=\lambda_m m_j,
\end{equation}
\noindent which can also be put into the form of \Eref{eq:Chebychev_rec}. Making this identification and a little algebra yields the following expression for the normalised excited states of the aggregate~\cite{Malyshev_PRB_51_1995,TKobayashiJAggregates2b},
\begin{equation}
|m\rangle = \sqrt{\frac{2}{n+1}}\sum_{j=1}^n\sin\left (\frac{jm\pi}{n+1}\right )|1_j\rangle.
\end{equation}
\section{Solving the Optical Bloch Equations}
The Optical Bloch Equations (OBEs) derived from the Liouville von-Neumann equation can be written in the compact form as,
\begin{equation}
\dot{\vec{\rho}} = \bar{L}\vec{\rho},
\end{equation}
\noindent where $\vec{\rho}$ is a vector of the density matrix elements. One may solve the OBEs by application of the Rotating Wave Approximation (RWA)~\cite{prl_111_043601_Dorfman} with subsequent use of a matrix inversion method, or numerically by way of a Runge-Kutta method, such as the RK10(8) method~\cite{Hairer}. In this section, both of these approaches are detailed, evaluated, and shown to be equivalent for an ensemble illuminated with a cosine potential.
\subsection{Matrix inversion method}
The matrix-inversion method relies on application of the rotating wave approximation (RWA)~\cite{arXiv13013585} to the perturbing potential, which enables one to write $\bar{L}(t,\omega)$ as a time-independent matrix, $\bar{L}(\omega)$. In this case, solutions for the density matrix elements can be written in the following form,
\begin{equation}
\rho_{mn}(t,\omega) = \sum_i c_{mn,i}e^{i\omega_{i}t},
\label{eq:rho_super}
\end{equation}
\noindent using the principle of superposition. The coefficients $c_i$ are determined from initial conditions and from the eigenvectors of $L(\omega)$. The angular frequencies $\omega_i$ are related to the eigenvalues $\lambda_i$ of the matrix $L(\omega)$ by $i\omega_i=\lambda_i$. $\lambda_i$ is complex with a negative real part in order to conserve probability. The RWA may be applied to a cosine potential, $G= \boldsymbol{d}\cdot\boldsymbol{E}_0~cos(\omega t)$. An advantage of this method is that computation time is very short and weak fields may be considered easily.\\
A restriction on the RWA is that only a field of one frequency may impinge upon the system in this method, since the time-independence of $\bar{L}$ must be preserved. In order to solve the OBEs for circumstances including a frequency-spread pulse, a numerical approach must be used instead, as we now indicate.
\subsection{Explicit Runge-Kutta methods}
For the general case where $\bar{L}$ cannot be written as time-independent \textit{e.g.} when the system is subjected to a pulse, a numerical method must be used to solve the OBEs. The OBEs have the general form $\dot{y}=f(t,y)$ which has a general solution written in discretised form,
\begin{equation}
y_{n+1} = y_n + h_n\sum^n_{i=1}{b_i k_i},\\
\label{eq:RK_solution}
\end{equation}
\noindent where,
\begin{equation}
k_i = f\left (t_n+h_na_i,y_n+h_n\sum^{j=i-1}_{j=1}{C_{ij}k_j} \right ).
\end{equation}
\noindent The estimated error at each step is evaluated as,
\begin{equation}
e_{n+1} \propto y_{n+1} - y^*_{n+1}.
\label{eq:RK45_error}
\end{equation}
\noindent The values $a_i$, $b_i$ and $C_{ij}$ all depend upon the specific method involved. The original 1st-order algorithm using this general method is the Euler method~\cite{Euler}, but a much more accurate method is the Runge-Kutta (RK4) method, which has been used previously to probe the dynamics of 2-level systems~\cite{Charron_JChemPhys_138_024108_2013}. This method suffers from being non-adaptive, in the sense that the step size is always taken to be constant. This makes numerical solutions for rapidly-changing behaviour unreliable. An improved approach is to dynamically allocate the step size between each iteration through a comparison of the estimated local error between the 4th and 5th-order solutions at each point. The yields the Runge-Kutta-Fehlberg (RK4(5)) method~\cite{Fehlberg}. The advantage of an adaptive numerical method such as the RK4(5) method over a non-adaptive method such as the RK4 method is that rapidly-changing behaviour can be modelled with greater accuracy. Local errors are also minimized and the resultant solution that one determines is numerically smoother~\cite{Tong_JCompAppMath_233_10561062_2009}. However, for our work the RK4(5) method is insufficient for producing numerically stable solutions for weak fields. The RK10(8) method (a 10th-order Runge-Kutta method with in-built error estimation by comparison to the 8th order)~\cite{Hairer} was tested and found to produce smoother output for insignificantly more computing time than that of the RK4(5) method.\\
The main advantage of a Runge-Kutta method over the RWA is that no terms in the Hamiltonian are neglected. As a result, any arbitrary potential can be modelled, not just a cosine potential. However, the main drawback of any Runge-Kutta method is that the computation time for the process is many times longer than using the RWA and matrix inversion. This longer timescale is sometimes prohibitive depending on the parameters involved. A further drawback is that for weak fields the solutions to the OBEs become stiff in time and numerical instability is encountered. This can be seen from the fact that with our calculations for TDBC illuminated with a $1~mW$ cosine
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figurea1a_a1b}
\caption{a). The relative occupancy of the excited state $\rho_{11}$, for a $1~mW$ laser potential with spot size $1.5~mm$. b). The relative occupancy of the excited state $\rho_{11}$, together with the coherence $\rho_{01}$ (real and imaginary parts shown) for a $10~MW$ laser with the same spot size.}
\label{fig:figurea1a_a1b}
\end{figure}
\noindent potential with spot size $1.5~mm$, the relative occupancy of the excited state $\rho_{11}$ does not exceed one part in $10^{10}$ and so $\rho_{00}$ remains more-or-less constant in time with value $\approx 1$ as shown in \Fref{fig:figurea1a_a1b} (conversely, the coherences oscillate in time at the applied frequency and are of order $10^{-6}$). Therefore, the populations are incredibly stiff and this is where explicit numerical methods fail. One way to avoid the resultant numerical instability is to confine this procedure to solving OBEs with strong fields. To achieve physically meaningful solutions in this circumstance, one would need to introduce other effects such as multi-exciton recombination and nonlinear effects. By neglecting these effects, similar solutions can be obtained to those of weak fields for the coherences, provided the field does not induce significant population inversion. As an alternative to this compromise, one can use an implicit Runge-Kutta method to solve stiff equations. However, the computation time for implicit Runge-Kutta methods was found to exceed that of the explicit methods by at least a factor of ten, making this approach ungainly.\\
In summary, our procedure is as follows: for weak fields the RWA is used. Solutions derived using this method are supported by the solutions obtained using the RK10(8) method for strong fields with multi-exciton and nonlinear effects neglected. We conclude that by careful use of the RWA, realistic behaviour can be simulated subject to the condition that only one frequency illuminates the system. For slowly time-varying strong fields or for two strong lasers, an explicit Runge-Kutta method may be used, but for weak time-varying fields an implicit Runge-Kutta method may be implemented successfully.
\section{The permittivity of a collection of randomly oriented non--interacting dipoles}
The quantum mechanical model discussed in the main text predicts the polarizability of an individual aggregate. A typical macroscopic sample consists of a large number of randomly oriented aggregates that can be treated as non-interacting. The permittivity of such a macroscopic sample has a simple relationship to the microscopic aggregate polarizability that depends on the dimensionality of the sample, which we derive in this section.\\
For a collection of $N$ dipoles with dipole moments $\bi{d}_{i}$ the polarization per unit volume $\bi{P}$ is given by,
\begin{equation}
\bi{P}=\frac{1}{V}\sum_{i=1}^{N}\bi{d}_{i}=\frac{N}{V}\left\langle\boldsymbol{\alpha}(\omega)\right\rangle\cdot\bi{E}=\boldsymbol{\chi}(\omega)\cdot\bi{E}.\label{polarization}
\end{equation}
\noindent In the second step of the above equation we assumed that the dipole moments are induced by an electric field that can be treated as uniform over the volume $V$, and that the dipoles have a tensor polarizability $\boldsymbol{\alpha}_{i}(\omega)$ that depends on the frequency of the field. The average value of the polarizability is defined as $\langle\boldsymbol{\alpha}(\omega)\rangle=N^{-1}\sum_{i}\boldsymbol{\alpha}_{i}(\omega)$. If we assume that the dipoles are of the same type then the polarizabilities $\boldsymbol{\alpha}_{i}(\omega)$ have the same magnitude $\alpha(\omega)$, and differ only due to the dipole orientation $\hat{\boldsymbol{n}}_{i}$. For a collection of such dipoles distributed in a $\mathcal{D}$ dimensional space,
\begin{eqnarray}
\langle\boldsymbol{\alpha}(\omega)\rangle&=\alpha(\omega)\langle\hat{\bi{n}}\otimes\hat{\bi{n}}\rangle\nonumber\\[10pt]
&=\alpha(\omega)\left(\langle n_{x}^{2}\rangle\hat{\bi{x}}\otimes\hat{\bi{x}}+\langle n_{y}^{2}\rangle\hat{\bi{y}}\otimes\hat{\bi{y}}+\dots\right)\nonumber\\[10pt]
&=\frac{\alpha(\omega)}{\mathcal{D}}\left(\hat{\bi{x}}\otimes\hat{\bi{x}}+\hat{\bi{y}}\otimes\hat{\bi{y}}+\dots\right)\label{average_alpha}
\end{eqnarray}
\noindent where to obtain the second and third lines we used isotropy to determine that quantities such as $\langle n_{x}n_{y}\rangle$ are zero and that $\langle n_{x}^{2}\rangle=\langle n_{y}^{2}\rangle=\dots$, and used the normalization of the unit vectors \(\hat{\bi{n}_{i}}\cdot\hat{\bi{n}}_{i}=1\) to determine that \(\langle n_{x}^{2}\rangle+\langle n_{y}^{2}\rangle+\dots=\mathcal{D}\langle n_{x}^{2}\rangle=1\). Combining Equations \eref{polarization} and \eref{average_alpha} we can then find the macroscopic permittivity $\varepsilon$ which is isotropic and equal to,
\begin{equation}
\varepsilon(\omega)=1+\frac{N\alpha(\omega)}{\mathcal{D}}
\end{equation}
\noindent with $\mathcal{D}=2$ for a planar sample, and $\mathcal{D}=3$ for a bulk one.
\section{Improved Analysis of Experimental Data}
To extract the complex relative permittivity $\varepsilon$ or equivalently, the complex refractive index $\tilde{n}=n+i\kappa$, of our thin film, we compared our experimental values of reflectance ($R_e$) and transmittance ($T_e$) at normal incidence with theoretical ones for each wavelength, following the theoretical framework outlined by Heavens~\cite{Heavens}. In this formalism, the function,
\begin{equation}
f(n,\kappa)=|T_t(n,\kappa)-T_{e}|+|R_t(n,\kappa)-R_{e}|,
\label{eq:fresnelresidual}
\end{equation}
\noindent must be minimised, where the subscript $t$ denotes theoretical values. This process relies upon perfect values of $R_e$ and $T_e$ and $n$ and $\kappa$ are `guessed' to lie in a sensible range. Eq.~\ref{eq:fresnelresidual} is minimised over this range. By this process, small errors in $R_e$ and $T_e$ coupled with rounding errors in the guessed range of values for $n$ and $\kappa$ can yield spurious final values of $n$ and $\kappa$. In order to improve this process, we made additional calculations for $\kappa$, varying the thickness of the film from $63-77 nm$, the range of experimental uncertainty. Our results are shown collectively in \Fref{fig:figurea2a_a2b}(a).\\
\begin{figure}[!htbm]
\centering
\includegraphics[width=\columnwidth]{figurea2a_a2b}
\caption{$\kappa$ for a $1.46 wt\%$ TDBC:PVA film, assuming a range of thicknesses before (a) and after (b) adjustment for these thicknesses.}
\label{fig:figurea2a_a2b}
\end{figure}
\noindent In this figure, we have kept the first two minimum values of Eq.~\ref{eq:fresnelresidual} for each wavelength. Therefore, the figure shows the physical and first spurious solutions. The gaps associated with taking a single thickness are evidence for the need to consider other thicknesses. Doing this produces a range of values for $\kappa$ for each wavelength. We then adjusted each value for the thickness taken by assuming for a given thickness $t$,
\begin{equation}
T\approx e^{-\kappa t}=e^{-\kappa't'},
\end{equation}
\noindent which leads to,
\begin{equation}
\kappa'=\frac{t}{t'}\kappa.
\end{equation}
\noindent This has the effect of deconvolution on our data, to produce \Fref{fig:figurea2a_a2b}(b). The two zero-values show where the process has still failed. Our final step was to eliminate the spurious values for $\kappa$, and use the Kramers-Kronig relations to find $n$, as shown in \Fref{fig:figurea3}.\\
\begin{figure}[!htbm]
\centering
\includegraphics[width=0.5\columnwidth]{figurea3}
\caption{The improved extracted values of the real (blue circle) and imaginary (black pluses) parts of the refractive index from experimental values.}
\label{fig:figurea3}
\end{figure}
\noindent This process produces much smoother values of $\tilde{n}$ than in our previous work. This is because we assumed previously that the true value of $\kappa$ lay in the middle of the two solutions found in \Fref{fig:figurea2a_a2b} and the input thickness was held constant. With this improved analysis by varying the input thickness, it can be seen that intermediate values between the two clear solutions in \Fref{fig:figurea2a_a2b} do not exist, and it becomes apparent that the upper solution is a spurious one.\\
\newpage
A single Lorentz oscillator model can be used to describe the permittivity of a material classically~\cite{Fox2,lebedev},
\begin{equation}
\varepsilon(\omega) = \varepsilon_m + \frac{f_0\omega_0^2}{\omega_0^2-\omega^2-i\omega\gamma_0}.
\end{equation}
\noindent The parameters for this model which fit our data in Fig.~\ref{fig:figurea3} best (assuming a host medium of $n_m=1.52$) are $\omega_0=2.11~eV$, $f_0=0.3$, $\gamma_0=46.1~meV$. The output of these for $\varepsilon$ and the complex refractive index $\tilde{n}$ are illustrated in \Fref{fig:figurea4a_a4b} together with our quantum fit and extracted data. It can be seen that the Lorentz model is not as close to the extracted data as the quantum model, particularly for the real parts of both $\varepsilon$ and $\tilde{n}$.
\begin{figure}
\includegraphics[width=\columnwidth]{figurea4a_a4b}
\caption{The real (blue) and imaginary (black) parts of the a). permittivity and b). complex refractive index of our $1.46~wt\%$ TDBC:PVA film. Solid lines correspond to results from Optical Bloch Equations and dashed lines correspond to results from a single-oscillator Lorentz model. The discrete data points correspond to our extracted values.}
\label{fig:figurea4a_a4b}
\end{figure}
\section*{References}
\section{Introduction}
Plasmonic nanoparticles exhibit optical field enhancement when localized surface plasmon polariton (SPP) resonances are excited~\cite{LRandE,Vollmer}. The strength of the enhancement depends sensitively on the nanoparticle's environment and geometry~\cite{Kelly_JPCB_2003_107_668}. The enhancement is a vital part of phenomena such as surface-enhanced Raman scattering~\cite{Stiles_AnnRevAnalChem_2008}, and finds application in areas such as biosensing~\cite{Willets_AnnuRevPhysChem_2007_58_267}, monitoring lipid membranes~\cite{Taylor_SciReps_2014_4_5940}, modifying molecular fluorescence~\cite{Kitson_PRB_1995_52_11441} and materials characterization~\cite{Isaac_APL_2008_93_241115}. Localized SPP resonances occur because of the way the free conduction electrons in metal particles respond to light. For many metals at optical frequencies their response is such that the permittivity is negative - a critical requirement if the nanoparticle is to support a plasmon mode.\\
However, metals are not the only materials to exhibit negative permittivity; materials doped with excitonic organic dye molecules are of interest for photonics~\cite{Saikin_Nanophotonics_2013_2_21} and may also possess negative permittivity over a small frequency range~\cite{Philpott_MolCrystLiqCryst_1979_50_139}. Interest in such materials as a means to support surface exciton-polariton (SEP) resonances has recently been rekindled~\cite{Gu_APL_2013_103_021104,Gentile_NL_2014_14_2339,Triolo_arxiv1503_07499}. An example of this class of material is a polymer doped with dye molecules. In a previous work we showed, through experiment and with the aid of a classical model, that polyvinyl alcohol (PVA) doped with TDBC molecules (\textit{5,6-dichloro-2-[[5,6-dichloro-1-ethyl-3-(4-sulphobutyl)-benzimidazol-2-ylidene]-propenyl]-1-ethyl-3-(4-sulphobutyl)-benzimidazolium hydroxide}, sodium salt, inner salt) may support localized surface exciton-polariton modes. We extracted the complex permittivity $\varepsilon(\omega)$ of this material from reflectance and transmittance measurements of thin films using a Fresnel approach~\cite{AandB}. TDBC was chosen because of its tendency to form J-aggregates: this leads to a narrowing of the optical resonance, making them interesting for strong coupling~\cite{Lidzey_Science_2000_288_1620,Dintinger_PRB_2005_71_035424, Torma_RepProgPhys_2015_78_013901}. More importantly in the present context, at sufficiently high concentrations materials doped with such molecules exhibit a negative permittivity; it is this negative permittivity that enables these materials to support localized resonances. In our previous work~\cite{Gentile_NL_2014_14_2339} a two-oscillator Lorentz model~\cite{Fox_OPS,lebedev} was used to calculate the electric field enhancement and field confinement around the nanoparticle supporting the resonance by use of Mie theory~\cite{Mie,B+H}. The field enhancement and confinement we predicted compared favorably with respect to gold nanospheres, albeit over a much narrower spectral range. Here we extend that earlier work by going beyond a simple classical Lorentz oscillator model. In doing so, we are able to explore new transient phenomena and develop a richer microscopic physical picture for the system.\\
In what follows we first outline the elements and assumptions of the quantum model we have used. We then compute the relative permittivity, $\varepsilon(\omega)$, using this model and compare our results with those obtained from a classical model. We next use our model to investigate the steady-state response of nanospheres of possessing this relative permittivity, with a focus on the nature of the localized surface exciton-polariton (LSEP) mode. The response of the same particle to a suddenly turned on applied optical field is then explored, and the occurrence of transient LSEP modes discussed.\\
\section{Theory}\label{sec:theory}
The key difference between the work we report here and previous work based on a bulk, macroscopic approach~\cite{Gentile_NL_2014_14_2339} is that here we develop an effective medium description from a quantum model of the relative permittivity $\varepsilon=1+\chi$, where $\chi$ is the susceptibility. To do so we assume that the molecules in our material can be represented as an ensemble of two-level quantum systems. For a general material, an applied electric field $\bi{E}$ induces a polarization in the material $\bi{P}\propto{\langle\bi{d}\rangle}$, where $\langle\bi{d}\rangle$ is the average dipole moment of each molecule (quantum system). Assuming linearity, $\bi{P}$ is a linear function~\cite{Mandel_Wolf} of $\bi{E}$, given by:
\begin{equation}
\bi{P} = \frac{1}{2}\varepsilon_0\bi{E}(\chi e^{-i\omega t}+\chi^*e^{i\omega t}) = N\langle\bi{d}\rangle
\label{eq:Polarisation1}
\end{equation}
\noindent where $N$ is the number density of quantum emitters, and $\bi{E}$ and $\bi{d}$ are generally time and frequency dependent. To find $\chi$ and hence $\varepsilon$ we need to find $\langle\bi{d}\rangle$. If we adopt a quantum picture, then $\langle\bi{d}\rangle$ becomes the expectation value of the dipole moment and can be computed from the trace of $\rho\bi{d}$, where $\rho$ is the density matrix of the system and $\bi{d}$ is the transition dipole matrix. The density operator $\hat{\rho}$ is defined as $\hat{\rho}=\sum_k p_k|k\rangle\langle k|$, where $p_k$ are the relative probabilities of finding a system element in state $|k\rangle$. In order to find $\rho$, a Hamiltonian that describes the system must be determined. In general, the Hamiltonian for an open quantum system can be expressed as~\cite{Skinner_JPhysChem_1986,Abramavicius_Wurfel_2011},
\begin{equation}
\hat{H}=\hat{H}_0+\hat{H}_B+\hat{H}_I,
\end{equation}
\noindent where $\hat{H}_0$ is the Hamiltonian of the isolated system, $\hat{H}_B$ describes the interaction of $\hat{H}_0$ with the bath, and $\hat{H}_I$ describes the interaction of $\hat{H}_0$ with the applied electric field.\\
For TDBC molecules in a PVA host medium, $\hat{H}_B$ should represent the $3n_m-6=129$ intramolecular~\cite{Harris_Vibrational} vibrational modes (where $n_m$ is the number of atoms \textit{per} molecule) with a multitude of intermolecular modes. These vibrational modes are responsible for induced decay and dephasing in the system~\cite{McCumber_JAP_1963,Harris_JCP_1977}, along with a small shift in the excited state energy of the molecules~\cite{Ambrosek_JPChmA_116_2012}. Rather than determining $\hat{H}_B$ directly we have made a commonly-used simplification, that of incorporating the effects of the bath (vibrationally induced decay and dephasing) phenomenologically by application of the dissipative Lindblad superoperator (see below) and by making the assumption that the small energy shift can be ignored~\cite{Skinner_JPhysChem_1986,Abramavicius_Wurfel_2011,TKobayashiJAggregates2b}.\\
For an ensemble of $n$ two-level emitters (molecules), $\hat{H}_0$ can be written as~\cite{Valleau_JCP_137_034109_2012,Ambrosek_JPChmA_116_2012,TKobayashiJAggregates2},
\begin{equation}
\hat{H}_0 = \hbar\omega_0|0\rangle\langle 0|+\sum_{i=1}^n\left (\hbar\omega^{(1)}_1|1_i\rangle\langle 1_i|+\mathop{\sum_{j=1}^n}_{j\neq i} J_{ij}|1_i\rangle\langle 1_j|\right ),
\label{eq:Hamiltonian_elec}
\end{equation}
\noindent where $|0\rangle$ is the ground state of the nanoparticle, and $|1_i\rangle$ represents a single exciton excited in the nanoparticle, localized on molecule $i$, with the other molecules in their ground states \textit{i.e.} $|1_i\rangle=|0_1,...,1_i,...0_n\rangle$. In this way, only a single exciton is permitted within the ensemble at any time. The first term in brackets in \Eref{eq:Hamiltonian_elec}, $\hbar\omega_1^{(1)}$, represents the average energy eigenvalue of a non-interacting molecule in the excited state (an exciton). The second term corresponds to inter-molecular coupling, with coupling energy $J_{ij}$. The coupling is taken to be F\"{o}rster (dipole-dipole) coupling~\cite{YongShengZhaoOrganicNanophotonics,Valleau_JCP_137_034109_2012} since we assume that the overlap between the wave functions of each site are small. The corresponding interaction Hamiltonian $\hat{H}_I$ modeled in the Schr\"{o}dinger picture~\cite{Parker_1994} and written in the same basis is,
\begin{equation}
\hat{H}_I=\sum^n_{i=1}\left (g_i^{*}|0\rangle\langle 1_i|+g_i|1_i\rangle\langle 0|\right ),
\label{eq:Hamiltonian_Int}
\end{equation}
\noindent where the coupling strength of the dipole to the external optical driving field is defined as $g_i=-\bi{E}(\bi{r}_i)\cdot\boldsymbol{\mu}_i$, where $\boldsymbol{\mu}_i$ is the exciton dipole moment.\\
Although thorough, a density matrix formed using Equations \eref{eq:Hamiltonian_elec}~\&~\eref{eq:Hamiltonian_Int} would have dimension $(n+1)\times (n+1)$, where $n$ is the number of molecules in the system. Given that $n$ can be several thousand for even a moderately-doped $100~nm$ diameter nanosphere, solving for such a large matrix would be computationally very demanding, despite considering only a single exciton in the ensemble. We therefore seek a simpler Hamiltonian for a TDBC-doped nanosphere which approximates the formalism above.\\
\noindent As a first step in this process, we identify that for an ensemble of aggregates (where the monomers within each aggregate are aligned with each other), the intra-aggregate coupling terms dominate~\cite{Knoester_2002}; this enables us to neglect the inter-aggregate coupling terms. By making this approximation, our approach to describe a nanoparticle doped with randomly distributed and randomly oriented aggregates is to first describe a Hamiltonian for a single aggregate, and then to take an orientational average.\\
\noindent The next step is to note that for a single aggregate, nearest-neighbour couplings dominate. The Hamiltonian matrix obtained under this approximation using \Eref{eq:Hamiltonian_elec} for a single aggregate containing $n$ monomers is,
\begin{equation}
H = \left (
\begin{array}{cccccc}
\hbar\omega_0 & 0 & 0 & 0 & \cdots & 0\\
0 & \hbar\omega_1^{(1)} & J & 0 & \cdots & 0\\
0 & J & \hbar\omega_1^{(1)} & J & \cdots & 0\\
0 & 0 & J & \hbar\omega_1^{(1)} & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & 0 & \cdots & \hbar\omega_1^{(1)}
\end{array}\right ),
\label{eq:Hamiltonian_elec_matrix}
\end{equation}
\noindent where $J$ is the nearest-neighbor interaction energy. The eigenvalues and eigenstates for this Hamiltonian matrix are derived in our Supporting Information. The first eigenstate is the ground state $|0\rangle$, with energy eigenvalue $\hbar\omega_0$. The second is a set of excited states where a single exciton is delocalized over the aggregate~\cite{Malyshev_PRB_51_1995,TKobayashiJAggregates2b},
\begin{equation}
|m\rangle=\sqrt{\frac{2}{n+1}}\sum^n_{j=1}\sin\left(\frac{jm\pi}{n+1}\right)|1_j\rangle,
\label{eq:excited_state_gen}
\end{equation}
\noindent where $1<m<n$. The single exciton transition dipole moment of the aggregate $\bi{d}_{01}(m)$ for mode $m$ is related to the transition dipole moment of the monomers $\boldsymbol{\mu}_{01}$ (assuming identical dipole moments) by~\cite{HochstrasserWhiteman_JCM_1972},
\begin{equation}
\bi{d}_{01}(m) = \boldsymbol{\mu}_{01}\sqrt{\frac{1-(-1)^m}{n+1}}\cot\left (\frac{\pi m}{2(n+1)}\right ).
\end{equation}
\noindent This implies that $\bi{d}_{01}(m)$ is zero for even values of $m$. Even for very modest aggregates, $\sim n>6$, the leading eigenstate ($|1\rangle$) gives rise to a transition dipole moment a factor of three stronger than the next eigenstate, \textit{i.e.} the leading eigenstate is the 'brightest'~\cite{PhysRevA_53_2711}. We can take the transition dipole moment of the aggregate as $\bi{d}_{01}\approx\bi{d}_{01}(1)$, and use the two states $|0\rangle$ and $|1\rangle$ as an approximation for the aggregate, where $|1\rangle$, using equation 6, is given by,
\begin{equation}
|1\rangle=\sqrt{\frac{2}{n+1}}\sum^n_{j=1}\sin\left(\frac{j\pi}{n+1}\right)|1_j\rangle.
\label{eq:excited_state_gen2}
\end{equation}
\noindent The eigenvalue of $|1\rangle$ is (\textit{c.f.} Supporting Information, Equation S6),
\begin{equation}
\hbar\omega_1 = \hbar\omega_1^{(1)}-2J\cos\left (\frac{\pi}{n+1}\right ).
\end{equation}
\noindent This allows us to write $\hbar\omega_1=\hbar\omega_1^{(1)}+\Delta$. The excitation energy of the aggregate is shifted from the monomer value by $\Delta$, and this shift arises from the interaction with other molecules in the aggregate. This energy shift has been observed elsewhere for aggregates~\cite{TKobayashiJAggregates,Ambrosek_JPChmA_116_2012}, and is loosely termed the `effect of aggregation'~\cite{YongShengZhaoOrganicNanophotonics}. The magnitude of $\Delta$ is typically hundreds of $meV$~\cite{Valleau_JCP_137_034109_2012}. Therefore, by considering only the ground state and this (brightest) excited state, \Eref{eq:Hamiltonian_elec} can be re-written as,
\begin{eqnarray}
\hat{H}_0&\approx\hbar\omega_0|0\rangle\langle 0|+(\hbar\omega^{(1)}_1+\Delta)|1\rangle\langle 1|\nonumber\\
&=\hbar\omega_0|0\rangle\langle 0|+\hbar\omega_1|1\rangle\langle 1|.
\label{eq:AggModEnergy}
\end{eqnarray}
\noindent The interaction Hamiltonian for the aggregate is written as,
\begin{equation}
\hat{H}_I = G(|0\rangle\langle 1|+|1\rangle\langle 0|),
\label{eq:potential}
\end{equation}
\noindent where $G=\bi{E}\cdot\bi{d}_{01}$ is the coupling strength of the electric field, $\bi{E}$, to the dipole moment, $\bi{d}_{01}$, of the aggregate. The Hamiltonian formed by adding \Eref{eq:AggModEnergy}~\&~\eref{eq:potential} can be applied to an ensemble of randomly-distributed aggregates by taking $G=\bi{E}\cdot\bi{\overline{d}}_{01}$, where $\bi{\overline{d}}_{01}=\bi{d}_{01}/\mathcal{D}$ is the orientational average of the aggregate dipole moments in the system of interest. Here, $\mathcal{D}$ is equal to either $2$ or $3$, corresponding to the number of spatial dimensions in the planar and bulk cases respectively (this orientational average is derived in our Supporting Information: see section 3).\\
Our goal now is to find an effective medium value of $\varepsilon$ at time $t$ and at the frequency of illumination, $\omega$. The first step is to note that $\hat{H}_I/|E(t)|$ defines the transition dipole matrix $\bi{d}$ for the system as a whole. Given that the density matrix ($\rho$) can be used to obtain the expectation value of an observable, we seek $\rho$ using the Liouville-von Neumann equation~\cite{Blum},
\begin{equation}
\dot{\rho}(t,\omega) = -\frac{i}{\hbar}[H,\rho(t,\omega)]+L_D\rho(t,\omega).
\label{eq:LvN}
\end{equation}
\noindent The first term in \Eref{eq:LvN} governs unitary evolution. The Lindblad dissipation superoperator~\cite{Schirmer_Solomon_PRA_70_022107_2004,Breuer_Petruccione_2002} $L_D$, is used to account for the decay and dephasing effects the bath has on the system. In this work, we assume the electron-phonon coupling to be weak at room temperature and for weak fields, and this enables the Born-Markov approximation upon which this formalism relies~\cite{PhysRevA.78.022106} to be used. The total dephasing rate of the transition $|0\rangle\leftrightarrow|1\rangle$ is $\Gamma_{01}$. This quantity is related to the population decay rate for the $|1\rangle\rightarrow|0\rangle$ decay channel, $\gamma_{01}$, and the pure dephasing rate, $\Gamma_{01}^{(d)}$, by~\cite{Schirmer_Solomon_PRA_70_022107_2004},
\begin{equation}
\Gamma_{01} = \frac{\gamma_{01}}{2}+\Gamma_{01}^{(d)}.
\label{eq:rates}
\end{equation}
\noindent The pure dephasing rate, $\Gamma_{01}^{(d)}$, arises from phase-changing interactions of the excitons with the environment~\cite{Wang_Chu_JChPhys_86_3225}, \textit{i.e.} the bath of vibrational modes. A less approximate approach could be adopted~\cite{PhysRevA.78.022106,Del_Pino_NJP_2015}, but assuming a simple rate for $\Gamma_{01}^{(d)}$ is sufficient for our present purposes, that of enabling an illustrative calculation to be carried out.\\
For our two-level system, \Eref{eq:LvN} is used to find $4$ coupled differential equations~\cite{Kavanaugh_Silbey}, the well-known Optical Bloch Equations~\cite{Allen_Eberly} (OBEs). Solving these for the applied cosine potential allows us to use the Rotating Wave Approximation (RWA)~\cite{prl_111_043601_Dorfman}, as detailed in our Supporting Information. To derive an expression for the permittivity, $\langle\bi{d}\rangle$ is determined using $\langle\bi{d}\rangle=\Tr(\rho\bi{d})$. For our aggregates, this value is equal to $\rho_{01}\overline{\bi{d}}_{01}+c.c.$. By choosing the forward-propagating electric field, \Eref{eq:Polarisation1} is re-arranged to give $\varepsilon$ for an ensemble of two-level molecules as,
\begin{equation}
\varepsilon(t,\omega)=\varepsilon_b+\frac{2N}{\varepsilon_0}\frac{|\bi{\overline{d}}_{01}|}{|\bi{E}|}\rho_{01}e^{i\omega t}.
\label{eq:chi}
\end{equation}
\noindent This expression is applicable to an ensemble of molecules with number density $N$, arranged in aggregates, distributed randomly spatially and orientationally (in two or three dimensions), in a medium of background permittivity $\varepsilon=\varepsilon_b$ in the single-exciton regime. This formula holds for weak fields, as we show below. At first glance, \Eref{eq:chi} may appear to diverge for the case where $|\bi{E}|\rightarrow 0$. The resolution to this is that $\rho_{01}$ is linear in $|\bi{E}|$ in this limit: this gives us a field independent permittivity in this limit as expected.
\section{Results and Discussion}
\noindent For our model we require the following parameters: $\bi{\overline{d}}_{01}$, $\hbar\omega_1$, $\gamma_{01}$ and $\Gamma_{01}^{(d)}$. We used our experimental reflectivity and transmittance data (for a $1.46~wt\%$ TDBC:PVA $70~nm$ film~\cite{Gentile_NL_2014_14_2339}) to determine that $\hbar\omega_1=2.11eV\equiv 588~nm$. This agrees with the values obtained by van Burgel~\cite{van_Burgel_JChemPhys_1995_102_20} and Valleau~\cite{Valleau_JCP_137_034109_2012} although it is a slight change from our previous work, where we indicated that the transition occurred at $2.10~eV$ ($590~nm$), with a (weaker) shoulder transition at $2.03~eV$ ($610~nm$). Our revised value follows from an improved Kramers-Kronig analysis of our original data, as outlined in our Supporting Information.\\
From photoluminescence measurements~\cite{Wang_JPCL_5_14331439_2014}, we took the decay rate of $|1\rangle$ to be $\gamma_{01}=1.15\times 10^{12}s^{-1}$ for the aggregate in a PVA host medium. Using Molinspiration~\copyright, we determined the molecular weight and the effective volume of the TDBC molecule. Together with the concentration of the solution, these quantities allowed us to determine the molecular number density to be $N=1.47\times 10^{25}~m^{-3}$. We were then able to estimate the transition dipole moment for TDBC molecules in aggregate form, $\langle\bi{\overline{d}}_{01}\rangle$, and the dephasing rate, $\Gamma^{(d)}_{01}$, by fitting the steady-state solutions for \Eref{eq:chi} to our experimental data for $\varepsilon(\omega)$ by adjusting $|\bi{\overline{d}}_{01}|$ and $\Gamma^{(d)}_{01}$. In this way we found the dipole moment to be $48$ debye (D). The TDBC-doped thin films from which the experimental data were obtained were produced by spin-coating~\cite{Gentile_NL_2014_14_2339}. Previous work to investigate the orientation of dipole moments in thin polymer films produced by spin-coating found that the dipole moments lie predominantly in the plane~\cite{Garrett_Barnes_JMO_2004_51_2287}. Assuming that the TDBC aggregates also lie in the plane of the spun films reported in~\cite{Gentile_NL_2014_14_2339}, then the value of $48$ D we have determined here is a two-dimensionally averaged value, implying that the on-axis dipole moment of an aggregate is $\bi{d}_{01}=97$ D, and the three-dimensionally averaged moment is $32$ D. This three-dimensionally averaged moment compares with the $24$ D estimated by van Burgel \textit{et al.}~\cite{van_Burgel_JChemPhys_1995_102_20} from experiments in solution (3-dimensional).\\
\noindent The dephasing rate, $\Gamma^{(d)}_{01}$, was found to be equal to $17~meV$, which is $\approx\kappa_BT$ as expected~\cite{Valleau_JCP_137_034109_2012}. To provide additional support for our value of $\Gamma^{(d)}_{01}$, we extracted and modeled $\varepsilon(\omega)$ from the reflectance and transmittance data from a $5.1~nm$ thick film obtained by Bradley \textit{et al.}~\cite{Bradley_AM_2005_17_1881}. We determined $\Gamma^{(d)}_{01}$ to be around $13~meV$, a value comparable with our own, bearing in mind that different bath spectral densities associated with differences in the host and substrate may change the value of $\Gamma^{(d)}_{01}$. Our results for $\varepsilon(\omega)$ against experimentally-determined data for our film are displayed in \Fref{fig:2levelTDBCdielectric}.
\begin{figure}[!htbm]
\includegraphics[width=\columnwidth]{figure1color}
\caption{Extracted relative permittivity $\varepsilon(\omega)$ from a $1.46~wt\%$ TDBC:PVA film (circles for real part and crosses for imaginary part)~\cite{Gentile_NL_2014_14_2339}. The theoretical fit for $\varepsilon(\omega)$ indicated by the solid and dashed lines corresponds to a concentration of $1.46~wt\%$ ($3.22~wt\%$) for a two (three) dimensional distribution of dipoles.}
\label{fig:2levelTDBCdielectric}
\end{figure}
\noindent The permittivity of the thin film shown in \Fref{fig:2levelTDBCdielectric} has been modeled assuming the dipole moments of the aggregates are randomly oriented in the plane of the film. In what follows we wish to look at the optical response of a nanoparticle. For generality, and to ensure we consider an isotropic system, we will consider the dipole moments of the aggregates to be randomly oriented in three dimensions. Making this assumption requires us to increase the number density of our molecules from $1.47\times 10^{25}~m^{-3}$ to $3.29\times 10^{25}~m^{-3}$. In this way, our nanoparticle will be comprised of a material that has the same permittivity as that shown in \Fref{fig:2levelTDBCdielectric}. In all the calculations that follow we use this number density, which corresponds to a concentration of TDBC in PVA of $3.22~wt\%$.\\
\subsection{Numerical Results: Steady-State}\label{sec:SteadyStateResults}
We now explore theoretically the Mie~\cite{Mie,B+H} absorption efficiency spectra $Q_{abs}(\omega)$ for a $100~nm$ diameter nanosphere of $3.22\%$ TDBC:PVA, assuming a volume distribution of dipole orientations, based on $\varepsilon(\omega)$ calculated using \Eref{eq:chi}. In practice, the applied optical field we model here might be a laser beam. For a $1~mW$ laser with a spot diameter of $1.5~mm$, the strength of the electric field of our incident optical field would be equal to $462~Vm^{-1}$; we assume this value here.\\
In a $100~nm$ diameter nanosphere of our material, there are on average $n=1.72\times 10^4$ molecules. Note that it is the number of molecules and by extension their number density, which is the important quantity (and is used for $N$ in \Eref{eq:chi}) rather than the number density of aggregates, since each molecule provides a potential site for exciton excitation. To check the validity of our assumption that multi-exciton and nonlinear effects~\cite{Wang_Chu_JChPhys_86_3225} can be neglected, we computed the maximum expectation value of the number of excitons in the nanosphere ($n_{ex}=\max(\rho_{11})n$) using Equations \eref{eq:AggModEnergy} and \eref{eq:potential} in \Eref{eq:LvN}. We found that $n_{ex}/n\ll 1$ holds for laser powers of up to $10^2~W$ with a spot size of $1.5~mm$. Given that our laser power is $1~mW$, we assumed that the single-exciton linear regime is sufficient to describe the system under this illumination power.\\
\begin{figure}[!htbm]
\centering
\includegraphics[width=\columnwidth]{figure2color}
\caption{Mie calculations for the absorption efficiency $Q_{abs}(\omega)$ in the steady-state for a $100~nm$ diameter nanosphere of $3.22~wt\%$ TDBC:PVA using the values for $\varepsilon(\omega)$ from experiment (dashed line), from two-level OBEs (solid line), and from our previous Lorentz oscillator model (dotted line). The material absorption coefficient $\kappa$ (imaginary part of the refractive index), normalized to unity, is also plotted for illustrative purposes (long dashed line). Inset: a 3D representation of the aggregated emitters (assuming brick-stone aggregation, with 15 molecules \textit{per} aggregate) randomly distributed in a $100~nm$ diameter nanosphere.}
\label{fig:figure2}
\end{figure}
\noindent In \Fref{fig:figure2} we plot the absorption efficiency $Q_{abs}(\omega)$ for a $100~nm$ diameter nanosphere, calculated for a variety of permittivities; in each case the absorption efficiency is calculated using Mie theory \cite{Mie,B+H}. Calculated values for $Q_{abs}$ based upon the permittivity obtained using our improved analysis of experimental data are shown in \Fref{fig:figure2} as a dashed line. Our quantum theoretical spectrum for $Q_{abs}$, using $\varepsilon(\omega)$ from \Eref{eq:chi}, is shown as the solid line. This theoretically derived spectrum provides a close match to the extracted data, most importantly for energies in the region of interest below $2.22~eV$. For energies exceeding $2.2~eV$, there is a limb in the extracted data (dashed curve) which might perhaps be attributed to inhomogeneous (non-Lorentzian) broadening which is not accounted for using the OBEs. Also displayed in \Fref{fig:figure2} is the result for $Q_{abs}$ using a best-fit classical Lorentz oscillator model (the parameters for which can be found in our Supporting Information) shown as a dotted line. It can be seen that the quantum model outlined in the present paper provides an improved fit to the experimental data. We attribute this to the inclusion of dephasing ($\Gamma_{01}^{(d)}$) in the model: if $\Gamma_{01}^{(d)}$ were set to zero, a Lorentz model would be recovered in the steady state, and the single damping term in the Lorentz model would have to accommodate both decay and dephasing. Therefore, by including dephasing, the actual physical value of the decay rate $\gamma_{01}$ can be included to achieve an accurate result for $\varepsilon$.\\
It is interesting to note a key feature shown by the data in \Fref{fig:figure2}: $Q_{abs}$, reaches its peak value at $2.16~eV$ ($574~nm$). This is in contrast to the absorption coefficient, $\kappa(\omega)$, which peaks at $2.12~eV$ ($586~nm$), shown as a long dashed line in \Fref{fig:figure2}. This difference in spectral position arises because the peak in $Q_{abs}$ is not due simply to absorption: rather, it is due to the excitation of a localized SEP mode~\cite{Gentile_NL_2014_14_2339}. Confirmation of this interpretation comes from two sources. First, in the quasistatic limit the polarizability of the nanosphere follows the Clausius-Mossotti condition, for which resonance occurs when $\varepsilon$ is real-valued and equal to $-2$ (when the nanosphere is in free space)~\cite{NandH}. From \Fref{fig:2levelTDBCdielectric} this can be seen to be approximately true for our absorbing $100~nm$ diameter nanosphere, as the permittivity value at the wavelength of peak absorption efficiency, $Q_{abs}$, is complex and equal to $\varepsilon=-2.251+1.728i$. This difference from $\varepsilon=-2$ originates from the fact that $\varepsilon=-2$ only gives the resonance condition if the imaginary part of $\varepsilon$ is zero; the complex nature of the permittivity changes the spectral location of the absorption peak. Second, $Q_{abs}$ near the peak goes well above unity: this is associated with field enhancement~\cite{Vollmer}, another signature of a resonant mode. The enhanced electric field in the vicinity of the nanosphere is illustrated graphically in \Fref{fig:figure3a_3b}(a), together with direction of power flow shown by the Poynting vector $\bi{S}$.\\
\begin{figure}
\includegraphics[width=\columnwidth]{figure3a_3b}
\caption{The time-averaged electric field strength normalized to the incident field strength (color plot) and the Poynting vector $\bi{S}$ (arrows) in the vicinity and on the surface of the $100~nm$ $3.22~wt\%$ TDBC:PVA nanosphere, with incident power flow along the positive z-direction. Data are calculated for incident photon energies of a). $2.16~eV=574~nm$ and b). $2.12~eV=586~nm$ corresponding to peak absorption efficiency and $\kappa$ respectively.}
\label{fig:figure3a_3b}
\end{figure}
\noindent In ~\Fref{fig:figure3a_3b}, the incident electric field is polarized in the x-direction. The Poynting vector arrows shown in the figure were calculated at starting points for which $z=-200~nm$ and $x=0~nm$, linearly spaced in the range $-200~nm\leq y\leq 200~nm$. Subsequent points for evaluation of the Poynting vector were taken at $10~nm$ steps in the direction of the Poynting vector at each point, resulting in the flux lines shown. The power flow in \Fref{fig:figure3a_3b}(a) shows that incident light is drawn towards and absorbed by the nanosphere for starting positions up to around $130~nm$ from the central position of the nanosphere. This demonstrates that at this energy, the nanosphere absorbs more light than the light geometrically striking it~\cite{BohrenAmJPhys_1983}, and hence $Q_{abs}>1$. In comparison, absorption at the transition energy, \textit{i.e.} at $2.11~eV$ is seen only as a shoulder mode in the absorption efficiency of the nanosphere (\Fref{fig:figure2}) and the efficiency does not exceed unity. The power flow around the nanosphere for the energy at which $\kappa$ peaks ($2.12~eV$) is shown in \Fref{fig:figure3a_3b}(b), and the enhancement of the field is much weaker than for excitation on resonance at $2.16~eV$.
\subsection{Numerical Results: Time Domain}\label{sec:TDBCtd}
We now turn our attention to the time domain. Our theoretical model for dynamic processes in two-level quantum systems subject to a perturbing cosine electric field is similar to models considered elsewhere~\cite{Fox,Foot,Slowik_PRB_88_195414_2013,OptComm_283_23532355_2010}, but here the observable of interest arises from the temporal evolution of the coherences of the density matrix, rather than the populations. The dynamics of a two-level ensemble subject to a pulse potential has been the subject of recent investigation~\cite{Sukharev_ACSNANO_8_807817_2014}, but here we investigate a rather different case: that of a cosine potential of fixed amplitude that is switched instantaneously on at some moment in time. We do this to provide an easily soluble model that illustrates the time-dependent phenomena we wish to discuss.\\
\begin{figure}[!htbm]
\centering
\includegraphics[width=\columnwidth]{figure4color}
\caption{Calculated $Q_{abs}(t,\omega)$ for a $100~nm$ diameter nanosphere of $3.22\%$ TDBC:PVA warming up a pure state at $t=0$ with a $1~mW$ laser set at five different detunings $\delta$ from the exciton transition.}
\label{fig:figure4}
\end{figure}
By using \Eref{eq:chi} to calculate $\varepsilon(t)$ for a given illumination frequency $\omega$ as before, $Q_{abs}(t,\omega)$ can be determined and its temporal behavior examined. To do this, we again use Mie theory. This is an approximation since the fields scattered in Mie theory are assumed to be instantaneous. Given that the dynamics seen in \Fref{fig:figure4} evolve over a few femtoseconds and that light propagates over a length scale three times the size of the nanoparticle during a single femtosecond, and that the nanoparticle is illuminated with an electric field of constant amplitude, this approximation is deemed to hold. Mie theory can therefore be used to give a quasi-instantaneous picture of the absorption.\\
$Q_{abs}(t)$ is shown in \Fref{fig:figure4} for five different detunings, $\delta=\hbar\omega_1-\hbar\omega$, from the transition at $\hbar\omega_1=2.11~eV$. We assume that all the molecules in the nanoparticle are initially in their ground state. At $t=0$ we turn on our field abruptly. We see that a steady-state response is attained after $>200~fs$, but interestingly, for $\delta=0.09~eV$, $Q_{abs}(t)$ repeatedly exceeds unity in spite of the steady-state value of $Q_{abs}$ being below unity at this detuning. Since $Q_{abs}(t)>1$ implies field enhancement, these data are indicative of a transient LSEP mode being present at early times. The time-dependent behavior comprises two contributions: the first is the oscillatory behavior arising from Rabi oscillations; the second is the transient effects associated with the sudden turning-on of the field. In the latter, the magnitude of the density matrix coherences exceed their steady-state values for tens of femtoseconds, resulting in larger values of $\varepsilon$ and hence different absorption properties to the steady-state.\\
If $\varepsilon$ passes through the Clausius-Mossotti condition for the nanoparticle as it approaches its steady-state value, $Q_{abs}$ exceeds unity, implying a transient LSEP mode. This is seen best for a detuning of $0.091~eV$ between $0-30~fs$. The Rabi oscillations follow the generalized Rabi frequency $\tilde{\Omega}_R$, given by~\cite{PhysRevLett_46_1192},
\begin{equation}
\tilde{\Omega}_R=\sqrt{\Omega_1^2+\delta^2},
\label{eq:generalRabi}
\end{equation}
\noindent where, $\Omega_1\ll\delta$, and $\tilde{\Omega}_R\rightarrow\delta$ in this case, $\Omega_1$ and $\delta$ are the Rabi frequency and the detuning respectively. These Rabi oscillations are naturally convoluted with the transient effects. This implies, together with the short timescales involved in the system, that it would be a challenge to see these transitory effects, but might perhaps be possible~\cite{Vasa_NatPhot_2013_7_128}. Critical to the transient LSEP lifetime is $\Gamma^{(d)}_{01}$. If this dephasing could be reduced without losing the transient negative permittivity that is essential for field enhancement (and field confinement), then the transient timescale of the system would be increased up to a maximum of $1/\gamma_{01}$. This corresponds to the picosecond regime for our TDBC:PVA system. Under this circumstance, transient LSEP modes would become more easily observable.\\
\section{Conclusions}\label{sec:conclusions}
We have re-evaluated the measurements reported in our previous work and have obtained an improved permittivity for our J-aggregate-doped $1.46~wt\%$ TDBC:PVA polymer film. Using a quantum-mechanical framework we have given support to our previous investigation based on a classical analysis~\cite{Gentile_NL_2014_14_2339}, that TDBC doped nanoparticles can exhibit a localized surface exciton-polariton (LSEP) mode. We have used a quantum model to show that these nanoparticles may also exhibit transient LSEP modes in the sub-picosecond regime. These results help strengthen the idea that molecular excitonic materials provide an interesting alternative upon which to base nanophotonics~\cite{Saikin_Nanophotonics_2013_2_21}. By using molecular materials the possibility of bottom-up approaches such as supramolecular chemistry and self-assembly can be brought to bear on the production of nanophotonic structures.
\ack{The work was supported in part by the UK Engineering and Physical Sciences Research Council, and in part by The Leverhulme Trust.}
\section*{References}
|
2302.12342
|
\section{Introduction}
This work is motivated by the question of what dynamical features of a
system are reflected by its action in first homology group. In the particular case of a toral
endomorphism, this action is represented by a homomorphism on $\mathbb{Z}^2$ and
we wish to understand what role this action plays in understanding which maps
are transitive.
It is clear that, by itself, knowledge of the action in homology cannot suffice to guarantee
transitivity, since every homotopy class contains maps with attractors. Some
extra structure is needed.
In previous works we have considered this question for maps which are
conservative \cite{A} or for which the non-wandering set is the whole of $\mathbb{T}^2$
\cite{WR}. In the present paper we consider the question in the setting of
partial hyperbolicity.
In this paper, an endomorphism is synonymous with non-invertible local diffeomorphism.
A partially hyperbolic endomorphism is a local
diffeomorphism $f:\mathbb{T}^2 \to
\mathbb{T}^2$ admitting an unstable cone-field $\mathcal{C}^u: p \mapsto
\mathcal{C}^u(p)$, where $\mathcal{C}^u(p)$ is a closed cone in
$T_{p}\mathbb{T}^2$, and the constants $\ell>0$ and $\lambda >1$ satisfying:
\begin{enumerate}[label=(\roman*)]
\item $\mathcal{C}^u$ is $df^{\ell}$-invariant, that is,
\[
Df^{\ell}(\mathcal{C}^u(p)) \subseteq
\mathrm{int}(\mathcal{C}^u(f^{\ell}(p)))\cup \{0\}
\]
where
$\mathrm{int}(\mathcal{C}^u(p))$ denotes the interior of
$\mathcal{C}^u(p)$;
\item for every $v \in \mathcal{C}^u(p), \ \ \|Df^{\ell}(v)\|\geq
\lambda\|v\|$.
\end{enumerate}
The action of an endomorphism in the first homology group is given by a $2
\times~2$ matrix with integer entries. We refer to this matrix (and the maps it induces on $\mathbb{R}^2$ and $\mathbb{T}^2$) as the \emph{linear part} of the endomorphism.
\begin{theorem}\label{thm-A}
Let $f$ be a partially hyperbolic endomorphism whose linear part has
integer eigenvalues $\lambda_1, \lambda_2$ with $|\lambda_1|\geq
|\lambda_2|>1$. Suppose that
\begin{align}\label{sv-exp}
|\det (Df_p)|>|\lambda_1| \ \ \text{for every p} \, \in
\mathbb{T}^2.
\end{align}
Then $f$ is transitive.
\end{theorem}
Condition \eqref{sv-exp} says that the Jacobian of $f$ at every point is larger
than the spectral radius of the linear part of $f$. An endmomorphism with
this property is said to be \emph{strongly volume expanding}.
Recall that a linear map is \emph{expanding} if all its eigenvalues have modulus greater than one.
Consequently, the condition $|\lambda_1| \geq | \lambda_2|>1$ will sometimes be referred to by saying that $f$ has \emph{expanding linear part with integer eigenvalues.}
Theorem \ref{thm-A} is similar in flavour to a theorem by Hertz, Ures and Yang
\cite{MR4419061} about partially hyperbolic diffeomorphisms on $\mathbb{T}^3$. Using
the hypothesis that $f$ is $C^2$ and a slightly weaker version of
\eqref{sv-exp} (they allow for equality in \eqref{sv-exp} in a set with zero
leaf volume along unstable leaves), they conclude that the strong stable and
unstable foliations are $C^1$ robustly minimal, which in particular implies
$C^1$ robust transitivity. Here we require less regularity but a
slightly stronger condition on the Jacobian than that of \cite{MR4419061}.
We point out that the strongly volume expanding assumption in
Theorem~\ref{thm-A}
cannot be relaxed to $|\det (Df_p)|>|\lambda_2|$, when $|\lambda_2|$ is
strictly smaller that $|\lambda_1|$. To see why, consider the following example.
\begin{ex} \label{invariant_stripe_example}
Let $F$ be the direct product of
two maps $f,g : S^1 \to S^1 $, where $f(x) = 3 x \mod 1$ and $g(x)$ a map
homotopic to $x \mapsto 2x \mod 1$, satisfying
\begin{enumerate}
\item $g(0) = 0$
\item $g'(0) <1$
\item $ \frac{2}{3} < g'(x) < 3, \ \forall x \in S^1$.
\end{enumerate}
Then $F$ has Jacobian larger than $2$ everywhere but is
clearly not transitive, as $g$ has an attractor at $0$.
\end{ex}
It should be noted that partial hyperbolicity and the strongly volume expanding condition are both persistent under $C^1$-perturbations. As a consequence:
\begin{corA}
Suppose that $f$ is a partially hyperbolic endomorphism whose linear part is expanding with integer eigenvalues. If $f$ is strongly volume expanding, then $f$ is $C^1$ robustly transitive.
\end{corA}
An interesting question is whether the property of being strongly volume
expanding in itself is enough to ensure transitivity. We suspect that the answer
is no. In any case, it is clear that the hypotheses of Theorem~\ref{thm-A} are merely a
sufficient condition for transitivity. It is far from being necessary.
\begin{comment}
\begin{ex}\label{ex1}
$f:\mathbb{T}^2 \to \mathbb{T}^2, \ \ f(x,y)=(f_1(x),f_2(y)),$
where $f_1(x)= 3x \,\, (\mathrm{mod} \,\, 1)$ and $f_2$ is a
deformation of the doubling map, $x \mapsto 2x \,\,
(\mathrm{mod} \,\, 1)$, around the fixed point $0 \in
\mathbb{S}^1$ which $0$ becomes an attractor. Then, its linear
part a diagonal matrix with $\lambda_1=3$ and $\lambda_2=2$.
Since $\{0\}\times\mathbb{S}^1$ is attractor, we have that $f$
is not transitive.
\end{ex}
\end{comment}
\subsection{Transitivity of specially partially hyperbolic endomorphisms}
\label{special}
Whenever $f$ is a partially hyperbolic endomorphism, we may define the
\emph{center direction} at a point $p$ by
\[E_p^c = \{ v \in T_x \mathbb{T}^2: Df_p^n( v) \notin \mathcal{C}^u(f^n(p)) \ \forall n \geq 0
\} \cup \{ 0\}.\]
In contrast to the case of partially hyperbolic diffeomorphisms, however, there
may not be a well defined unstable direction.
More precisely, given a choice of pre-orbit $\hat{p} = ( \ldots, p_{-2}, p_{-1},
p_0)$ of $p$, i.e. a sequence of points in $\mathbb{T}^2$ satisfying $p_0 = p$ and
$f(p_{i-1}) = p_i$ for every $i \geq 0$, we define the direction
\begin{equation} \label{unstable_direction}
\hat{E}_{\hat{p}}^u = \bigcap_{n \geq 0} Df^n(\mathcal{C}^u(p_n)) .
\end{equation}
In general, $\hat{E}_{\hat{p}}^u$ will depend on the particular choice of pre-orbit
$\hat{p}$. In the exceptional case where it doesn't, we say that
$f$ is a \emph{specially partially hyperbolic endomorphism} and write $E_p^u =\hat{E}_{\hat{p}}^u$. In this case, $E_p^u$ can easily be shown to be $f$-invariant
and continuous.
For specially partially hyperbolic endomorphisms we are able to give a full
characterization of transitivity both in terms of conjugacy and in terms of
absence of periodic or wandering annuli. By an \textit{annulus} we mean an open
subset $\mathbb{A}$ of $\mathbb{T}^2$ homeomorphic to $(-1,1)\times
S^1$. We say that an annulus $\mathbb{A}$ is \textit{periodic} if
there is $n\geq 1$ such that $f^n(\mathbb{A})=\mathbb{A}$; and it is
\textit{wandering} if $f^n(\mathbb{A})\cap \mathbb{A}=\emptyset$ for every $n
\geq 1$.
\begin{theorem}\label{thm-B} Let $f$ be a specially partially hyperbolic
endomorphism with linear part $A$. Suppose that $A$ has integer eigenvalues
$|\lambda_1| > |\lambda_2| >1$. Then the following are equivalent:
\begin{enumerate}[label=\rm{\alph*)}]
\item $f$ is transitive; \label{f_transitive}
\item $f$ topologically conjugated to $A$; \label{f_conjugated}
\item $f$ admits neither a periodic nor a wandering annulus. \label{f_no_annulus}
\end{enumerate}
\end{theorem}
When they exist, periodic and wandering annuli are necessarily saturated by
unstalbe leaves. We can therefore restate Theorem~\ref{thm-B} as:
\begin{corB} Let $f$ be a specially partially hyperbolic
endomorphism with linear part $A$ having eigenvalues
$|\lambda_1| > |\lambda_2| >1$. Then one of the following holds:
\begin{enumerate}[label=\rm{\alph*)}]
\item $f$ is transitive and topologically conjugated to $A$;
\label{not_trans}
\item $f$ is not transitive and there is a periodic or wandering
annulus saturated by the unstable foliation.
\label{no_saturated_annulus}
\end{enumerate}
\end{corB}
Note that, in virtue of being a direct product,
Example~\ref{invariant_stripe_example} is in fact specially partially
hyperbolic, so it serves as an example for the non-transitive case in
Theorems~\ref{thm-B} (and B'). In that example, the product of the basin of the
attractor for $g$ with a circle is a periodic annulus.
\subsection*{Injectivity points} If $f$ endomorphism having expanding linear
part $A$ (here, the eigenvalues are not necessarily integers) then there is a
surjective continuous map $h:\mathbb{T}^2 \to \mathbb{T}^2$ such that
\begin{align} \label{semiconj}
h\circ f= A\circ h.
\end{align}
The existence of $h$ was proved by Franks in \cite{MR0271990} for
diffeomorphisms with hyperbolic linear part, but the proof can be easily
adapted to endomorphisms with expanding linear part. (We remark that if the
linear part is a hyperbolic endomorphism, such a map may not exist. See
\cite{2104.01693}.) The map $h$ is called a \textit{semiconjugacy} from $f$ to
$A$. When $h$ is a homeomorphism we say that it is a \textit{conjugacy} between $f$ and $A$.
We say that a point $p \in \mathbb{T}^2$ is an \textit{injectivity point of
$h$} when $h(q)=h(p)$ implies that $q=p$. Equivalently, $h^{-1}(h(p))=\{p\}$.
Injectivity poins will play an important role in the proofs of Theorems
\ref{thm-A} and \ref{thm-B}. Indeed, we are going to use the strongly volume
expanding codition together with Blichfedt's Theorem (see section \ref{section3}) to show that, under
the assumptions of Theorem \ref{thm-A}, the set of injectivity points is dense
in $\mathbb{T}^2$. Then transitivity follows from the following (possibly folcloric)
theorem:
\begin{theorem}\label{folkloric}
Let $f$ be an endomorphism having an expanding linear part $A$. If the
injectivity points of $h$ are dense in $\mathbb{T}^2$ then $f$ is
transitive.
\end{theorem}
In the proof of Theorem \ref{thm-B} we prove that either the set of injectivity
poins is the whole of $\mathbb{T}^2$, or there exists an invariant annulus $\mathbb{A}$
which is collapsed by the semi-conjugacy to a curve.
\subsection*{Acknowledgments}
We would like to thank Rafael Potrie and Enrique Pujals for their fruitful
comments suggestions.
\section{Preliminaries and proof of Theorem \ref{folkloric}} \label{section1}
The aim of this section is to bring together some general facts about
endomorphisms having expanding linear part and, in the end, to prove
Theorem~\ref{folkloric}.
\subsection{Some facts about the semi-conjugacy}
Let us establish some notation. We denote by
$(\mathcal{K}(\mathbb{T}^2),d_{\mathcal{H}})$ the metric space of all compact
subsets of $\mathbb{T}^2$ endowed with the \textit{Hausdorff distance}. Let
$f:\mathbb{T}^2 \to \mathbb{T}^2$ be an endomorphism having expanding linear
part $A$ and let $h$ be the semi-conjugacy defined by \eqref{semiconj}.
We define $h^{-1}:\mathbb{T}^2 \to \mathcal{K}(\mathbb{T}^2)$ by $x \mapsto h^{-1}(x)$
and set $\phi: \mathbb{T}^2 \to \mathcal{K}(\mathbb{T}^2)$ by $x \mapsto h^{-1} (h(x))$.
Throughout this work, we denote by $\tilde{\ast}$ the lift of $\ast$ to the
universal cover $\mathbb{R}^2$. Here $\ast$ may be a point, a map, a set or a foliation.
The only exception is the matrix $A$ that will
denote both the linear endomorphism on $\mathbb{T}^2$ and its lift to $\mathbb{R}^2 \to \mathbb{R}^2$.
An endomorphism $f$ has linear part $A$ if and only if
\begin{equation}
\tilde{f}(\tilde{x}+v) = \tilde{f}(\tilde{x})+Av
\label{lift_property}
\end{equation}
for every $\tilde{x} \in \mathbb{R}^2$ and $v \in \mathbb{Z}^2$.
The lift $\tilde{h}$ of the semi-conjugacy defined by \eqref{semiconj} is a surjective continuous map satisfying
\begin{align}\label{lift-semi}
\tilde{h}\circ \tilde{f} = A\circ \tilde{h} \ \ \mathrm{and} \ \ \|\tilde{h}-id\|<\kappa
\end{align}
for some $\kappa > 0$.
Let $\pi: \mathbb{R}^2 \to \mathbb{T}^2$ be the natural projection.
It is easy to see that $\tilde{\phi}$ (the lift of $\phi$) is given by the map $\tilde{x} \mapsto \tilde{h}^{-1}(\tilde{h}(\tilde{x}))$
and that $\tilde{f}(\tilde{\phi}(\tilde{x}))
= \tilde{\phi}(\tilde{f}(\tilde{x}))$ for every $\tilde{x} \in \mathbb{R}^2$.
\begin{prop}\label{prop-Phi}
Let $f: \mathbb{T}^2 \to \mathbb{T}^2$ be an endomorphism with expanding linear part $A$. Then the following hold:
\begin{enumerate}[label=$\mathrm{(\alph*)}$]
\item \label{characterization} There is $r>0$ such that
\[\tilde{\phi}(\tilde{x}) =\bigcap_{k \geq 0}
\tilde{f}^{-n_k}(B(\tilde{f}^{n_k}(\tilde{x}),r)),\]
for each $\tilde{x} \in \mathbb{R}^2$ and each sequence
$n_k \to \infty$.
\item \label{growing_ball} There exists $r_0$ and $k\geq 1$ such that
$\tilde{f}^k(B(\tilde{x}, r)) \supset \overline{B(\tilde{f}^k (\tilde{x}), r)}$
for every $\tilde{x} \in \mathbb{R}^2$ and $r>r_0$, where $B(\tilde{x}, r)$
is the ball of radius $r$ centred at $\tilde{x}$.
\item \label{tphi_connected} For each $\tilde{x} \in \mathbb{R}^2$,
$\tilde{\phi}(\tilde{x})$ is a connected set.
\item \label{tldh_connected} For each $\tilde{x}$, $\tilde{h}^{-1}(\tilde{x})$ is connected.
\item \label{preimage_of_connected} For each compact connected
set $\mathcal{C}$ in $\mathbb{T}^2$, the set
$\tilde{h}^{-1}(\mathcal{C})$ is connected.
\end{enumerate}
\end{prop}
\begin{proof} The inclusion ``$\supset$'' in \ref{characterization} holds for
every $r>0$. This follows by noting that iterates of any two points in
the set on the right remain a bounded distance from one another. Since
the linear part is expanding, this can only happen if they have the
same image under $\tilde{h}$.
The inclusion ``$\subset$'' in \ref{characterization} holds for any
$r>2\kappa$ where $\kappa>0$ is chosen such a way that
$\|\tilde{h}-id\| \leq \kappa$. To see this, let $\tilde{y} \in \tilde{\phi}(\tilde{x})$.
Then $\tilde{h}(\tilde{y}) = \tilde{h}(\tilde{x})$ and, for $n\geq 0$,
\begin{align*}
\tilde{h}(\tilde{f}^n(\tilde{y})) =A^n(\tilde{h}(\tilde{y}))
=A^n(\tilde{h}(\tilde{x})) =\tilde{h}(\tilde{f}^n(\tilde{x})).
\end{align*}
Hence
\begin{align*} \|\tilde{f}^n(\tilde{y})-\tilde{f}^n(\tilde{x})\| \leq
\|\tilde{f}^n(\tilde{y})-\tilde{h}(\tilde{f}^n(\tilde{y}))\| +
\|\tilde{h}(\tilde{f}^n(\tilde{x}))-\tilde{f}^n(\tilde{x})\| < r,
\end{align*} and we conclude that $\tilde{y} \in \bigcap_{n\geq 0}
\tilde{f}^{-n}(B(\tilde{f}^n(\tilde{x}),r))$.
Item \ref{growing_ball} holds because $A$ is expanding and $\tilde{f}$ is a
bounded distance from $A$.
To show \ref{tphi_connected}, fix $k$ and $r$ such that
\ref{growing_ball} holds. If necessary, increase $r$ so that
\ref{characterization} holds as well. Consider the sets $D_n(r) =
\tilde{f}^{-n}(B(\tilde{f}^{n}(\tilde{x}),r))$. From \ref{characterization} we have
that $\tilde{\phi}(\tilde{x}) = \bigcap_{k \geq 0} D_{kn}$. Now,
\[\tilde{f}^{k(n+1)}(\overline{D_{k(n+1)}}) = \overline{B(\tilde{f}^{k(n+1)}(\tilde{x}),
r)} \subset \tilde{f}^k (B(\tilde{f}^{nk}(\tilde{x}), r)) = \tilde{f}^{k(n+1)}(D_{nk}),\] so
that $\overline{D_{k(n+1)}} \subset D_{nk}$. Hence $\tilde{\phi}(\tilde{x})$ can be
written as $\bigcap_{n \geq 0} \overline{D_{nk}}$. In other words,
$\tilde{\phi}(\tilde{x})$ is the intersection of
a decreasing sequence of comact connected sets, so it is itself
connected.
Item \ref{tldh_connected} is an immediate consequence of \ref{tphi_connected}.
We prove \ref{preimage_of_connected} by contradiction. First note that
$\tilde{h}^{-1}(\mathcal{C})$ is necessarily compact, since $\tilde{h}$ is a
bounded distance from the identity. Suppose that
$\tilde{h}^{-1}(\mathcal{C})$ is not connected. Then there are disjoint
compact sets $A$ and $B$ such that $\tilde{h}^{-1}(\mathcal{C})=A\cup B$.
Hence $\mathcal{C}=\tilde{h}(A)\cup \tilde{h}(B)$ with both $\tilde{h}(A)$ and
$\tilde{h}(B)$ compact. Now, since $\mathcal{C}$ is connected, there exists
some point $p \in \tilde{h}(A) \cap \tilde{h}(B)$. But then $\tilde{h}^{-1}(p)$ can
be written as the disjoint union $(\tilde{h}^{-1}(p) \cap A) \cup
(\tilde{h}^{-1}(p) \cap B)$, both of which are closed. That is absurd.
\end{proof}
\begin{cor}\label{cor-connected}
Let $f: \mathbb{T}^2 \to \mathbb{T}^2$ be an endomorphism with expanding linear part $A$. Then the following hold:
\begin{enumerate}[label=$\mathrm{(\alph*)}$]
\item For each $p \in \mathbb{T}^2$, the set $h^{-1}(p)$ is a
connected set.
\item For each closed connected set $\mathcal{C}$ in
$\mathbb{T}^2$, the set $h^{-1}(\mathcal{C})$ is
connected.
\item For each $p \in \mathbb{T}^2$, $f(\phi(p))=\phi(f(p))$.
\end{enumerate}
\end{cor}
\subsection{Upper semicontinuity}
Recall that a map $\psi: \mathbb{T}^2 \to \mathcal{K}(\mathbb{T}^2)$ is \emph{upper
semicontinuous} if given any $x \in \mathbb{T}^2$ and any open set $U$ in $\mathbb{T}^2$
containing $\psi(x)$ there exists $\delta>0$ such that if $\psi(y) \subset U$
whenever $y \in B_\delta (x)$. Recall that $h^{-1}$ is a map from $\mathbb{T}^2$ to
$\mathcal{K}(\mathbb{T}^2)$.
\begin{prop}\label{usc-prop} The map $h^{-1}$ is upper semicontinous. Furthermore,
if $h^{-1}(q)$ is a unitary set then $q$ is a continuity point of $h^{-1}$.
\end{prop}
The proof of Proposition \ref{usc-prop} is straightforward and holds for any continus map
$h: \mathbb{T}^2 \to \mathbb{T}^2$, i.e. it has nothing to do with $h$ being a semi-conjugacy.
As an immediate consequence we have:
\begin{lemma} \label{lemma:open}
Let $p$ pe an injectivity point of $h$. Then, given any neighbourhood
$U$ of $p$, $h(U)$ has non-empty interior and $h(p)$ is an interior
point of $h(U)$.
\end{lemma}
Another consequence of Proposition \ref{usc-prop} is:
\begin{cor}
The map $\phi$ is upper semicontinuous. If $\phi(p)=\{p\}$ then $p$ is
continuity point of $\phi$.
\end{cor}
In other words, $\phi(p)=\{p\}$ if and only if $p$ is an
injectivity point of $h$. Furthermore, the set of
continuity points of $\phi$ is a residual (dense $G_\delta$) set in $\mathbb{T}^2$ because
of:
\begin{sc-lemma}
Given $M$ a Baire space, $N$ a compact metric space and $\Phi:M \to
\mathcal{K}(N)$ an upper semicontinous map, the set of all
continuity points of $\Phi$ is a residual (i.e. dense $G_\delta$) set in $M$.
\end{sc-lemma}
The proof can be found in \cite{MR0259835}.
\subsection{Proof of Theorem~\ref{folkloric}}
In order to obtain transitivity it suffices to prove that for every open
set $U$ in $\mathbb{T}^2$ there is some $n\geq 1$ such that $f^n(U)$ is
dense in $\mathbb{T}^2$.
To this end, fix some arbitrary non-empty open set $U \subset \mathbb{T}^2$. By
hypothesis, $U$ contains an injectivity point, $p$ say. By
Lemma~\ref{lemma:open}, $h(U)$ has non-empty interior. Since $A$ (the
linear part of $f$) is expanding, there exists $n \geq 1$ such that
$A^n(h(U)) = h(f^n(U)) = \mathbb{T}^2$. Now, any set $S$ with the property that
$h(S) = \mathbb{T}^2$ must contain every injectivity point. Hence $f^n(U)$
contains every injectivity point and must therefore be dense in $\mathbb{T}^2$.
\subsection{Changing coordinates} In all that follows, we will assume that the
linear part of $f$ has expanding linear part with integer eigenvalues.
More specifically, we are going to assume that the linear part is
represented by a lower triangular matrix of the form
\begin{equation} \label{lower-triangular}
A = \left( \begin{matrix}
\lambda_1 & 0 \\
\mu & \lambda_2
\end{matrix} \right)
\end{equation}
where $|\lambda_1| \geq | \lambda_2| > 1$ are the (integer) eigenvalues of $A$
and $\mu$ is some integer.
There is no loss of generality in doing that.
\begin{lemma}\label{canonical_form} Let $A$ be a $2$ by $2$ matrix with integer
matrix with
eigenvalues $\lambda_1, \lambda_2 \in \mathbb{Z}$. Then there exists $P \in
\SL2Z$ such that $P^{-1}A P $ is of the form \eqref{lower-triangular}
for some $\mu \in \mathbb{Z}$.
\end{lemma}
\begin{proof}
Since $A$ has integer eigenvalues, there exists $v \in \mathbb{Z}^2$ such that
$Av = \lambda_2 v$. Without loss of generality we may suppose that the
components $v_1, v_2$ of $v$ are coprime. Let $p, q$ be such that $p
v_1 + q v_2 = 1$ and take
\begin{equation*}
P = \left( \begin{matrix}
q & v_1 \\
-p & v_2
\end{matrix} \right).
\end{equation*}
Then $P^{-1}AP$ is of the form \eqref{lower-triangular}.
\end{proof}
\section{Proof of Theorem~\ref{thm-A}} \label{section3}
Throughout all of this section we are going to assume that the endomorphism $f$ is as in the hypothesis of Theorem~\ref{thm-A}. In other words,
$f$ is partially hyperbolic having linear part $A$ with integers eigenvalues
$\lambda_1, \lambda_2$ satisfying $|\lambda_1| \geq |\lambda_2|\geq 2$. By
Lemma~\ref{canonical_form}, we can (and do) assume that $A$ is of
the form $\eqref{lower-triangular}$. Moreover, $f$ is assumed to be strongly volume
expanding, that is, $$|\det(Df_x)|>|\lambda_1| \, \, \text{for every} \ \ x \in
\mathbb{T}^2.$$
We fix a constant $\lambda$ satisfying
\begin{align*}
|\lambda_1| < \lambda < |\det (Df_{x})| \quad \forall x \in \mathbb{T}^2.
\end{align*}
In particular, we have that
\begin{align}\label{volume_expansion}
Leb(\tilde{f}^n(B))\geq \lambda^n Leb(B)
\end{align}
for every measurable $B\subset \mathbb{R}^2$.
\subsection{Strongly volume expanding endomorphisms}
The aim of this section is to use strongly volume expanding property and Blicheldt's Theorem as tools to produce homology in two linearly independent directions for large iterates of an open set
Recall that an open set $U \subset \mathbb{T}^2$ is called \emph{essential} if it
contains a loop $\gamma$ such that its homotopy class $[\gamma]$ is non-zero in $\pi_1(\mathbb{T}^2) \cong \mathbb{Z}^2$. Similarily, we
define $U$ to be \emph{doubly essential} if it contains loops $\gamma$ and $\sigma$ such that $[\gamma]$ and $[\sigma]$ are linearly independent.
\begin{lemma} \label{doubly-essential}
Let $f$ be a strongly volume expanding endomorphism on $\mathbb{T}^2$ with
linear part of the form \eqref{lower-triangular}. Then,
given any open set $U \subset \mathbb{T}^2$, there exists $n \geq 0$ such that
$f^n(U)$ is doubly essential. Moreover, $U$ contains a loop $\gamma$
such that $[\gamma]$ is an eigenvector corresponding to $\lambda_2$.
\end{lemma}
The proof is based on a classical theorem about the geometry of numbers.
\begin{thm}[Blichfeldt's Theorem \cite{MR1500976}] \label{blichfeldt} Let $B \subseteq
\mathbb{R}^2$ be a Lebesgue measurable set such that $Leb(B) \geq N$
for some
positive integer $N$. Then there exist $x_0, \ldots, x_N$ in $B$ such
that
$x_i-x_0 \in \mathbb{Z}^n$ for every $i=1, \ldots, N$.
\end{thm}
\begin{proof}[Proof of Lemma~\ref{doubly-essential}]
Let $\tilde{U}$ be a lift of $U$ to $\mathbb{R}^2$.
Pick some square $Q$ of the form $v+(0,1)^2$ such that $\tilde{U}_0 = \tilde{U} \cap Q$
is non-empty (and therefore has positive measure). If $\tilde{U}_0$ is not connected we replace it with one
of its connected components. Since $\tilde{h}$ satisfies \eqref{lift-semi},
$\tilde{h}(\tilde{U}_0)$ must be contained in some square with side $2 \kappa+1$.
Note that $A^n$ is of the form
\begin{equation*}
\left(\begin{array}{cc}
\lambda_1^n & 0 \\
\mu_n & \lambda_2^n
\end{array}
\right),
\end{equation*}
where $\frac{\mu_n}{\lambda_1^n} \to 0$ as $n
\to~+\infty$.
Hence $\tilde{h}(\tilde{f}^n(\tilde{U}_0)) = A^n (\tilde{h}(\tilde{U}_0))$ is contained in a square of
side $2 \lambda_1^n (2 \kappa + 1)$ when $n$ is sufficiently large.
Again, by \eqref{lift-semi}, $\tilde{f}^n(\tilde{U}_0)$ must then be contained in some
square with side $L=2 \lambda_1^n (2 \kappa + 1) + 2\kappa$.
Take $n$ large enough so that $\lambda^n > (L+1) / Leb(\tilde{U}_0)$. By
\eqref{volume_expansion}, $\tilde{f}^n(\tilde{U}_0)$ then has Lebesgue measure
greater than $L+1$. In particular, there is some $\tilde{x} \in \mathbb{R}^2$ such
that $\tilde{x}+\tilde{f}^n(\tilde{U}_0)$ intersects $\mathbb{Z}^2$ in at least $\ell+1$ points,
where $\ell$ is the integer part of $L$. Upon possibly adding an
element of $\mathbb{Z}^2$ to $\tilde{x}$, we may assume that \[ (\tilde{x} + \tilde{f}^n(\tilde{U}_0))
\cap \mathbb{Z}^2 \subset \{1, \ldots, \ell\}^2.\] Summing up, the intersection
of $\tilde{x}+\tilde{f}^n(\tilde{U}_0)$ with $\mathbb{Z}^2$ consists of at least $\ell+1$ points
and is contained in $\{1, \ldots, \ell \}^2$. Hence, by the pigeon hole
principle, there is a line $\{1, \ldots, \ell\} \times \{i\}$
containing two points $\tilde{x}_1, \tilde{x}_2$ of the intesection. Similarily,
there is a column $\{j\} \times \{1, \ldots, \ell\}$ containing two
points $\tilde{y}_1, \tilde{y}_2$ of the
intersection. Since $\tilde{f}^n(\tilde{U}_0)$ is (path) connected, there is a
curve $\tilde{\sigma}$ in $\tilde{f}^n(\tilde{U}_0)$ from $\tilde{x}_1-\tilde{x}$ to $\tilde{x}_2-\tilde{x}$.
Similarily, there is a curve $\tilde{\gamma}$ in $\tilde{f}^n(\tilde{U}_0)$ from
$\tilde{y}_1-\tilde{x}$ to $\tilde{y}_2-\tilde{x}$. Their projections $\sigma, \gamma$ are
closed loops in $f^n(U)$. By construction, $[\sigma]$ is a (non-zero)
multiple of $e_1$ and $[\gamma]$ is a (non-zero) multiple of $e_2$ when
considered as elements of the fundamental group of $\mathbb{T}^2$.
\end{proof}
It is worth pointing out that so far in this section we have not used yet that
$f$ is a partially hyperbolic endomorphism.
\subsection{Dynamical coherence}
A partially hyperbolic endomorphism on $\mathbb{T}^2$ is said to be \emph{dynamically
coherent} if there exists an invariant $C^0$ foliation with $C^1$ leaves tangent to $E^c$. When it
exists, such a foliation is called a \emph{center foliation} of $f$ and its
leaves are called \emph{center leaves}. If $f$ and $g$ are two dynamically
coherent partially hyperbolic endomorphisms, we say that $f$ and $g$ are
\emph{leaf conjugate} if there exists a homeomorphism $\psi: \mathbb{T}^2 \to \mathbb{T}^2$
mapping center leaves of $f$ to center leaves of $g$. A \emph{periodic center
annulus} is an annulus $\mathbb{A} \subset \mathbb{T}^2$ such that $f^n(\mathbb{A}) = \mathbb{A}$
for some $n \geq 1$ whose boundary consists of either one or two $C^1$ circles
tangent to the center direction.
\begin{thm}[Hall and Hammerlindl \cite{HH-classification}] \label{HH-class}
Let $f:\mathbb{T}^2 \to \mathbb{T}^2$ be a partially hyperbolic endomorphism which does not admit a periodic center annulus. Then $f$ is dynamically coherent and leaf conjugate to $A$.
\end{thm}
\begin{rmk}
In general, a partially hyperbolic endomorphism is not necessarily
dynamically coherent, even when having expanding linear part. An
example was given in \cite{HH-Incoherent} with linear part is as in
$\eqref{lower-triangular}$.
\end{rmk}
Henceforth let $f$ be an endomorphism as in the standing hypotheses of this section.
\begin{lemma} \label{leaf_conj_dyn_coh}
The map $f$ is dynamically
coherent and leaf conjugated to its linear part.
\end{lemma}
\begin{proof}
By Theorem \ref{HH-class}, it suffices to show that $f$ does not admit
an periodic center annulus.
Lemma~\ref{doubly-essential} implies that any open set must
become doubly essential after a sufficient number of iterations. But no iterate of a periodic center annulus is doubly essential.
\end{proof}
\begin{rmk}
It is proved in \cite{HH-classification} that absence of a periodic center annulus implies that the eigenvalues $\lambda_1$ and $\lambda_2$ of $A$ are distinct real numbers.
\end{rmk}
Let us fix some notation. The center foliation will be denoted by
$\mathcal{F}^c$ and its lift to $\mathbb{R}^2$ by $\widetilde{\mathcal{F}}^c$. The
eigenspace of $A$ corresponding to $\lambda_2$ will be denoted by $E_A^c$ and
the foliation of $\mathbb{R}^2$ into straight lines parallel to $E_A^c$ by $\tilde{\mathcal{A}}^c$.
Define $E_A^u$ and $\tilde{\mathcal{A}}^u$ analogously with $\lambda_1$ in place of
$\lambda_2$. The foliations $\tilde{\mathcal{A}}^c$ and $\tilde{\mathcal{A}}^u$ descend to foliations of
cirlces on $\mathbb{T}^2$, denoted by $\mathcal{A}^c$ and $\mathcal{A}^u$, respectively. Whenever we
have a foliation $\mathcal{F}$ of some manifold $M$ (i.e. $\mathbb{T}^2$ of $\mathbb{R}^2$ in this paper), $\mathcal{F}(p)$ denotes
the leaf that passes through $p \in M$.
Let $\pi^c$ be the projection from $\mathbb{R}^2$ to $E_A^c$ whose the kernel is $E^u_A$ and $\pi^u$ the projection to
$E_A^u$ whose whose the kernel is $E^c_A$. We say that a foliation $\widetilde{\mathcal{F}}$ of $\mathbb{R}^2$ is at a bounded distance from $\tilde{\mathcal{A}}^c$
(respectively $\tilde{\mathcal{A}}^u$) if there is some $M>0$ such that the length of
$\pi^u(\mathcal{L})$ (resp. $\pi^c(\mathcal{L})$) is smaller than $M$ for every $\mathcal{L} \in \widetilde{\mathcal{F}}$. Similarily, if $\widetilde{\mathcal{F}}$ descends to $\mathcal{F}$ on $\mathbb{T}^2$, we say that $\mathcal{F}$ is a bounded distance from $\mathcal{A}^c$ (resp. $\mathcal{A}^u$) if $\widetilde{\mathcal{F}}$ is a bounded distance from $\tilde{\mathcal{A}}^c$ (resp. $\tilde{\mathcal{A}}^u$).
Recall that a partially hyperbolic endomorphism may not have an invariant
distribution in the unstable cone $\mathcal{C}^u$. (When it does, we say that the
endomorphism is specially partially hyperbolic, see section \ref{special}.) On
the other hand, by lifting the cone field to $\mathbb{R}^2$ one obtains an
$\tilde{f}$-invariant splitting $\tilde{E}^c \oplus \tilde{E}^u$ in the usual way. These
integrate to foliations $\widetilde{\mathcal{F}}^c$ and $\widetilde{\mathcal{F}}^u$. The first of these is simply the
lift of $\mathcal{F}^c$, but the latter is (in general) not the lift of any foliation
at all, since there is no guarantee that $\widetilde{\mathcal{F}}^u$ be invariant under
translations of elements of $\mathbb{Z}^2$.
Under our current hypotheses, $f$ is leaf conjugated to $A$ (Lemma~\ref{leaf_conj_dyn_coh}), so $\mathcal{F}^c$ is a foliation by circles. Furthermore, $\widetilde{\mathcal{F}}^c$ is at a bounded distance from $\tilde{\mathcal{A}}^c$ and
the foliations $\widetilde{\mathcal{F}}^c$ and $\widetilde{\mathcal{F}}^u$ have
global product sturcure, meanding that any pair of leaves from them intersect
in exactly one point (see \cite[Lemma~3.8 and Proposition~4.8]{HH-classification}).
\begin{lemma}\label{c-interval}
The leaves of $\widetilde{\mathcal{F}}^c$ are
sent by $\tilde{h}$ into lines of $\tilde{\mathcal{A}}^c$.
\end{lemma}
\begin{proof}
Since $\widetilde{\mathcal{F}}^c$ is at a bounded distance from $\tilde{\mathcal{A}}^c$, there is a
constant $R>0$ such that for every $\tilde{p} \in \mathbb{R}^2$ we can find a
line $\mathcal{L} \in \tilde{\mathcal{A}}^c$ such that the leaf
$\widetilde{\mathcal{F}}^c(\tilde{p})$ is contained in $R$-neighbourhood of
$\mathcal{L}$, which is a $R$-vertical strip. By \eqref{lift-semi}, we
have that $\|A^n\circ \tilde{h}-\tilde{f}^n\|<\kappa$ for each integer $n$ and,
thus, $A^n(\tilde{h}(\widetilde{\mathcal{F}}^c(\tilde{p})))$ is contained in
$(R+\kappa)$-vertical strip.
Now, suppose that $\tilde{q} \in \widetilde{\mathcal{F}}^c(\tilde{p})$ and that $\tilde{h}$ sends $\tilde{p}$ and $\tilde{q}$
to $(x_1, x_2)$ and $(y_1, y_2)$ in $\mathbb{R}^2=E^u_A\oplus E^c_A$
respectively, with $x_1\neq y_1$. Then $$|\pi^u(A^n(x_1, x_2)) - \pi^u(A^n(y_1,
y_2))| =|\lambda_1|^n |x_1-y_1|$$ gets arbitrarily large as $n$
grows, contradicting that $A^n(\tilde{h}(\widetilde{\mathcal{F}}^c(\tilde{p})))$ is contained in
a $(R+\kappa)$-vertical strip. That proves that $\tilde{h}$ sends leaves of
$\widetilde{\mathcal{F}}^c$ to lines in $\tilde{\mathcal{A}}^c$.
\end{proof}
\begin{lemma}\label{c_leaves}
The map $\tilde{h}$ send distinct leaves of $\widetilde{\mathcal{F}}^c$ to distinct lines of $\tilde{\mathcal{A}}^c$. Consequently, for each $\tilde{p} \in \mathbb{R}^2$, $\tilde{\phi}(\tilde{p})$ is contained in $\widetilde{\mathcal{F}}^c(\tilde{p})$.
\end{lemma}
\begin{proof}
By Lemma~\ref{c-interval}, the map $h$ sends leaves of $\widetilde{\mathcal{F}}^c$ to
lines of $\tilde{\mathcal{A}}^c$. Suppose that $\tilde{h}$ sends two distinct
leaves of $\widetilde{\mathcal{F}}^c$ to the same line $\mathcal{L}$ of $\tilde{\mathcal{A}}^c$.
Then, taking $\mathbb{A}_n$ as the interior of $\tilde{h}^{-1}(A^{n}(\mathcal{L}))$, we get a family of annuli satisfying
$f(\mathbb{A}_n)= \mathbb{A}_{n+1}$. That contradicts Lemma~\ref{doubly-essential}
since $\mathbb{A}_n$ is almost vertical for every $n$. In particular, we get
that $\tilde{\phi}(\tilde{p}) = \tilde{h}^{-1}(\tilde{h}(\tilde{p})) \subseteq \widetilde{\mathcal{F}}^c(\tilde{p})$.
Otherwise, there would be two distinct leaves of $\widetilde{\mathcal{F}}^c$ sent to the
same line of $\tilde{\mathcal{A}}^c$ by $\tilde{h}$.
\end{proof}
In the proof of Theorem~\ref{thm-A} it will be convenient to reduce the
argument to the case in which $f$ is a skew-product. This can always be done
--- at least at the cost of sacrificing differentiability. Indeed, by
Lemma~\ref{leaf_conj_dyn_coh}, $f$ is leaf conjugated to its linear part $A$.
Let us denote the leaf conjugacy by $\psi$. Then the map $g = \psi \circ
f\circ\psi^{-1}$ preserve the foliation of $\mathbb{T}^2$ into vertical circles (the
center leaves of the map $A$), and is therefore a skew product. It is clear
that the map $h_g = h \circ \psi^{-1}$ is a semi-conjugacy from $g$ to $A$.
\begin{rmk}
Although it is not stated explicitly in \cite{HH-classification}, it
can be read from the proofs that the leaf conjugacy $\psi: \mathbb{T}^2 \to \mathbb{T}^2$
is homotopic to the identity and $g = \psi f \psi^{-1}$ is of the
form $g(x,y) = (\lambda_1 x, \tau_x(y))$, where $\tau_x : S^1 \to S^1$
is a continuous family of differentiable maps of degree
$\lambda_2$. Since $\psi$ and $h$ are homotopic to the identity, so is
$h_g$.
\end{rmk}
Let us make some remarks about the semi-conjugacy $h_g$. We note hat $p$ is
an injectivity point of $h$ if and only if $\psi(p)$ is an injectivity
point of $h_g$. Moreover, by Lemma~\ref{c_leaves}, if $q$ is not an
injectivity point of $h_g$, then $h_g^{-1}(h_g(q))$ is a closed vertical line
segment. Taking $\phi_g: \mathbb{T}^2 \to \mathcal{K}(\mathbb{T}^2)$ as
$\phi_g(q)=h_g^{-1}(h_g(q))$, we also have that $p$ is a continuity point of
$\phi$ if and only if $\psi(p)$ is a continuity point of $\phi_g$.
We should also point out that for each non-empty open set $U$ there is $n\geq 1$ such
that $g^n(U)$ is doubly essential. This is simply because $f^n\psi^{-1}(U)$ is doubly essential (by
Lemma~\ref{doubly-essential}) and $ g^n(U) = \psi f^n\psi^{-1}(U)$.
We are now set up to prove Theorem~\ref{thm-A}.
\subsection{Proof of Theorem~\ref{thm-A}}
To prove the theorem, we will show that the injectivity points of $h$
is a residual set in $\mathbb{T}^2$ and use Theorem~\ref{folkloric} to conclude
that $f$ is transitive. Now, $\psi$ is a homeomorphism sending
injectivity points of $h$ to injectivity points of $h_g$ and vice
versa. Hence it suffices to show genericity of the set of injectivity
points of $h_g$.
Suppose then that $\tilde{x} = (x_1, x_2) \in \mathbb{R}^2$ is a continuity point of
$\tilde{\phi}_g$ such that $\tilde{\phi}_g(\tilde{x}) \neq \{ \tilde{x}\}$. By
Lemma~\ref{c_leaves}, $\tilde{\phi}_g(\tilde{x})$ is then a line segment of the
form $\{x_1\} \times [a,b]$ with $b>a$. Note that $x_2$ must be in the
interior of the interval $[a,b]$, since if it were not, moving $x_2$
outside of $[a,b]$ would cause $\tilde{\phi}_g(\tilde{x})$ to undergo a
discontinuity.
Let $\epsilon> 0$ be such that $(x_2-2 \epsilon, x_2+2 \epsilon)
\subset [a,b]$ and let $\tilde{\gamma}(t) = (x_1+t, x_2)$, $t \in (-\delta,
\delta)$ be a horizontal curve passing through $\tilde{x}$. By taking
$\delta$ sufficiently small we have that
\[ \tilde{U} = (x_1-\delta, x_1+\delta) \times (x_2-\epsilon, x_2+\epsilon)
\subset \bigcup_{\tilde{y} \in \operatorname{Im} \tilde{\gamma}} \tilde{\phi}_g(\tilde{y}).\]
This is because we are assuming that $\tilde{\phi}_g$ is continuous on $\tilde{x}$.
By Lemma \ref{doubly-essential} there is some $n \geq 1$ such
that $\tilde{g}^n(\tilde{U})$ contains two points which differ by a
(non-zero) multiple of $e_2$. But $g^n(\tilde{U})$ is a union of
vertical segments, so that means that $\tilde{U}$ contains a vertical
segment of length greater than or equal to one. In other words,
there is some $t_0 \in (-\delta, \delta)$ such that $g^n$ maps
$\{x_1+t_0\} \times (x_2-\epsilon, x_2+\epsilon)$ to a set of
the form $\{z\} \times \mathbb{T}$. By Corollary~\ref{cor-connected}
(c), $\phi_g(x_1+t_0, x_2)$ must then contain $\{x_1+t_0\}
\times \mathbb{T}$. But this is absurd, since $\{\phi_g(x_1+t_0, y): y
\in \mathbb{T}\}$ partitions $\{x_1+t_0\}\times \mathbb{T}$.
We conclude that any continuity point of $\tilde{\phi}_g$ must be a point of
injectivity. It then follows by the Semi-continuity Lemma that
injectivity points form a residual set and the proof is complete. \qed
\section{Proof of Theorem~\ref{thm-B}}\label{section2}
In what follows we shall fix a specially partially hyperbolic endomorphism \linebreak
$f:\mathbb{T}^2 \to \mathbb{T}^2$ and $\lambda_1, \lambda_2 \in \mathbb{Z}$ with $|\lambda_1|>|\lambda_2|>1$ as the eigenvalues of $A$.
Since the unstable direction (defined by \eqref{unstable_direction}) is independent of the past,
$f$ has a non-trivial invariant splitting
\begin{equation}
T_p\mathbb{T}^2=E^c\oplus E^u
\end{equation}
such that for all $p \in \mathbb{T}^2$ and all unit vectors $v \in E^c_p$ and $w \in E^u_p$,
\begin{equation*}
\| Df(v)\|<\|Df(w)\| \ \ \text{and} \ \ \|Df(w)\|>1.
\end{equation*}
Such an endomorphism always has a foliation tangent to the
unstable bundle $E^u$. Indeed this follows by applying the classical arguments
of Hirsh, Pugh and Shub to the lift and then projecting to the torus (or
whatever be the manifold under consideration). Let us denote by $\mathcal{F}^u$
the foliation tangent to $E^u$ and call it the \textit{unstable foliation}. As
explained in \cite[Section 4.A]{Potrie}, every leaf of $\widetilde{\mathcal{F}}^u$ is at a
uniformly bounded distance from a subspace of $\mathbb{R}^2$. This subspace is an eigenspace of $A$, so it must be either $E^c_A$ or $E^u_A$.
Although every specially hyperbolic endomorphism has an unstable foliation, it does not necessarily have a central one.
Indeed, in \cite{He-Shi-Wang} there is an example of a \emph{dynamically
incoherent} specially partially hyperbolic endomorphism (whose linear part is
not expanding).
\subsection{Unstable and center foliations}\label{uc-foliation}
Two important concepts for understanding foliations on $\mathbb{T}^2$ are Reeb components and Tannuli.
A \emph{Reeb component} of a foliation $\mathcal{F}$ on $\mathbb{T}^2$ is an annulus $\mathbb{A}$ such
that the restriction of $\mathcal{F}$ to the closure of $\mathbb{A}$ is homoeomorphic to one of the following:
\begin{enumerate}
\item the foliation on $[-1,1]\times \mathbb{S}^1$
induced by the foliation on $[-1, 1] \times \mathbb{R}$ given by the
lines $\{-1\} \times \mathbb{R}$ and $\{1\} \times \mathbb{R}$, along with the
graphs of the functions $ x \mapsto \exp(1/(1-x^2))+y$ with $y
\in \mathbb{R}$.
\item the foliation on $\mathbb{T}^2$ induced by the foliation on $S^1 \times
\mathbb{R}$ obtained by identifying $\{-1\} \times \mathbb{R}$ with $\{1\} \times
\mathbb{R}$ in case (1).
\end{enumerate}
A \emph{Tannulus component} (or simply \emph{tannulus}) is defined analogoulsy,
substituting the functions $x \mapsto \exp(1/(1-x^2))+y$ with $x \mapsto
\tan(\pi x/2) + y$. See Figures~\ref{fig.Reeb} and \ref{fig.tan}.
By the classification of foliations on $\mathbb{T}^2$ (see
\cite[Proposion~4.3.2]{Foliation-partA}), if a foliation does not admit Reeb
components then it is a suspension of a circle homeomorphism. Such a foliation may or may not contain a tannulus component.
\begin{figure}[htb!]
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[scale=0.8]{Reeb_component.pdf}
\caption{Reeb component}
\label{fig.Reeb}
\end{minipage}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[scale=0.8]{Tannulus.pdf}
\caption{Tannulus}
\label{fig.tan}
\end{minipage}
\end{figure}
\begin{rmk}
A foliation on $\mathbb{T}^2$ may have
infinitely many tannuli but it can have at most
finitely many Reeb components. See \cite{Foliation-partA}.
\end{rmk}
Our goal in this section is to prove:
\begin{prop} \label{unstable_tannuli_vs_DC}
If $\mathcal{F}^u$ does not admit a tannulus, then $f$ is dynamically coherent.
Moreover, $h$ sends leaves of $\mathcal{F}^c$ onto leaves of $\mathcal{A}^c$ and leaves
of $\mathcal{F}^u$ onto leaves of $\mathcal{A}^u$.
\end{prop}
A main ingredient is the following very general topological lemma.
\begin{lemma} \label{no_periodic_annulus}
Let $f: \mathbb{T}^2 \to \mathbb{T}^2$ be a self-cover. If there exists an annulus $\mathbb{A}$
and $n \geq 1$ such that $\mathbb{A} = f^{-n}(\mathbb{A})$,
then the linear part of $f$ has an eigenvalue $\pm 1$.
\end{lemma}
Since we are assuming that $f$ has expanding linear part, Lemma
\ref{no_periodic_annulus} implies that there cannot be a \emph{backward}
invariant annulus.
The proof of Lemma~\ref{no_periodic_annulus} follows
by the arguments used in \cite{A} and
\cite{WR-thesis}. In short, if $\mathbb{A}$ is a periodic annulus with $f^{-n}(\mathbb{A}) =
\mathbb{A}$, then the restriction of $f^n$ to $\mathbb{A}$ is a self-cover of degree
$\lambda_1^n \cdot \lambda_n^2$. At the same time, if $i: \mathbb{A} \to \mathbb{T}^2$ is the
inclusion map, then $i_\star$ sends the fundamental group of $\mathbb{A}$ to a
subgroup of $\mathbb{Z}^2$ of the form $G = \{k v: k \in \mathbb{Z}\} \subset \mathbb{Z}^2$, where $v \in
\mathbb{Z}^2$ is an eigenvalue of the linear part of $f$. The action $f$ on $G$ produces
a subgroup whose index is on the one hand equal to $\lambda_1^n \cdot
\lambda_2^n$, and on the other equal to $\lambda_i^n$, where $\lambda_i$ is the
eigenvalue associated to $v$. Hence the other eigenvalue must be $\pm 1$.
The proof of Proposition \ref{unstable_tannuli_vs_DC} consists of five steps,
organized here as Lemmas~\ref{u-foliation} to \ref{lema:leaf}.
The first step is to establish that $\mathcal{F}^u$ is necessarily a suspension.
\begin{lemma}\label{u-foliation}
The unstable foliation $\mathcal{F}^u$ has no Reeb component.
\end{lemma}
\begin{proof}
Suppose by contradiction that $\mathcal{F}^u$ contains a Reeb component
$\mathbb{A}\subseteq \mathbb{T}^2$. Then, by
\cite[Lemma~2.2]{He-Shi-Wang}, there is an integer $n>0$ such that
$f^{-n}(\mathbb{A})=\mathbb{A}$.
But that is impossible according to Lemma~\ref{no_periodic_annulus},
since we are assuming that $f$ has expanding linear part.
\end{proof}
As we mentioned above, it follows from the classification of foliations on $\mathbb{T}^2$
that $\mathcal{F}^u$ is a suspension.
Moreover, $\widetilde{\mathcal{F}}^u$ has rational slope since its
leaves are a bounded distance from an eigenspace of $A$. Thus by the
classification of foliations on $\mathbb{T}^2$, either $\widetilde{\mathcal{F}}^u$ has a tannulus
or all the leaves of $\widetilde{\mathcal{F}}^u$ are circles.
\begin{lemma} \label{curva_transversal}
Let $\mathcal{F}$ be a foliation of $\mathbb{T}^2$ in which every leaf is a
circle. Then every leaf of $\mathcal{F}$ represents the same non-zero element
$v$ in $\mathbb{Z}^2$ (the fundamental group of $\mathbb{T}^2$). Suppose, moreover, that $\gamma$ is a closed $C^1$ curve
transverse to $\mathcal{F}$. Then $[\gamma]$ is not a multiple of $v$.
\end{lemma}
\begin{proof}
Let $\mathcal{L}$ be a leaf of $\mathcal{F}$ and write $v = [\mathcal{L}]$. That $v$
is non-zero can be deduced from the Poincaré-Benedixon Theorem (a
foliation of $\mathbb{R}^2$ cannot have a compact leaf). If $\mathcal{L}'$ is another
leaf then $[\mathcal{L}']$ must be equal to $v$, for else
$\mathcal{L}$ and $\mathcal{L}'$ would intersect.
Fix some lift $\tilde{\gamma}:[0,1] \to \mathbb{R}^2$ of $\gamma$ and extend it
periodically to $\tilde{\Gamma}:\mathbb{R} \to \mathbb{R}^2$. We claim that $\mathcal{L}$
intersects (the image of) $\tilde{\Gamma}$. Indeed, this also follows from the
Poincaré-Benedixon Theorem since if it were not true, then the vector field
tangent to $\widetilde{\mathcal{F}}$ would exhibit a singularity.
We now observe that $\tilde{\Gamma}(t+k)= \tilde{\Gamma}(t) + k [\gamma]$ for every $k \in \mathbb{Z}^2$ so that the image of $\tilde{\Gamma}$ is invariant under translation by $[\gamma]$.
Similarily, $\mathcal{L}$ is invariant by translation of $v$. Hence $[\gamma]$ cannot
be a multiple of $v$. For if it were then $\mathcal{L}$ and $\tilde{\Gamma}$ would have infinitely many intersections.
\end{proof}
\begin{lemma} \label{no_center_annulus}
Suppose that the foliation $\mathcal{F}^u$ has no tannulus. Then $f$ does not
admit a periodic center annulus.
\end{lemma}
\begin{proof}
Suppose there is a periodic center annulus
$\mathbb{A}$. Then the two boundaries of $\mathbb{A}$ are closed curves represented by
some non-zero vector $v \in \mathbb{Z}^2$. Since $\mathcal{F}^u$ has no tannulus,
its leaves are circles. By Lemma \ref{curva_transversal} these circles are all
represented by some $u \in \mathbb{Z}^2$ with $u$ and $v$ not collinear. Consequently, every
unstable leaf must intersect both boundaries of $\mathbb{A}$ (if the boundaries coincide we consider the lift).
Denote by $\delta>0$ the greatest length of an unstable arc inside $\mathbb{A}$ joining
its two boundaries.
Now take some unstable segment $\gamma$ in the of $\mathbb{A}$. It has some
iterate $f^n(\gamma)$ whose length is larger than $\delta$. Thus $f^n(\gamma)$
must intersect $\partial \mathbb{A}$. But that is absurd since $\mathbb{A}$ is
$f$-invariant.
\end{proof}
As a consequence of Lemma \ref{no_center_annulus} and Theorem~\ref{HH-class}, $f$ is dynamically coherent
and leaf conjugate to $A$.
\begin{lemma} \label{if_no_tannuli}
Suppose that the foliation $\mathcal{F}^u$ has no tannulus. Then $\mathcal{F}^c$ and $\mathcal{F}^u$ are foliations by circles. Their lifts $\widetilde{\mathcal{F}}^c$ and $\widetilde{\mathcal{F}}^u$ are a bounded distance from $\tilde{\mathcal{A}}^c$ and $\tilde{\mathcal{A}}^u$, respectively.
\end{lemma}
\begin{proof}
Suppose that $\mathcal{F}^u$ has no tannulus. Then the leaves of $\widetilde{\mathcal{F}}^u$ are circles and, by Lemma~\ref{no_center_annulus}, $f$ has no center annulus. Hence, by Theorem~\ref{HH-class}, we conclude that the leaves of $\mathcal{F}^c$ are circles. Moreover, by \cite{HH-classification}[Lemma~3.8], they are at a bounded distance from the circles in $\mathcal{A}^c$.
Recall that every leaf of $\widetilde{\mathcal{F}}^u$ is bounded distance from a translation of an
eigenspace of $A$. Then, as the leaves of $\mathcal{F}^u$ are circles and it is
transverse to any leaf of $\mathcal{F}^c$, we can conclude by
Lemma~\ref{curva_transversal} that this eigenspace cannot be $E_A^c$. So it has to be $E_A^u$.
\end{proof}
A consequence of Lemmas~\ref{if_no_tannuli} is that the restriction of $\pi^c$
(resp. $\pi^u$) to $\widetilde{\mathcal{F}}^c(\tilde{p})$ (resp. $\widetilde{\mathcal{F}}^u(\tilde{p})$) is onto, so
$\widetilde{\mathcal{F}}^c(\tilde{p})$ and $\widetilde{\mathcal{F}}^u(\tilde{p})$ intersect each other. By the Poincaré-Bendixson
Theorem, we conclude that they intersect each other exactly
once. In other words, $\widetilde{\mathcal{F}}^c$ and $\widetilde{\mathcal{F}}^u$ have global product structure
and are \textit{quasi-isometric}. That is,
\begin{align}\label{eq1}
\exists a,b>0 \ \ \text{such that} \ \ d_{\widetilde{\mathcal{F}}^{\ast}}(\tilde{p},\tilde{q})\leq
a\|\tilde{p}-\tilde{q}\|+b,
\end{align}
where $d_{\widetilde{\mathcal{F}}^{\ast}}(\tilde{p},\tilde{q})$ denotes the distance between $\tilde{p}$ and $\tilde{q}$
along of a leaf of $\widetilde{\mathcal{F}}^{\ast}$, for $\ast = c, u$.
To conclude the proof of Proposition~\ref{unstable_tannuli_vs_DC}, it remains
to show the following.
\begin{lemma}\label{lema:leaf}
Suppose that the foliation $\mathcal{F}^u$ has no tannulus. The map $\tilde{h}$ sends leaves of $\widetilde{\mathcal{F}}^c$ onto leaves of $\tilde{\mathcal{A}}^c$ and leaves of $\widetilde{\mathcal{F}}^u$ onto leaves of $\tilde{\mathcal{A}}^u$.
\end{lemma}
\begin{proof}
The first part follows from an identical argument of the proof in Lemma~\ref{c-interval}, since $A$ is expanding and, by Lemma~\ref{if_no_tannuli}, $\widetilde{\mathcal{F}}^c$ and $\widetilde{\mathcal{F}}^u$ are a bounded distance from $\tilde{\mathcal{A}}^c$ and $\tilde{\mathcal{A}}^u$, respectively.
\end{proof}
Lemmas~\ref{no_center_annulus} and \ref{lema:leaf} conclude the proof of Proposition~\ref{unstable_tannuli_vs_DC}.
\subsection{The semi-conjugacy}\label{semiconjugacy}
Next we state a lemma analogous to Lemma~\ref{c_leaves}.
Recall that the standing hypotheses in
Lemma~\ref{c_leaves} are those of Section~\ref{section3} while here we assume
the standing hypotheses of Section~\ref{section2}.
\begin{lemma}\label{lemma-phi}
Suppose that $\mathcal{F}^u$ has no tannulus.
Then the map $\tilde{h}$ sends distinct leaves of $\widetilde{\mathcal{F}}^c$ to distinct lines of $\tilde{\mathcal{A}}^c$. Consequently, for each $\tilde{p} \in \mathbb{R}^2$, $\tilde{\phi}(\tilde{p})$ is contained in $\widetilde{\mathcal{F}}^c(\tilde{p})$.
\end{lemma}
\begin{proof}
Suppose by contradiction that distinct leaves of $\widetilde{\mathcal{F}}^c$ intersects
$\tilde{\phi}(\tilde{p})$, say $\widetilde{\mathcal{F}}^c_1$ and $\widetilde{\mathcal{F}}^c_2$. Then $\tilde{h}$ sends
$\widetilde{\mathcal{F}}^c_1$ and $\widetilde{\mathcal{F}}^c_2$ onto the same line in $\tilde{\mathcal{A}}^c$. Consequently,
for every $\tilde{q}_1 \in \widetilde{\mathcal{F}}^c_1$ and $\tilde{q}_2 \in \widetilde{\mathcal{F}}^c_2$, we have
$\pi^u(\tilde{h}(\tilde{q}_1))=\pi^u(\tilde{h}(\tilde{q}_2))$ and so
$\|\pi^u(\tilde{f}^n(\tilde{q}_1))-\pi^u(\tilde{f}^n(\tilde{q}_2))\|$
is uniformly bounded for every $n$. By the global product structure, we
can get $\tilde{q}_1$ and $\tilde{q}_2$ in the same leaf of $\widetilde{\mathcal{F}}^u$. Since
$\widetilde{\mathcal{F}}^u$ is at a bounded distance from $\tilde{\mathcal{A}}^c$, we have that
$\|\pi^c(\tilde{f}^n(\tilde{q}_1))-\pi^c(\tilde{f}^n(\tilde{q}_1))\|$ is also uniformly bounded
for
every $n$. Hence $\|\tilde{f}^n(\tilde{q}_1)-\tilde{f}^n(\tilde{q}_2)\|$ is
uniformly bounded for every $n$. But that is impossible since $\tilde{q}_1$ and $\tilde{q}_2$
are in the same unstable leaf which is quasi-isometric.
\end{proof}
As a consequence of the proof in Lemma~\ref{lemma-phi}, we obtain
\begin{align}\label{eq2}
\sup_n\|\pi^u(\tilde{f}^n(\tilde{p}))-\pi^u(\tilde{f}^n(\tilde{q}))\|=+\infty
\end{align}
whenever $\tilde{p} \neq \tilde{q}$ lie on the same leaf of $\widetilde{\mathcal{F}}^u$.
\begin{lemma}\label{cor:stripe}
Suppose that $\mathcal{F}^u$ has no tannulus.
If $\phi(p) \neq \{p\}$, then the interior of $h^{-1}(\mathcal{A}^u(h(p)))$ is an annulus which
is either wandering or periodic for $f$.
\end{lemma}
\begin{proof}
Since $\mathcal{F}^u$ has no tannulus,
the leaves of $\mathcal{F}^u$ are cirlces so we may consider fibres of a
trivial bundle $ \pi : \mathbb{T}^2 \to S^1$ whose fibers are the leaves of $\mathcal{F}^u$.
The set $\phi(p)$ is a segment transversal to the fibres and $h$
sends $\mathcal{F}^u(x)$ to $\mathcal{A}^u(h(p))$ for every $x \in \phi(p)$. Hence
$h^{-1}(\mathcal{A}^u(h(p))$ is equal to $\pi^{-1}(\pi(\phi(p)))$.
\end{proof}
\subsection{Proof of Theorem~\ref{thm-B}}
The implication (\ref{f_conjugated}) $\Longrightarrow$ (\ref{f_transitive}) is
obvious. To see why (\ref{f_transitive}) $\Longrightarrow$
(\ref{f_no_annulus}), first note that a transitive map may not have a wandering
open set of any kind. Suppose that $f$ has a periodic annulus $\mathbb{A} = f^n(\mathbb{A})$ for
some $n\geq 1$. Then, by transitivity of $f$, we must have $f^{-n}(\mathbb{A}) = \mathbb{A}$. Indeed, if it were not so, $f^{-n}(\mathbb{A})$
would consist of a union of several annuli, some of which would be wandering.
But Lemma~\ref{no_center_annulus} says that it is impossible to have a backwards
invariant annulus when the linear part is expanding.
It remains to show that (\ref{f_no_annulus}) implies (\ref{f_conjugated}).
Note that $h$ is a congacy between $f$ and $A$ if and only if $\phi(p) = \{p\}$ for every $p
\in \mathbb{T}^2$. (A continuous bijection on a compact space is a homeomorpism.) Thus,
by Lemma~\ref{cor:stripe}, it suffices to show that if $f$ does not admit a wandering or periodic
annulus, then $\mathcal{F}^u$ does not admit a tannulus. Suppose it does admit a
tannulus $\mathbb{A}$. Then $f^n(\mathbb{A})$ would be a tannulus for every $n\geq 0$. Moreover,
$\mathbb{A}$ and $f^n(\mathbb{A})$ must either coincide or be disjoint. Hence $\mathbb{A}$ must be
either wandering or periodic.
\section{An example} Here we present a non-trivial example of an endomorphism
satifying the hypotheses of Theorem \ref{thm-A}. More precisely, we construct
a $C^\infty$ local diffeomorphism $f: \mathbb{T}^2 \to \mathbb{T}^2$ satisfying
\begin{enumerate}
\item the linear part of $f$ is $A = \left( \begin{smallmatrix} 5 & 0
\\ 0 & 2 \end{smallmatrix} \right)$,
\item $\det Df(x,y)>5$ for every $(x,y) \in \mathbb{T}^2$ \label{sve}
\item $f$ is partially hyperbolic, and
\item $f$ has a hyperbolic fixed point with stable index $1$ and is
therefore $C^1$ persistently not conjugated to $A$.
\label{not_conjugated}
\end{enumerate}
By Theorem~\ref{thm-A}, $f$ is robustly transitive. The example
is a skew-product, but all properties are robust, so the
construction leads implicitly to examples which are not
skew-products. They are, however, topologically conjugated to
skew-products. But that is unavoidable according to
\cite{HH-classification} (see Theorem \ref{HH-class}).
Here's the construction. Let $\alpha: S^1 \to S^1$ and $\beta: \mathbb{T}^2 \to \mathbb{T}$ be
given by
\begin{align}
\alpha(x) &= 5x + \frac{\sin(2 \pi x) }{2 \pi} \\
\beta(x,y) &= 2y - (1+\epsilon)\cos^{2}(\pi x) \frac{ \sin(2 \pi y)}{2 \pi}
\end{align}
and take $f(x,y) = (\alpha(x), \beta(x,y))$. Clearly $f$ is a well defined
$C^\infty$ map on $\mathbb{T}^2$ homotopic to $A$. That it is a local diffeomorphism
will follow as soon as we have proved item (\ref{sve}) above. The derivative
of $f$ at $(0,0)$ is given by
\[\left(\begin{matrix}
6 & 0 \\
0 & 1-\epsilon
\end{matrix}\right)\] which is hyperbolic with stable index $1$ for every
$\epsilon>0$. This property persists under $C^1$ perturbations and guarantees
that neither $f$ nor its neighbours are conjugated to $A$. To se why
(\ref{sve}) holds, note that the
Jacobian
\begin{equation*}
J(x,y) = | \det Df(x,y)| = \left( 5+\cos(2 \pi x) \right)
\left( 2-(1+\epsilon)\cos^2(\pi x)\cos(2\pi y)\right)
\end{equation*}
is $C^\infty$ on $\mathbb{T}^2$ and that
\begin{equation*}
\partial_y J = 2 \pi (1+\epsilon)(5+\cos(2 \pi x))
\cos^2(\pi x) \sin(2 \pi y)
\end{equation*}
vanishes only on $x=\frac{1}{2}$, $y=\frac{1}{2}$, and $y=0$. It therefore suffices to
check that $J$ is greater than $5$ along these three curves.
\begin{itemize}
\item On $x = \frac{1}{2}$ we have $J(\frac{1}{2}, y) \equiv 8$.
\item On $y=\frac{1}{2}$ we have $J(x,\frac{1}{2}) = \left( 5+\cos(2 \pi x) \right)
\left(2+(1+\epsilon)\cos^2(\pi x) \right) \geq 8$.
\item On $y=0$ we have \begin{align*} J(x,0) & =
\left(5+\cos(2 \pi x) \right)
\left( 2-(1+\epsilon) \cos^2(\pi x) \right) \\
& = 6+2 \sin^2(\pi x) (2- \sin^2(\pi x))
- \epsilon \cos^2(\pi x) (5+\cos(2 \pi x)) \\
& \geq 6- 6 \epsilon,
\end{align*}
which is greater that $5$ for every $\epsilon < 1/6$. That
proves (\ref{sve}).
\end{itemize}
Finally let us verify that $f$ is partially hyperbolic. For that, fix
some $p \in \mathbb{T}^2$ and let $(u_1, u_2)= Df_p (1,1)$, $(w_1,w_2) = Df_p (1,-1)$.
We claim that
\begin{equation}
u_1 = w_1 \geq 4 \label{u_1},
\end{equation}
\begin{equation}
1-\epsilon \leq u_2 \leq 3+\epsilon, \label{u_2}
\end{equation}
and
\begin{equation}
-3-\epsilon \leq w_2 \leq 1+\epsilon.
\label{w_2}
\end{equation}
Once that is shown, it follows that the cone
\begin{equation*}
S = \{ (v_1, v_2) \in \mathbb{R}^2\setminus \{(0,0)\} : |v_1| \geq |v_2| \}
\end{equation*}
is strictly $Df_p$-invariant at every $p \in \mathbb{T}^2$ as long as
$\epsilon<1$.
The estimate in \eqref{u_1} also shows that vectors in $S$ are expanded
by $Df_p$ by a factor of at least $2 \sqrt{2}$. This is because
\begin{equation*}
\max_{0 \leq t \leq 1} \|t(1,1) + (1-t)(1,-1)\| = \sqrt{2}
\end{equation*}
while
\begin{equation*}
\min_{0\leq t \leq 1} \|t Df_p (1,1) + (1-t) Df_p (1,-1)\|
\geq 4
\end{equation*}
for every $p$, and every $v \in S$ is a multiple of a vector of this
type.
It remains to prove \eqref{u_1}, \eqref{u_2}, and \eqref{w_2}.
For that, let us write $p = (x,y)$.
Then inequality \eqref{u_1} is immediate, as
\begin{equation}
u_1 = w_1 = \partial_x \alpha(x) = 5+\cos(2 \pi x).
\label{horizontal_expansion}
\end{equation}
The inequalities in \eqref{u_2} follows by rewriting $u_2$ as
\begin{align*}
u_2 & = \partial_x \beta(x,y) + \partial_y \beta(x,y) \\
& = \left(\epsilon + 1\right) \sin{\left(\pi x \right)}
\sin{\left(2 \pi y \right)} \cos{\left(\pi x \right)} \\
& + 2 - \left(\epsilon + 1\right) \cos^{2}{\left(\pi x \right)}
\cos{\left(2 \pi y \right)} \\
& = 2- ( \cos(2 \pi y) + \cos(2 \pi (x+y)))/2 \\
& - \epsilon \cos(\pi x) \cos(\pi(x+2y)).
\end{align*}
One can rewrite $w_2$ in a similar fashion to obtain \eqref{w_2}.
\bibliographystyle{alpha}
|
1511.05341
|
\section{Introduction}
Homomorphic encryption introduced by
Rivest, Adleman and Dertouzos \cite{RAD78} in 1978, is a useful cryptographic primitive because it can translate
an operation on the ciphertexts into an operation on the corresponding plaintexts.
The property is useful for some applications, such as e-voting, watermarking and secret sharing schemes. For example, if an additively homomorphic
encryption is used in an e-voting scheme, one can obtain an encryption of the sum of all
ballots from their encryption. Consequently, it becomes possible that a single decryption will reveal the result
of the election. That is, it is unnecessary to decrypt all ciphertexts one by one.
A fully homomorphic encryption (FHE) is defined as a scheme which allows anyone to perform arbitrarily
computations on encrypted data, despite not having
the secret decryption key.
In 2009, Gentry \cite{G09} proposed a FHE scheme over ideal lattices, which is capable of evaluating some functions in the encrypted domain.
Since then, the primitive has interested many researchers.
\subsection{Related works}
Homomorphic encryption schemes supporting either addition or multiplication operations (but not both) had been intensively studied, e.g., Goldwasser-Micali encryption \cite{GM82}, ElGamal encryption \cite{E84}, and Paillier encryption \cite{P99}. The Gentry encryption \cite{G09} is a fully homomorphic encryption
scheme, which makes it possible to evaluate some functions in the encrypted domain. After that, some new FHE schemes appeared.
At Eurocrypt'10, Gentry, Halevi and Vaikuntanathan \cite{GHV10} proposed a FHE scheme based on the Learning With Error (LWE) problem. In 2010, van Dijk, et al. \cite{DGHV10} constructed a simple FHE scheme using only elementary modular arithmetic.
At Crypto'11, a FHE scheme working over integers with shorter public keys and a FHE scheme based on ring-LWE were presented by Coron et al. \cite{C11}, Brakerski and Vaikuntanathan \cite{BV11}, separately. At FOCS'11, a FHE scheme based on standard LWE by Brakerski and Vaikuntanathan \cite{BV12,BV14}, and a FHE scheme
using depth-3 arithmetic circuits by Gentry and Halevi \cite{GH11}, have interested many audiences. In 2012, Brakerski, Gentry and Vaikuntanathan \cite{BGV12} designed a leveled FHE scheme without bootstrapping. At Eurocrypt'13, Cheon, et al. \cite{C13} investigated the problem of batching FHE schemes over integers.
In 2013, Brakerski, Gentry and Halevi \cite{BGH13} discussed the problem of packing ciphertexts in LWE-based
homomorphic encryption.
In 2015, Castagnos and Laguillaumie \cite{CL15} proposed a linearly homomorphic encryption scheme whose
security relies on the hardness of the decisional Diffie-Hellman problem. Recently, Cheon and Kim \cite{CK15} introduced a hybrid homomorphic encryption
which combines public-key encryption and somewhat homomorphic encryption in order to reduce the storage requirements
for some applications.
FHE makes it possible to enable secure storage and computation on the cloud. However, current homomorphic encryption
schemes are still inefficient. For example, key generation in Gentry's FHE scheme takes from 2.5 seconds to 2.2 hours \cite{GHE11}.
A recent implementation required 36 hours for a homomorphic evaluation of AES \cite{GHS12}. One of the most remarkable things about these implementations is that
the computations did not involve common arithmetic expressions and relational expressions.
\subsection{Our contributions}
In this paper, we want to stress that any computations performed on encrypted data are constrained to the encrypted domain (finite fields or rings).
This restriction makes the primitive useless for most computations involving common arithmetic expressions, logical expressions and relational expressions.
It is only applicable to the computations related to modular arithmetic.
Some researchers have neglected the differences between common arithmetic and modular arithmetic, and falsely claimed that FHE enables arbitrary computations on encrypted data.
We here reaffirm that cryptography uses modular arithmetic a lot in order to obscure and dissipate the redundancies in a plaintext message, not to perform any numerical calculations.
We revisit the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme \cite{DGHV10} and Nuida-Kurosawa FHE scheme \cite{NK15} under the client-server computing model.
The former encrypts bit by bit. The latter works over the encrypted domain $\mathbb{Z}_Q$, where $Q$ is a prime.
We find that in the Dijk-Gentry-Halevi-Vaikuntanathan scheme the server can not decide the carries by the encrypted data, and in the Nuida-Kurosawa scheme it is impossible to find an invertible transformation $\mathcal{T}$ from the real number set $\mathbb{R}$ to the field $\mathbb{Z}_Q$. Therefore, in both schemes the server can not return right values to the client even though the server is asked to help to evaluate the simple function $f(x, y)=x+y$.
In view of the limitations mentioned above, we believe it might be a false claim that FHE is of great importance to cloud computing.
To the best of our knowledge, it is the first time to concretely discuss FHE schemes under the client-server computing model.
\section{The real goal of using modular arithmetic in cryptography}
Any calculation needs an describing expression, which consists of variables, constants and operators. There are three kinds of expressions: arithmetic expressions, logical expressions and relational expressions. Arithmetic operators include addition $(+)$, substraction $(-)$, multiplication ($*$), division $(/)$, integer-division $(\setminus)$, modulus (Mod), and so on.
Like common arithmetic, modular arithmetic is commutative, associative, and distributive.
Suppose that $a, b$ are in the decrypted domain $\mathbb{Z}_p$ where $p$ is a prime, $E(\cdot)$ is a fully homomorphic encryption algorithm, and $D(\cdot)$ is the corresponding decryption algorithm. Then the following properties are obvious.
$$D(E(a)+E(b))=D(E(a+b))=a+b \mod p$$
$$D(E(a)\cdot E(b))=D(E(ab))=ab \mod p.$$
Generally,
$$ a+b \not=(a+b \mod p), \qquad ab\not= (ab \mod p) $$
$$ a<b \not\Longrightarrow E(a)<E(b), \qquad E(a)<E(b) \not\Longrightarrow a<b$$
We here want to stress that cryptography uses modular arithmetic a lot, because it can obscure the relationship between the plaintext and the ciphertext, and dissipate the redundancy of the plaintext by spreading it out over the ciphertext. It is well known that confusion and diffusion are the two basic techniques for obscuring the redundancies in a plaintext message. They could frustrate attempts to study the ciphertext looking for redundancies and statistical patterns.
Practically speaking, the real goal of using modular arithmetic in cryptography is to \emph{obscure and dissipate the redundancies in a plaintext message, not to perform any numerical calculations.}
To see this, we will have a close look at two typical FHE schemes proposed by van Dijk et al. \cite{DGHV10}, Nuida and Kurosawa \cite{NK15}.
The former encrypts bit by bit. The encrypted domain for the latter is $\mathbb{Z}_Q$, where $Q$ is a prime.
\section{Analysis of Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme under the client-server computing model}
\subsection{Description of Dijk-Gentry-Halevi-Vaikuntanathan scheme}
At Eurocrypt 2010, van Dijk et al. \cite{DGHV10} constructed an FHE scheme.
For convenience, we here only describe the symmetric version of the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme as follows.
\textbf{KeyGen}($\lambda$): For a security parameter $\lambda$, pick an odd number $p \in[2^{\lambda-1}, 2^{\lambda})$ and set it as the secret key.
\textbf{Encrypt}($p, m$): Given a bit $m\in\{0, 1\}$, compute the ciphertext as
$$ c = pq + 2r + m$$ where the integers $q, r$ are chosen at random in some other
prescribed intervals, such that $2r$ is smaller than $p/2$ in absolute value.
\textbf{Decrypt}($p, c$): $m=(c \mod p) \mod 2$.
\textbf{Additively homomorphic property (under the modulus)}: If $c_1 = pq_1 + 2r_1 + m_1$ and $c_2 = pq_2 + 2r_2 + m_2$, then $m_1+m_2=(c_1+c_2\mod p)\mod 2$.
\textbf{Multiplicatively homomorphic property (under the modulus)}: If $c_1 = pq_1 + 2r_1 + m_1$ and $c_2 = pq_2 + 2r_2 + m_2$, then $m_1\cdot m_2=(c_1 \cdot c_2\mod p)\mod 2$.
Notice that these homomorphic properties hold only on the condition that computations are constrained by the prescribed modulus $p, 2$.
This restriction makes the scheme impossible to deal with any numerical calculations without knowing the modulus.
\subsection{An example for Dijk-Gentry-Halevi-Vaikuntanathan scheme}
Suppose that one client sets $p=7919$ as his secret key.
He has two numbers $a=5$, $b=3$, and wants a server to help him to compute $c=a+b$.
Now, he encrypts two numbers $a$ and $b$ as follows (see Table 1).
\begin{center}
\begin{tabular}{l|ccc}
\hline
$a=5$& \fbox{1}& \fbox{0}& \fbox{1} \\ \hline
& $7919\times 1325+2\times 57+1 $ & $7919\times 3168+2\times 49+0 $ & $7919\times 5247+2\times 63+1 $ \\
& \fbox{10492790} & \fbox{25087490} & \fbox{41551120} \\
\hline \hline
$b=3$& & \fbox{1}& \fbox{1} \\ \hline
& & $7919\times 5538+2\times 85+1 $ & $7919\times 6214+2\times 74+1 $ \\
& & \fbox{43855593} & \fbox{49208815} \\
\hline
\end{tabular}\vspace*{3mm}
Table 1: Ciphertexts of 5 and 3 w.r.t. the secret key $7919$
\end{center}
The client sends two ciphertexts
$$ \underbrace{\fbox{10492790},\fbox{25087490},\fbox{41551120}}_x \
\mbox{and}\ \underbrace{\fbox{43855593},\fbox{49208815}}_y$$
to a server and asks the server to compute the function
$$f(x, y)=x+y.$$
Hence, the server may return the values
$$ \fbox{10492790},\fbox{68943083},\fbox{90759935} $$
to the client. Thus, the client decrypts the returned values as follows
$$(10492790 \mod p) \mod 2= 1,$$
$$(68943083 \mod p) \mod 2= 1,$$
$$(90759935 \mod p) \mod 2= 0,$$
and obtains the number $(110)_2=6$, not the right number $8$. See the following Table 2 for the process.
\begin{center}
\begin{tabular}{|lcl|}
\hline
Client & & Server \\ \hline
Input: $p=7919$, & & $f(x, y)=x+y$\\
\hspace*{10mm} $a=5$, $b=3$ & & \\ \hline
Encryption: $3 \rightarrow \underbrace{\fbox{43855593},\fbox{49208815}}_y $,&&\\
$5\rightarrow \underbrace{\fbox{10492790},\fbox{25087490},\fbox{41551120}}_x$. && \\
& $\xlongrightarrow{x, y}$ & \\
& $\xlongleftarrow{\hat c}$ & $ f(x, y)=\underbrace{\fbox{10492790},\fbox{68943083},\fbox{90759935}}_{\hat c}$\\
Decryption: $\hat c \rightarrow \underbrace{\fbox{1},\fbox{1},\fbox{0}}_c$ & & \\
\hline
\end{tabular}
\vspace*{3mm}
Table 2: An example for the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme
\end{center}
What is the problem with this process?
The returned values miss all carries because \emph{the server can not decide the carries by the encrypted data.}
\textbf{Remark 1}. One might argue that the client himself can construct a Boolean circuit which contains the carries and send the circuit to the server.
The argument is unreasonable because the client is assumed to be of weak computational capability. If the client can construct such a Boolean circuit, then
he can directly evaluate the circuit, instead of asking a server to help him to evaluate it.
\section{Analysis of Nuida-Kurosawa FHE scheme under the client-server computing model}
In the Dijk-Gentry-Halevi-Vaikuntanathan FHE scheme, the message space is $\mathbb{Z}_2$.
The scheme is very inefficient because it has to generate 256 or more bits in order to mask one bit.
At Eurocrypt 2015, Nuida and Kurosawa \cite{NK15} extended the scheme to the message space $\mathbb{Z}_Q$ where $Q$ is any prime. We here only describe the symmetric version of Nuida-Kurosawa FHE scheme as follows.
\subsection{Description of Nuida-Kurosawa FHE scheme}
\textbf{KeyGen}($\lambda$): For a security parameter $\lambda$, pick an odd number $p \in[2^{\lambda-1}, 2^{\lambda})$ and a prime $Q$. Set $p$ as the secret key ($Q$ is published).
\textbf{Encrypt}($p, m$): Given a message $m\in \mathbb{Z}_Q$, compute the ciphertext as
$$ c = pq + Qr + m$$ where the integers $q, r$ are chosen at random in some other
prescribed intervals, such that $Qr$ is smaller than $p/2$ in absolute value.
\textbf{Decrypt}($p, c$): $m=(c \mod p) \mod Q$.
\textbf{Additively homomorphic property (under the modulus)}: If $c_1 = pq_1 + Qr_1 + m_1$ and $c_2 = pq_2 + Qr_2 + m_2$, then $m_1+m_2=(c_1+c_2\mod p)\mod Q$.
\textbf{Multiplicatively homomorphic property (under the modulus)}: If $c_1 = pq_1 + Qr_1 + m_1$ and $c_2 = pq_2 + Qr_2 + m_2$, then $m_1\cdot m_2=(c_1 \cdot c_2\mod p)\mod Q$.
\subsection{An example for Nuida-Kurosawa FHE scheme}
Suppose that one client sets $p=22801763489$ as his secret key and sets $Q=15485863$.
He has two numbers $a=0.1$, $b=2.3$, and wants a server to help him to compute $c=a+b$.
First, he has to transform $a=0.1$, $b=2.3$ into integers $\bar a, \bar b$ such that $\bar a, \bar b\in \mathbb{Z}_Q$. Denote the transformation by $\mathcal{T}$.
Second, he encrypts $\bar a, \bar b$ and obtains the corresponding ciphertexts $\hat a, \hat b$. Third, he sends $\hat a, \hat b$ to
a server. The server then takes $\hat a, \hat b$ as the inputs of the function $f(x, y)=x+y$. Finally, the server returns $\hat c=f(\hat a, \hat b)$ to the client.
See the following Table 3 for the process.
\begin{center}
\begin{tabular}{|lcl|}
\hline
Client & & Server \\
Input: $p=22801763489$, $Q=15485863$;& & $f(x, y)=x+y$\\
\hspace*{10mm} $a=0.1$, $b=2.3$ & & \\ \hline
Transformation $\mathcal{T}$: $ a\rightarrow \bar a$, $ b\rightarrow \bar b$. & & \\
\hspace*{14mm} such that $\bar a, \bar b\in \mathbb{Z}_Q$. & & \\
Encryption: $\bar a\rightarrow\hat a$, $\bar b\rightarrow \hat b$.
& $\xlongrightarrow{\hat a, \hat b} $ & \\
& $\xlongleftarrow{\hat c}$ & Computation $ f(\hat a,\hat b)\rightarrow \hat c$\\
Decryption: $\hat c \rightarrow \bar c$ & & \\
Inverse Transformation $\mathcal{T}^{-1}$: $\bar c \rightarrow c. $ & & \\
\hline
\end{tabular}
\vspace*{3mm}
Table 3: An example for the Nuida-Kurosawa FHE scheme
\end{center}
What is the problem with this process? \emph{It is impossible to find an invertible transformation $\mathcal{T}$ from the
real number set $\mathbb{R}$ to the field $\mathbb{Z}_Q$.}
Note that most encryption algorithms must run over some finite field or ring. One has to transform all inputting characters into integers in the field or ring.
That means an invertible encoding algorithm is necessary for any encryption scheme.
This condition is easily satisfied if all inputting characters are indeed viewed as characters. But when some inputting characters are viewed as real numbers and they are used for some arithmetic computations, it is impossible to find such an invertible encoding algorithm that maps any real number to an integer in a prescribed field or ring.
\begin{center}
\begin{tabular}{|cc|cc|}
\hline
character & ASCII code & character & ASCII code \\ \hline
0 & 48 & 6 & 54 \\
1 & 49 & 7 & 55 \\
2 & 50 & 8 & 56 \\
3 & 51 & 9 & 57 \\
4 & 52 & $\cdot$ & 250 \\
5 & 53 & & \\
\hline
\end{tabular}\end{center}
We here describe a possible encryption-decryption process for the real numbers $0.1$ and $2.3$. The ASCII coding method will map $0.1, 2.3$ to two integers in the field $\mathbb{Z}_{15485863}$.
\begin{center}
\begin{tabular}{|c|}
\hline
$a=0.1\xlongrightarrow{\mbox{ASCII}} \fbox{48}\fbox{250}\fbox{49}\xlongrightarrow{\mathcal{T}}\bar a=48\times 256^2+250\times 256+49=3209777\xlongrightarrow{q=3215964,r=13} $\\
$\hat a= 73329650721664392\xlongrightarrow{\mod p, \mod Q} \bar a=3209777
\xlongrightarrow{\mathcal{T}^{-1}}\fbox{48}\fbox{250}\fbox{49}\xlongrightarrow{\mbox{ASCII}} 0.1 $ \\ \hline
$b=2.3\xlongrightarrow{\mbox{ASCII}} \fbox{50}\fbox{250}\fbox{51}\xlongrightarrow{\mathcal{T}}\bar b=50\times 256^2+250\times 256+51=3340851\xlongrightarrow{q=6490231,r=9} $\\
$\hat b= 147988712393689577\xlongrightarrow{\mod p, \mod Q}\bar b = 3340851
\xlongrightarrow{\mathcal{T}^{-1}}\fbox{50}\fbox{250}\fbox{51}\xlongrightarrow{\mbox{ASCII}} 2.3 $
\\ \hline
\end{tabular}\end{center}
If a server performs the operator of addition on the encrypted data, $\hat a, \hat b $, then it gives
$$ \hat c= \hat a+\hat b = 73329650721664392+147988712393689577= 221318363115353969.$$
The server returns the value to the client. The client will obtain
$$\bar c= (221318363115353969 \mod p) \mod Q=6550628.$$
Notice that
$$6550628=99\times 256^2+244\times 256+100\xlongrightarrow{\mathcal{T}^{-1}} \fbox{99}\fbox{244}\fbox{100}.$$
It does not correspond to the wanted number $2.4$ when ASCII coding method is used.
\section{FHE is not applicable to client-server computing}
Cloud computing refers to the practice of transferring computer services such as computation or data storage to other redundant offsite locations available on the Internet, which allows application software to be operated using internet-enabled devices. It benefits one from the existing technologies and paradigms, even though he is short of deep knowledge about or expertise with them. The cloud aims to cut costs, and helps the users focus on their core business instead of being impeded by IT obstacles. Usually, cloud computing adopts the client-server business model.
What computations do you want to outsource privately? Backup your phone's contacts directory to the cloud? Ask the cloud to solve a mathematic problem in your homework? Do a private web search? $\cdots$. It seems obvious that the daily computational tasks are rarely constrained by some prescribed modulus. Moreover,
the client-server computing model can not deal with relational expressions which are defined over plain data, not over encrypted data. This is because
$$ a<b \not\Longrightarrow E(a)<E(b), \qquad E(a)<E(b) \not\Longrightarrow a<b.$$
In view of this weakness of FHE and the flaws of two typical schemes mentioned above, we think, FHE is not applicable to cloud computing.
\textbf{Remark 2}. The problem that what computations are worth delegating privately by individuals and companies to untrusted devices or servers remains untouched. We think the cloud computing community has not yet found a good for-profit model convincing individuals to pay for this or that computational service.
\section{Conclusion}
We reaffirm the role of modular arithmetic in modern cryptography and
show that FHE is not applicable to cloud computing because any FHE scheme does work over some encrypted domains. When two decrypted number are added, one cannot decide the carries
without knowing the secret decryption key. Moreover, there is no an invertible transformation from the
real number set to the encrypted domain which makes it impossible to tackle numerical calculations.
We think the primitive of FHE might be of little importance to client-server computing.
|
1101.5031
|
\section{Introduction}
In recent years semileptonic kaon decays ($\rm K_{\ell 3}, \ell=e, \mu$)
have attracted renewed interest \cite{WG1-2010}.
These decays provide the most accurate and theoretically cleanest way
to measure $\vert V_{us} \vert$~ and can give stringent constraints on new physics
scenarios by testing for possible violations of CKM unitarity and
lepton universality.
The hadronic matrix element of these decays is described by two
dimensionless form factors $f_{\pm}(t)$ where $t=(p_{K}-p_{\pi})^2$
is the four--momentum squared transferred to the lepton system.
These form factors are one of the input (through the phase space
integral) needed to determine $\vert V_{us} \vert$.
In the matrix element $f_{-}$ is multiplied by the lepton mass
and therefore its contribution can be neglected in $\rm K_{e3}$
decays.
$\rm K_{\mu3}$ decays instead are usually described in terms of
$f_{+}$ and the scalar form factor $f_{0}$ defined as:
\begin{equation}
f_{0}(t) = f_{+}(t) + t/(m^{2}_{K} - m^{2}_{\pi}) f_{-}(t), \nonumber
\end{equation}
$f_{+}$ and $f_{ 0}$ are related to the vector ($1^{-}$) and scalar
($0^{+}$) exchange to the lepton system, respectively.
By construction $f_{0}(0)=f_{+}(0)$ and since $f_{+}(0)$ is not
directly measurable it is customary to factor out $f_+^{K^0 \pi^-}(0)$
and normalize to this quantity all the form factors so that:
\begin{equation}
\bar f_+(t) = \frac{f_+(t)}{f_+(0)},
\ \bar f_0(t) = \frac{f_0(t)}{f_+(0)},
\ \bar f_+(0) = \bar f_{0} (0) = 1. \nonumber
\label{eq:Normff}
\end{equation}
There exist many parametrizations of the $\rm K_{\ell 3}$ form factors
in the literature, a widely known and most used is the Taylor
expansion:
\begin{equation}
\bar f_{+,0}(t) = 1 +
\lambda'_{+,0} \frac{t}{m_{\pi^\pm}^2} +
\frac{1}{2}\lambda''_{+,0} \left(\frac{t}{m_{\pi^\pm}^2}\right)^2, \nonumber
\label{eq:Taylor}
\end{equation}
where $\lambda'_{+,0}$ and $\lambda''_{+,0}$ are the slope and
the curvature of the form factors, respectively.
The disadvantage of such kind of parametrization is related \cite{franzini}
to the strong correlations that arise between parameters.
These forbid the experimental determination of $\lambda''_{0}$
experimentally, although, at least a quadratic expansion would be needed to
correctly describe the form factors.
This problem is avoided by parametrizations which, applaying
physical constraints, reduce to one the number of parameters used.
A typical example is the pole one:
\begin{equation}
\bar f_{+,0}(t) = \frac{M_{V,S}^2}{M_{V,S}^2-t}, \nonumber
\label{eq:pole}
\end{equation}
where the dominance of a single resonance is assumed and the
corresponding pole mass $M_{V,S}$ is the only free parameter.
More recently a parametrization based on dispersion techniques
has been proposed \cite{stern}:
\begin{eqnarray}
\bar{f}_{+}(t) = \exp\Bigl{[}\frac{t}{m^{2}_{\pi}}(\Lambda_{+}
+ \mathrm{H(t)})\Bigr{]},~~~~~~~~~
\bar{f}_{0}(t) = \exp\Bigl{[}\frac{t}{\Delta_{K\pi}}
(\mathrm{lnC}-\mathrm{G(t)})\Bigr{]}. \nonumber
\label{eq:dispersive}
\end{eqnarray}
The parameter C is the value of the scalar form factor at the
Callan--Treiman point, $f_{0}(t_{CT})$, where
$t_{CT} = \Delta_{K\pi} = m_{K}^{2} - m_{\pi}^{2}$.
It can be used to test the existence of right handed quark
currents coupled to the standard W boson.\\
\section{The NA48/2 experimental set--up}
The NA48/2 experiment at CERN/SPS was primarly designed to measure
the CP violating asymmetry in $\rm K^{\pm} \to 3\pi$ decays.
The layout of beams and detectors is shown on Fig.~\ref{fig:na48-2}
\begin{figure}[htb]
\centering
\includegraphics[height=5.cm]{na48_2_beam.eps}
\caption{Schematic side view of the NA48/2 beam line, decay volume and
detectors.}
\label{fig:na48-2}
\end{figure}
Two simultaneous $\rm K^+$ and $\rm K^-$ beams were produced by 400 GeV
primary protons impinging on a beryllium target. A system of magnets
and collimators selected particles, with average momentum of 60~GeV,
of both positive and negative charge. At the entrance of the decay
volume, a 114~m long vacuum tank, the $\rm K^+$ flux was $\sim2.3\times10^6$
per pulse of 4.5~s duration and the $\rm K^+/ \rm K^-$ ratio was about 1.8.
The NA48 detector is described in detail elsewhere \cite{NA48detector},
the main component used in this analysis are:\\
-- a magnetic spectrometer (DCH), designed to measure the momentum of
charged particles, it consisted of a magnet dipole located
between two sets of drift chambers. The obtained momentum resolution was
$\sigma(p)/p(\%) = 1 \oplus 0.044~p$ ($p$ in GeV/c);\\
-- a charged hodoscope (Hodo), made of two perpendicular segmented planes
of scintillators, it triggered the detector readout. The time resolution
was $\sim 150~$ps;\\
-- a liquid krypton electromagnetic calorimeter (LKr) of 27 radiation
lengths and e\-ner\-gy resolution of
$\sigma(E)/E(\%) = 3.2/\sqrt{E} \oplus 9.0/E \oplus 0.42$ ($E$ in GeV);\\
-- a muon system (MUV) consisting of three planes of alternating
horizontal and vertical scintillator strips, each plane was shielded
by a 80~cm thick iron wall. The inefficiency of the system was at
the level of one per mill and the time resolution was below 1~ns.
The data used for this analysis were collected in 2004 during a dedicated
run with a special trigger setup,
lower intensity
and a reduced momentum spread.
\section{ $\rm K^{\pm}_{\mu3}$~ event selection}
$\rm K^{\pm}_{\mu3}$~events are selected by requiring a track in DCH and at least
two clusters (photons) in LKr that are consistent with a $\pi^0$
decay. The track has to be inside the geometrical acceptance of the
detector, satisfy vertex and timing cuts and have $p >$ 10 GeV/c to
ensure proper efficiency of MUV system. In order to be identified
as a muon the track has to be associated in space and time to a MUV
hit and have $E/p < 0.2$, where $E$ is the energy deposited in the
calorimeter and $p$ the track momentum.
Finally a kinematical constraint is applied requiring the missing mass
squared (in the $\mu$ hypothesis) to satisfy: $| m_{\nu} | <$ 10~MeV$^2$.
Background from $\rm K^{\pm} \to \pi^{\pm} \pi^0$ events with
charged $\pi$ that decays in flight are suppressed by using a
combined cut on the invariant mass $m_{\pi^{\pm} \pi^0}$ and the
$\pi^0$ transverse momentum.
This cut reduces to 0.6\% the contamination but causes a
loss of statistics of about 24\%.
Another source of background is due to
$\rm K^{\pm} \to \pi^{\pm} \pi^0 \pi^0$ events with $\pi$
decaying in flight and a $\pi^0$ not reconstructed, the estimated
contamination amounts to about 0.1\% and no specific cut is applied.
The selected $\rm K^{\pm}_{\mu3}$~sample amounts to about 3.4$\times 10^6$ events.
\section{Fitting procedure and preliminary results}
To extract the form factors a fit is performed to the Dalitz
plot density. The Dalitz plot is subdivided into
5$\times$5 MeV$^2$ cells, those crossed by the kinematical
border are not used for the fit.
The raw density must be corrected for acceptance and resolution,
residual background, and the distortions induced by radiative effects.
The results of the fit for quadratic, pole and dispersive
parametrizations, are listed in Table~\ref{table:fit_results}.
\begin{table}[h]
\begin{center}
\begin{tabular}{cccc}
\hline
\rule{0mm}{5mm}Quadratic ($\times 10^{3}$) &
$\lambda^{'}_{+}$ & $\lambda^{''}_{+}$ & $\lambda_{0}$ \\
& 30.3$\pm$2.7$\pm$1.4 & 1.0$\pm$1.0$\pm$0.7 & 15.6$\pm$1.2$\pm$0.9 \\ \hline
\rule{0mm}{5mm}Pole (MeV/c$^2$) &
$m_V$ & $m_S$ & \\
& 836$\pm7\pm9$ & 1210$\pm$25$\pm$10 & \\ \hline
\rule{0mm}{5mm}Dispersive ($\times 10^{3}$) &
$\Lambda_{+}$ & $\ln C$ & \\
& 28.5$\pm$0.6$\pm$0.7$\pm$0.5 & 188.8$\pm$7.1$\pm$3.7$\pm$5.0 & \\ \hline
\end{tabular}
\caption{ NA48/2 preliminary form factors fit results for quadratic,
pole and dispersive parametrizations. The first error is statistical,
the second systematical. The theo\-re\-ti\-cal uncertainty \cite{stern}
has been evaluated and added to dispersive results.}
\label{table:fit_results}
\end{center}
\end{table}
The comparison with the results of $\rm K_{\mu 3}$ quadratic fit as
reported by recent experiments \cite{WG1-2010} is shown in
Fig.~\ref{fig:kmu3ff}.
The $1 \sigma$ contour plots are displayed both for $\rm K^{0}_{\mu 3}$
decays (KLOE, KTeV and NA48) and charged $\rm K$ (ISTRA studied
$\rm K^{-}_{\mu3}$ only), our high precision measurement is the
first to use both $\rm K^+$ and $\rm K^-$ particles.
We find a quadratic term in the expansion of the vector form
factor compatible with zero and a slope of the scalar form factor
larger with respect to NA48 case \cite{NA48kmu3ff} and in agreement
with other measurements.
For this preliminary evaluation of systematic uncertainty
we have changed by small amounts the cuts defining the vertex quality
and the geometrical acceptance, we applyed variations to the
values of pion and muon energies in the kaon cms, we increased
$\pi \to \mu$ background and took into account the differences in the
results of two independent analyses that are realized in parallel.
\begin{figure}[h]
\centering
\vskip -1.0cm
\includegraphics[height=11.cm]{plot_kmu3ff.eps}
\vskip -4.5cm
\caption{Quadratic fit results for $\rm K_{\mu 3}$ ($\rm K_{L}$ for neutral
and $\rm K^{\pm}$ for charged) decays.
The ellipses are the 1 $\sigma$ contour plot.
For comparison also the $\rm K_{e3}$ fit from FlaviaNet
WG1 is shown.}
\label{fig:kmu3ff}
\end{figure}
\section{$\rm K^{\pm}_{e3}$ form factors and future perspectives at NA62}
Using the same data sample we are also investigating $\rm K^{\pm}_{e3}$~decays. Their
selection is similar to that of $\rm K^{\pm}_{\mu3}$, a track and a $\pi^0$ being
required. The electron ID is achieved by demanding $0.95<E/p<1.05$,
this results in a $\rm K^{\pm}_{e3}$~sample of 4.2$\times 10^6$ events.
Since these decays are described by only one form factor the problems
related to the correlations between parameters are greatly reduced.
Furthermore background issues are less critical given that to fake these
decays $\pi^{\pm} \pi^0$ events with a $\pi^{\pm}$ having $E/p>0.95$
are needed. For these reasons, results of higher precision with respect
to $\rm K^{\pm}_{\mu3}$~analysis are expected from this measurement.\\
The NA62 experiment, using the same beam line and detector of NA48/2,
collected in 2007 data for the measurement of
$R_K = \Gamma(\rm K_{e2})/\Gamma(\rm K_{\mu2})$ and made tests for the future
$\rm K^{+} \to \pi^{+} \nu~ \overline \nu$ experiment. The data collected
contain also $\rm K^{+}_{e3}/\rm K^{+}_{\mu 3}$ samples of
$\simeq 40/20\times 10^6 $ events.
A special $\rm K_L$ run was also taken, it provides
$\rm K^{0}_{e 3}$ and $\rm K^{0}_{\mu 3}$ sample
of about $4\times 10^6$ events.
With this statistics, high precision measurements of the form factors for
all $\rm K_{\ell 3}$ channels will be done by NA62, providing important
inputs to further reduce the uncertainty on $\vert V_{us} \vert$~.
|
1101.5259
|
\section{Introduction}
Let $(M,g)$ be a closed, n-dimensional Riemannian manifold and $T^{1}M$ the unit tangent bundle of $M$ considered as a closed Riemannian manifold with the Sasaki
metric. Let $X:M\longrightarrow T^{1}M$ be a unit vector field defined on $M$, regarded as a smooth section on the unit tangent bundle $T^{1}M$. The volume of $X$ was defined in [8] by $\mathrm{vol}(X):=\mathrm{vol}(X(M))$, where $\mathrm{vol}(X(M))$ is the volume of the submanifold $X(M)\subset T^{1}M$. Using an orthonormal local frame $\left\{e_{1}, e_{2},\ldots,e_{n-1}, e_{n}=X\right\}$, the volume of the unit vector field $X$ is given by
\begin{eqnarray*}
\mathrm{vol}(X)=\int_{M} (1+\sum\limits_{a=1}^{n}\left\|\nabla_{e_{a}}X\right\|^{2}+\sum\limits_{a<b}\left\|\nabla_{e_{a}}X\wedge\nabla_{e_{b}}X\right\|^{2}+\ldots
\\
\ldots+\sum\limits_{a_{1}<\cdots<a_{n-1}}\left\|\nabla_{e_{a_{1}}}X\wedge\cdots\wedge\nabla_{e_{a_{n-1}}}X\right\|^{2})^{1/2} \nu_{_{M}}(g)
\end{eqnarray*}
and the energy of the vector field $X$ is given by
\begin{eqnarray*}
\mathcal{E}(X)=\frac{n}{2}\mathrm{vol}(M)+\frac{1}{2}\int_{M}\sum\limits_{a=1}^{n}\left\|\nabla_{e_{a}}X\right\|^{2}\nu_{_{M}}(g)
\end{eqnarray*}
The Hopf vector fields on $\mathbb{S}^{3}$ are unit vector fields tangent to the classical Hopf
fibration $\pi:\mathbb{S}^{3}\longrightarrow \mathbb{S}^{2}$ with fiber homeomorphic to $\mathbb{S}^{1}$.
\\
The following theorems gives a characterization of Hopf flows as absolute minima of volume and energy functionals:
\theorem [\textbf{[8]}] {The unit vector fields of minimum volume on the sphere $\mathbb{S}^{3}$ are precisely
the Hopf vector fields and no others.}
\normalfont
\theorem [\textbf{[1]}] {The unit vector fields of minimum energy on the sphere $\mathbb{S}^{3}$ are precisely
the Hopf vector fields and no others.}
\\
\\
\normalfont We prove in this paper the following boundary version for these Theorems:
\theorem {Let $U$ be an open set of the three-dimensional unit sphere $\mathbb{S}^{3}$ and let $K\subset U$ be a compact set. Let $\vec{v}$ be an unit vector field on $U$ which coincides with a Hopf flow $H$ along the boundary of K. Then $\mathrm{vol}(\vec{v})\geq \mathrm{vol}(H)$ and $\mathcal{E}(\vec{v})\geq \mathcal{E}(H)$.}
\\
\\
\normalfont
Other results for higher dimensions may be found in [2], [5], [7] and [8].
\section{Preliminaries}
Let $U\subset\mathbb{S}^{3}$ be an open set. We consider a compact set $K\subset U$. Let $H$ be a Hopf vector field on $\mathbb{S}^{3}$ and let $\vec{v}$ be an unit vector field defined on $U$. We also consider the map $\varphi_{t}^{\vec{v}}:U\longrightarrow \mathbb{S}^{3}(\sqrt{1+t^{2}})$ given by $\varphi_{t}^{\vec{v}}(x)=x+t\vec{v}(x)$. This map was introduced in [10] and [3].
\lemma {For $t>0$ sufficiently small, the map $\varphi_{t}^{\vec{v}}$ is a diffeomorphism.}
\normalfont
\proof A simple application of the identity perturbation method $\square$
\\
\\
In order to find the Jacobian matrix of $\varphi_{t}^{\vec{u}}$, we define the unit vector field $\vec{u}$
\begin{eqnarray*}
\vec{u}(x):=\frac{1}{\sqrt{1+t^{2}}}\vec{v}(x)-\frac{t}{\sqrt{1+t^{2}}}x
\end{eqnarray*}
Using an adapted orthonormal frame $\left\{e_{1},e_{2},\vec{v}\right\}$ on a neighborhood $V\subset U$, we obtain an adapted orthonormal frame on $\varphi_{t}^{\vec{v}}(V)$ given by $\left\{\bar{e}_{1},\bar{e}_{2},\vec{u}\right\}$, where $\bar{e}_{1}=e_{1}$, $\bar{e}_{2}=e_{2}$.
\\
\\
In this manner, we can write
\begin{eqnarray*}
d\varphi_{t}^{\vec{v}}(e_{1})\!\!\!&=&\!\!\!\left\langle d\varphi_{t}^{\vec{v}}(e_{1}),e_{1}\right\rangle e_{1}+\left\langle d\varphi_{t}^{\vec{v}}(e_{1}),e_{2}\right\rangle e_{2}+\left\langle d\varphi_{t}^{\vec{v}}(e_{1}),\vec{u}\right\rangle \vec{u}\\
d\varphi_{t}^{\vec{v}}(e_{2})\!\!\!&=&\!\!\!\left\langle d\varphi_{t}^{\vec{v}}(e_{2}),e_{1}\right\rangle e_{1}+\left\langle d\varphi_{t}^{\vec{v}}(e_{2}),e_{2}\right\rangle e_{2}+\left\langle d\varphi_{t}^{\vec{v}}(e_{2}),\vec{u}\right\rangle \vec{u}\\
d\varphi_{t}^{\vec{v}}(\vec{v})\!\!\!&=&\!\!\!\left\langle d\varphi_{t}^{\vec{v}}(\vec{v}),e_{1}\right\rangle e_{1}+\left\langle d\varphi_{t}^{\vec{v}}(\vec{v}),e_{2}\right\rangle e_{2}+\left\langle d\varphi_{t}^{\vec{v}}(\vec{v}),\vec{u}\right\rangle \vec{u}
\end{eqnarray*}
Now, by Gauss' equation of immersion $\mathbb{S}^{3}\hookrightarrow \mathbb{R}^{4}$, we have
\begin{eqnarray*}
d\vec{v}(Y)=\nabla_{Y}\vec{v}-\left\langle \vec{v},Y\right\rangle x
\end{eqnarray*}
for every vector field $Y$ on $\mathbb{S}^{3}$, and then
\begin{eqnarray*}
\left\langle d\varphi_{t}^{\vec{v}}(e_{1}),e_{1}\right\rangle = \left\langle e_{1}+td\vec{v}(e_{1}),e_{1}\right\rangle = 1 + t\left\langle \nabla_{e_{1}}\vec{v},e_{1}\right\rangle
\end{eqnarray*}
Analogously, we can conclude that
\begin{eqnarray*}
\left\langle d\varphi_{t}^{\vec{v}}(e_{1}),e_{2}\right\rangle \!\!\!&=&\!\!\!t\left\langle \nabla_{e_{1}}\vec{v},e_{2}\right\rangle\\
\left\langle d\varphi_{t}^{\vec{v}}(e_{2}),e_{1}\right\rangle \!\!\!&=&\!\!\! t\left\langle \nabla_{e_{2}}\vec{v},e_{1}\right\rangle\\
\left\langle d\varphi_{t}^{\vec{v}}(e_{2}),e_{2}\right\rangle \!\!\!&=&\!\!\! 1+t\left\langle \nabla_{e_{2}}\vec{v}, e_{2}\right\rangle\\
\left\langle d\varphi_{t}^{\vec{v}}(e_{1}),\vec{u}\right\rangle \!\!\!&=&\!\!\! 0\\
\left\langle d\varphi_{t}^{\vec{v}}(e_{2}),\vec{u}\right\rangle \!\!\!&=&\!\!\! 0\\
\left\langle d\varphi_{t}^{\vec{v}}(\vec{v}),\vec{u}\right\rangle \!\!\!&=&\!\!\! \sqrt{1+t^{2}}
\end{eqnarray*}
By applying the notation $h_{ij}(\vec{v}):=\left\langle \nabla_{e_{i}}\vec{v},e_{j}\right\rangle$ ($i,j=1,2$), the determinant of the Jacobian matrix of $\varphi_{t}^{\vec{v}}$ can be express in the form
\begin{eqnarray*}
\det(d\varphi_{t}^{\vec{v}})=\sqrt{1+t^{2}}(1+\sigma_{1}(\vec{v}).t+\sigma_{2}(\vec{v}).t^{2})
\end{eqnarray*}
where, by definition,
\begin{eqnarray*}
\sigma_{1}(\vec{v})\!\!\!&:=&\!\!\!h_{11}(\vec{v})+h_{22}(\vec{v})\\ \sigma_{2}(\vec{v})\!\!\!&:=&\!\!\!h_{11}(\vec{v})h_{22}(\vec{v})-h_{12}(\vec{v})h_{21}(\vec{v})
\end{eqnarray*}
\section{Proof of Theorem 1.3}
The energy of the vector field $\vec{v}$ (on $K$) is given by
\begin{eqnarray*} \mathcal{E}(\vec{v}):=\frac{1}{2}\int_{K}\left\|d\vec{v}\right\|^{2}=\frac{3}{2}\mathrm{vol}(K)+\frac{1}{2}\int_{K}\left\|\nabla \vec{v}\right\|^{2}
\end{eqnarray*}
Using the notations above, we have
\begin{eqnarray*} \mathcal{E}(\vec{v})\!\!\!&=&\!\!\!\frac{3}{2}\mathrm{vol}(K)+\frac{1}{2}\int_{K}[(\sum\limits_{i,j=1}^{2}(h_{ij})^{2}+(\left\langle \nabla_{\vec{v}}\vec{v},e_{1}\right\rangle)^{2}+(\left\langle \nabla_{\vec{v}}\vec{v},e_{2}\right\rangle)^{2}]
\end{eqnarray*}
and then
\begin{eqnarray*}
\mathcal{E}(\vec{v})&\geq&\!\!\! \frac{3}{2}\mathrm{vol}(K)+\frac{1}{2}\int_{K}\sum\limits_{i,j=1}^{2}(h_{ij})^{2}\\
\!\!\!&\geq&\!\!\! \frac{3}{2}\mathrm{vol}(K)+\frac{1}{2}\int_{K}2(h_{11}h_{22}-h_{12}h_{21})\\
\!\!\!&=&\!\!\!\frac{3}{2}\mathrm{vol}(K)+\int_{K}\sigma_{2}(\vec{v})
\end{eqnarray*}
On the other hand, by change of variables theorem, we obtain
\begin{eqnarray*} \mathrm{vol}[\varphi_{t}^{H}(K)]=\int_{K}\sqrt{1+t^{2}}(1+\sigma_{1}(H).t+\sigma_{2}(H).t^{2})=\delta\cdot\mathrm{vol}(\mathbb{S}^{3}(\sqrt{1+t^{2}}))
\end{eqnarray*}
where $\delta:=\mathrm{vol}(K)/\mathrm{vol}(\mathbb{S}^{3})$.
\\
(Remark that $\sigma_{1}(H)$ and $\sigma_{2}(H)$ are constant functions on $\mathbb{S}^{3}$, in fact, we have $\sigma_{1}(H)=0$ and $\sigma_{2}(H)=1$, by a straightforward computation shown in [6]).
\\
\\
Suppose now that $\vec{v}$ is an unit vector field on $K$ which coincides with a Hopf vector field $H$ on the boundary of $K$. Then, obviously
\begin{eqnarray*}
\mathrm{vol}[\varphi_{t}^{\vec{v}}(K)]=\mathrm{vol}[\varphi_{t}^{H}(K)]
\end{eqnarray*}
Therefore, we obtain
\begin{eqnarray*} \mathrm{vol}[\varphi_{t}^{\vec{v}}(K)]\!\!\!&=&\!\!\!\int_{K}\sqrt{1+t^{2}}(1+\sigma_{1}(\vec{v}).t+\sigma_{2}(\vec{v}).t^{2})\\
&=&\!\!\!\delta\cdot\mathrm{vol}(\mathbb{S}^{3}(\sqrt{1+t^{2}}))=[\mathrm{vol}(K)](1+t^{2})^{3/2}
\end{eqnarray*}
By identity of polynomials, we conclude that
\begin{eqnarray*}
\int_{K}\sigma_{2}(\vec{v})=\mathrm{vol}(K)
\end{eqnarray*}
and consequently
\begin{eqnarray*}
\mathcal{E}(\vec{v})\geq \frac{3}{2}\mathrm{vol}(K)+\mathrm{vol}(K)=\mathcal{E}(H)
\end{eqnarray*}
Now, observing that
\begin{eqnarray*}
\mathrm{vol}(H)= 2\mathrm{vol}(K), \ \ \ \ \int_{K}\sigma_{2}(\vec{v})=\mathrm{vol}(K)\ \ \ and \ \ \ \sum\limits_{i,j=1}^{2} h_{ij}^{2}(\vec{v})\geq 2\sigma_{2}(\vec{v})
\end{eqnarray*}
we can obtain an analogue of this result for volumes
\begin{eqnarray*}
\mathrm{vol}(X)\!\!\!&=&\!\!\!\int_{K}\sqrt{1+\sum\limits h_{ij}^{2}+[\det(h_{ij})]^{2}+\cdots}
\\
&\geq&\!\!\!\int_{K}\sqrt{1+2\sigma_{2}+\sigma_{2}^{2}}
\\
&=&\!\!\!\int_{K}(1+\sigma_{2})=2\mathrm{vol}(K)=\mathrm{vol}(H)\ \square
\end{eqnarray*}
\section{Final remarks}
\begin{enumerate}
\item If $K$ is a spherical cap (the closure of a connected open set
with round boundary of the three unit sphere), the theorem provides a
``boundary version" for the minimalization theorem of
energy and volume functionals on [1] and [8].
\\
\item The ``Hopf boundary" hypothesis is essential. In fact, if there is no constraint for the unit vector field $\vec{v}$ on $\partial K$, it is possible to construct vector fields on ``small caps" such that $\left\|\nabla \vec{v}\right\|$ is small on $K$ (exponential maps may be used on that construction). A consequence of this is that $\mathcal{E}(\vec{v})$ and $\mathrm{vol}(\vec{v})$ are less than volume and energy of Hopf vector fields respectively.
\\
\item The results of this paper may, possibly, be extended for the energy of solenoidal unit vector fields in the higher dimensional case ($n=2k+1$). We intend to treat this subject in a forthcoming paper.
\\
\item We express our gratitude to Prof. Jaime Ripoll for helpful
conversation concerning the final draft of our paper.
\end{enumerate}
|
1101.5312
|
\section{Introduction}
In the context of the gauge/gravity duality, consistent Kaluza--Klein truncations of 10- and 11-dimensional supergravity have proved to be powerful solution-generating tools. A truncation of the higher-dimensional spectrum on some compact manifold is said to be consistent if all solutions of the truncated, lower-dimensional theory are also solutions to the full, higher-dimensional theory. Since they provide an effective lower-dimensional model with a restricted number of degrees of freedom, consistent truncations allow to tackle several problems in a setup which is much simpler than the original theory, guaranteeing at the same time the lifting of the solutions. As an example, consistent truncations to 5 dimensions provide a convenient framework for describing the renormalization group flow of the dual 4-dimensional quantum field theories, with the fifth coordinate playing the role of an energy scale.
A further, more formal, motivation to study consistent truncations is that they represent the main tool to investigate which lower-dimensional supergravities can be connected to string theory.
Recently, there has been a revival of interest in consistent truncations, mainly motivated by the intense research activity towards a holographic description of strongly coupled condensed matter phenomena, such as superconductivity and quantum critical points exhibiting non-relativistic scale invariance.
A limitation of these holographic models is that most of them are {\it ad hoc} constructions built in a bottom-up approach, while in order to have full control on the gauge/gravity correspondence a rigorous embedding into string theory is needed. To achieve this, one should find a consistent truncation of 10-dimensional supergravity to the desired lower-dimensional model. A crucial, non-trivial feature required by these applications is that the truncation preserve massive and/or charged Kaluza--Klein modes (see in particular \cite{MaldMartTach, GauntlettKimVarelaWaldram, Gubser1, Gauntlett:2009dn}).
Here, we will review the basic ideas underlying some recently found consistent truncations of higher-dimensional supergravity with massive modes. Often, the consistency of a truncation relies on some symmetry under which the preserved modes are invariant, and such that the truncated non-invariant modes are never generated in the equations of motion. In the cases of interest for us, the compact manifold admits a $G$-structure whose intrinsic torsion is also $G$-invariant. The modes preserved by the truncation are chosen to be precisely all the singlets under $G$. When, as it will be the case for us, among the invariant fields there is at least one spinor, and the invariant bosonic fields can be reconstructed by taking appropriate spinor bilinears, one can define a truncation ansatz which preserves a fraction of supersymmetry. A clear advantage in this case is that one disposes of a powerful organizing principle: since supersymmetric theories are very constrained, just a few data need to be provided in order to fully specify the truncated model.
In the following, we illustrate how these ideas work in practice by presenting a consistent truncation of type IIB supergravity on arbitrary squashed Sasaki--Einstein manifolds, leading to gauged $\mathcal N=4$ supergravity in five dimensions \cite{IIBonSE} (see also \cite{LiuEtAl, GauntlettVarelaSE, SkenderisTaylorTsimpis} for closely related work, and \cite{GauntlettKimVarelaWaldram, ExploitingN=2, BuchelLiu, GauntlettVarela07} for earlier supersymmetric results). As we will briefly describe, the resulting 5-dimensional model exhibits quite remarkable features, such as the presence of both massless and massive modes, as well as tensor fields dual to vectors charged under a non-abelian gauge group. Moreover, the retained Kaluza--Klein modes capture the universal pure gauge sector of the dual 4-dimensional super Yang--Mills theories.
\section{Type IIB supergravity on squashed Sasaki-Einstein manifolds}
A regular (respectively, quasi-regular) Sasaki--Einstein manifold $Y$ can be seen as a U(1) fibration over a K\"ahler--Einstein base manifold (respectively, orbifold) $B_{\rm KE}\,$:
\begin{equation}\label{SEmetric}
ds^2(Y) \,=\, ds^2(B_{\rm KE}) + \eta\otimes\eta\,,
\end{equation}
where $\eta$ is the globally defined 1-form specifying the fibration.
All 5-dimensional Sasaki--Einstein manifolds are also endowed with a real 2-form $J$ and a complex 2-form $\Omega$, both globally defined. These satisfy the algebraic constraints
\begin{equation}\label{eq:AlgConstr}
\eta\,\lrcorner
\, J = \eta\,\lrcorner
\, \Omega = 0\,,\qquad \Omega\wedge J \,=\, \Omega\wedge \Omega\,=\,0\,,\qquad \Omega\wedge \overline\Omega \,=\,2\,J\wedge J \,=\,4\,{\rm vol}(B_{\rm KE})\,,
\end{equation}
as well as the differential conditions
\begin{equation}
\label{SEstructure}
d\eta \,=\, 2J\;,\quad\qquad d \Omega = 3 i\, \eta \wedge \Omega\,.
\end{equation}
The relations (\ref{eq:AlgConstr}) imply that the structure group of the 5-dimensional manifold, which generically is SO(5), is reduced to SU(2), with the forms $\eta$, $J$, $\Omega$ being SU(2) singlets. The conditions (\ref{SEstructure}) constrain the torsion of the SU(2) structure, which is required to also be an SU(2) singlet, and constant. Starting from the forms, the metric on the 4-dimensional subspace transverse to $\eta$ can be reconstructed by identifying a complex structure $I$ with respect to which $\Omega$ is of type $(2,0)$, and taking the product $J I$. We also have the Hodge duality relations
\begin{equation}\label{eq:*SEforms}
* \eta = {\rm vol}(B_{\rm KE})\;,\quad\qquad *J \,=\, J\wedge \eta\,,\qquad *\Omega \,=\, \Omega\wedge \eta\,.
\end{equation}
Our ansatz for the dimensional reduction is defined by writing down the most general expression for the metric and the various tensor fields of type IIB supergravity in terms of the forms characterizing the Sasaki--Einstein structure.
In doing this, we actually consider a class of internal metrics which is more general than (\ref{SEmetric}): we allow for an overall volume parameter (the ``breathing mode''), as well as for a parameter modifying the relative size of the U(1) fibre with respect to the size of the K\"ahler--Einstein base (the ``squashing mode'').
Hence the class of spaces on which we are reducing is the one of {\it squashed} Sasaki--Einstein manifolds.
Specifically, our truncation ansatz for the 10-dimensional metric in the Einstein frame is \cite{MaldMartTach}:
\begin{equation}\label{eq:10dmetric}
ds^2 = e^{-\frac{2}{3}(4U+V)}ds^2(M)\,+\,e^{2U}ds^2(B_{\rm KE})\,+\,e^{2V}(\eta+ A)\otimes (\eta+ A)\,,
\end{equation}
where $A$ is a 1--form on the external 5-dimensional spacetime $M$, while $U$ and $V$ are scalars on $M$, which combined parameterize the breathing and the squashing modes of the compact manifold.
Regarding the form fields of type IIB supergravity, let us for instance consider the NSNS 2-form $B$. We take the general expansion
\begin{equation}
\label{eq:Bcov}
B \,=\, b_2 + b_1 \wedge(\eta + A) + b^{J} J+ {\rm Re}(b^\Omega\,\Omega)\,,
\end{equation}
where $b_2$ is a 2-form, $b_1$ a 1-form, $b^J$ a real scalar, and $b^\Omega$ a complex scalar on $M$. The other type IIB form fields are expanded along the same lines. Finally, the dilaton $\phi$ and the Ramond-Ramond scalar $C_0$ are assumed to be independent of the internal coordinates.
The fact that the system of differential forms $\{\eta,\,J,\,\Omega\}$ is closed under the various operations appearing in the higher-dimensional equations of motion (exterior derivative, wedge product, Hodge star) ensures the consistency of the truncation. Indeed, one can plug the truncation ansatz in the 10-dimensional equations of motion, and check that they reduce to 5-dimensional equations; in particular, the dependence on the internal coordinates drops out. In parallel, one reduces the type IIB action by performing the integral over the internal space (care is needed in implementing the self-duality constraint on the Ramond-Ramond 5-form $F_5$, see discussion in \cite{IIBonSE}). Then one verifies that the resulting 5-dimensional action provides precisely the 5-dimensional equations that were obtained by reducing the 10-dimensional equations. This proves the consistency of the truncation.
As a final remark, we note that an ansatz expressed in terms of the invariant forms $\{\eta,\, J,\,\Omega\}$ of a generic SU(2) structure on the compact manifold would in general not be enough for having a consistent truncation. Indeed, while these forms are singlets under the structure group, their derivatives, providing the intrinsic torsion of the SU(2) structure, would generically contain non-singlet contributions, and this would a priori spoil consistency. Hence the simplicity of (\ref{SEstructure}), namely the fact that only singlets (with constant coefficients) appear on the right hand side, is as essential as (\ref{eq:AlgConstr}). Besides the case of Sasaki--Einstein structures in odd dimension, there are further examples of geometries characterized by a $G$-structure whose intrinsic torsion is also $G$-invariant, so that a truncation ansatz expressed in terms of the $G$-invariant tensors is consistent: for instance, the 7-dimensional weak-$G_2$ manifolds \cite{GauntlettKimVarelaWaldram}, the 6-dimensional Nearly-K\"ahler manifolds \cite{KashaniPoor:2007tr}, as well as the special holonomy manifolds, whose $G$-structure has vanishing torsion.
A situation in which the conditions just described are fulfilled is when the compact manifold is a coset space $\mathcal G/\mathcal H$. In this case, the structure group can be identified with $\mathcal H$, and the ansatz based on the singlets of the structure group can be rephrased as an ansatz invariant under the left-action of $\mathcal G$. Consistent truncations of supergravity on coset spaces have been studied in \cite{ExploitingN=2, T11reduction}. In \cite{T11reduction} (see also \cite{Bena:2010pr}), a specific example of 5-dimensional Sasaki--Einstein manifold was considered, namely the $T^{1,1}= (SU(2)\times SU(2))/U(1)$ coset space (also known as the base of the conifold). The consistent truncation outlined above can in this case be substantially enhanced, and provides the supersymmetric completion of a more limited non-supersymmetric truncation on $T^{1,1}$ \cite{PT-BHM}, containing several physically relevant conifold solutions, such as the Klebanov--Strassler and the Maldacena--Nu\~nez ones~\cite{KSandMN}.
\section{Gauged $\mathcal N=4$ supergravity description}
The 5-dimensional model stemming from the procedure described above can be understood in the framework of gauged $\mathcal N=4$ supergravity \cite{DHZ, SchonWeidner}. In this section we present just its most relevant features, referring to \cite{IIBonSE} for the expression of the complete bosonic action.\footnote{A detailed study of the inclusion of the fermionic sector has been performed in \cite{FermionicConsTrunc}.} Actually, thanks to the constraints dictated by half-maximal supersymmetry, in order to completely describe the model one needs to specify just a few data, namely the number of vector multiplets and the embedding tensor describing how the gauge group is embedded into the global symmetry group. Let us discuss them in turn.
The expectation of having $\mathcal N=4$ supersymmetry is first of all motivated by the gravitino ansatz. Type IIB supergravity contains two Majorana--Weyl gravitini of the same chirality $\Psi_M^{\alpha}$, where $\alpha=1,2$, and $M$ is a 10-dimensional spacetime index.
To define the truncation ansatz for these fields, we exploit the fact that the SU(2) structure condition implies the existence of two globally defined spinors $\zeta^{1}, \zeta^{2}$ on the internal manifold, being one the charge conjugate of the other. These are related to the forms $\eta,\,J,\,\Omega$ via appropriate spinor bilinears.
We use the two spinors to expand the 5-dimensional spacetime components of the 10-dimensional gravitini as
\begin{equation}
\Psi^\alpha_\mu \,=\, \psi_\mu^{\alpha\,1} \otimes \zeta^1 + \psi_\mu^{\alpha\,2} \otimes \zeta^2\,.
\end{equation}
The 5-dimensional fields $\psi_\mu^{\alpha\,1},\,\psi_\mu^{\alpha\,2}$ can then be combined into four gravitini $\psi_\mu^i$, $i=1,\ldots,4$, satisfying the symplectic-Majorana condition
\begin{equation}
\overline\psi{}_{\mu\,i} \equiv (\psi^i_{\mu})^{\dagger} \gamma^0= \Omega_{ij}(\psi_{\mu}^j)^TC\, ,
\end{equation}
where $\Omega_{ij}$ is the USp(4) invariant form while $C$ is the charge conjugation matrix. This provides the gravitino content of $\mathcal N=4$ supergravity.
Turning to the 5-dimensional bosonic sector, one can verify that it organizes in ${\cal N} = 4$ multiplets, with all the couplings respecting supersymmetry. For instance, by using a solvable parameterization one verifies that the target manifold of the scalar sigma-model is the prescribed homogeneous space
\begin{equation}
{\cal M}_{\rm scal} = {\rm SO}(1,1) \times \frac{{\rm SO}(5,n)}{{\rm SO}(5)\times {\rm SO}(n)}\,,
\end{equation}
with $n=2$, which corresponds to a model with two vector multiplets. The counting and the couplings of the vector fields agree with this, though one has to take into account some complications due to the gauging. Indeed, while ungauged $\mathcal N=4$ supergravity in 5 dimensions contains eight vector fields, in our model we find four vectors and four 2-forms. The latter are seen as the Poincar\'e duals of the missing four vectors, the dualization being required by the gauging at hand.
In order to fully specify the gauging, one computes the embedding tensor mapping the gauge group generators into the generators of the duality group, which in the present case is SO(1,1) $\times$ SO(5,2). This determines the various additional couplings in the lagrangian with respect to the ungauged case, including the scalar potential, as well as the fermionic shifts appearing in the supersymmetry transformations (for the embedding tensor formalism we refer to \cite{Samtleben:2008pe} and references therein). In our case, the embedding tensor has components $f_{MNP}=f_{[MNP]}$ and $\xi_{MN}=\xi_{[MN]}$, where the indices $M,N,P=1,\ldots 7$ run in the fundamental of SO$(5,2)$. These can be determined by studying the gauge-covariant derivative of the scalars, and we find
\begin{equation}
\nonumber f_{125} = f_{256} = f_{567} = - f_{157} = -2,
\end{equation}
\begin{equation}\label{eq:OurEmbTensor}
\xi_{34}= -3\sqrt 2,\qquad\qquad \xi_{12}=\xi_{17}=-\xi_{26}= \xi_{67} = -\sqrt 2\, k\,.
\end{equation}
The higher-dimensional origin of the $f$-components is found in the geometric flux associated with the non-closure of the form $\eta$ on the internal manifold, while $\xi_{34}$ can be traced back to the non-closure of $\Omega$.
The remaining non-zero $\xi$-components are proportional to the constant $k$ parameterizing the internal Ramond-Ramond 5-form flux, $F_5^{\rm flux} = k\, J\wedge J\wedge \eta\,$.
By studying the commutation relations of the generators identified by the embedding tensor, one can infer that the gauge group is given by the product of U(1) with the three-dimensional Heisenberg group. The 10-dimensional origin of this gauge symmetry is found in part in the reparameterization invariance of the spacetime and in part in the shift symmetry of the type IIB form fields.
Finally, we notice that taking the limit of vanishing fluxes ($d\eta = d\Omega = k = 0$), we obtain a consistent truncation of type IIB supergravity on $K3 \times S^1$ to ungauged $\mathcal N=4$ supergravity coupled to two vector multiplets.
\section{The holographic picture}
In this section we put our consistent truncation in the perspective of the gauge/gravity correspondence.
We start from the 5-dimensional scalar potential, which reads
\begin{equation}\label{eq:ScalPot}
\begin{array}{rcl}
\mathcal V &=&\displaystyle -\,12 \,{ e}^{-\frac{14}{3}U-\frac{2}{3}V} +2\, { e}^{-\frac{20}{3}U + \frac{4}{3}V} + \frac{9}{2}\, { e}^{-\frac{20}{3}U-\frac{8}{3}V -\phi}|b^\Omega|^2 \\[4mm]
&&\displaystyle+\, \frac{9}{2}\, { e}^{-\frac{20}{3}U-\frac{8}{3}V+ \phi}|c^\Omega - C_0 b^\Omega|^2 + \;{ e}^{-\frac{32}{3}U-\frac{8}{3}V} \big[3\,{\rm Im}\big(b^\Omega\,\overline{c^\Omega}\big) + k\big]^2,
\end{array}
\end{equation}
where $c^\Omega$ is a complex scalar arising from the Ramond-Ramond 2-form potential.
Studying $\mathcal V$ one finds two extrema. Choosing the RR flux $k=2$, these are
\begin{equation}
U \,=\, V \,=\, b^\Omega \,=\, c^\Omega \,=\, 0\,,\quad{\rm with}\;{\rm arbitrary}\; \phi, \;C_0\,,
\label{susyvacuum}
\end{equation}
and
\begin{equation}
e^{4U}=e^{-4V}=\frac{2}{3}\,,\qquad b^\Omega=\frac{e^{i \theta+\phi/2}}{\sqrt{3}}\,\,,\qquad c^\Omega=b^\Omega \tau\,, \qquad \tau \equiv (C_0 + i\,e^{-\phi})\,,
\label{nonsusyvacuum}
\end{equation}
where $\phi$, $C_0$ and $\theta$ are moduli of the solution. The cosmological constant $\Lambda \equiv \langle \mathcal V \rangle$ is negative in both cases ($\Lambda = -6$ for the first and $\Lambda = - \frac{27}{4}\,$ for the second), hence we have Anti-de Sitter vacua. The first extremum has $\mathcal N=2$ supersymmetry, and lifts to the standard AdS$_5\times\,$Sasaki--Einstein$_5$ solution of type IIB supergravity.
The second extremum is non-supersymmetric, and corresponds to an AdS$_5$ solution of type IIB supergravity with squashed internal metric, originally found in \cite{RomansIIBsols}.
By studying the mass spectrum of the field fluctuations about these backgrounds, one can implement the standard AdS/CFT dictionary and deduce the anomalous dimensions of the dual CFT operators. For the non-supersymmetric extremum we find some irrational conformal dimensions; since the square masses are all positive, the vacuum is stable at least with respect to the modes kept in the truncation. The results for the supersymmetric vacuum are summarized in table \ref{GaugeGravityTable}. By studying the dual spectrum (for instance in the well-known cases of $S^5$ and $T^{1,1}$), we see that we are keeping just flavor singlets, which are built in terms of the gauge superfield $W_\alpha$ in the $\mathcal N=1$ super Yang-Mills theory. Since this is a fermionic superfield, we can construct just a finite number of non-vanishing combinations, which precisely match the degrees of freedom of the gravity model. We conclude that our truncation describes the large $N$ limit of the universal gauge sector of $\mathcal N=1$ super Yang--Mills theories in 4 dimensions.
As noticed in \cite{SkenderisTaylorTsimpis}, on the field theory side the consistency of our truncation translates into the fact that the set of operators appearing in table~\ref{GaugeGravityTable} is closed under the operator product expansion (at least in the large $N$ limit). It would be interesting to explore to what extent this is a general feature of consistent truncations with a field theory dual.
\renewcommand{\arraystretch}{1.2}
\begin{table
\begin{center}
$ \begin{array}{rcccccccl}\hline
\mathcal N=2\: {\rm multiplet} && {\rm field\: fluctuations} && m^2 &\phantom{s}& \Delta & \phantom{s}& {\rm dual\; operators} \\\hline
\rule{0pt}{3ex}
{\rm gravity} && \begin{array}{c} A - 2a_1^J \\ g_{\mu\nu} \end{array}&& \begin{array}{c} 0 \\ 0 \end{array} && \begin{array}{c} 3\\ 4 \end{array} && {\rm Tr}(W_{\alpha}\overline W_{ \dot\alpha})+\ldots\\ \hline
{\rm universal \;hyper} && \begin{array}{c} b^\Omega - i \,c^\Omega \\ \phi\,,\;\;C_0 \end{array} && \begin{array}{c} -3 \\ 0 \end{array} && \begin{array}{c} 3 \\ 4 \end{array} && {\rm Tr} (W^2) +\ldots \\ \hline
&& b_1,\;c_1 && 8 && 5 && \\
{\rm massive\; gravitino} && a_2^\Omega && 9 && 5 && {\rm Tr}(W^2\overline{W}_{\dot \alpha}) + \ldots \\
&& b_2,\;c_2 && 16 && 6 && \\ \hline
{\rm massive \;vector} && \begin{array}{c} U - V\\ A + a_1^J\\ b^\Omega + i \,c^\Omega \\ 4U + V \end{array}&&\begin{array}{c} 12\\ 24 \\ 21\\ 32 \end{array} &&\begin{array}{c} 6\\ 7 \\ 7\\ 8 \end{array} && {\rm Tr}(W^2\overline W{}^2) + \ldots \\ \hline
\end{array}$
\caption{Mass eigenstates of the type IIB fields in our truncation on the supersymmetric AdS$_5\times\,$Sasaki--Einstein$_5$ background, and their dual superfield operators. Since the vacuum has $\mathcal N=2$ supersymmetry, the field fluctuations organize in $\mathcal N=2$ multiplets. We provide the mass eigenstates entering in each multiplet, together with their mass eigenvalues $m^2$, the conformal dimension $\Delta$ of the corresponding dual operators, and the dual superfields accommodating the single operators. The mass eigenvalues are evaluated choosing the flux $k=2$, which yields a unit AdS radius.
The vectors $b_1,\,c_1,\, A + a_1^J$ acquire a mass via a St\"uckelberg mechanism (the notation is the one of \cite{IIBonSE}).} \label{GaugeGravityTable}
\end{center}
\end{table}
\begin{acknowledgement}
The authors are supported in part by the ERC Advanced Grant no. 226455, ``Supersymmetry, Quantum Gravity and Gauge Fields'' (\textit{SUPERFIELDS}) and by the Fondazione Cariparo Excellence Grant ``String-derived supergravities with branes and fluxes and their phenomenological implications''.\end{acknowledgement}
|
2205.06634
|
\section{Introduction}
The purpose of this work is to study
translation planes that arise from a scattered linear set
of maximum rank in a finite projective line by replacing a related hyper-regulus.
Early investigations of this kind, concerning
the scattered linear sets of pseudoregulus type in $\PG(1,q^t)$, can be found
in \cite{LuPo01}.
The focus in that paper was on the theory of blocking sets,
but it is worth noticing that the planes constructed in \cite{LuPo01}
only depend on linear sets in the projective line.
Linear sets
generalize the notion of subgeometry of a projective space and
have ubiquitous applications in finite
geometry, being connected with many different
objects, such as blocking sets, two-intersection sets, complete caps, translation
spreads of the Cayley Generalized Hexagon, translation ovoids of polar
spaces, semifield flocks, finite semifields and rank metric codes
(see e.g. \cite{LaVV15,Po10,Sh16}).
The construction carried out in this article is also motivated by some results in \cite{JhJo08}, where
the authors introduce a new class of hyper-reguli in a Desarguesian spread, and then
investigate the translation planes obtained by the replacement of such hyper-reguli.
They prove that such planes are neither Andr\'e nor generalized Andr\'e planes.
What follows is
a description both of the main results and
of the structure of this paper.
In Section \ref{Notations} some terminology and notation are introduced.
In Section \ref{s:spaq}
a quasifield $\mathcal Q_f$ and a translation plane $\mathcal A_f$ are suitably associated
with any scattered
$\F_q$-linearized polynomial $f(x)\in\F_{q^t}[x]$, $q>2$.
Section \ref{S.equiv} is devoted to the question of
isomorphism between translation planes corresponding to distinct scattered polynomials;
a characterization of isomorphic translation planes is provided
(Theorems \ref{t:main} and \ref{th:pseudoreg}).
More precisely, given a scattered linearized polynomial $f(x)\in\F_{q^t}[x]$,
denote by $U_f$ and $L_f$ the $\F_q$-subspace of $\F_{q^t}^2$ and the $\F_q$-linear set in
$\PG(1,q^t)$ associated with $f(x)$, respectively.
Let $f(x), f'(x)\in\F_{q^t}[x]$ be scattered linearized polynomials;
then ${\mathcal A}_f$ and ${\mathcal A}_{f'}$ are isomorphic if and only if
$f(x)$ and $f'(x)$ are equivalent, that is,
$U_f$ and $U_{f'}$ belong to the same orbit under the action of $\GaL(2,q^t)$.
The number of known scattered linearized polynomials is
constantly growing and also includes classes of infinitely many
extensions of a field $\F_q$: see e.g.\ \cite{BaZaZu20,LoMaTrZh21,LoZa21}
and the references therein. By virtue of the main
result of this paper (Theorem \ref{t:main}), this leads to as many
distinct translation planes.
As is known \cite{CsZa16}, two $\F_q$-subspaces of $\F_{q^t}^2$ defining the same scattered linear
set of maximum rank in $\PG(1,q^t)$ not necessarily are in the same orbit under the action of
$\GaL(2,q^t)$.
This led to the notion of $\GaL$-class of a linear set $L$
\cite{CsMaPo18} as the maximum number $c_\Gamma(L)$
of $\F_q$-subspaces defining $L$ and belonging to distinct orbits under
the action of $\GaL(2,q^t)$.
As a consequence of Theorem \ref{t:main}, any scattered linear set $L$ of maximum rank in $\PG(1,q^t)$
gives rise to $c_\Gamma(L)$
pairwise non-isomorphic translation planes.
In Section \ref{s:LP} the discussion is focused on the number and the automorphism
group of the
translation planes associated with Lunardon-Polverino polynomials, which are described
by means of the results in \cite{JhJo08}.
\section{Notation and preliminaries}\label{Notations}
Let $q$ be a power of a prime $p$, and $r,t\in\mathbb N$, $r>0$, $t>1$.
Let $U$ be an $r$-dimensional $\F_q$-subspace of the $(dt)$-dimensional vector space
$V(\F_{q^t}^d,\F_q)$.
For a set $S$ of field elements (or vectors), we denote by $S^*$ the set of non-zero elements
(non-zero vectors) of $S$.
The following subset of $\PG(d-1,q^t)=\PG(\F_{q^t}^d,\F_{q^t})$
\[
L_U=\{\langle v\rangle_{\F_{q^t}}\colon v\in U^*\}
\]
is called \emph{$\F_q$-linear set} (or just \emph{linear set}) of \emph{rank $r$}.
The linear set $L_U$ is \emph{scattered} if it has the maximum possible size related to the
given $q$ and $r$, that is, $\#L_U=(q^r-1)/(q-1)$.
Equivalently, $L_U$ is scattered if and only if
\begin{equation}\label{e:scattered}
\dim_{\F_q}\left(U\cap\langle v\rangle_{\F_{q^t}}\right)\le1\quad\mbox{ for any }v\in\F_{q^t}^d.
\end{equation}
Any $\F_q$-subspace $U$ satisfying \eqref{e:scattered} is a \emph{scattered $\F_q$-subspace}.
Clearly, any $\F_q$-linear set in $\PG(d-1,q^t)$ of rank greater than $t(d-1)$ coincides
with $\PG(d-1,q^t)$.
The scattered linear sets of rank as large as
possible are called of \emph{maximum rank}. By \cite{BlLa00}, any scattered $\F_q$-linear set in
$\PG(d - 1, q^t )$ has rank at most $td/2$; for $td$ even the bound is sharp~\cite{CsMaPoZu17}.
In this note only $\F_q$-linear sets of maximum rank in $\PG(1,q^t)$ are dealt with.
They are associated with linearized polynomials.
An \emph{$\F_q$-linearized polynomial} in $\F_{q^t}[x]$ is of type $f(x)=\sum_{i=0}^ka_ix^{q^i}$ ($k\in\mathbb N$).
If $a_k\neq0$, then $k$ is the \emph{$q$-degree} of $f(x)$.
It is well-known that the $\F_q$-linearized polynomials of $q$-degree less than $t$ in $\F_{q^t}[x]$
are in one-to-one correspondence with the endomorphisms of the vector space
$V(\F_{q^t},\F_q)$.
An $\F_q$-linearized polynomial $f(x)\in\F_{q^t}[x]$ is called \emph{scattered} if for any $y,z\in\F_{q^t}$
the condition $zf(y)-yf(z)=0$ implies that $y$ and $z$ are $\F_q$-linearly dependent.
Let $f(x)\in\F_{q^t}[x]$ be an $\F_q$-linearized polynomial and define
\[U_f=\{(x,f(x))\colon x\in\F_{q^t}\},\]
and $L_f=L_{U_f}$.
Such $L_f$ is an $\F_q$-linear set of maximum rank of $\PG(1,q^t)$, and is scattered if and only if
$f(x)$ is.
Two $\F_q$-linearized polynomials $f(x)$ and $f'(x)$ in $\F_{q^t}[x]$ are said to be \emph{equivalent}
when a $\kappa\in\GaL(2,q^t)$ exists such that $U_f^\kappa=U_f'$ \cite{CsMaPo18}.
Since $\PGL(2, q^t )$ acts $3$-transitively on $\PG(1, q^t )$ and
since the size of an $\F_q$-linear set of $\PG(1, q^t)$ is at most $(q^t - 1)/(q - 1)$, if
$q > 2$, then
any linear set of maximum rank is projectively equivalent to an $L_U$ such that
\[
\left\{\langle(1,0)\rangle_{\F_{q^t}},\langle(0,1)\rangle_{\F_{q^t}},\langle(1,1)\rangle_{\F_{q^t}}\right\}\cap L_U=\emptyset.
\]
Therefore,
a linearized polynomial $f(x)$ exists such that $U=U_f$ and $\ker f=\{0\}=\ker(f-\id)$.
By abuse of notation, $L_f$ will also denote the set $\{f(x)/x\colon x\in\F_{q^t}^*\}$ of the
nonhomogeneous projective coordinates of the points belonging to the set $L_f$.
According to the convention above for $f(x)$, $0,1\notin L_f$.
A (right) \emph{quasifield} is an ordered triple $(Q,+,\circ)$, where $Q$ is a set,
$(Q,+)$ is an abelian group, $(Q^*,\circ)$ is a loop, $(x+y)\circ m=x\circ m+y\circ m$
for any $x,y,m\in Q$, and for any $a,b,c\in Q$ with $a\neq b$,
the following equation in the
unknown $x\in Q$ has a unique solution: $x\circ a=x\circ b+c$.
The \emph{kernel} of the quasifield $(Q,+,\circ)$ is
\begin{gather*}
K(Q)=\{k\in Q\colon k\circ(x+y)=k\circ x+k\circ y\mbox{ and }
k\circ(x\circ y)=(k\circ x)\circ y\\ \mbox{ for any }x,y\in Q\}.
\end{gather*}
The kernel of $Q$ is a skewfield and $Q$ is a left vector space over $K(Q)$
\cite{An54}\cite[Chapter 1]{Kn95}.
A quasifield satisfying the right distributive property is a \emph{semifield}; an associative quasifield is a
\emph{nearfield}.
From now on, any quasifield is assumed to be finite.
Therefore, $K(Q)$ is a finite field.
A \emph{partial planar spread} of a vector space $\mathbb V$ is a collection ${\mathcal B}$ of at least
three subspaces
such that $\mathbb V=V\oplus V'$ for any $V,V'\in{\mathcal B}$ with $V\neq V'$.
If ${\mathcal B}$ is a partial planar spread of $\mathbb V$ such that
for any $v\in\mathbb V$ there exists a $V\in{\mathcal B}$ such that $v\in V$, then ${\mathcal B}$ is a
\emph{planar spread} of $\mathbb V$.
The \emph{Desarguesian planar spread} of the $(2t)$-dimensional $\F_q$-vector space $\F_{q^t}^2$ is
\[
{\mathcal D}=\left\{\langle v\rangle_{\F_{q^t}}\colon v\in\left(\F_{q^t}^2\right)^*\right\}.
\]
Let $(Q,+,\circ)$ be a finite quasifield.
Define $V_\infty=\{0\}\times Q$, and
\[
V_m=\{(x,x\circ m)\colon x\in Q\}\mbox{ for }m\in Q.
\]
Then ${\mathcal B}(Q)=\{V_m\colon m\in Q\cup\{\infty\}\}$ is a planar spread of the $K(Q)$-vector space
$Q^2$.
Conversely, for any planar spread ${\mathcal B}$ a quasifield $Q$ exists such that ${\mathcal B}={\mathcal B}(Q)$.
Let ${\mathcal B}$ be a planar spread of a vector space $\mathbb V$ over $\F_{q^t}$.
Other planar spreads can be constructed by using the following technique of \emph{net replacement}.
If ${\mathcal B}'$ and ${\mathcal B}''$ are distinct partial planar spreads of $\mathbb V$ such that
${\mathcal B}'\subseteq{\mathcal B}$, and
\[
\bigcup_{V'\in{\mathcal B}'}V'=\bigcup_{V''\in{\mathcal B}''}V'',
\]
then $({\mathcal B}\setminus{\mathcal B}')\cup{\mathcal B}''$ is a planar spread of $\mathbb V$.
In the special case where $\dim_{\F_q}\mathbb V=2t$ and $\#{\mathcal B}'=\#{\mathcal B}''=(q^t-1)/(q-1)$,
the partial spreads ${\mathcal B}'$ and ${\mathcal B}''$ are called \emph{hyper-reguli} \cite[Chapter 19]{BiJhJo07}.
Such hyper-reguli are also called \emph{replacement set} of each other.
In some of the literature the definition requires that $\dim_{\F_q}(V'\cap V'')=1$ for any
$V'\in{\mathcal B}'$ and $V''\in{\mathcal B}''$, e.g.~\cite{JhJo08}.
In \cite{LuPo01}, Lunardon and Polverino show that $\{\langle(x,x^q)\rangle_{\F_{q^t}}\colon x\in\F_{q^t}^*\}$
is a hyper-regulus of $\F_{q^t}^2$, contained in ${\mathcal D}$, leading to a net replacement.
Actually the same arguments as in \cite[Section 3]{LuPo01} lead to a hyper-regulus
starting from
any scattered $\F_q$-linear set of maximum rank in $\PG(1,q^t)$.
This fact is folklore and addressed in Section \ref{s:spaq}.
The \emph{translation plane ${\mathcal A}({\mathcal B})$ associated with the planar spread} ${\mathcal B}$ of $\mathbb V$
is the plane whose points are the elements of $\mathbb V$ and whose
lines are the cosets of the elements of ${\mathcal B}$ in the group $(\mathbb V,+)$.
Given a quasifield $Q$, the \emph{translation plane ${\mathcal A}(Q)$ associated with $Q$}
is the translation plane ${\mathcal A}({\mathcal B}(Q))$ associated with ${\mathcal B}(Q)$.
In other words, the lines
are represented by the equations of type $x=b$ and $y=x\circ m+b$ ($m,b\in Q$).
If ${\mathcal A}(Q)$ and ${\mathcal A}(Q')$ ($Q$ and $Q'$ two quasifields) are isomorphic
translation planes, then $K(Q)$ and $K(Q')$ are isomorphic fields.
Hence the kernel of $Q$ is a geometric invariant of the plane ${\mathcal A}(Q)$,
also called the \emph{kernel of the translation plane ${\mathcal A}(Q)$}.
See \cite{Kn95} for a purely geometric definition.
Let $\N_{q^t/q}(m)=m^{(q^t-1)/(q-1)}$ denote the \emph{norm} of $m\in\F_{q^t}$ over $\F_q$.
An \emph{Andr\'e $q$-plane} is associated to the planar spread of $\F_{q^t}^2$ obtained from
${\mathcal D}$ by replacing each partial spread of type
\[
{\mathcal B}'_{\xi}=\{\langle(1,m)\rangle_{\F_{q^t}}\colon \N_{q^t/q}(m)=\xi\}\qquad(\xi\in\F_q^*)
\]
with
\[
{\mathcal B}''_{\mu(\xi)}=\left\{\{(x,x^{\mu(\xi)}m)\colon x\in\F_{q^t}\}\colon \N_{q^t/q}(m)=\xi\right\},
\]
where $\mu:\F_q^*\rightarrow\Gal(\F_{q^t}/\F_q)$ is any map.
Actually, $\mu(\xi)=\id$ means no replacement for that value of $\xi$.
Any ${\mathcal B}'_{\xi}$ is called \emph{Andr\'e $q$-net}.
If for $\xi\in\F_q^*$, the map $\mu(\xi)$ is $x\mapsto x^{q^s}$,
the partial spread ${\mathcal B}''_{\mu(\xi)}$ is an \emph{Andr\'e $q^s$-replacement}
\cite[Definition 16.1]{BiJhJo07}.
\section{Scattered polynomials and quasifields}\label{s:spaq}
In this section $f(x)\in\F_{q^t}[x]$ is a scattered $\F_q$-polynomial, $q>2$, satisfying $0,1\notin L_f$.
\begin{proposition}\label{p:quasicorpo}
Let $q>2$.
Let $Q_f=\F_{q^t}$ be endowed with the sum of $\F_{q^t}$, and define
\begin{equation}\label{e:quasicorpo}
x\circ m=\begin{cases}
xm&\mbox{ if }m\notin L_f,\\
h^{-1}f(hx)&\mbox{ if }m\in L_f\mbox{ and }f(h)-mh=0,\ h\neq0
\end{cases}
\end{equation}
for any $x,m\in Q_f$.
Then $(Q_f,+,\circ)$ is a quasifield, and $K(Q_f)=\F_q$.
\end{proposition}
\begin{proof}
To verify the quasifield axioms is a routine proof.
Next, note that $x\circ m=xm$ for any $x\in\F_q$ and $m\in Q_f$.
Furthermore, the map $\rho_m: Q_f\rightarrow Q_f$ defined by
\begin{equation}\label{e:rhom}
\rho_m(x)=x\circ m
\end{equation}
is $\F_q$-linear for any $m\in Q_f$.
Then for any $x\in\F_q$ and $m,n\in Q_f$, it holds that
$x\circ(m+n)=x(m+n)=xm+xn=x\circ m+x\circ n$, and
$x\circ(m\circ n)=x(m\circ n)=x\rho_n(m)=\rho_n(xm)=(x\circ m)\circ n$.
This implies $\F_q\subseteq K(Q_f)$.
Since $\F_{q^t}\setminus L_f$ is not a subgroup of $(\F_{q^t},+)$, two elements
$m,n\in\F_{q^t}\setminus L_f$ exist such that $w=m+n$ belongs to $L_f$.
Let $x\in Q_f\setminus\F_q$, and assume $x\circ w=xw$.
On the other hand, $x\circ w=h^{-1}f(hx)$ with $f(h)-wh=0$ for some $h\neq0$.
Hence
\[
\frac{f(hx)}{hx}=w=\frac{f(h)}h
\]
that together with $\langle h\rangle_{\F_q}\neq\langle hx\rangle_{\F_q}$ contradicts the assumption that
$f(x)$ is scattered.
As a consequence,
\[
x\circ(m+n)\neq x(m+n)=x\circ m+x\circ n
\]
and this implies $x\notin K(Q_f)$.
\end{proof}
Let ${\mathcal B}_f={\mathcal B}(Q_f)$ and ${\mathcal A}_f$ be the planar spread of $Q_f^2$ associated with $Q_f$
and the related translation plane, respectively.
\begin{remark}\label{r:generalizzabile}
The elements of ${\mathcal B}_f$ distinct from $V_\infty$ are of two types:
\begin{enumerate}
\item
if $m\in\F_{q^t}\setminus L_f$, then $V_m=\langle(1,m)\rangle_{\F_{q^t}}$;
\item
if $m\in L_f$ and $f(h)-mh=0$, $h\neq0$, then
\[
V_m=\{(h^{-1}x,h^{-1}f(x))\colon x\in\F_{q^t}\}=h^{-1}U_f.
\]
\end{enumerate}
In particular, $U_f=V_{f(1)}$.
\end{remark}
Note that $\{V_m\colon m\in L_f\}$ and $\{\langle(1,m)\rangle_{\F_{q^t}}\colon m\in L_f\}$
are hyper-reguli covering the same vector set.
Their union in the related projective space is a hypersurface \cite{LaShZa15}.
\begin{definition}\cite[Definition 3.1]{LuMaPoTr14}\label{d:pseudo}
Let $s\in\mathbb N^*$ and $\omega\in\F_{q^t}$ such that $(s,t)=1$, and $\N_{q^t/q}(\omega)\neq0,1$.
Any subset of $\PG(1,q^t)$ projectively equivalent to $L_{g_s}$ where $g_s(x)=\omega x^{q^s}$ is
called an \emph{$\F_q$-linear set of pseudoregulus type}.
\begin{remark}
Note that the definition above has been rephrased in order to obtain $1\notin {L_{g_s}}$.
In fact, $1\in {L_{g_s}}$ would imply that $y\in{\mathbb F}_{q^t}^*$ exists such that
$\omega y^{q^s-1}=1$, hence $\N_{q^t/q}(\omega)=1$, a contradiction.
The reason for such a choice is that we want the assumptions stated at the beginning of this section
to be satisfied.
All linear sets of pseudoregulus type in $\PG(1,q^t)$ are projectively equivalent by definition.
In case $L_f$ is a linear set of pseudoregulus type and $0\in L_f$ or $1\in L_f$,
then~\eqref{e:quasicorpo} does not define a quasifield, although ${\mathcal B}_f$ is still a planar spread.
\end{remark}
\end{definition}
The linear set $L_{g_s}$ is scattered for any $s\in\mathbb N^*$ such that $(s,t)=1$ and
for any $\omega\in\F_{q^t}^*$.
Under such assumptions,
$L_{g_s}=\{m\in\F_{q^t}\colon \N_{q^t/q}(m)=\N_{q^t/q}(\omega)\}$.
If $m\in L_{g_s}$ and $g_s(h)-mh=0$, $h\neq0$, then $m=\omega h^{q^s-1}$, and
$h^{-1}g_s(hx)=x^{q^s}m$.
Therefore, the product in $Q_{g_s}$ is
\[
x\circ m=\begin{cases} xm&\mbox{ if }\N_{q^t/q}(m)\neq \N_{q^t/q}(\omega),\\
x^{q^s}m&\mbox{ if }\N_{q^t/q}(m)=\N_{q^t/q}(\omega).\end{cases}
\]
If $\N_{q^t/q}(m)=\N_{q^t/q}(\omega)$, then $m=h^{q^s-1}\omega$ implies
\[
V_m=h^{-1}\{(x,x^{q^s}\omega)\colon x\in\F_{q^t}\}.
\]
Therefore, $V_\omega=U_{g_s}$.
For $s=1$, ${\mathcal B}_{g_s}$ is mapped by the projectivity of matrix
$\begin{pmatrix}1&0\\ 0&\omega^{-1}\end{pmatrix}$ into the spread used in \cite{LuPo01}
in order to construct an Andr\'e translation plane
which is therefore isomorphic to ${\mathcal A}_{\omega x^q}$ (see also Theorem \ref{t:an}).
The plane ${\mathcal A}_{g_s}$ associated with the polynomial described in Definition
\ref{d:pseudo} is the Andr\'e
$q$-plane obtained by one $q^s$-Andr\'e net replacement.
If there is no need for a quasifield framework, one can omit the condition $0,1\notin L_f$
and introduce, in view of Remark \ref{r:generalizzabile},
the more general definition of a translation plane associated to a scattered polynomial
as follows.
\begin{definition}
Let $f(x)\in\F_{q^t}[x]$ be a scattered $\F_q$-linearized polynomial.
Let ${\mathcal B}_f$ be the collection of the following subspaces of $\F_{q^t}^2$:
\[
\langle(1,m)\rangle_{\F_{q^t}}\ (m\in\F_{q^t}\setminus L_f), \qquad hU_f \ (h\in\F_{q^t}^*),
\qquad\{0\}\times\F_{q^t}.
\]
The \emph{translation plane associated with $f$}, also denoted by ${\mathcal A}_f$,
is ${\mathcal A}_f={\mathcal A}({\mathcal B}_f)$.
\end{definition}
\section{Equivalence of the translation planes associated with scattered polynomials}
\label{S.equiv}
In this section $f(x),f'(x)\in\F_{q^t}[x]$ are two scattered $\F_q$-linearized polynomials.
The proof concerning the equivalence between the corresponding translation planes
will be based on the following restriction of the
Andr\'e's isomorphism theorem to the finite case.
\begin{theorem}\cite{An54}\cite[Theorem 1.18]{Kn95}\label{t:an}
Let ${\mathcal B}_1$ and ${\mathcal B}_2$ be planar spreads of $\F_{q^t}^2$, and ${\mathcal A}_i={\mathcal A}({\mathcal B}_i)$,
$i=1,2$.
Assume moreover that $F$ is the kernel of ${\mathcal A}_1$.
Let $\alpha:\F_{q^t}^2\rightarrow\F_{q^t}^2$ be bijective.
Then $\alpha$ induces an isomorphism between ${\mathcal A}_1$ and ${\mathcal A}_2$ if and only if
there exists an $F$-semilinear mapping $\lambda:\F_{q^t}^2\rightarrow\F_{q^t}^2$ and a vector
$u\in\F_{q^t}^2$ such that $\lambda({\mathcal B}_1)={\mathcal B}_2$ and $\alpha(v)=\lambda(v)+u$
for all $v\in\F_{q^t}^2$.
\end{theorem}
\begin{theorem}\label{t:main}
Let $q>3$.
A bijection $\alpha:\F_{q^t}^2\rightarrow\F_{q^t}^2$ is an isomorphism between ${\mathcal A}_f$ and ${\mathcal A}_{f'}$
if and only if $\alpha(v)=\lambda(v)+u$ for some $u\in\F_{q^t}^2$ and
$\lambda$ is an $\F_{q^t}$-semilinear bijective map satisfying $\lambda(U_f)=h'U_{f'}$ with
$h'\in\F_{q^t}^*$.
Therefore, the planes ${\mathcal A}_f$ and ${\mathcal A}_{f'}$ are isomorphic if and only if $U_f$ and $U_{f'}$
belong to the same orbit under the action of $\GaL(2,q^t)$.
\end{theorem}
\begin{proof}
Assume that ${\mathcal A}_f$ and ${\mathcal A}_{f'}$ are isomorphic.
By Theorem \ref{t:an}, a $\lambda\in\GaL(2t,q)$ exists such that $\lambda({\mathcal B}_f)={\mathcal B}_{f'}$.
There exists an element of $\GaL(2,q^t)$ mapping $U_f$ to $U_g$ where $g(x)$ is a
scattered linearized polynomial such that $0,1\not\in L_g$.
Such semilinear map transforms ${\mathcal B}_f$ into ${\mathcal B}_g$.
Then Theorem \ref{t:an} and Proposition \ref{p:quasicorpo} imply that the kernel of ${\mathcal A}_f$
is $\F_q$.
By Proposition \ref{p:quasicorpo}, $\lambda$ is an $\F_q$-semilinear map, with companion
automorphism $\sigma$, say.
Both partial spreads ${\mathcal B}_{f}\cap{\mathcal D}$ and ${\mathcal B}_{f'}\cap{\mathcal D}$ have size $q^t+1-(q^t-1)/(q-1)$,
and since $q>3$ there
are three distinct elements $X_1$, $X_2$, $X_3$ of
${\mathcal B}_f\cap{\mathcal D}$ which are mapped by $\lambda$ into elements
of ${\mathcal B}_{f'}\cap{\mathcal D}$.
Correspondingly, there are $v_i,w_i\in\F_{q^t}^2$, $i=1,2,3$, such that $v_3=v_1+v_2$, and
\[
\langle v_i\rangle_{\F_{q^t}}\in{\mathcal B}_f,\quad \lambda(\langle v_i\rangle_{\F_{q^t}})=\langle w_i\rangle_{\F_{q^t}},\quad i=1,2,3.
\]
For $i=1,2$ a map $\lambda_i:\,\F_{q^t}\rightarrow\F_{q^t}$ exists such that
$\lambda(xv_i)=\lambda_i(x)w_i$ for $x\in\F_{q^t}$.
Clearly $\lambda_i$ is an $\F_q$-semilinear map with companion automorphism $\sigma$.
As a consequence, for any $x\in\F_{q^t}$,
\begin{equation}\label{e:coord0}
\lambda(xv_3)=\lambda_1(x)w_1+\lambda_2(x)w_2\in\langle w_3\rangle_{\F_{q^t}}.
\end{equation}
The vectors $w_i$, $i=1,2,3$ are pairwise $\F_{q^t}$-linearly independent, hence a $c\in\F_{q^t}^*$
exists such that $\langle w_3\rangle_{\F_{q^t}}=\langle w_1+c w_2\rangle_{\F_{q^t}}$.
This implies $\lambda_2(x)=c\lambda_1(x)$ for any $x$, and
\begin{equation}\label{e:coord}
\lambda(\xi v_1+\eta v_2)=\lambda_1(\xi)w_1+c\lambda_1(\eta)w_2\quad\mbox{for any }\xi,\eta\in\F_{q^t}.
\end{equation}
Assume now that $n\in\F_{q^t}^*$ and $z_n\in(\F_{q^t}^2)^*$ satisfy
\begin{equation}\label{e:topica}
\langle v_1+nv_2\rangle_{\F_{q^t}}\in{\mathcal B}_f,\qquad \lambda(\langle v_1+nv_2\rangle_{\F_{q^t}})=\langle z_n\rangle_{\F_{q^t}}.
\end{equation}
By \eqref{e:coord} and \eqref{e:topica}, for any $x\in\F_{q^t}$, $\lambda_1(x)$ and $\lambda_1(nx)$
are the coordinates of the vector $w(x)=\lambda(x(v_1+nv_2))$ in $\langle z_n\rangle_{\F_{q^t}}$ with respect to
the basis $\{w_1,cw_2\}$.
Since $w(x)$ ranges in a one-dimensional subspace of $\F_{q^t}^2$,
\[
\begin{vmatrix}\lambda_1(x)&\lambda_1(nx)\\ \lambda_1(1)&\lambda_1(n)\end{vmatrix}=0
\]
for all $x\in\F_{q^t}$.
Hence the $\F_q$-semilinear map $\rho(x)=\lambda_1(x)/\lambda_1(1)$ satisfies the condition
\begin{equation}\label{e:topica2}
\rho(nx)=\rho(n)\rho(x),\quad\mbox{for all }x\in\F_{q^t}.
\end{equation}
The elements of ${\mathcal B}_f$ which are also in the Desarguesian spread are precisely
\[ q^t+1-\frac{q^t-1}{q-1}. \]
All of them except $\langle v_2\rangle_{\F_{q^t}}$ are of type $\langle v_1+nv_2\rangle_{\F_{q^t}}$, $n\in\F_{q^t}$.
Since $\lambda$ is a bijection between ${\mathcal B}_f$ and ${\mathcal B}_{f'}$ and
$\#({\mathcal B}_{f'}\setminus{\mathcal D})=(q^t-1)/(q-1)$, the number of elements of type
$\langle v_1+nv_2\rangle_{\F_{q^t}}$ satisfying \eqref{e:topica} is at least
\[
N=q^t-2\frac{q^t-1}{q-1},
\]
and $q>3$ implies $N>q^{t-1}$.
Then there exists an $\F_q$-basis $\{n_1,n_2,\ldots,n_t\}$ of $\F_{q^t}$ such that
the condition \eqref{e:topica} is satisfied for any $n=n_i$, $i=1,2,\ldots,t$.
From \eqref{e:topica2}, for any $y,x\in\F_{q^t}$, $y=\sum_{i=1}^ty_in_i$ ($y_i\in\F_q$, $i=1,2,\ldots,t$),
\[
\rho(yx)=\sum_iy_i^\sigma\rho(n_ix)=\sum_iy_i^\sigma\rho(n_i)\rho(x)=\rho(y)\rho(x).
\]
As a consequence, $\rho$ is an automorphism of the field $\F_{q^t}$.
Condition \eqref{e:coord} yields
\[
\lambda(\xi v_1+\eta v_2)=
\lambda_1(1)\left(\rho(\xi)w_1+\rho(\eta)cw_2\right)\quad\mbox{for any }\xi,\eta\in\F_{q^t}.
\]
So, $\lambda\in\GaL(2,q^t)$.
Then $\lambda({\mathcal D})={\mathcal D}$, and since $\lambda({\mathcal B}_f)={\mathcal B}_{f'}$, one obtains
$\lambda({\mathcal B}_f\setminus{\mathcal D})={\mathcal B}_{f'}\setminus{\mathcal D}$.
The elements of ${\mathcal B}_f\setminus{\mathcal D}$ are of type $hU_f$, $h\in\F_{q^t}^*$, and the
elements of ${\mathcal B}_{f'}\setminus{\mathcal D}$ are of type $h'U_{f'}$, $h'\in\F_{q^t}^*$.
Then an $h'\in\F_{q^t}^*$ exists such that $\lambda(U_f)=h'U_{f'}$.
Conversely, assume that $\lambda(U_f)=h'U_{f'}$ for some $h'\in\F_{q^t}^*$ and
$\lambda\in\GaL(2,q^t)$ with companion
automorphism $\sigma$.
This implies $\lambda(hU_f)=h^\sigma h' U_{f'}$ for any $h\in\F_{q^t}^*$, hence
$\lambda({\mathcal B}_f\setminus{\mathcal D})={\mathcal B}_{f'}\setminus{\mathcal D}$.
If $\langle v\rangle_{\F_{q^t}}\in{\mathcal B}_f$, then
$\langle v\rangle_{\F_{q^t}}\cap U_f=\{0\}$, and consequently
$\lambda(\langle v\rangle_{\F_{q^t}})\cap U_{f'}=\{0\}$.
Since the elements of ${\mathcal D}$ which intersect trivially $U_{f'}$ belong to ${\mathcal B}_{f'}$,
it follows $\lambda(\langle v\rangle_{\F_{q^t}})\in{\mathcal B}_{f'}$.
As a result, the assumptions of Theorem \ref{t:an} are satisfied and $\alpha$ is an
isomorphism between ${\mathcal A}_f$ and ${\mathcal A}_{f'}$.
\end{proof}
\begin{corollary}\label{c:nt} Assume $q>3$.
If $\alpha$ is an automorphism of ${\mathcal A}_f$, and $\lambda(v)=\alpha(v)-\alpha(0)$
for any $v\in\F_{q^t}^2$, then $\lambda$ maps elements of ${\mathcal B}_f\cap {\mathcal D}$ into elements of
${\mathcal B}_f\cap {\mathcal D}$, and elements of ${\mathcal B}_f\setminus{\mathcal D}$ into elements of ${\mathcal B}_f\setminus{\mathcal D}$.
\end{corollary}
\begin{corollary} Assume $q>3$.
The plane ${\mathcal A}_f$ is neither a semifield plane nor a nearfield plane.
\end{corollary}
\begin{proof}
By Theorem \ref{t:main} it may be assumed again that $0,1\notin L_f$.
If ${\mathcal A}={\mathcal A}({\mathcal B})$, ${\mathcal B}$ a planar spread of $\mathbb V$, is a semifield plane,
then there is
an $S\in{\mathcal B}$ and a group contained in $\GaL(\mathbb V,K({\mathcal A}))$ preserving ${\mathcal B}$ and acting
transitively on ${\mathcal B}\setminus\{S\}$ \cite{Os77}.
Similarly, if ${\mathcal A}$ is a nearfield plane,
then there are distinct
$S,T\in{\mathcal B}$ and a group contained in $\GaL(\mathbb V,K({\mathcal A}))$ preserving ${\mathcal B}$ and acting
transitively on ${\mathcal B}\setminus\{S,T\}$ \cite{Os77}.
As a consequence, if ${\mathcal A}_f$ is a semifield or a nearfield plane, a
$\lambda\in\GaL(\F_{q^t}^2,\F_q)$ preserving ${\mathcal B}_f$ exists
mapping some element of ${\mathcal D}$ into an element of ${\mathcal B}_f\setminus{\mathcal D}$.
This contradicts Corollary~\ref{c:nt}.
\end{proof}
\begin{definition}\cite{CsMaPo18}
Let $L_U$ be an $\F_q$-linear set of $\PG(1, q^t)$ of maximum rank with
maximum field of linearity ${\mathbb F}_q$; that is, $L_U$ is not an ${\mathbb F}_{q^r}$-linear set for any $r>1$.
Then $L_U$ is said of \emph{$\GaL$-class
$c=c_\Gamma(L)$} if $c$ is the largest integer
such that there exist $\F_q$-subspaces $U_1, U_2, \ldots, U_c$ of $\F_{q^t}^2$ with
$L_{U_i} = L_U$ for $i\in\{1, 2, \ldots, c\}$
and there is no $\kappa\in\GaL(2, q^t )$ such that $U_i = \kappa(U_j)$
for each $i\neq j$, $i, j\in\{1, 2, \ldots, c\}$.
\end{definition}
\begin{corollary}\label{c:class}
Let $q>3$.
Any scattered $\F_q$-linear set $L_U$ of maximum rank in $\PG(1,q^t)$ gives rise to
$c_\Gamma(L)$
pairwise nonisomorphic translation planes.
\end{corollary}
As a consequence of the results in \cite{CsZa16}, the linear set of pseudoregulus type in
$\PG(1,q^t)$ is of $\GaL$-class $\phi(t)/2$, where $\phi(t)$ is the totient function.
The sets $U_{x^{q^s}}$ and $U_{x^{q^{s'}}}$ are in a common orbit under the
action of $\GaL(2,q^t)$ if and only if $s\equiv \pm s'\Mod t$.
This implies the following result.
\begin{theorem}\label{th:pseudoreg}
Let $q>3$.
If $f(x)= x^{q^s}$, $f'(x)= x^{q^{s'}}$ are polynomials in $\F_{q^t}[x]$,
and $(s,t)=1=(s',t)$, then ${\mathcal A}_f$ and ${\mathcal A}_{f'}$
are isomorphic if and only if $s\equiv \pm s'\Mod t$.
\end{theorem}
\begin{remark}
The main argument in the proof of Theorem \ref{t:main} can be generalized to obtain
the following result.
\begin{theorem}
Let ${\mathcal B}_1$ and ${\mathcal B}_2$ be planar spreads of $V(\F_{q^t}^2,\F_q)$,
obtained with $\ell_1$ and $\ell_2$ hyper-reguli replacements from the Desarguesian spread ${\mathcal D}$,
respectively.
Assume $\ell_1+\ell_2\le q-2$.
If $\lambda\in\GaL(2t,q)$ satisfies $\lambda({\mathcal B}_1)={\mathcal B}_2$, then $\lambda$ is $\F_{q^t}$-semilinear.
\end{theorem}
\end{remark}
\section{Translation planes associated with the Lunardon-Polverino polynomials}\label{s:LP}
The \emph{Lunardon-Polverino polynomial}, or \emph{LP-polynomial}, of indices $b\in\F_{q^t}^*$ and
$s\in\mathbb N$ is
\[
P_{b,s}=x^{q^s}+x^{q^{t-s}}b.
\]
Such a polynomial is scattered if \cite{LuPo01} and only if \cite{Za19} $\N_{q^t/q}(b)\neq1$
and $(s,t)=1$.
In this case, for $t>3$ the related scattered linear set is not of pseudoregulus type
\cite{LuPo01}.
\begin{theorem}\cite[Theorem 4.1]{JhJo08}
Consider the following set of subspaces of $V(\F_{q^t}^2,\F_q)$:
\begin{equation}\label{e:fhr}
\mathcal{H}_{b,s}= \left \{ V_{b,d,s} \colon d\in\F_{q^t}^*\right\},
\end{equation}
where
\begin{equation}\label{e:fhrsub}
V_{b,d,s}=\left\{(x,x^{q^s}d^{1-q^s}+x^{q^{t-s}}d^{1-q^{t-s}}b)\colon x\in\F_{q^t} \right\}
\end{equation}
and $b^{(q^t-1)/(q^{(s,t)}-1)}$ is not equal to 1.
Then
\begin{enumerate}[(1)]
\item
this set is a replacement set for a hyper-regulus of $V(\F_{q^t}^2,\mathbb F_{q^{(s,t)}})$;
\item
$V_{b,d,s}$, $d \in {\mathbb F}_{q^{(s,t)}}$, is an $\mathbb F_{q^{(s,t)}}$-subspace and lies over the Desarguesian spread
in the sense that the subspace intersects exactly $(q^t-1)/(q^{(s,t)}-1)$ components in
1-dimensional $\mathbb F_{q^{(s,t)}}$-subspaces;
\item
if $t/(s,t)>3$, the hyper-regulus of components that this subspace lies over is not an Andr\'e
hyper-regulus.
\end{enumerate}
\end{theorem}
\begin{definition}\cite[\textit{Definition 4.2 \& Corollary 4.3}]{JhJo08}
Any hyper-regulus contained in the Desarguesian spread of $V(\F_{q^t}^2,\mathbb F_{q^{(s,t)}})$
that has a replacement set of the form \eqref{e:fhr} shall be called \emph{fundamental hyper-regulus}.
\end{definition}
When $(s,t)=1$, the hyper-regulus associated with
a scattered LP-polynomial turns out to be a fundamental hyper-regulus.
As a matter of fact,
\[
V_{b,d,s}=d\{(x,x^{q^s}+bx^{q^{t-s}})\colon x\in\F_{q^t}\}=dU_{P_{b,s}}.
\]
As a consequence, the results in \cite{JhJo08} in particular describe the planes
${\mathcal A}_{P_{b,s}}$ with $\N_{q^t/q}(b)\neq1$ and $(s,t)=1$.
In the next theorem:
\begin{itemize}
\item [-] an \emph{affine central collineation} is a collineation of the affine plane whose projective
extension is a central collineation, and whose axis is an affine line;
\item [-] the \emph{kernel homology group of the associated Desarguesian plane} contains all
collineations of type $(x,y)\mapsto(ax,ay)$ with $a\in\F_{q^t}^*$; note that such maps are collineations
but not central collineations of ${\mathcal A}_{P_{b,s}}$;
\item [-] a \emph{coaxis} of an affine central collineation is any affine line through the
center;
\item [-] given two groups $G$ and $H$ of affine homologies and two lines
$\ell$, $m$, if $\ell$ is axis of any element in $G$ and coaxis of any element in $H$,
and furthermore $m$ is axis of any element in $H$ and coaxis of any element in $G$, then
$G$ and $H$ are called \emph{symmetric affine homology groups};
\item [-] an \emph{elation} of an affine translation plane is intended to have proper axis.
\end{itemize}
\begin{theorem}\cite[Theorem 6.1]{JhJo08}\label{t:jj}
Assume that $q > 3$, $\N_{q^t/q}(b)\neq0,1$ and $(s,t)=1$. Then the following hold:
\begin{enumerate}[(1)]
\item
If $t>3$ is odd, then the plane ${\mathcal A}_{P_{b,s}}$ admits no affine central collineation
group and the full collineation group in $\GL(2, q^t )$ has order $(q^t - 1)$ and is the
kernel homology group of the associated Desarguesian plane.
\item
If $t>3$ is even, then the plane ${\mathcal A}_{P_{b,s}}$ admits symmetric affine homology
groups of order $q + 1$ but admits no elation group.
The full collineation group
in $\GL(2, q^t )$ has order $(q + 1)(q^t - 1)$, and is the direct product of the kernel
homology group of order $(q^t - 1)$ by a homology group of order $q + 1$.
\end{enumerate}
\end{theorem}
\begin{corollary}\cite[Corollary 6.2]{JhJo08}.
The translation plane ${\mathcal A}_{P_{b,s}}$, $t>3$, $q > 3$, $\N_{q^t/q}(b)\neq0,1$ and $(s,t)=1$, has
kernel $\F_q$ and cannot be a generalized Andr\'e or an Andr\'e plane.
\end{corollary}
In \cite[Section 2]{LoMaTrZh21}, the authors investigate the $\GaL$-equivalence of the set
of type $U_{P_{b,s}}$.
Rephrasing Proposition 2.3 there,
the result reads as follows:
\begin{proposition}\cite{LoMaTrZh21}
Assume $s,s'<t/2$, and $b,b'\in\F_{q^t}$ with $\N_{q^t/q}(b)\neq1\neq\N_{q^t/q}(b')$.
Then the subspaces $U_{P_{b,s}}$ and $U_{P_{b',s'}}$ are in the same orbit under the
action of $\GaL(2,q^t)$ if and only if $s=s'$, and there exist $z\in\F_{q^t}$ and an automorphism
$\sigma$ of $\F_{q^t}$
such that
\begin{equation}\label{e:ejj}
b'=b^\sigma z^{q^{2s}-1}.
\end{equation}
As a consequence, if $q=p^e$, under the action of $\GaL(2,q^t)$ there are at least
\begin{equation}\label{e:atleast} N_{q,t}=
\begin{cases}
\frac{q-2}{e}\,\frac{\phi(t)}2&\mbox{ for odd }t\\
\frac{q^2-1-(q+1)}{2e}\,\frac{\phi(t)}2&\mbox{for even }t
\end{cases}
\end{equation}
orbits of subspaces of type $U_{P_{b,s}}$.
\end{proposition}
The subspaces $U_{P_{b,s}}$ and $U_{P_{b^{-1},t-s}}$ are in the same orbit under
the action of $\GL(2,q^t)$.
Since for $q>3$ two planes ${\mathcal A}_f$ and ${\mathcal A}_{f'}$ are isomorphic if and only if
$U_f$ and $U_{f'}$ are in the same orbit under the action of $\GaL(2,q^t)$, one obtains
by \eqref{e:ejj}:
\begin{theorem}\label{t:nesatto}
The number of pairwise non-isomorphic planes of type ${\mathcal A}_{P_{b,s}}$, $t>3$, $q > 3$,
equals
\begin{enumerate}[(1)]
\item the number of the orbits in $\F_q\setminus\{0,1\}$ under the action of the Galois group
$\Gal(\F_q/\F_p)$ over the prime subfield of $\F_q$, if $t$ is odd, or
\item the number of the orbits in ${\mathbb F}_{q^2}\setminus\{x^{q-1}\colon x\in{\mathbb F}_{q^2}\}$
under the action of $\Gal({\mathbb F}_{q^2}/\F_p)$, if $t$ is even.
\end{enumerate}
\end{theorem}
\begin{corollary}\label{c:disaccordo}
For $q>3$ and $t>3$ there are at least $N_{q,t}$ mutually non-isomorphic
translation planes of order $q^t$ and kernel $\F_q$ that may be obtained from a Desarguesian spread
by replacement of a hyper-regulus of the type $\mathcal{H}_{b,s}$ with $(s,t)=1$.
\end{corollary}
\begin{remark} Corollary \ref{c:disaccordo} does not agree with
\cite[Theorem 5.5 \& Corollary 5.6]{JhJo08}.
It is opinion of the authors of this paper that the argument at \cite[p.96, line 11]{JhJo08}
is incorrect.
As a counterexample, let $t>3$ be odd, $q=p^e$, $p$ a prime, $e>1$, $0<s<t$ such that $(s,t)=1$;
furthermore, take $t$ and $e$ coprime with $q-2$.
In \cite{JhJo08} it is asserted that there are at least $(q-2)/(te,q-2)=q-2$ non-isomorphic
planes of type ${\mathcal A}_{P_{b,s}}$.
See also at p.97, line 5 from the bottom, where the same assertion is involved.
However, by Theorem \ref{t:nesatto},
the number of such planes is less than $q-2$.
\end{remark}
\section*{Acknowledgment}
The authors are grateful to Guglielmo Lunardon for sowing the seeds of this work,
and for his valuable suggestions and comments.
|
2202.00780
|
\section{INTRODUCTION}
Magnonics is an emerging field aiming for the future low-loss wave-based computation \cite{Barman2021,Mahmoud2020,Demidov2017,Grundler2016,Chumak2015,Lenk_2010,Serga2010}. Among the splendid magnonic functionalities, chirality and non-reciprocity serve as the basic building blocks \cite{Chen2021,Szulc2020,Grassi2020,Lan2015,Jamali2013} for the integrated magnonic circuits since the spin precession is innately chiral \cite{Pirro2021,Kruglyak2021,Kruglyak_2010}. The non-reciprocity can root in the magneto-dipolar interaction via, for example, the well known Damon-Eshbach (DE) geometry \cite{Damon1961,Camley1987,An2013,Kwon2016}, bilayer magnet and inhomogeneous thin film \cite{Ishibashi2020,GallardoPRAppl2019,AnPRAppl2019,Gladii2016,GallardoNJP2019,Borys2021,MacedoAEM2021,SadovnikovPRB2019}, and magnetic heterostructure in the presence of magneto-elastic or magneto-optic coupling \cite{TatenoPRAppl2020,Shah2020,ZhangPRAppl2020,WangPRL2019}. However, with the isotropic exchange interaction dominating in the microscale region \cite{WangNE2020,Chumak2014,Mohseni2019}, the dipolar effects, followed by the induced non-reciprocity, are vanishingly small \cite{Wong2014}. The non-reciprocity can also emerge in the chiral edge states of elaborately devised topological magnetic materials or spin-texture arrays, which are robust to defects and disorders \cite{WangPRAppl2018,WangPRB2017,LiPR2021,LiPRB2018,MookPRB2014,Shindou2013}. But it requires specific lattice designs and complicated couplings between atoms or elements, and the confined magnon (the quantum of spin wave) channels at the edges reduce the usage of the magnetic systems. Another origin for the non-reciprocity comes from the Dzyaloshinskii-Moriya interaction (DMI) \cite{Udvardi2009}. Yet, the effect is negligibly weak in ferromagnetic insulators, like yttrium iron garnet (YIG, Y$_3$Fe$_5$O$_{12}$) \cite{WangPRL2020}. Additional heavy metal structures can introduce a sizable DMI \cite{Bouloussa2020,Hrabec2020,GallardoPRL2019}, but inevitably bring remarkably increased damping and Joule heating \cite{SunPRL2013}.
To realize an efficient excitation of the non-reciprocal short-wavelength dipolar-exchange or even pure exchange spin waves (SWs) in ferromagnetic insulators for miniaturizing magnonic devices, several promising methods have been suggested \cite{AuAPL2012Res,AuAPL2012Nano,ChenPRB2019,ChenACS2020,WangNR2020,Sushruth2020, FrippPRB2021}. Conventionally, the coherent SW excitation harnesses the microwave antennas with the exciting field linearly polarized and uniform across the film thickness. Since the in-plane component of microwave fields dominantly contributes to the excitations, it is solely accounted in the analysis \cite{Dmitriev1988,Schneider2008,Demidov2009,Kasahara2017}. By contrast, the dynamic fields generated by micro-magnetic structures are not only highly localized at interfaces favoring the short-wavelength SWs excitation \cite{Yu2016,Liu2018,Che2020}, but also polarized with complex chiralities. Yu {\it et al.} have reported an analysis for the chiral pumping (excitation) of exchange magnons in YIG into (from) the proximate magnetic wires via directional dipolar interactions \cite{YuPRB2019,YuPRL2019}. A selection rule is adopted that circular magnons and photons with the same (opposite) chiralities are allowed (forbidden) to interact \cite{ZhangPRAppl2020}. One critical issue noteworthily rises how the microwave fields with contrary chirality excite the propagating SWs.
In this work, we theoretically investigate the propagating SWs in ferromagnetic films excited by microwave fields with generic chiralities. We find that the left-hand microwave can drive SWs because of the ellipticity mismatch between microwave and dynamic magnetization, which extrapolates the aforementioned selection rule for the magnon-photon conversion. Since the contributions of the in-plane and out-of-plane components of left- (right-) hand microwave fields are destructive (constructive) superposed, we introduce an analog to the common and differential signals in the differential amplifier. Surprisingly, we find a compensation frequency where no SWs can be excited by left-hand microwaves with certain ellipticity. We propose a proof-of-concept strategy for generating non-reciprocal SWs via applying the left-hand local microwave unevenly across the film thickness. A directional mutual demagnetizing factor is suggested to understand the emerging switchable SW chirality that depends on the microwave frequency. This proposal makes full use of the magnetic structures without breaking the symmetry of the dispersion relations and increasing the damping, which is superior to other methods. Our work lays a foundation of employing the chiral excitation for magnonic diodes in nano scales.
The paper is organized as follows. In Sec. \ref{Chiral}, we present the characteristics of the chiral excitation of SWs via a combination of theoretical analysis and numerical simulations. The strategy for nonreciprocal SWs excitations is proposed and demonstrated in Sec. \ref{Nonreciprocal}. Discussions and conclusions are drawn in Sec. \ref{Conclusion}.
\section{CHARACTERISTICS OF SPIN WAVES DRIVEN BY CHIRAL EXCITATIONS}\label{Chiral}
\subsection{MODELLING AND DISPERSION RELATION}\label{MODEL}
We consider a ferromagnetic layer YIG with thickness $d$ extended in the $x-z$ plane and magnetized along $z$ direction by the bias magnetic field ${\bf H}_0={H}_0 {\bf z}$ (see Fig. \ref{fig0}). The microwave field ${\bf h}_{\rm rf}$ for the SWs excitation is centered at $x=0$ and located in the region with width $w$. {\color{red}The characteristics of SWs propagating along $x$ direction, i.e., the DE geometry \cite{Damon1961}, are investigated.} In the calculations, we set $d =$ 40 nm, $H_0 =$ 52 mT, and $w =$ 10 nm if not stated otherwise. The narrow excitation width ensures ${\bf h}_{\rm rf}$ comprises multiple wave vector in a wide range within $2\pi/w$ \cite{Sushruth2020,Stancil_2009}. Micromagnetic simulations are performed using MuMax3 \cite{Vansteenkiste2014} to verify the derived theories. {\color{red}The systems are meshed by cells with dimensions equal to $2\times2\times100~{\rm nm^3}$. Periodic boundary conditions ($\rm {PBC}\times200$) in the $z$ direction are applied, which means that the film is practically infinite in the $z$ direction. Absorbing boundary conditions are applied by adding the attenuating areas (not shown in the figure) where the $\alpha$ gradually increases to 0.25 to avoid the reflection at the two ends of simulated systems. }
\begin{figure
\centering
\includegraphics[width=0.40\textwidth]{model0.pdf}\\
\caption{Schematic of the chiral excitation of SWs. The chiral microwave field ${\bf h}_{\rm rf}$ is locally applied in the patched green region. The SWs are propagating along $x$ direction indicated by the hollow arrows.} \label{fig0}
\end{figure}
\begin{figure
\centering
\includegraphics[width=0.40\textwidth]{model1.pdf}\\
\caption{(a) SW dispersion relation of the film obtained from micromagnetic simulations. {\color{red}The dashed line represents the theoretical result by Eq. (\ref{DispRela})}. Inset: profiles {\color{red}of SWs with various wavevectors} across the thickness at 4 GHz. (b) Frequency dependence of the dynamic magnetization ellipticity ($|\varepsilon_m|$). The solid curve is from Eq. (\ref{epsilon_m}). Circles are micromagnetic simulations. The dashed line indicates $|\varepsilon_m| = 1$. Inset: Spatial distribution of normalized dynamic magnetization [$m_{x(y)}/M_s$] at 4 GHz at an arbitrary time slot. The blue (red) curves represent the $x$ ($y$) component.} \label{fig1}
\end{figure}
The magnetization dynamics is governed by Landau-Lifshitz-Gilbert (LLG) equation
\begin{equation} \label{LLG}
\frac{\partial {\bf M}}{\partial t} = -\gamma \mu _0{\bf M}\times {\bf H}_{\rm eff} + \frac{\alpha}{M_s} {\bf M} \times \frac{\partial {\bf M}}{\partial t},
\end{equation}
where $\gamma$ is the gyromagnetic ratio, $\mu _0$ is the vacuum permeability, $\alpha\ll1$ is the dimensionless Gilbert damping constant, $M_{s}$ is the saturated magnetization, ${\bf M}={\bf m}+M_s {\bf z}$ is the magnetization with ${\bf m} = m_x {\bf x}+m_y{\bf y}$ the dynamic component, and ${\bf H}_{\rm eff} = {\bf H}_0+{\bf h}_{\rm rf}+{\bf h}_{ex}+{\bf h}_d$ with ${\bf h}_{\rm rf}= h_x {\bf x} + h_y {\bf y}$ the microwave field, ${\bf h}_{ex}=(2A_{ex}/\mu _0M_s^2)\nabla ^2{\bf m}$ the exchange field where $A_{ex}$ is the exchange constant, and ${\bf h}_d$ being the dipolar field satisfying the magneto-static Maxwell's equations $\nabla \cdot ({\bf h}_{d}+{\bf m}) = 0$ and $\nabla \times {\bf h}_{d} = 0$. The magnetic parameters of YIG are $M_{s} = 1.48\times 10^5$ A/m, $A_{ex} = 3.1\times 10^{-12}$ J/m, and $\alpha =5\times 10^{-4}$ \cite{Stancil_2009}. The free boundary conditions at the top and bottom surfaces require $\partial m_{x(y)}/\partial y \bigg |_{y=0,-d} =0$ \cite{ZhangPRB2021}. Thus, only the first unpinned mode exists in the low frequency band due to the ultra thin thickness \cite{Mohseni2019,Demokritov2001}, whose profile of dynamic magnetization is uniform across the thickness as shown in the inset of Fig. \ref{fig1}(b). We assume a plane-wave form ${\bf m} = {\bf m}_0 e^{j(\omega t-k_xx)}$ with ${\bf m}_0= m_{x0} {\bf x}+m_{y0} {\bf y}$ and $m_{x(y)} = m_{x0(y0)}e^{j(\omega t-k_xx)}$. Substituting these terms into Eq. (\ref{LLG}) and adopting the linear approximation \cite{Kalinikos1986}, we obtain
\begin{subequations} \label{LLGExpWithRF}
\begin{align}
\label{LLGExpWithRFa}j\omega m_x + (j \alpha \omega + \omega_y) m_y&= \omega_M h_{y}, \\
\label{LLGExpWithRFb}-(j \alpha \omega + \omega_x)m_x + j\omega m_y &= -\omega_M h_{x},
\end{align}
\end{subequations}
where $\omega_{x} = n_x\omega_M+\omega_H+\omega_{ex}$ and $\omega_{y} = n_y\omega_M+\omega_H+\omega_{ex}$, with $\omega_M=\gamma \mu_0M_s$, $\omega_H=\gamma \mu_0 H_0$, and $\omega_{ex}=(2\gamma A/M_s)k_x^2$. The demagnetizing factors $n_x$ and $n_y$ of ${\bf h}_d=-n_xm_x {\bf x}-n_ym_y {\bf y}$ are given by (see Appendix \ref{SelfDE} for detailed derivation)
\begin{equation} \label{DemagFac}
n_x=1-n_y=1-\frac{1-e^{-|k_x| d}}{|k_x| d}.
\end{equation}
The non-zero $m_x$ and $m_y$ in Eqs. (\ref{LLGExpWithRF}) requires the determinant
of the coefficient matrix equal to zero, which gives the dispersion relation
\begin{equation} \label{DispRela}
\omega = \sqrt{\omega_{x}\omega_{y}}.
\end{equation}
{\color{red}To verify the theoretical dispersion relation, we perform the simulation using YIG film with 50 $\mu$m long and the excitation with $w = 10$ nm for a broad wave vector range. The excitation is applied using a ``sinc" function ${\bf h}_{\rm rf}(t)=h_0 \sin [\omega _f(t-t_0)]/[\omega _f(t-t_0)] {\bf x}$ with the cut-off frequency $\omega _f/2\pi=50~\rm {GHz}$, $t_0 = 0.5 ~\rm {ns}$, and $h_0 = 1~\rm {mT}$. The total simulation time is 200 ns, and the results record the dynamic normalized magnetization ($m_y/M_s$) evolution as a function of time and position along $x$ direction. The dispersion relations were obtained through the two-dimensional FFT (2D-FFT) operation on $m_y/M_s$ \cite{Kumar2011}. Figure \ref{fig1}(a) presents a good agreement between the theory and the full micromagnetic simulations. The SW dispersion relation obtained from simulation shows only one band exists in the low frequency range from 3 to 8 GHz, whose profile across the thickness is uniform. Factors $n_x$ and $n_y$ in Eq. (\ref{DemagFac}) are uniquely describing the SW mode with uniform transverse profile [inset of Fig. \ref{fig1}(a)], quite different with those of other SW modes with much higher frequencies \cite{Kalinikos1986}.}
\subsection{ELLIPTICITY}\label{ellipticity}
Solving Eqs. (\ref{LLGExpWithRF}), we obtain
\begin{subequations} \label{LLGExpRF}
\begin{align}
\label{LLGExpRFa}m_x&= \chi_{y}(k_x,\omega) h_{x} +j\kappa(k_x,\omega) h_{y}, \\
\label{LLGExpRFb}m_y&=-j\kappa(k_x,\omega) h_{x} +\chi_{x}(k_x,\omega) h_{y},
\end{align}
\end{subequations}
where
\begin{subequations} \label{Permi}
\begin{align}
\label{Permia}\chi _{x}(k_x,\omega) = -\frac{(\omega_x + j \alpha \omega)\omega_M}{\omega^2-(\omega_x + j \alpha \omega)(\omega_y + j \alpha \omega)}, \\
\label{Permib}\chi _{y}(k_x,\omega) = -\frac{(\omega_y + j \alpha \omega)\omega_M}{\omega^2-(\omega_x + j \alpha \omega)(\omega_y + j \alpha \omega)}, \\
\label{Permic}\kappa(k_x,\omega) = -\frac{\omega \omega_M}{\omega^2-(\omega_x + j \alpha \omega)(\omega_y + j \alpha \omega)}. &
\end{align}
\end{subequations}
{\color{red}The coefficients $\chi_{x}(k_x,\omega)$, $\chi_{y}(k_x,\omega)$ and $\kappa(k_x,\omega)$ possess the same denominator, whose absolute value takes the minimum when the dispersion relation Eq. (\ref{DispRela}) is satisfied. It means that even though the microwave field comprises multiple wave vector components within $2\pi/w$ \cite{Sushruth2020,Stancil_2009}, only the SWs with $k_x$ and $\omega$ satisfying Eq. (\ref{DispRela}) can be efficiently excited. Substituting Eq. (\ref{DispRela}) into Eqs. (\ref{Permi}) and neglecting the higher-order terms, the magnetic parameters reduce to}
\begin{subequations} \label{LLGExpRF}
\begin{align}
\label{LLGExpRFa}m_x&= \chi_{y} h_{x} +j\kappa h_{y}, \\
\label{LLGExpRFb}m_y&=-j\kappa h_{x} +\chi_{x} h_{y},
\end{align}
\end{subequations}
with
\begin{subequations} \label{PermiApx}
\begin{align}
\label{PermiApxa}\chi _{x}& = -\frac{j\omega_x \omega_M}{\alpha (\omega_x + \omega_y )\sqrt {\omega_x \omega_y}}, \\
\label{PermiApxb}\chi _{y}& = -\frac{j\omega_y \omega_M}{\alpha (\omega_x + \omega_y )\sqrt {\omega_x \omega_y}}, \\
\label{PermiApxc}\kappa & = -\frac{j\omega_M}{\alpha (\omega_x + \omega_y )}.
\end{align}
\end{subequations}
We obtain the ratio between the $x$ and $y$ components of the dynamic magnetization as
\begin{equation} \label{epsilon_m}
\varepsilon_m=\frac{m_x}{m_y}=j\sqrt{\frac{\omega_{y}}{\omega_{x}}}
=j\sqrt{\frac{n_y\omega_M+\omega_H+\omega_{ex}}{n_x\omega_M+\omega_H+\omega_{ex}}}.
\end{equation}
Equation (\ref{epsilon_m}) delivers following features of spin precessions: (i) the imaginary unit $j$ in $\varepsilon_m$ implies spin precessions in ferromagnetic films are always right-hand polarized, as shown in the inset of Fig. \ref{fig1}(b) where $m_y$ drops behind $m_x$ for $1/4$ wavelength regardless of their propagating directions. (ii) The $\varepsilon_m$ is irrelevant to the amplitudes or phases of $h_x$ and $h_y$. In the exchange limit of $k_x\rightarrow\infty$, $\varepsilon_m\rightarrow j$ indicates the SWs are perfectly right-circularly polarized \cite{YuSpringer2021}. Meanwhile, in the dipolar-exchange region where the dipolar effect is comparable to exchange interaction, $\varepsilon_m$ varies with factors $n_x$ and $n_y$, which rely on $(\omega, k_x)$, as shown in Fig. \ref{fig1}(b).
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.48\textwidth]{model2.pdf}\\
\caption{ SW amplitudes ($|{\bf m}|$) normalized by the maximal value excited by (a) the right- and (b) left-hand chiral microwave fields with the same power density but different ellipticities ranging from 0.9 to 1.1. Solid curves are calculated based on Eqs. (\ref{LLGExpRF}). Insets in (a) and (b) depict the schematics of constructive and destructive superposition of the contributions of $h_x$ and $h_y$, in analog to the common and different signals in differential amplifiers, respectively. Symbols are from micromagnetic simulations. Illustrations of the left-hand chiral photon-magnon conversion (c) below, (d) at and (e) above $\omega_c$ {\color{red}, respectively}. The blue wavy arrays and circles {\color{red}represent} the {\color{red}microwave} fields with thickness indicating the intensity, {\color{red} where the blue arrowed circles represent the chirality. The blue glowing backgrounds indicate the converted microwave energy}. The dots and circles represent the spins and their precession cones, respectively.} \label{fig2}
\end{figure}
\subsection{INTENSITY SPECTRA}\label{ChiExci}
Below, we investigate SW amplitudes $|{\bf m}| =\sqrt{m_x^2+m_y^2}$ dependence on the microwave field chiralities. Specifically, we inspect the typical cases that $\varepsilon_h = h_x/h_y$ is purely imaginary where {\color{red}$h_x = j|\varepsilon_h|h_y$} and {\color{red}$h_x = -j|\varepsilon_h|h_y$} represent respectively the right- and left-handed polarization with $\left |\varepsilon_h \right |$ being their ellipticity. In micromagnetic simulations, the excitation is applied using the function ${\bf h}_{\rm rf}(t) = h_{x0}\sin (\omega t) {\bf x} + h_{y0} \sin (\omega t\pm \pi/2) {\bf y}$, with ``+" (``$-$") for the left (right) hand polarization. We fix $h_0 = \sqrt{h_{x0}^2+h_{y0}^2}=0.1~{\rm mT}$ to ensure the same RF power density with different ellipticities. The results record the dynamic normalized magnetization ($m_x/M_s$ and $m_y/M_s$) evolution as a function of time and space. The amplitude spectra of the SWs excited by the right- and left-hand polarized microwaves with $|\varepsilon_h|$ ranging from 0.9 to 1.1 are plotted in Figs. \ref{fig2}(a) and \ref{fig2}(b), respectively.
The chiral excitation of SWs possesses the following features. Firstly, the complex parameters $\chi_{x}$, $\chi_{y}$ and $\kappa$ expressed by Eqs. (\ref{PermiApx}) take the same phase factor. Therefore, Eqs. (\ref{LLGExpRF}) manifest that the contribution of $h_y$ to ${\bf m}$ is delayed by the phase of $\pi/2$ compared to that of $h_x$. Consequently, they are superposed destructively (constructively) in the case of left- (right-) hand excitation. Simulation results confirm this point that the left-hand excited SW intensities are weaker than its right-hand counterpart. We thus introduce an analog to the differential amplifier in electronic systems, in which the dual inputs are separately amplified, subtracted (added) and output as different (common) mode signals \cite{Yawale2022}, as illustrated by the insets in Figs. \ref{fig2}(a) and \ref{fig2}(b). The dual inputs, amplifying factors and outputs are in comparison with ${\bf h}_{\rm rf}$, the complex parameters ($\chi_{x}$, $\chi_{y}$ or $\kappa$) and ${\bf m}$, respectively. And $\varepsilon_h$ reflects the ratio between the dual inputs. {\color{red} Then we obtain
\begin{equation} \label{DiffAmpMod}
\left[
\begin{aligned}
m_{x} \\
m_{y}
\end{aligned}
\right]
=
\left[
\begin{aligned}
jh_y(\kappa\pm|\varepsilon_h|\chi_y)\\
h_y(\chi_x\pm|\varepsilon_h|\kappa)
\end{aligned}
\right],
\end{equation}
with ``+'' and ``$-$'' for the results of left- and right-handed excitations respectively, which certify the validation of the differential amplifier model.
}
Secondly, using the differential amplifier model, it can be explained that the left-hand excited SW spectra are much more sensitive to the variation of $\varepsilon_h$ than the right-hand excited ones since the differential (common) mode signal is sensitive (irresponsive) to the tiny variation ($\varepsilon _h$) of the dual inputs ($h_x$ and $h_y$). It is observed that the curves describing different $\varepsilon_h$ in Fig. \ref{fig2}(a) are almost merged, while those in Fig. \ref{fig2}(b) are well separated. {\color{red}Moreover, the intensity of left-hand excited SWs is much weaker than that of their right-hand counterparts. Especially the former drops to almost one tenth of the latter at high frequencies.}
Lastly, even though the two pairs of amplifying factors [($\chi_{y}$, $\kappa$) and ($\kappa$, $\chi_{x}$)] for the outputs $m_x$ and $m_y$ are different, their ratios are both $1/|\varepsilon_m|$. Mathematically, we can substitute Eqs. (\ref{PermiApx}) and Eq. (\ref{epsilon_m}) into Eqs. (\ref{LLGExpRF}), and obtain {\color{red} a more generalized expression for $\bf m$}
\begin{equation} \label{Chiral_Exci}
\left[
\begin{aligned}
m_{x} \\
m_{y}
\end{aligned}
\right]
=-\frac{\omega_m h_y\left(\varepsilon_h \varepsilon_m -1 \right)}{\alpha\left(\omega_x+\omega_y\right)}
\left[
\begin{aligned}
&1 \\
1&/\varepsilon_m
\end{aligned}
\right].
\end{equation}
It suggests a compensation frequency ($\omega_c$) when $\varepsilon_h\varepsilon_m=1$. Equation (\ref{epsilon_m}) indicates $\Im\left(\varepsilon_m\right)>0$, therefore only the the left-hand microwaves with $\Im\left(\varepsilon_h\right)<0$ support $\omega_c$, at (below and above) which the microwaves with any intensities are unable (able) to excite any SWs, as illustrated by Figs. \ref{fig2}(c), \ref{fig2}(d) and \ref{fig2}(e). The equality of the ratios is also the prerequisite for treating SWs as scalar variables in previous researches \cite{VasilievJAP2007,KostylevPRB2007,DemokritovPRL2004}. This finding broadens the selection rule for photon-magnon conversion, which is instructive to the chiral magneto-optic and -acoustic effects \cite{ZhangPRAppl2020,XuSciAdv2020}. However, $\omega_c$ cannot exist for arbitrary $\varepsilon_h$ because $|\varepsilon_m|$ given by Eq. (\ref{epsilon_m}) can only take values from 0.91 to 2.14 for the present model parameters. Consequently, $\omega_c$ can only emerge with $\varepsilon_h$ in the range from 0.47 to 1.09. Analytical and numerical results indeed confirm this point that the curve for $|\varepsilon_h| = 1.1$ (the green one) in Fig. \ref{fig2}(b) cannot intersect with $x$ axis.
\section{STRATEGY FOR NONRECIPROCAL SPIN WAVES}\label{Nonreciprocal}
The above discussions indicate that using the left-hand excitation is essential to generate the non-reciprocal SWs {\color{red}due to its high sensitivity of the spectra to $\varepsilon_h$. We only need to slightly alter the ellipticity ($\varepsilon_h^+$ and $\varepsilon_h^-$) of microwave fields for exciting the forward and backward propagating SWs with quite different spectra (superscripts ``$+$'' and ``$-$'' are used to label the forward and backward parameters, respectively, same hereinafter). In comparison, the right-hand exciting cases require the $\varepsilon_h^+$ and $\varepsilon_h^-$ to be dramatically varied for substantially different spectra. Hence, the left-handed excitation brings remarkable convenience for designing the method on nonreciprocity.} One critical technique is to differentiate $\varepsilon_h^+$ and $\varepsilon_h^-$. In the multi-layer structure, the dynamic mutual dipolar effect between layers has been demonstrated to be directionally dependent \cite{Henry2016}. So, one natural issue arises if we can introduce the mutual dipolar field combined with ${\bf h}_{\rm rf}$ to differentiate $\varepsilon_h^+$ and $\varepsilon_h^-$. Here, we investigate the method by applying microwave field unevenly across the film thickness. This idea is in contrast to preceding works, where {\color{red}additional micro-magnets outside YIG films are indispensable to serve as the spin wave source } and the effective exciting polarizations are simply circular with directionally opposite chirality, resulting in the nonreciprocity \cite{WangNR2020,ChenPRB2019,YuPRB2019,YuPRL2019,YuSpringer2021}. For simplification, ${\bf h}_{\rm rf}$ is uniformly applied only on the top part of the film with thickness $d_1$ {\color{red}and width $w$}, as shown in Fig. \ref{fig3}(a). The SW characteristics dependence on the excitation is investigated by varying $d_1$. {\color{red}The spin wave information in the exciting area along thickness is extracted by calculating $|\bf {m}|$ at every mesh grid, averaged in the exciting area with width $w$ and normalized by the maximal value.} We estimate the different dynamic magnetizations (${\bf m}_1= m_{x,1} {\bf x}+m_{y,1} {\bf y}$ and ${\bf m}_2= m_{x,2} {\bf x}+m_{y,2} {\bf y}$) along $d_1$ and $d_2=d-d_1$ that introduces the mutual demagnetizing field, whose simulated SW amplitudes are shown in Fig. \ref{fig3}(b). Even though the minor inhomogeneity appears at the interface, they are approximated to be transversely uniform in the following analysis. The part in dashed red box taking bilayer structure is regarded as the SW source. In this case, the dipolar fields are composed of two components: the self demagnetizing field ${\bf h}_{d,p}=-n_{x,p} m_{x,p} {\bf x}-n_{y,p} m_{y,p} {\bf y}$ where $n_{x(y),p}$ is given by Eq. (\ref{DemagFac}) with $n_{x(y)} \rightarrow n_{x(y),p}$ and $d\rightarrow d_{p}$, and the mutual demagnetizing field ${\bf h}_{d,pq}=h_{d,x,pq} {\bf x}+h_{d,y,pq} {\bf y}$ [($p,q$) = (1,2) or (2,1)]. Here $h_{d,x(y),pq}$ satisfy the following identity (see Appendix \ref{MutDE} for detailed derivation)
\begin{equation} \label{DipFldMutil}
\left[
\begin{aligned}
h_{d,x,pq} \\
h_{d,y,pq}
\end{aligned}
\right]=-n_{pq}
\left[
\begin{matrix}
1 & j\sign{k_{x}}(q-p) \\
j\sign{k_{x}}(q-p) & -1
\end{matrix}
\right]
\left[
\begin{aligned}
m_{x,p} \\
m_{y,p}
\end{aligned}
\right],
\end{equation}
with
\begin{equation} \label{GenDemFac}
n_{pq}=
\frac{\left(1-e^{-|k_{x}|d_p}\right)\left(1-e^{-|k_{x}|d_q}\right)}{2|k_{x}|d_q}.
\end{equation}
The ${\bf h}_{d,pq}$ (${\bf h}_{d,p}$) is directionally dependent (independent) according to Eqs. (\ref{DipFldMutil}) [Eq. (\ref{DemagFac})]. Hence, ${\bf h}_{d,pq}$ rather than ${\bf h}_{d,p}$ contributes to the non-reciprocity. In addition, ${\bf h}_{d,21}^{+}$ (${\bf h}_{d,21}^{-}$) and ${\bf h}_{d,12}^{+}$ (${\bf h}_{d,12}^{-}$) are contrarily circular polarized with different intensities, as sketched in the inset of Fig. \ref{fig3}(a) [as proved by Eqs. (\ref{DeMut}) in Appendix \ref{MutDE}]. The net effective mutual field ${\bf h}_{d,{\rm mut}}$ on the entire film is therefore given by
\begin{widetext}
\begin{equation} \label{NetDeMut}
\begin{aligned}
{\bf h}_{d,{\rm mut}}=\frac{{\bf h}_{d,12}d_2+{\bf h}_{d,21}d_1}{d}
=-n_{\rm mut}\left\{
\begin{aligned}
\left[(m_{x,1}+m_{x,2})+j\sign {k_x}(m_{y,1}-m_{y,2})\right] {\bf x}
+\left[j\sign {k_x}(m_{x,1}-m_{x,2})-(m_{y,1}+m_{y,2})\right] {\bf y}
\end{aligned}
\right\}.
\end{aligned}
\end{equation}
\end{widetext}
with
\begin{equation} \label{NetDeMutFac}
\begin{aligned}
n_{\rm mut} = \frac{\left(1-e^{-|k_{x}|d_1}\right)\left(1-e^{-|k_{x}|d_2}\right)}{2|k_x|d}.
\end{aligned}
\end{equation}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.40\textwidth]{model3.pdf}\\
\caption{(a) The schematic for the non-reciprocal SW excitation using left-hand chiral microwave field applied on the top part of the film in the patched blue area. Inset shows the precession cones of the mutual dipolar fields induced by the forward and backward propagating SWs in each layer, with the amplitude indicated by the radius. (b) Simulated SW amplitudes in red box for $d_1 = 10$ nm at 4.6 GHz.} \label{fig3}
\end{figure}
Following conclusions can thus be drawn. Firstly, the non-reciprocity disappear if $d_1 = 0$ or $d_2 = 0$, which causes $n_{\rm mut}=0$ and ${\bf h}_{d,{\rm mut}}=0$. It was confirmed that the SWs propagating along opposite directions share the same amplitude with the uniform excitation across the thickness, as shown in Fig. \ref{fig1}(b).
Secondly, since ${\bf h}_{d,{\rm mut}}$ is determined by ${\bf m}_{1}$ and ${\bf m}_{2}$, its role is to tune the two gains in the differential amplifier [insets of Figs. \ref{fig2}(a) and \ref{fig2}(b)], equivalent to varying $\varepsilon_h$ of the input microwave ${\bf h}_{\rm rf}$. As the variation of $\varepsilon_h$ is directional with ${\bf h}_{d,{\rm mut}}$, the intensity spectra are well separated for the forward and backward SWs, as plotted in Fig. \ref{fig4} (a). Even though ${\bf h}_{d,{\rm mut}}$ is frequency dependent, simulation results can still be well fitted using Eqs. (\ref{LLGExpRF}) and $h_x = \varepsilon_h^{+(-)}(d_1) h_y$ with $\omega_c$ satisfying $\varepsilon_h^{+(-)}(d_1)\varepsilon_m = 1$, where $\varepsilon_h^{+(-)}(d_1)$ is the effective ellipticity to be determined. The fitted $|\varepsilon_h^{+(-)}(d_1)|$ is plotted in the inset of Fig. \ref{fig4}(a) {\color{red}, where the goodness of all fittings is greater than $91\%$}. Representatively, we obtain $|\varepsilon_h^{+}(10~{\rm nm})|=1.05$ and $|\varepsilon_h^{-}(10~{\rm nm})|=0.93$, corresponding to $\omega_c/2\pi = 5.1$ and 6.7 GHz, with the dynamic magnetization presented in upper and lower panels of Fig. \ref{fig4}(b), respectively. {\color{red}Sacrificing the efficiency of excitations with the amplitude one order lower than that in the inset of Fig. \ref{fig1}(b), we can obtain theoretically switchable non-reciprocities and one hundred percentage (perfect) unidirectionality. This is advantageous over many other strategies \cite{XuSciAdv2020}.}
Thirdly, the difference between $\varepsilon_h^{+}$ and $\varepsilon_h^{-}$ and the separation of the forward and backward SW intensity spectra approaches the maximum at $d_1=d_2=d/2$, meeting the maximal value condition of $n_{\text{mut}}$ in Eq. (\ref{NetDeMutFac}). {\color{red} Notwithstanding, the value of $|\varepsilon_h^{+}(d_1 = 20 {\rm nm})|=1.13$ exceeds the range from 0.47 to 1.09. It} indicates that {\color{red} no $\omega_c$ would present in the forward SW spectra as discussed in Sec. \ref{ChiExci} . Consequently,} the perfect backward SW propagation without any forward SW cannot be achieved.
\begin{figure*}[htbp!]
\centering
\includegraphics[width=0.96\textwidth]{model4.pdf}\\
\caption{(a) Spectra of the forward (green) and backward (orange) SWs amplitudes with excitation depths $d_1=20~{\rm nm}$. The intensities are normalized with the maximal value. Symbols are numerical simulations and the curves are fitting results. The inset of (a) shows the fitted $\varepsilon_h^{+}$ and $\varepsilon_h^{-}$ dependence on $d_1$. (b) Simulated $m_{x}/M_s$ and $m_{y}/M_s$ distribution at two compensation frequencies, $\omega/2\pi=5.1~{\rm and ~6.7~ GHz}$, respectively. (c) Simulated forward (green symbols) and backward (orange symbols) SW spectra under the right-hand (diamonds) and linear (circles) excitations unevenly applied across the film thickness ($d_1 = 10$ nm).} \label{fig4}
\end{figure*}
Lastly, for completeness, we perform the simulations applying right-hand and linear microwave on the top part of the films with $d_1 = 10$ nm. The forward and backward SW spectra are presented in Fig. \ref{fig4}(c). {\color{red}Following features are observed. (i) Both spectra} are not well separated, indicating {\color{red}that $\bf h_{\rm {rf}}$ can induce non-reciprocity, but the effect is not significant. It can be understood in this configuration since ferromagnetic films are much thinner than the spin wavelengths \cite{YuPRB2019,YuPRL2019}. (ii)} The forward SWs are always stronger than the backward ones in the whole frequency band, implying that DE mechanism induced nonreciprocity cannot be switched by tuning frequencies since it is merely dependent on the surface normal and static magnetization directions \cite{Kwon2016,An2013}. {\color{red}In conclusion, the switchable and perfect non-reciprocities do not appear in the spectra of right-hand and linear excitation, reconfirming that they are the unique features of the left-hand excited SWs.}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.4\textwidth]{model5.pdf}\\
\caption{Simulated $|\bf{m}^-|/|\bf{m}^+|$ dependence on the decay length of the exponential profiled left-handed excitations at 4.6 GHz. The line connecting the symbols guide the trend. The inset schematically shows the simulated structure with $d = 40$ nm. The microwave field is applied on dashed box area with exponential intensity profile indicated by the patched blue area. } \label{fig5}
\end{figure}
Finally, we note that the key for exciting non-reciprocal SWs is the introduction of non-zero ${\bf h}_{d,\rm mut}$, induced by the asymmetrically distributed ${\bf m}$ across thickness, which can be simply excited by uneven profiled ${\bf h}_{\rm rf}$. Such fields can be generated by the resonant spin nano-oscillators with various structures, like nano-disks \cite{Demidov2012} or nano-wires \cite{Safranski2017}{\color{red}, which are less sharp than that proposed in the above analysis}. {\color{red} Even so, ${\bf h}_{\rm rf} $ with more gradual uneven profiles can also excite non-reciprocal SWs. To verify this point, we perform simulations using left-handed ${\bf h}_{\rm rf}$ with exponential profiles and $|\varepsilon_h| = 1$, as shown in the inset of Fig. \ref{fig5}. The intensity dependence on the thickness is described by $h_0(y) = h_0(\lambda)e^{-y/\lambda}$, where $h_0(\lambda)$ is determined by $\int_{-d}^{0} h_0(y)dy = h_0d ~ (h_0 = 0.1 \rm{mT})$ to ensure the same power intensity. The ratio $|\bf{m}^-|/|\bf{m}^+|$ depending on the decay length $\lambda$ is plotted in Fig. \ref{fig5}. When $\lambda$ is shorter than $d$, the ratio is observed to rise dramatically with the increase of $\lambda$ due to the rapid decrease of the uneven degree of the exciting field. It generally converges to $100 \%$ with $\lambda\rightarrow\infty$, corresponding to the case without the non-reciprocity.} Furthermore, the switched non-reciprocity at two compensation frequencies contributes additional methodology for magnonic frequency division multiplexing, broadening the strategy for designing magnonic circuits \cite{ZhangAPL2019}.
\section{DISCUSSION AND CONCLUSION}\label{Conclusion}
In summary, we investigated the propagating dipolar-exchange SWs excited by chiral microwaves in ferromagnetic thin films. We showed that the left-hand microwave can excite non-reciprocal SWs in the condition of ellipticity mismatch. When the left-hand microwave is unevenly applied across the film thickness, we observed a SW chirality switching by tuning the microwave frequency. Our findings shine a new light on the photon-magnon conversion and pave the way toward engineering the nano-scaled chiral microwave field for the realization of the diode-like functionalities in magnonics.
\section{ACKNOWLEDGEMENTS}
\begin{acknowledgments}\label{Acknowledgments}
We thank Y. Henry for helpful discussions. This work was supported by the National Natural Science Foundation of China (NSFC) (Grants No. 12074057, No. 11604041, and No. 11704060). Z.Z. acknowledges the financial support of the China Postdoctoral Science Foundation under Grant No. 2020M673180. Z.W. was supported by the China Postdoctoral Science Foundation under Grant No. 2019M653063 and the NSFC (Grant No. 12204089). Z.-X.L. acknowledges financial support from the China Postdoctoral Science Foundation (Grant No. 2019M663461) and the NSFC (Grant No. 11904048).
\end{acknowledgments}
\begin{appendix}
\section*{APPENDIX}
We investigate the dipolar effects induced by the SWs propagating in the ultra thin magnetic film. The dipolar field in the whole space is calculated. The self and mutual demagnetizing factors are figured out in Sec. \ref{SelfDE} and Sec. \ref{MutDE}, respectively.
We considered a magnetic film extended infinitely along $x$ and $z$ directions, located from $y = -d$ to 0 and labelled as $L_i$. The SWs takes the form ${\bf m}_i = {\bf m}_{0,i} e^{j(\omega t-k_xx)}=m_{x,i} {\bf x}+m_{y,i} {\bf y}$ with ${\bf m}_{0,i}= m_{x0,i} {\bf x}+m_{y0,i} {\bf y}$. The dynamic magnetization ${\bf m}_i$ and the correspondingly induced dipolar field ${\bf h}_{d,i}$ satisfy the magnetostatic equations
\begin{subequations} \label{MaxEqs}
\begin{align}
\label{MaxEqsa}\nabla \cdot ({\bf h}_{d,i}+{\bf m}_i) = 0, \\
\label{MaxEqsb}\nabla \times {\bf h}_{d,i} = 0. &
\end{align}
\end{subequations}
Introducing the scale potential $\psi_{m,i}$, we have
\begin{equation} \label{Magn_Pot}
{\bf h}_{d,i}=-\nabla \psi_{m,i}.
\end{equation}
Then Eq. (\ref{MaxEqsa}) becomes Poisson equation
\begin{equation} \label{PoiEq}
\nabla^2 \psi_{m,i} = -\rho_{i},
\end{equation}
where $\rho_{i}$ is the effective magnetic-charge density, given as
\begin{equation} \label{MagChg}
\rho_{i} = -\nabla \cdot {\bf m}_i.
\end{equation}
One crucial step is to find the solution of $\psi_{m,i}$ in Eq. (\ref{PoiEq}). We note that there are two contributions to $\psi_{m,i}$ in magnetic materials: effective \emph{volume} magnetic-charge density $\rho_{m,i}$ and effective \emph{surface} magnetic-charge density $\sigma_{m,i}$ \cite{Jackson1962}.
Firstly, we calculate the contribution of $\rho_{m,i}$. Inside the film, $\rho_{m,i}=-\nabla \cdot {\bf m}_i=jk_xm_{x,i}$ is induced by the $x$ component of $m_i$ \cite{Henry2016}. To begin with, we consider a tiny sheet of film located at position $y = y_0$ with thickness $dy_0$, whose surface magnetic charge density is $\sigma_{0,i} = \rho_{m, i}dy_0$ [see Fig. \ref{SMfig1}(a)]. The magnetostatic potential $\psi_{m, i}(\sigma_{0,i}, y_0, {\bf r}, t)$ induced by $\sigma_{0,i}$ is periodic (evanescent) along $x$ ($y$) direction, while its maximum locates at $y = y_0$ and satisfy Laplace equation $\nabla^2\psi_{m, i}(\sigma_{0,i}, y_0, {\bf r}, t) = 0$ \cite{Bailleul2011}. Then the solution can be expressed as
\begin{equation} \label{PotEle}
\psi_{m, i}(\sigma_{0,i}, y_0, {\bf r}, t)
= \psi_{m0,i}(\sigma_{0,i})e^{-|k_x(y-y_0)|}e^{j(\omega t-k_xx)}.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{SMfig1.pdf}\\
\caption{Schematics of the effective (a) \emph{volume} and (b) \emph{surface} magnetic-charges. The patched yellow parts represent the differential elements} \label{SMfig1}
\end{figure}
The next important step is to find out the value of $\psi_{m0,i}$. Note that the boundary condition (continuity of $B_y$) of the tiny sheet is given as
\begin{equation} \label{BdC}
h_{y,i}(\sigma_{0,i},y_0^+,{\bf r},t)-h_{y,i}(\sigma_{0,i},y_0^-,{\bf r},t) = \sigma_{0,i}.
\end{equation}
Using Eq. (\ref{Magn_Pot}), we have
\begin{equation} \label{DipField}
\begin{aligned}
h_{y,i}(\sigma_{0,i},y_0,{\bf r},t)
& = -\frac{\partial}{\partial y} \psi_{m,i}(\sigma_{0,i},y_0,{\bf r},t)\\
& = \left \{
\begin{aligned}
|k_x|\psi_{m0,i}e^{|k_x|(y-y_0)}e^{j(\omega t-k_xx)},&y\geq y_0\\
|k_x|\psi_{m0,i}e^{-|k_x|(y-y_0)}e^{j(\omega t-k_xx)},&y<y_0\\
\end{aligned}
\right .
\end{aligned}
\end{equation}
Therefore, we have
\begin{equation} \label{SolPot}
2|k_x|\psi_{m0,i}(\sigma_{0,i})e^{j(\omega t-k_xx)}=\sigma_{0,i}.
\end{equation}
The magneto-static potential induced by the sheet at $y = y_0$ can be expressed as
\begin{equation} \label{SolvedPot}
\psi_{m,i}(\sigma_{0,i},y_0,{\bf r},t)
=\frac{j\sign {k_x}m_{x0,i}e^{j(\omega t-k_xx)}}{2}e^{-|k_x(y-y_0)|}dy_0.
\end{equation}
Correspondingly, the dipolar magnetic field ${\bf h}_{d,i}(\sigma_{0,i},y_0,{\bf r},t)$ derived from $\psi_{m,i}(\sigma_{0,i},y_0,{\bf r},t)$ is given as
\begin{equation} \label{SolvedDipFld}
\begin{aligned}
& {\bf h}_{d,i}(\sigma_{0,i},y_0,{\bf r},t)\\
= &\frac{j\sign {k_x}m_{x,i}}{2}e^{-|k_x(y-y_0)|}
\left [-\sign{k_x} {\bf x}+j\sign{y-y_0} {\bf y}\right ]dy_0.
\end{aligned}
\end{equation}
The dipolar field ${\bf h}_{d,i}(\rho_{m,i},{\bf r},t)$ induced by $\rho_{m,i}$ at any position ${\bf r}$ is given
\begin{widetext}
\begin{equation} \label{AnyVolFld}
\begin{aligned}
{\bf h}_{d,i}(\rho_{m,i},{\bf r},t)& =\frac{1}{2}\int_{-d}^{0}j\sign {k_x}m_{x,i}e^{-|k_x(y-y_0)|}\left [-\sign{k_x} {\bf x}+j\sign{y-y_0} {\bf y}\right ]dy_0\\
&= \left \{
\begin{aligned}
& \frac{m_{x,i}}{2}e^{-|k_x|y}\left (1-e^{-|k_x|d}\right )\left [- {\bf x}+j\sign {k_x} {\bf y}\right ],&y\geq0\\
& -\frac{m_{x,i}}{2}\left [2-e^{-|k_x|(y+d)}-e^{|k_x|y}\right ] {\bf x}+\frac{jm_{x,i}}{2}\sign {k_x}\left [e^{|k_x|y}-e^{-|k_x|(y+d)}\right] {\bf y},&-d\leq y<0\\
& \frac{m_{x,i}}{2}e^{|k_x|y}\left (e^{|k_x|d}-1\right )\left [- {\bf x}-j\sign {k_x} {\bf y}\right ],&y<-d\\
\end{aligned}
\right.
\end{aligned}
\end{equation}
\end{widetext}
Next, we calculate the contribution of $\sigma_{m,i} = {\bf m}_{i}\cdot {\bf {n}}$, located only at the position $y = 0$ and $y = -d$ with ${\bf {n}}$ the unit vector normal to the surface. They are equal to $m_{y, i} = m_{y0, i} e^{j(\omega t-k_xx)}$ and $-m_{y, i}$, where the minus sign comes from the opposite directions of the top and bottom surfaces.Following the steps from Eqs. (\ref{PotEle}) to (\ref{SolvedPot}), we obtain the magneto-static potential induced by $\sigma_{m,i}$
\begin{subequations} \label{SurfPot}
\begin{align}
\label{SurfPota}\psi_{m,i}(\sigma_{m,i},0,{\bf r},t) &= \frac{m_{y0,i}}{2|k_x|}e^{j(\omega t-k_xx)}e^{-|k_xy|}, \\
\label{SurfPotb}\psi_{m,i}(\sigma_{m,i},-d,{\bf r},t) &= -\frac{m_{y0,i}}{2|k_x|}e^{j(\omega t-k_xx)}e^{-|k_x(y+d)|}.
\end{align}
\end{subequations}
The dipolar field ${\bf h}_{d, i}(\sigma_{m, i}, {\bf r}, t)$ induced by $\sigma_{m, i}$ at any position $\bf r$ is given \cite{Henry2016}
\begin{widetext}
\begin{equation} \label{AnySurfFld}
\begin{aligned}
{\bf h}_{d,i}(\sigma_{m,i},{\bf r},t) &= -\nabla \left [\psi_{m,i}(\sigma_{m,i},0,{\bf r},t)+\psi_{m,i}(\sigma_{m,i},-d,{\bf r},t)\right]\\&=\left\{
\begin{aligned}
&\frac{m_{y,i}}{2}e^{-|k_x|y}\left (1-e^{-|k_x|d}\right )\left [j\sign {k_x} {\bf x}+ {\bf y}\right ],&y\geq0\\
&-\frac{j m_{y,i}}{2}\left [e^{|k_x|y}-e^{-|k_x|(y+d)}\right ]\sign {k_x} {\bf x}+\frac{m_{y,i}}{2}\left [-e^{-|k_x|(y+d)}-e^{|k_x|y}\right ] {\bf y},&-d\leq y<0\\
&\frac{m_{y,i}}{2}e^{|k_x|y}\left (e^{|k_x|d}-1\right )\left [-j\sign {k_x} {\bf x}+ {\bf y}\right ].&y<-d\\
\end{aligned}
\right.
\end{aligned}
\end{equation}
\end{widetext}
Finally, we obtain the dipolar magnetic field ${\bf h}_{d,i}({\bf r},t) = {\bf h}_{d,i}(\rho_{m,i},{\bf r},t)+{\bf h}_{d,i}(\sigma_{m,i},{\bf r},t)$ in the whole space
\begin{widetext}
\begin{equation} \label{AnyDipFld}
{\bf h}_{d,i}({\bf r},t) =\left\{
\begin{aligned}
&\frac{1}{2}e^{-|k_x|y}\left (1-e^{-|k_x|d}\right )\left\{\left [-m_{x,i}+j\sign {k_x}m_{y,i}\right ] {\bf x}+\left [j\sign {k_x}m_{x,i}+m_{y,i}\right ] {\bf y}\right\},&y\geq0\\
& \left\{\left [e^{-|k_x|(y+d)}+e^{|k_x|y}-2\right ]\frac{m_{x,i}}{2}+\frac{j\sign {k_x}}{2}\left [e^{-|k_x(y+d)|}-e^{-|k_xy|}\right ]m_{y,i}\right \} {\bf x}&\\
&+\left \{\frac{j\sign {k_x}}{2}\left [e^{|k_x|y}-e^{-|k_x|(y+d)}\right ]m_{x,i}-\left [e^{-|k_xy|}+e^{-|k_x(y+d)|}\right ]\frac{m_{y,i}}{2}\right\} {\bf y},&-d \leq y<0\\
&\frac{1}{2}e^{|k_x|y}\left (e^{|k_x|d}-1\right )\left \{\left [-m_{x,i}-j\sign {k_x}m_{y,i}\right ] {\bf x}+\left [-j\sign {k_x}m_{x,i}+m_{y,i}\right ] {\bf y}\right \},&y<-d.\\
\end{aligned}
\right.
\end{equation}
\end{widetext}
\subsection{Self demagnetizing factors} \label{SelfDE}
When calculating the demagnetizing factor of a single layer with thickness $d_i$, we care about the region $-d_i<y<0$. The demagnetizing factors inside the film are defined as the ratios between the average dipolar field and the magnetization
\begin{subequations} \label{DemFacDef}
\begin{align}
\label{DemFacDefa}-n_{x,i}m_{x,i}=\frac{1}{d}\int_{-d}^{0} {\bf x}\cdot{\bf h}_{d,i}({\bf r},t)dy, \\
\label{DemFacDefb}-n_{yx}m_{x,i}=\frac{1}{d}\int_{-d}^{0} {\bf y}\cdot{\bf h}_{d,i}({\bf r},t)dy, \\
\label{DemFacDefc}-n_{xy}m_{x,i}=\frac{1}{d}\int_{-d}^{0} {\bf x}\cdot{\bf h}_{d,i}({\bf r},t)dy, \\
\label{DemFacDefd}-n_{y,i}m_{x,i}=\frac{1}{d}\int_{-d}^{0} {\bf y}\cdot{\bf h}_{d,i}({\bf r},t)dy. &
\end{align}
\end{subequations}
We obtain
\begin{subequations} \label{DemFac}
\begin{align}
\label{DemFaca}n_{x,i}=1-&n_{y,i}=1-\frac{1-e^{-|k_x|d_i}}{|k_x|d_i} \\
\label{DemFacb}n_{xy}&=n_{yx}=0.
\end{align}
\end{subequations}
The net self-induced dipolar field ${\bf h}_{d,self,i}=h_{d,self,x} {\bf x}+h_{d,self,y} {\bf y}$ by the SWs can be evaluated
\begin{equation} \label{DipFldSelf}
\left[
\begin{aligned}
h_{d,self,x} \\
h_{d,self,y}
\end{aligned}
\right]=
-\left[
\begin{matrix}
n_{x,i} & 0 \\
0 & n_{y,i}
\end{matrix}
\right]
\left[
\begin{aligned}
m_{x,i} \\
m_{y,i}
\end{aligned}
\right].
\end{equation}
The ratio $\varepsilon_{hd}$ between $h_{d,x}$ and $h_{d,y}$ is given as
\begin{equation} \label{EllipSelf}
\varepsilon_{hd}=\frac{h_{d,x}}{h_{d,y}}=\frac{n_{x,i}m_{x,i}}{n_{y,i}m_{y,i}},
\end{equation}
indicating that the chirality of the self-induced dipolar field depends on the chirality of the dynamic magnetization.
\subsection{Mutual demagnetizing factors}\label{MutDE}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.4\textwidth]{SMfig2.pdf}\\
\caption{Schematic of the bilayer consisted of $L_1$ (purple) and $L_2$ (green) with SWs ${\bf m}_1$ and ${\bf m}_2$ inside, respectively.} \label{SMfig2}
\end{figure}
In this part, we consider the dipolar effects between the two adjacent layers labelled as $L_1$ and $L_2$, as shown in Fig. \ref{SMfig2}. They are located from $y = -d_1$ to 0 and from $y=-d$ to $-d_1$, respectively. For simplification, we denote $d_2 = d-d_1$. The SWs propagating inside take the form ${\bf m}_p = {\bf m}_{0,p} e^{j(\omega t-k_{x,p}x)} = m_{x,p} {\bf x}+m_{y,p} {\bf y}$ with $p = 1,2$. According to Eq. (\ref{AnyDipFld}), the dipolar field induced by ${\bf m}_1$ and acting on $L_2$$\left [-(d_1+d_2)< y < -d_1\right ]$ is given as
\begin{widetext}
\begin{equation} \label{DipFld12}
\begin{aligned}
{\bf h}_{d,12}({\bf r},t)
= \frac{1}{2}e^{|k_{x,1}|y}\left (e^{|k_{x,1}|d_1}-1\right )\left\{\left [-m_{x,1}-j\sign{k_{x,1}}m_{y,1}\right ] {\bf x}
+\left [-j\sign{k_{x,1}}m_{x,1}+m_{y,1}\right ] {\bf y}\right \}.
\end{aligned}
\end{equation}
\end{widetext}
The average dipolar field acting on $L_2$ can be evaluated by introducing the mutual demagnetizing factors $n_{x12}$, $n_{xy12}$, $n_{yx12}$ and $n_{y12}$
\begin{subequations} \label{GenDemFacDef12}
\begin{align}
\label{GenDemFacDef12a}-n_{x12}m_{x,1}-n_{xy12}m_{y,1}=\frac{1}{d_2}\int_{-(d_1+d_2)}^{-d_1} {\bf x}\cdot{\bf h}_{d,12}({\bf r},t)dy, \\
\label{GenDemFacDef12b}-n_{yx12}m_{x,1}-n_{y12}m_{y,1}=\frac{1}{d_2}\int_{-(d_1+d_2)}^{-d_1} {\bf y}\cdot{\bf h}_{d,12}({\bf r},t)dy. &
\end{align}
\end{subequations}
We obtain
\begin{subequations} \label{GenDemFac12}
\begin{align}
\label{GenDemFac12a}&n_{x12}=-n_{y12}
=\frac{\left(1-e^{-|k_{x,1}|d_1}\right)\left(1-e^{-|k_{x,1}|d_2}\right)}{2|k_{x,1}|d_2}, \\
\label{GenDemFac12b} &n_{xy12}=n_{yx12}=j\sign{k_{x,1}}n_{x12}.
\end{align}
\end{subequations}
The dipolar field induced by ${\bf m}_2$ and acting on $L_1 (-d_1<y<0)$ is given as
\begin{widetext}
\begin{equation} \label{DipFld21}
{\bf h}_{d,21}({\bf r},t)
=\frac{1}{2}e^{-|k_{x,2}|(y+d_1)}\left (1-e^{-|k_{x,2}|d_2}\right )\left\{\left [-m_{x,2}+j\sign{k_{x,2}}m_{y,2}\right ]{\bf x}
+\left [j\sign{k_{x,2}}m_{x,1}+m_{y,2}\right ]{\bf y}\right \}.
\end{equation}
\end{widetext}
Similarly, we introduce $n_{x21}$, $n_{xy21}$, $n_{yx21}$ and $n_{y21}$
\begin{subequations} \label{GenDemFacDef21}
\begin{align}
\label{GenDemFacDef21a}-n_{x21}m_{x,1}-n_{xy21}m_{y,1}=\frac{1}{d_1}\int_{-d_1}^{0} {\bf x}\cdot{\bf h}_{d,21}({\bf r},t)dy, \\
\label{GenDemFacDef21b}-n_{yx21}m_{x,1}-n_{y21}m_{y,1}=\frac{1}{d_1}\int_{-d_1}^{0} {\bf y}\cdot{\bf h}_{d,21}({\bf r},t)dy. &
\end{align}
\end{subequations}
We obtain
\begin{subequations} \label{GenDemFac21}
\begin{align}
\label{GenDemFac21a}&n_{x21}=-n_{y21}
=\frac{\left(1-e^{-|k_{x,2}|d_1}\right)\left(1-e^{-|k_{x,2}|d_2}\right)}{2|k_{x,2}|d_1}, \\
\label{GenDemFac21b} &n_{xy21}=n_{yx21}=-j\sign{k_{x,2}}n_{x21}.
\end{align}
\end{subequations}
Finally, the mutual net dipolar field ${\bf h}_{d,12}=h_{d,x,12} {\bf x}+h_{d,y,12} {\bf y}$ and ${\bf h}_{d,21}=h_{d,x,21} {\bf x}+h_{d,y,21} {\bf y}$ can be evaluated
\begin{equation} \label{DipFldMut12}
\left[
\begin{aligned}
h_{d,x,12} \\
h_{d,y,12}
\end{aligned}
\right]=-n_{x12}
\left[
\begin{matrix}
1 & j\sign{k_{x,1}} \\
j\sign{k_{x,1}} & -1
\end{matrix}
\right]
\left[
\begin{aligned}
m_{x,1} \\
m_{y,1}
\end{aligned}
\right],
\end{equation}
and
\begin{equation} \label{DipFldMut21}
\left[
\begin{aligned}
h_{d,x,21} \\
h_{d,y,21}
\end{aligned}
\right]=-n_{x21}
\left[
\begin{matrix}
1 & -j\sign{k_{x,2}} \\
-j\sign{k_{x,2}} & -1
\end{matrix}
\right]
\left[
\begin{aligned}
m_{x,2} \\
m_{y,2}
\end{aligned}
\right].
\end{equation}
The ratio $\varepsilon_{hd12}$ ($\varepsilon_{hd21}$) between $h_{d,x,12}$ and $h_{d,y,12}$ ($h_{d,x,21}$ and $h_{d,21,y}$) is given as
\begin{subequations} \label{EllipMut}
\begin{align}
\label{EllipMuta}\varepsilon_{hd12}=\frac{m_{x,1}+j\sign{k_{x,1}}m_{y,1}}{j\sign {k_{x,1}}m_{x,1}-m_{y,1}}=-j\sign{k_{x,1}}, \\
\label{EllipMutb} \varepsilon_{hd21}=\frac{m_{x,2}-j\sign{k_{x,2}}m_{y,2}}{-j\sign {k_{x,2}}m_{x,2}+m_{y,2}}=j\sign{k_{x,2}}.
\end{align}
\end{subequations}
indicating that the chirality of the mutual dipolar field depends on the signs of the wave vectors. The net mutual demagnetizing field can be estimated as
\begin{widetext}
\begin{equation} \label{DeMut}
\begin{aligned}
{\bf h}_{d,mut}&=\frac{{\bf h}_{d,12}d_2+{\bf h}_{d,21}d_1}{d}\\
&=-\frac{\left(1-e^{-|k_{x,2}|d_1}\right)\left(1-e^{-|k_{x,2}|d_2}\right)}{2|k_x|d}\left\{
\begin{aligned}
\left[(m_{x,1}+m_{x,2})+j\sign {k_x}(m_{y,1}-m_{y,2})\right] {\bf x} +\left[j\sign {k_x}(m_{x,1}-m_{x,2})-(m_{y,1}+m_{y,2})\right] {\bf y}
\end{aligned}
\right\}.
\end{aligned}
\end{equation}
\end{widetext}
Here, we note that in the main text, the dynamic magnetization ${\bf m}_1$ and ${\bf m}_2$ satisfy the boundary condition ${\bf m}_1={\bf m}_2\bigg |_{y=-d_1}$ \cite{Verba2020}, which gives $k_{x,1} = k_{x,2} = k_{x}$.
\end{appendix}
|
astro-ph/0703029
|
\section{Introduction}\label{sec:intro}
\setcounter{footnote}{11}
In the current paradigm of hierarchical galaxy formation, massive galaxies are built up
through a series of major and minor merger events \citep{sea78,whi78}.
Observations of galaxies at high redshift show that merging systems
are common \citep[e.g.,][]{abr96,lef00,con03,lot07}, and large-scale
simulations of galaxy formation in
a cosmological context have successfully reproduced many of the observed
properties of galaxies and galaxy clusters \citep[e.g.,][]{spr05,cro06}.
A consequence of hierarchical galaxy formation is that galactic stellar
halos should be at least partially composed of the tidal debris from past
accretion events. Numerical simulations and semi-analytic models of
stellar halo formation have made great strides toward understanding
the properties of halos that are built up through tidal stripping of
merging systems \citep[e.g.,][]{joh96,joh98,hel99,hel00,bul01,bul05}.
Detailed comparisons between observations and simulations are needed
to determine the fraction
of stellar halos that are made up of tidal debris and to better
understand the formation of galaxies in general.
Recent discoveries of tidal streams in the stellar halos of
the Milky Way (MW) and Andromeda (M31) galaxies
are providing the most direct and detailed observational constraints on theories of stellar halo
formation. Among the most prominent of these substructures are the Sagittarius
stream \citep{iba94,maj03,new03}, the
Monoceros stream \citep{yan03,roc03}, and the Magellanic stream \citep{mat74}
in the MW, and the giant southern stream
\citep[GSS;][]{iba01} in M31. Additional substructures have been identified in
M31 that are also likely remnants of past mergers, such as the Northeast shelf
\citep{fer02,iba04,far06}, a secondary cold component in the same physical location as the giant
southern stream \citep{kal06a}, and the
various substructures identified with the disk of M31 \citep{iba05}.
Tidal disruption has also been observed in M31's satellite galaxies M32 and NGC 205
\citep{cho02}.
In addition to providing insight into theoretical models of stellar halo
formation, the observed properties of tidal streams can be used to constrain the the galactic
potential in which they are found if sufficient
phase-space information is available
\citep[e.g.,][]{joh99,joh02,pen06}. Several attempts have been made to model the
mass distribution of the MW using observed substructure,
most of which have focused on the orbital properties of the Sagittarius stream
\citep[e.g.,][]{iba01a,hel04,mar-del04,joh05,law05,fel06}.
The GSS
has been the focus of detailed modeling in M31, spurred on by recent
observations. Imaging and photometry
have revealed the physical extent of the GSS
\citep{fer02,mccon03,fer06} and provided line-of-sight distances
at various points along it \citep{mccon03}, while spectroscopy
has yielded the mean line-of-sight velocity and velocity
dispersion of stream stars as a function of position
\citep{iba04,guh06,kal06a}. The availability of this phase-space
information has motivated
several groups to model the orbit of the progenitor of the GSS
\citep{iba04,fon06,far06,far07}. However, \citet{far06} concluded
that the degree to which the GSS can be used to constrain M31's mass
distribution is limited by
the current measurement uncertainties in the distance to the stream
and the lack of a clearly identified compact stellar concentration that might
correspond to the dense remnant core of the stream's progenitor galaxy.
Further observational
constraints on the orbit of the progenitor of the GSS, such
as the identification of tidal debris from other pericentric
passages, are needed to make progress.
Towards this end, \citet[][hereafter F07]{far07} show that several of the observed
features in M31 can
be explained as the forward continuation
of the GSS. Their model makes predictions that
can be tested by observations, including the stellar velocity distributions
in the Northeast
and Western shelves and the presence of a weaker shelf on the
eastern side of the galaxy. This last shelf is expected to be most
easily visible near the southeastern minor axis of M31\footnote{Although this
shell feature is expected to span an $\sim 180$$^\circ$\ range in position angle, covering
the eastern half of the galaxy, it is expected to be most easily observable in the southeast
due to overlap with the much denser Northeast shelf and M31's disk elsewhere
(see Fig.~\ref{fig:xieta_sim}).}.
This paper presents new substructure along M31's southeastern minor axis at
the expected location of the F07 Southeast shelf and displaying the distinct
kinematic profile predicted by their orbital model. The substructure
was discovered in the course of an on-going Keck/DEIMOS spectroscopic study of the
dynamics and metallicity of RGB stars in the inner spheroid and outer halo of
M31 \citep[see][and references therein]{gil06}. The portion of the
inner spheroid studied here appeared to be relatively undisturbed in
earlier star count maps \citep{fer02} and radial velocity surveys
\citep{rei02, kal06a}.
The photometric and spectroscopic data used in this analysis are described in
\S\,\ref{sec:data}. The criteria for selection of
M31 RGB stars are discussed in \S\,\ref{sec:sample}.
The kinematics of the RGB population
(first the combined data set and then the individual fields)
are characterized in \S\,\ref{sec:kin}.
The spatial trends and general properties of the dynamically hot
spheroid and cold substructure populations are discussed in
\S\,\ref{sec:veldisp} and \S\,\ref{sec:coldpop}, respectively.
The physical origin of the cold substructure is
explored in
\S\,\ref{sec:origin}, including its likely relation to M31's GSS.
The relevance
of the newly discovered cold substructure to
the \citet{bro03} discovery of an intermediate-age population in the spheroid of M31
is discussed \S\,\ref{sec:intm_age}.
The main conclusions of the paper are summarized in \S\,\ref{sec:concl}.
\section{Data}\label{sec:data}
The data set discussed in this paper is drawn from photometry and spectroscopy of several fields
on/near the southeastern minor axis of M31. The locations of the
fields are shown in Figures~\ref{fig:roadmap} and \ref{fig:cfht}. They span a
range of projected radial
distances from M31's center of $R_{\rm proj}\sim 9$ to 30~kpc (Table~\ref{tab:masks}).
A brief explanation of the data sets and reduction is included below. A more
detailed discussion of the
observational strategy and data reduction methods employed in our M31 survey can be
found in \citet{guh06}, \citet{kal06a,kal06b}, and \citet{gil06}.
\begin{figure}
\epsscale{0.85}
\plotone{f1.eps}
\caption{
Sky positions of the fields discussed in this paper.
The blue square represents the position and area of the CFHT/MegaCam image
(Fig.~\ref{fig:cfht}).
The red rectangles approximate
the size and position angle of the DEIMOS spectroscopic slitmasks. The three
masks nearest the outer ellipse are in field a0.
The remaining masks are identified in Figure~\ref{fig:cfht}.
The star count map comes from \citet{iba05}, and is in
standard M31-centric coordinates ($\xi, \eta$). The outer ellipse represents
a 55 kpc radius along the major axis, with a flattening of 3:5.
The major and minor axes of M31 are indicated by straight lines.
The giant southern
stream is the obvious overdensity of stars south of M31's center.
}
\label{fig:roadmap}
\end{figure}
\subsection{Photometry}\label{sec:phot}
Photometry and astrometry for the majority of the fields analysed in this paper
were derived from MegaCam images in the
$g'$ and $i'$ bands obtained with
the 3.6-m Canada-France-Hawaii Telescope (CFHT)\footnote{MegaPrime/MegaCam is
a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope
which is operated by the National Research Council of Canada, the Institut
National des Science de l'Univers of the Centre National de la Recherche
Scientifique of France, and the University of Hawaii.}.
The program SExtractor \citep{ber96} was used for object
detection, photometry, and morphological classification (via the {\tt stellarity}
parameter).
The instrumental $g'$ and $i'$ magnitudes were
transformed to Johnson-Cousins $V$ and $I$ magnitudes based on observations
of Landolt photometric standard stars \citep{kal06a}.
Photometry and astrometry for a0, the outermost field discussed in this paper,
were derived by \citet{ost02} from images obtained with the Mosaic camera
on the Kitt Peak National Observatory (KPNO)\footnote{Kitt Peak National
Observatory of the National Optical Astronomy Observatory is operated by the
Association of Universities for Research in Astronomy, Inc., under cooperative
agreement with the National Science Foundation.} 4-m telescope
in the Washington System $M$ and
$T_2$ bands and the intermediate-width DDO51\ band. This combination
of filters allows photometric selection of stars that are likely to be
M31 red giant branch (RGB) stars rather than MW dwarf stars \citep[e.g.,][]{pal03, maj05}.
The DDO51\ filter
is centered at a wavelength of 5150~\AA\ with a width of $\sim 100$~\AA, and
includes the surface-gravity sensitive Mg\,$b$ and MgH stellar absorption
features which are strong in dwarf stars but weak in RGB stars. Based on a
star's position in the ($M-DDO51$) versus ($M-T_2$) color-color diagram, it
is assigned a probability of being an M31 RGB star.
Johnson-Cousins $V$ and $I$ magnitudes were derived from the $M$ and
$T_2$ magnitudes using the photometric transformation relations in \citet{maj00}.
Use of DDO51\ photometry to screen for likely M31 RGB stars increases
the efficiency of the spectroscopic observations,
suppressing the number of selected MW dwarf stars by a factor of $\approx 3$
in a0 (Guhathakurta et al. 2007, in prep).
\begin{figure}
\plotone{f2.eps}
\caption{
Starcount map derived from CFHT/MegaCam photometry in a single pointing
with the 36-CCD mosaic (\S\,\ref{sec:phot}). The orientation of this map is
the same as Figure~\ref{fig:roadmap}.
The size and positions of the masks designed from the CFHT/MegaCam photometry
are
shown as red rectangles. The white square shows the position and approximate
orientation of the \citet{bro03} deep HST/ACS observations (\S\,\ref{sec:intm_age}).
The three a0 masks were
based on photometry from the KPNO 4-m telescope and are to the southeast (bottom left), beyond the limit of
this image. There is an apparent edge in the density of star counts in the image,
running from the upper left to the lower right and passing through field f123;
this feature will be discussed in \S\,\ref{sec:gss}.
}
\label{fig:cfht}
\end{figure}
\subsection{Spectroscopy}
\subsubsection{Slitmask Design and Observations}\label{sec:slit_design}
Objects in fields covered by the CFHT/MegaCam images were selected for
Keck/DEIMOS spectroscopy based on $I$ magnitude and
the SExtractor morphological criterion {\tt stellarity} \citep{kal06a}.
Objects in field a0 were selected on the basis of $I$ magnitude and
morphological criteria (DAOPHOT parameters {\tt chi} and {\tt sharp}), prioritized according to their
probability of being an M31 RGB star (based on $M$, $T_2$, and
DDO51\ photometry as described in \S\,\ref{sec:phot}). Pre-selection of likely M31 RGB
stars is vital for efficient study of the sparse outer parts of the M31 halo. The
inner fields (such as the fields drawn from the CFHT/MegaCam images) contain
a relatively high
surface density of M31 RGB stars, so the RGB to MW
dwarf star ratio is high even without DDO51-based pre-selection of RGB candidates.
For the purposes of most of the analysis in this paper, data are classified according
to fields, rather than spectroscopic masks. In general, a ``field'' refers to
the area covered by a single CFHT/MegaCam CCD ($\sim 15$$'$$\times$6\farcm5, Fig.~\ref{fig:cfht}); there
can be one or more overlapping
DEIMOS masks (16$'$$\times$4$'$) in a single field. For example,
masks H11\_1 and H11\_2 are both part of field H11.
There are two exceptions to this field/mask scheme: (1) the a0 field refers to a single Mosaic pointing,
which covers a 35$'$$\times$35$'$\ area, and (2)
two of the f130 masks each straddle a couple of adjacent MegaCam CCDs,
but since they were chosen to overlap with the f130\_1 mask, they have
been labelled f130\_2 and f130\_3.
Fields were observed using the Keck~II telescope and the DEIMOS instrument with
the 1200~line~mm$^{-1}$ grating. Most of the fields were observed in Fall~2005 [f109, f115, f116,
f123, f130 (1 mask), and f135]. The a0 masks were observed in Fall~2002
and 2004, the H11 masks in 2004, and the last two of the three f130
masks in Fall~2006
(Table~\ref{tab:masks}).
The central wavelength for most masks was $\rm\lambda7800~\AA$, yielding a
spectral coverage of approximately $\lambda\lambda6450$--$\rm9150~\AA$. The only exceptions were
the a0\_1 and a0\_2 masks observed in 2002, which had a central wavelength of
$\rm\lambda8550~\AA$
and a spectral coverage of approximately $\lambda\lambda7200$--$\rm9900~\AA$.
The 1200~line~mm$^{-1}$ grating has a dispersion of $\rm0.33~\AA$~pix$^{-1}$;
the scale in the spatial direction is 0$\farcs$12~pix$^{-1}$, and the effective scale in
the dispersion direction is 0$\farcs$21~pix$^{-1}$. Slits had a width of
1$''$. The spectral resolution for a star observed in typical $0.8''$
seeing conditions is about 1.3~\AA\ FWHM. Each mask was observed for a
total of 1~hour,
with the exception of field f109 which was observed for 3~hours.
\begin{deluxetable*}{rcrrrlrr}
\tablecolumns{8}
\tablewidth{0pc}
\tablecaption{Details of Spectroscopic Observations and Basic Results.}
\tablehead{\multicolumn{1}{c}{Mask} & \multicolumn{1}{c}{Projected} & \multicolumn{2}{c}{Pointing center:} &
\multicolumn{1}{c}{PA} & \multicolumn{1}{c}{Date of} &
\multicolumn{1}{c}{\#\ Sci.} & \multicolumn{1}{c}{\#\ of M31} \\
& \multicolumn{1}{c}{Radius} & \multicolumn{1}{c}{$\alpha_{\rm J2000}$} &
\multicolumn{1}{c}{$\delta_{\rm J2000}$} & \multicolumn{1}{c}{($^\circ$E of N)}
& \multicolumn{1}{c}{Obs. (UT)} & \multicolumn{1}{c}{targets\tablenotemark{a}} &
\multicolumn{1}{c}{Stars\tablenotemark{a,\rm b}} \\
& \multicolumn{1}{c}{(kpc)} & \multicolumn{1}{c}{($\rm^h$:$\rm^m$:$\rm^s$)} &
\multicolumn{1}{c}{($^\circ$:$'$:$''$)} & &}
\startdata
f109\_1 & 9 & 00:45:46.75 & +40:56:53.8 & 23.90 & 2005 Aug 29 & 208 & 169 \\
H11\_1 & 12 & 00:46:21.02 & +40:41:31.3 & 21.0 & 2004 Sep 20 & 139 & 89 \\
H11\_2 & 12 & 00:46:21.02 & +40:41:31.3 & $-21.0$ & 2004 Sep 20 & 138 & 88 \\
f116\_1 & 13 & 00:46:54.53 & +40:41:29.5 & 22.60 & 2005 Aug 28 & 199 & 149 \\
f115\_1 & 15 & 00:47:32.71 & +40:42:00.9 & $-20.0$ & 2005 Aug 28 & 191 & 114 \\
f123\_1 & 18 & 00:48:05.57 & +40:27:16.3 & $-20.0$ & 2005 Aug 28 & 171 & 104 \\
f135\_1 & 18 & 00:46:24.88 & +40:11:35.5 & $-27.0$ & 2005 Aug 29 & 146 & 99\\
f130\_1 & 22 & 00:49:11.97 & +40:11:45.3 & $-20.0$ & 2005 Aug 28 & 108 & 52 \\
f130\_2 & 23 & 00:49:37.49 & +40:16:07.0 & 90.0 & 2006 Nov 21 & 115 & 43 \\
f130\_3 & 20 & 00:48:34.59 & +40:16:07.0 & 90.0 & 2006 Nov 22 & 124 & 41 \\
a0\_1 & 31 & 00:51:51.32 & +39:50:21.4 & $-17.9$ & 2002 Aug 16 & 89 & 25 \\
a0\_2 & 31 & 00:51:29.59 & +39:44:00.8 & 90.0 & 2002 Oct 12 & 89 & 32 \\
a0\_3 & 29 & 00:51:50.46 & +40:07:00.9 & 0.0 & 2004 Jun 17 & 90 & 26 \\
\enddata
\tablenotetext{a}{A number of targets were observed on two different masks.
Therefore, the total number of unique science targets/M31 RGB stars in fields H11, f130
and a0 is less than the reported number. There are 18 M31 RGB
stars with duplicate observations (2 in H11, 8 in f130, and 8 in a0), thus the
total number of unique M31 RGB stars is 1013.}
\tablenotetext{b}{The number of M31 RGB stars is defined as the number of stars
that are identified as secure and marginal M31 RGB stars by the \citet{gil06}
diagnostic method, wihout the use of the radial velocity diagnostic ($\langle L_i\rangle_{\rm v\!\!\!/}$$>0$, \S\,\ref{sec:sample}).}
\label{tab:masks}
\end{deluxetable*}
\subsubsection{Spectroscopic Data Reduction}\label{sec:dataredux}
The spectra were reduced and analyzed using a modified version of the {\tt spec2d}
and {\tt spec1d} software developed by the DEEP2 team at the University of
California, Berkeley\footnote{{\tt
http://astron.berkeley.edu/$\sim$cooper/deep/spec2d/primer.html},
\newline\indent
{\tt http://astron.berkeley.edu/$\sim$cooper/deep/spec1d/primer.html}
}; these
routines perform standard spectral reduction steps, including flat-fielding,
night-sky emission line removal, and extraction of the two-dimensional spectra.
Reduced one-dimensional spectra are cross-correlated with a library of template
stellar spectra to determine the redshift of the object. Each
spectrum was visually inspected and assigned a quality code
based on the number and quality of absorption lines. Spectra with at least
two spectral features (even if one of them is marginal) are
considered to have secure redshift measurements. A heliocentric correction
is applied to the measured radial velocities based on the sky position of the
mask and the date and time of the observation.
The heliocentric velocities are not corrected for the changing
component of solar motion across our fields; our innermost and outermost
fields are separated by 1.5$^\circ$\ along the southeastern minor axis,
which corresponds to only a 1.6~\,km~s$^{-1}$\ velocity change.
Spectra in field a0 were reduced using the original reduction techniques briefly
outlined above. Detailed discussions of the spectral reduction techniques,
quality determination, and S/N measurements used in our survey can be found in
\citet{guh06} and \citet{gil06}; the typical velocity error in this field
is 15~\,km~s$^{-1}$. Our cross-correlation procedure has since been improved and
is described below. An offset of $+20$~\,km~s$^{-1}$\ has been applied to the
a0 data to make them consistent with the results of this new
cross-correlation procedure.
Spectra in the remainder of the fields were reduced using several
improvements to the reduction pipeline. A greater number of stellar
templates are used for the spectral cross-correlation, and the template
library has been expanded to include spectral templates from the Keck II
telescope's Echellete Spectrograph and Imager (ESI) and DEIMOS in addition to
the existing Sloan Digital Sky Survey (SDSS) spectral templates. The ESI and
DEIMOS templates were included because they more closely match the resolution
of the observed spectra. The position of the atmospheric A-band in the
observed spectrum is used to correct the observed radial velocity for
imperfect centering of the star in the slit \citep[Simon \& Geha 2007, in
prep;][]{soh07}. The improvement to the spectral templates and the A-band
correction allows us to reduce our velocity measurement errors relative to
our previous reductions. The median velocity error for the data presented in
this paper is 4.6~\,km~s$^{-1}$, estimated from the cross-correlation output routine
and confirmed by repeat measurements of individual stars on overlapping DEIMOS
masks.
Spectroscopic data in fields H11 and a0 have been presented in previous
papers \citep{kal06a,bro03,bro06}.
The radial velocity sample for H11 presented in this paper contains
$\approx50$\% more M31 RGB stars than the previously published sample,
due to the recent recovery of spectra from two CCDs that were not included in
these earlier papers and the
improvements in the data reduction process described above.
\section{Selecting a Sample of M31 Red Giants}\label{sec:sample}
The largest source of contaminants in our spectroscopic survey are foreground
MW dwarf stars. Background galaxies are easy to identify and remove
from the sample on the basis of their spectra and redshifts. However, the
radial velocity distribution of MW dwarf stars overlaps that of M31 RGB stars,
making identifying individual stars as M31 red giants or MW dwarfs problematic. We use the diagnostic method detailed in \citet{gil06} to separate M31 RGB
stars from MW dwarf star contaminants. The method uses empirical probability
distribution functions to estimate the likelihood a given star is an M31 red
giant based on five photometric and spectroscopic diagnostics:
\begin{itemize}
\item The radial velocity of the star.
\item
Photometry in the
$M$, $T_2$, and (surface-gravity sensitive) DDO51\ bands
\item The measured equivalent width of the Na\,{\smcap i}\ doublet at 8190\AA\ combined with the $(V-I)_0$\ color of the star.
\item The position of the star in an ($I,\,V-I$)\ color magnitude diagram with respect to theoretical RGB isochrones.
\item A comparison of the star's photometric vs. spectroscopic metallicity estimates.
\end{itemize}
The DDO51\ diagnostic is only used for field a0, which is the only field in the
present work for which DDO51\ photometry is available
(\S\,\ref{sec:phot}). The likelihoods for each diagnostic are
combined in a weighted average for each star to determine the overall
likelihood, $\langle L_i\rangle$, the star is an M31 RGB or MW dwarf star (\S\,A.1).
Based on the overall likelihood, each star is
identified as either a secure M31 RGB star ($\langle L_i\rangle$$>0.5$, or $>3\times$ more likely to be an M31 RGB star than
an MW dwarf) or secure MW dwarf star
($\langle L_i\rangle$$<-0.5$), or a marginal M31 RGB star ($0<$$\langle L_i\rangle$$<0.5$
) or marginal MW dwarf star ($-0.5<$$\langle L_i\rangle$$<0$).
An advantage of the diagnostic method for the present analysis is
the ability to select a sample that is chosen independently of radial velocity.
Since radial velocity is only one of a number of diagnostics which are used to
determine the nature of an individual star,
the
likelihood method
(even with the inclusion of the radial velocity diagnostic)
presents a significant improvement over the use of velocity cuts to select samples for
kinematical analysis by reducing the sensitivity of the
sample to velocity. However, by using the likelihood
method \textit{without} the radial velocity diagnostic (resulting in overall
likelihoods $\langle L_i\rangle_{\rm v\!\!\!/}$), we are able to select
a sample of M31 red giants that is completely \textit{independent} of the radial velocities
of the stars.
Fig. \ref{fig:velhist_bias} presents the radial velocity distribution of stars
selected as M31 red giants based on the diagnostic method with (\textit{shaded histograms}) and
without (\textit{thick open histograms})
radial velocity included, for multiple $\langle L_i\rangle$\ and $\langle L_i\rangle_{\rm v\!\!\!/}$\ thresholds. For reference, the radial velocity distribution of
all stars with successful velocity measurements is also shown
(\textit{thin open histograms}); the MW dwarf star contaminants form the
secondary peak centered at $v_{\rm hel}\approx -50$~\,km~s$^{-1}$.
The M31 RGB
distributions are similar,
with the sample that includes the radial velocity diagnostic showing a systematic
deficiency of stars at radial velocities near 0~\,km~s$^{-1}$.
The effect of the
radial velocity diagnostic on the overall likelihood ($\langle L_i\rangle$\ and $\langle L_i\rangle_{\rm v\!\!\!/}$) distributions is discussed
in \S\,A.1.
\begin{figure}
\plotone{f3.eps}
\caption{
The radial velocity distributions of samples with (\textit{shaded/dotted histograms})
and without (\textit{thick open histograms}) the radial velocity diagnostic included in the
overall likelihood calculation. The radial velocity distribution of all stars
with successful radial velocity measurements is shown for comparison (\textit{thin open histograms}); the MW dwarf star contaminants form the peak at $v_{\rm hel}\approx -50$~\,km~s$^{-1}$. \textit{Top:} Stars designated as secure
([$\langle L_i\rangle$,$\langle L_i\rangle_{\rm v\!\!\!/}$]$>0.5$) M31 red giants only. \textit{Middle:} Stars designated as marginal and
secure M31 red giants ([$\langle L_i\rangle$,$\langle L_i\rangle_{\rm v\!\!\!/}$]$>0$). \textit{Bottom:} Stars
designated as marginal MW dwarfs, marginal M31 red giants, and secure M31 red giants ([$\langle L_i\rangle$,$\langle L_i\rangle_{\rm v\!\!\!/}$]$>-0.5$).
The distributions are similar, but there is a deficiency of
stars at velocities near 0~\,km~s$^{-1}$\ in the sample that includes radial
velocity as a diagnostic.
}
\label{fig:velhist_bias}
\end{figure}
The M31 RGB samples identified by their $\langle L_i\rangle$\ values
have a minimal amount of MW dwarf star contamination, but are
also kinematically biased against stars with velocities near 0~\,km~s$^{-1}$\
(\S\,A.1). The M31 RGB samples identified by their $\langle L_i\rangle_{\rm v\!\!\!/}$\ values
have a slightly larger amount of dwarf
contamination (particularly evident in the bottom panel of Figure~\ref{fig:velhist_bias}),
but the underlying M31 RGB population is kinematically unbiased.
The RGB sample used in this paper is defined as stars that are designated
as secure and marginal M31 red giants by the diagnostic
method, with the radial velocity likelihood \textit{not} included in the
calculation of a star's overall likelihood of being an M31 RGB star (i.e., $\langle L_i\rangle_{\rm v\!\!\!/}$$>0$).
The number of M31 RGB stars in each field is listed in Table~\ref{tab:masks}.
The $\langle L_i\rangle_{\rm v\!\!\!/}$$>0$ threshold maximizes the completeness of the underlying, \textit{kinematically
unbiased} M31 RGB population,
but introduces an overall MW dwarf star contamination of 5\% to the sample
(\S\,A.2). The contamination is largely constrained to $v_{\rm hel}>-150$~\,km~s$^{-1}$\ due to the velocity distribution of MW dwarf stars, and its effect
on the measured parameters of the M31 RGB sample is quantified in
\S\,A.3.
\section{Stellar Kinematics in M31's Southeast Minor-Axis Fields}\label{sec:kin}
The data presented in this paper span a range in projected radial distance
from the center of M31 of 9 to 30~kpc, along the southeastern minor axis. We
refer to this region as the ``inner spheroid,'' even though it has
traditionally been referred to as the ``halo'' of M31.
This region departs from the classical picture of a stellar halo formed from
observations of the
Milky Way: M31's inner spheroid is about $10\times$ more metal-rich than the
MW's halo \citep{mou86,dur04} and follows
a de~Vaucouleurs $r^{1/4}$ surface density profile \citep{pri94,dur04},
while the surface density profile of the MW halo follows an $r^{-2}$ power law. In other words,
M31's inner spheroid appears to be a continuation of its
central bulge. Recent observations have discovered an outer stellar ``halo''
in M31 which is relatively metal-poor \citep{kal06b, cha06}, has a surface density
profile that follows an $\sim r^{-2}$ power law \citep{guh05, irw05}, and has
been detected out to $R_{\rm proj}=165$~kpc \citep{gil06}.
These observations
imply that the
spheroid of M31 has two components: an inner, de~Vaucouleurs profile spheroid, and an outer, power-law profile halo.
We thus use the term \textit{inner spheroid} to distinguish the region
$R_{\rm proj}\sim 9$\,--\,30~kpc from the canonical central
bulge and the newly discovered stellar halo of M31. The outer limit of this
region is well-defined;
a break in the surface brightness profile of M31 RGB stars has been observed at
$R_{\rm proj}\sim 20-30$~kpc \citep{guh05,irw05}, and there is observational
evidence that the crossover between the
predominantly metal-rich population of the inner spheroid and the
predominantly metal-poor population of the outer halo occurs at
$R_{\rm proj}\sim 30$~kpc \citep{kal06b}. The inner limit of this
region is arbitrary, as the relationship between this component and
the central bulge of M31 is not yet clear.
The rest of this section characterizes the line of sight velocity
distribution of stars in the inner spheroid of M31 through
maximum-likelihood fits of Gaussians to the combined data set
(\S\,\ref{sec:anal_comb}) and to individual fields (\S\,\ref{sec:anal_ind}).
Gaussians provide a convenient means of fitting
for multiple kinematical components in the data and characterizing their
mean velocity and velocity dispersion. The true shape of the velocity
distribution of a structural component in M31 is likely to be different
from a pure Gaussian. However, given the limited sample size and the absence
of any specific physical model, the choice of Gaussians seems appropriate.
\subsection{Maximum-Likelihood Fits to the Velocity Distribution of the Combined Data Set}\label{sec:anal_comb}
\begin{figure}
\plotone{f4.eps}
\caption{
The radial velocity distribution of the M31 RGB inner spheroid population.
A maximum-likelihood analysis was used to fit an analytic function to
the distribution. (\textit{a}) The best fit single Gaussian has parameters $\langle v\rangle^{\rm sph}$$=-287$~\,km~s$^{-1}$
and $\sigma^{\rm sph}_{\rm v}$=117~\,km~s$^{-1}$. A $\chi^2_{\rm red}$ test rules out the single-Gaussian fit at a very high
confidence level. (\textit{b}) The best constrained double-Gaussian fit
(Table~\ref{table:rvfits}) is shown as a solid curve, with the narrow component
and wide components displayed separately as dotted and dashed curves, respectively.
}
\label{fig:vel_all}
\end{figure}
Figure~\ref{fig:vel_all} shows the combined radial velocity distribution of
M31 RGB stars from all eight fields along the minor axis, ranging from 9
to 30~kpc in projected radial distance from the center of M31.
Fits to the radial velocity distribution were made using a maximum-likelihood
technique; the best-fit single (\textit{a}) and double (\textit{b})
Gaussians are displayed in Figure~\ref{fig:vel_all}.
A reduced $\chi^2$ analysis rules out the single-Gaussian fit, as the probability
is $<\!\!<\!\!1\%$ that the observed radial velocities were drawn from the best-fit
distribution. The observed velocity distribution is well fit
by a sum of two Gaussians (solid curve, panel \textit{b} of Fig.~\ref{fig:vel_all}), composed of a wide Gaussian (dashed curve) centered at $\langle v\rangle^{\rm sph}$=
$-287.2^{+8.0}_{-7.7}$~\,km~s$^{-1}$, with a width of
$\sigma^{\rm sph}_{\rm v}$$=128.9^{+7.7}_{-6.9}$~\,km~s$^{-1}$, and a narrow Gaussian (dotted curve)
centered at $\langle v\rangle^{\rm sub}$=$-285.4^{+12.8}_{-12.4}$~\,km~s$^{-1}$\ with a width of
$\sigma^{\rm sub}_v$=$42.2^{+12.5}_{-14.3}$~\,km~s$^{-1}$, which comprises $19^{+9}_{-8}$\% of
the total population (quoted errors represent the 90\% confidence
limits from the maximum-likelihood analysis). Due to the
MW dwarf star contaminants in the M31 RGB sample (\S\,\ref{sec:sample}), the true
$\langle v\rangle^{\rm sph}$\ value of the wide Gaussian component is 15 to 20~\,km~s$^{-1}$\ more
negative than the best-fit value (\S\,A.3), making it consistent with
the systemic velocity of M31 ($v_{\rm sys}=-300$~\,km~s$^{-1}$).
The kinematically hot component
corresponds to the inner spheroid of M31 (quantities related to this component are
denoted with the subscript ``sph''), while the kinematically cold
component corresponds to substructure in the inner spheroid (denoted with
the subscript ``sub''); the discussion
of these components
and their properties will be deferred to \S\,\ref{sec:veldisp} and
\S\,\ref{sec:coldpop}, respectively. The wider of the two Gaussian
components in the double Gaussian fit to the combined data set is hereafter
referred to as $G^{\rm sph}(v)$.
Figure~\ref{fig:errors_all} shows error estimates from the maximum-likelihood
analysis (in the form
of $\Delta\chi^2 \equiv \chi^2 - \chi_{\rm min}^2$ curves) for the five
double-Gaussian parameters. The best-fit value of each parameter is
marked as well as the 90\% confidence limits.
The $\Delta\chi^2$ curves have a deep minimum for all five parameters, an
indication that the double-Gaussian model is a good description of
the observed radial velocity distribution of inner spheroid
stars.
\begin{figure}
\plotone{f5.eps}
\caption{
Results of the maximum-likelihood analysis for the double-Gaussian
fit to the combined M31 RGB inner spheroid sample. The optimal value of
each parameter is marked with an arrow, and the 90\% confidence limits from
the maximum-likelihood analysis are
marked with dashed lines. The upper limit of the y-axis represents the
99\% confidence limits. The parameter $\Delta\chi^2\equiv \chi^2-\chi_{\rm min}^2$
is plotted as a function of
(\textit{a}) mean velocity of the cold substructure component (narrow Gaussian),
$\langle v\rangle^{\rm sub}$, (\textit{b}) velocity dispersion of the cold component, $\sigma^{\rm sub}_v$,
(\textit{c}) mean velocity of the M31 inner spheroid (wide Gaussian), $\langle v\rangle^{\rm sph}$,
(\textit{d}) velocity dispersion of the M31 inner spheroid, $\sigma^{\rm sph}_{\rm v}$, and
(\textit{e}) fraction of the total M31 RGB population in the cold component,
$N_{\rm sub}/N_{\rm tot}$.
The horizontal arrow in panel (\textit{c})
represents the correction to the $\langle v\rangle^{\rm sph}$\ value necessary to offset the
bias caused by MW dwarf contamination at $v_{\rm hel}>-150$~\,km~s$^{-1}$\ (\S\,A.3).
}
\label{fig:errors_all}
\end{figure}
\begin{deluxetable*}{lrrrrr}
\tablecolumns{6}
\tablewidth{0pc}
\tablecaption{Radial Velocity Distributions: Best Fit Gaussian Parameters.}
\tablehead{\multicolumn{1}{c}{Field} & \multicolumn{5}{c}{Best fit Gaussian Parameters\tablenotemark{a}} \\
& \multicolumn{2}{c}{Cold Component} & \multicolumn{2}{c}{Hot Spheroid~\tablenotemark{b}}
& \multicolumn{1}{c}{Fraction}\\
& \multicolumn{1}{c}{$\langle v\rangle^{\rm sub}$} & \multicolumn{1}{c}{$\sigma^{\rm sub}_v$}
& \multicolumn{1}{c}{$\langle v\rangle^{\rm sph}$} & \multicolumn{1}{c}{$\sigma^{\rm sph}_{\rm v}$}
& \multicolumn{1}{c}{$N_{\rm sub}/N_{\rm tot}$} }
\startdata
All fields & $-285.4^{+12.8}_{-12.4}$ & $42.2^{+12.5}_{-14.3}$ & $-287.2^{+8.0}_{-7.7}$ & $128.9^{+7.7}_{-6.9}$ & $0.19^{+0.09}_{-0.08}$ \\
f109 & ... & ... & $-274.5^{+15.4}_{-15.3}$ & $120.7^{+11.7}_{-10.1}$ & ...\\
H11 & $-294.3^{+17.3}_{-17.6}$ & $55.5^{+15.6}_{-12.7}$ & $-287.2$ & 128.9 & $0.44^{+0.16}_{-0.16}$\\
f116 & $-309.4^{+19.2}_{-17.5}$ & $51.2^{+24.4}_{-15.0}$ & $-287.2$ & 128.9 & $0.44^{+0.22}_{-0.17}$ \\
f115 & ... & ... & $-270.9^{+18.6}_{-18.6}$ & $120.1^{+14.4}_{-12.0}$ & ... \\
f123 & $-279.4^{+5.1}_{-4.6}$ & \tablenotemark{c}~$10.6^{+6.9}_{-5.0}$ & $-287.2$ & 128.9 & $0.31^{+0.11}_{-0.11}$ \\
f135 & ... & ... & $-315.1^{+21.3}_{-21.3}$ & $127.8^{+16.5}_{-13.6}$ & ...\\
f130 & ... & ... & $-259.8^{+19.3}_{-19.2}$ & $131.5^{+14.8}_{-12.5}$ & ...\\
a0 & ... & ... & $-299.2^{+25.2}_{-25.2}$ & $131.5^{+29.5}_{-21.6}$ & ...\\
\enddata
\tablenotetext{a}{A double Gaussian fit is presented for the combined
sample, as it is a poor fit to a single Gaussian (\S\,\ref{sec:anal_comb}).
Constrained double Gaussian fits are presented for three of the fields
(H11, f116, and f123), with the wider component held fixed (adopting
the fit to the combined sample). Single Gaussian fits are presented
for the remaining five fields. The reader is referred to
\S\,\ref{sec:anal_ind} for details of the fits to individual fields.
Errors quoted
represent the 90\% confidence limits from the maximum-likelihood analysis.}
\tablenotetext{b}{The M31 RGB sample used in this analysis was chosen to ensure a high degree of completeness, and thus suffers from some MW dwarf contamination (\S\,A.2). The MW dwarf star contaminants are largely at $v_{\rm hel}>-150$~\,km~s$^{-1}$, and cause the best-fit $\langle v\rangle^{\rm sph}$\ values to be biased towards more positive velocities. The true $\langle v\rangle^{\rm sph}$\ values of the M31 RGB population are 15 to 20~\,km~s$^{-1}$\ more negative than listed here. The effect on $\sigma^{\rm sph}_{\rm v}$\ is negligible (\S\,A.3).}
\tablenotetext{c}{The median velocity error for the stars in f123 is 4.6~\,km~s$^{-1}$. The estimated intrinsic velocity dispersion of the cold component in f123 after accounting for velocity measurement error is 9.5~\,km~s$^{-1}$.}
\label{table:rvfits}
\end{deluxetable*}
\subsection{Maximum-Likelihood Fits to the Velocity Distributions of Individual Fields}\label{sec:anal_ind}
As discussed in the previous section, the combined data set shows definite
evidence of a kinematically cold component centered at about $-300$~\,km~s$^{-1}$\
that comprises a significant fraction (19\%) of the total population. We next
investigate which of the fields in our data set are the main contributors to
this cold population. Figure~\ref{fig:velhist} shows
velocity histograms for each of the eight fields analyzed in this paper.
While most of the fields display hints of substructure--- in the form of one
or more small possible peaks in their velocity distribution that may be
marginally significant relative to the (substantial) Poisson noise--- we are
specifically interested in judging each field's contribution to the cold
component at $\sim-300$~\,km~s$^{-1}$. For this purpose, we compare the data in each
field to three sets of Gaussian fits. We describe each of the fits below,
summarize the results of the fits, and then list the quantitative details of
the fits for each field.
\begin{figure}
\plotone{f6.eps}
\caption{
Velocity histograms for each of the individual fields with best-fit
Gaussians overlaid. Fields that did not show clear evidence of substructure
(f109, f115, f130, f135 and a0, \S\,\ref{sec:anal_ind}) are shown with
the Gaussian component
from the double-Gaussian fit to the
full sample ($G^{\rm sph}(v)$, \textit{solid curves}), as well as the best-fit single Gaussian to
the individual field (\textit{dotted curves}).
Fields with evidence of substructure (H11, f116,
f123), are shown with both their best-fit single (\textit{dotted curves})
and double (\textit{solid curves}) Gaussians. For the constrained double-Gaussian
fits, both the narrow and broad ($G^{\rm sph}(v)$) Gaussian components are
shown (\textit{dashed curves}) scaled to their relative contributions.
}
\label{fig:velhist}
\end{figure}
First, the radial velocity distribution in each field is compared to the
Gaussian $G^{\rm sph}(v)$\ defined by $\langle v\rangle^{\rm sph}$$=-287.2$~\,km~s$^{-1}$,
$\sigma^{\rm sph}_{\rm v}$$=128.9$~\,km~s$^{-1}$\ (\S\,~\ref{sec:anal_comb}) using the reduced $\chi^2$
statistic ($\chi^2_{\rm red}$). Fields f109, f115, and a0 are consistent with being drawn from
$G^{\rm sph}(v)$, and so are ruled out as significant
contributors to the $\sim-300$~\,km~s$^{-1}$\ cold component.
The rest of the fields are at least marginally inconsistent with being drawn
from $G^{\rm sph}(v)$.
Second, a maximum-likelihood single-Gaussian
fit is performed on the radial velocity distribution in each field and
compared to the data.
Fields f123 and f135 are inconsistent with their respective best-fit single
Gaussians (based on the $\chi^2_{\rm red}$ statistic) and are therefore
suspected to contain substructure. The remaining fields (f109, H11, f116,
f115, f130, and a0) are consistent with their best-fit single Gaussians. For
fields f109, f115, f130, and a0, the best-fit Gaussians are consistent with
$G^{\rm sph}(v)$, and they are thus ruled out as significant contributors to the
$\sim-300$~\,km~s$^{-1}$\ cold component. Fields f116 and H11 are suspected to
contain substructure because
their best-fit Gaussians are significantly narrower than $G^{\rm sph}(v)$. Previous
kinematic studies of M31's inner spheroid, including this one, have found a
ubiquitously broad distribution of radial velocities ($v_{\rm hel}\approx0$
to $\approx-600$~\,km~s$^{-1}$; \S\,~\ref{sec:veldisp}).
Thus, the anomalously narrow single Gaussian fits in
fields f116 and H11 cause us to suspect them of being contributors to the
$\sim-300$~\,km~s$^{-1}$\ cold component.
Third, we carry out a maximum-likelihood fit to all fields using a
constrained double Gaussian, with $G^{\rm sph}(v)$\ as the fixed wide Gaussian component.
Fields H11, f116, and f123 are well fit by a constrained double Gaussian
(based on the $\chi^2_{\rm red}$ statistic and well-defined minima for the
variable Gaussian parameters). These three fields are significant
contributors to the $\sim-300$~\,km~s$^{-1}$\ cold component, and the constrained
double-Gaussian fit is adopted as the preferred fit in the subsequent
discussion. The best-fit cold component in the constrained double-Gaussian
fit to f130 is centered at $\sim-50$~\,km~s$^{-1}$, but this likely represents
residual contamination by MW dwarf stars (\S\,\ref{sec:sample},
\S\,A). The remaining fields (f109, f115,
f135, and a0) are poor fits to a constrained double-Gaussian
in that the Gaussian parameters do not have well-defined minima.
In summary:
\begin{itemize}
\item{Three fields, H11, f116, and f123 are identified as significant
contributors to the $\sim-300$~\,km~s$^{-1}$\ cold component.}
\item{Although field f135 shows evidence of substructure, it is a poor fit
to a constrained double-Gaussian and is not a definite contributor to
the $\sim-300$~\,km~s$^{-1}$\ cold component. (However, the fit may be confused by
the presence of multiple cold components; \S\,\ref{sec:f135}.)}
\item{Four fields, f109, f115, f130, and a0, are not significant
contributors to the $\sim-300$~\,km~s$^{-1}$\ cold component.}
\end{itemize}
Each field is discussed individually below, and Table~\ref{table:rvfits}
summarizes the preferred fits (single or double-Gaussian) to the velocity
distributions in each field.
\noindent\textit{Field f109}: The data in this field are consistent with being drawn
from the Gaussian $G^{\rm sph}(v)$\
(\S\,\ref{sec:anal_comb}). The best-fit single Gaussian to the data in
this field has parameters $\langle v\rangle$$=-274.5$~\,km~s$^{-1}$\ and $\sigma=120.7$~\,km~s$^{-1}$, and the data
are also consistent with being drawn from this distribution.
\noindent\textit{Field H11}: The probability the data in this field are drawn
from the Gaussian $G^{\rm sph}(v)$\
is $P<1$\%, thus the data are inconsistent with this distribution.
The best-fit single Gaussian to the data in
this field has parameters $\langle v\rangle$$=-291.1\pm 11.6$~\,km~s$^{-1}$\ and $\sigma=106.2^{+8.7}_{-7.7}$~\,km~s$^{-1}$, and the data
are consistent with being drawn from this distribution. However, the best-fit value
of $\sigma$ in this field is anomalously low compared to the value of
$\sigma^{\rm sph}_{\rm v}$\ determined from the double-Gaussian fit to the combined data set:
the two values are inconsistent at the $\sim 3.5\sigma$ level. This
suggests that there is a kinematically cold population in this field,
and a comparison of the data to the constrained double-Gaussian fit to
this field
($\langle v\rangle^{\rm sub}$$=-294.3$~\,km~s$^{-1}$, $\sigma^{\rm sub}_v$$=55.5$~\,km~s$^{-1}$, $N_{\rm sub}/N_{\rm tot}=0.44$)
returns a $\chi^2_{\rm red}$
that is significantly smaller than the $\chi^2_{\rm red}$ of the single
Gaussian fit.
\noindent\textit{Field f116}: The data in this field are inconsistent
with being drawn from the Gaussian $G^{\rm sph}(v)$. The best-fit single
Gaussian to the data in
this field has parameters $\langle v\rangle$$=-292.9^{+11.5}_{-11.6}$~\,km~s$^{-1}$\ and $\sigma=97^{+8.7}_{-7.5}$~\,km~s$^{-1}$, and the data
are consistent with being drawn from this distribution. As in field H11, the
best-fit value of $\sigma$ is inconsistent with $\sigma^{\rm sph}_{\rm v}$\ from the combined
data set, at the $\sim 4.2\sigma$ level. A comparison of the data to the
constrained double-Gaussian
fit to this field ($\langle v\rangle^{\rm sub}$$=-309.4$~\,km~s$^{-1}$, $\sigma^{\rm sub}_v$$=51.2$~\,km~s$^{-1}$,
$N_{\rm sub}/N_{\rm tot}=0.44$) returns a significantly smaller $\chi^2_{\rm red}$
than that of the single-Gaussian fit.
\noindent\textit{Field f115}: The data in this field are consistent with being drawn
from the Gaussian $G^{\rm sph}(v)$, as well as the best-fit single Gaussian
($\langle v\rangle^{\rm sph}$$=-270.9$~\,km~s$^{-1}$, $\sigma^{\rm sph}_{\rm v}$$=120.1$~\,km~s$^{-1}$).
\noindent\textit{Field f123}: The probability that the data in this field are
drawn from the Gaussian $G^{\rm sph}(v)$\ is $P<\!\!< 1$\%.
The probability that the data are drawn
from the best-fit single Gaussian ($\langle v\rangle^{\rm sph}$$=-270.9$~\,km~s$^{-1}$, $\sigma^{\rm sph}_{\rm v}$$=120.1$~\,km~s$^{-1}$) in
this field is also $<\!\!< 1$\%. The data in this field are strongly inconsistent with being drawn from any single Gaussian, but they
are consistent with the constrained double-Gaussian fit ($\langle v\rangle^{\rm sub}$$=-279.4$~\,km~s$^{-1}$, $\sigma^{\rm sub}_v$$=10.6$~\,km~s$^{-1}$, $N_{\rm sub}/N_{\rm tot}=0.31$). The median velocity error for the stars in field f123 is 4.6~\,km~s$^{-1}$, therefore the intrinsic velocity dispersion of the cold component in this field is estimated to be 9.5~\,km~s$^{-1}$.
\noindent\textit{Field f135}: The data in this field are inconsistent with being drawn
from the Gaussian $G^{\rm sph}(v)$\ ($P\sim 1$\%). The best-fit single Gaussian to the data in
this field has parameters $\langle v\rangle^{\rm sph}$$=-315.1$~\,km~s$^{-1}$\ and $\sigma^{\rm sph}_{\rm v}$$=127.8$~\,km~s$^{-1}$, but the data
are also inconsistent with being drawn from this distribution ($P\lesssim1$\%).
However, the maximum-likelihood analysis was unable to constrain a double-Gaussian
fit to any reasonable degree of certainty--- i.e., the error estimates on the
parameters show no strong global minima. Field f135 is therefore treated
as a field without a {\it definite\/} detection of substructure, although in
\S\,\ref{sec:f135} we discuss the possible presence of multiple
kinematically-cold components in this field.
\noindent\textit{Field f130}: The data in this field are inconsistent with being drawn
from the Gaussian $G^{\rm sph}(v)$\ ($P\lesssim 1$\%), but are consistent with being drawn
from the best-fit single Gaussian ($\langle v\rangle^{\rm sph}$$=-259.8$~\,km~s$^{-1}$,
$\sigma^{\rm sph}_{\rm v}$$=131.5$~\,km~s$^{-1}$). The best-fit single Gaussian for this field has
parameters which are consistent with $G^{\rm sph}(v)$\ (Table~\ref{table:rvfits}).
\noindent\textit{Field a0}: The data in this field are consistent with
being drawn from the Gaussian $G^{\rm sph}(v)$\, as well as the
best-fit single Gaussian ($\langle v\rangle^{\rm sph}$$=-299.2$~\,km~s$^{-1}$, $\sigma^{\rm sph}_{\rm v}$$=131.5$~\,km~s$^{-1}$).
\bigskip
We now have a characterization of the kinematical properties of the
combined data set and the individual fields, and have identified the fields
which significantly contribute to the $\sim-300$~\,km~s$^{-1}$\ cold component discovered
in the combined data set (H11, f116, and f123). The number of stars in fields
H11, f116, and f123 which are associated with the cold component comprise
17\% of the total number of M31 RGB stars in the sample, confirming that
these fields are the primary contributors to the cold component.
Next, we discuss the trends in the properties of
the dynamically hot and cold populations (\S\,\ref{sec:veldisp}
and \S\,\ref{sec:coldpop}, respectively), followed by possible
physical interpretations of
the cold component (\S\,\ref{sec:origin}).
\section{Velocity Dispersion of M31's Virialized Inner Spheroid}\label{sec:veldisp}
As discussed in \S\,\ref{sec:kin}, the radial velocity distribution of the
combined data set contains a significant cold component at $v_{\rm hel}\sim -300$~\,km~s$^{-1}$. An analysis of the
kinematical profile of the individual fields shows that this is due to the
presence of a significant amount of substructure in three of the fields (\S\,\ref{sec:anal_ind}).
The kinematically hot component in the double
Gaussian fit to the combined data set presumably represents the
underlying virialized inner spheroid of M31.
The velocity dispersion of the spheroid based on a maximum-likelihood double
Gaussian fit to the
combined data set (1013 M31 RGB stars) is $\sigma^{\rm sph}_{\rm v}$ = 128.9~\,km~s$^{-1}$.
We have combined all of the fields to get a more robust estimate of the
velocity dispersion of the inner spheroid, since the individual fields suffer
from small number statistics. However, the dynamical quantities
($\langle v\rangle^{\rm sph}$, $\sigma^{\rm sph}_{\rm v}$) may have radial dependencies. The velocity
dispersion of spheroids is expected to decrease with increasing radius
\citep[e.g.,][]{nav96,nav04,mam05,dek05}. Rotation of the spheroid or
the tangential motion of M31 could result in a radial dependency of $\langle v\rangle^{\rm sph}$,
but since our fields are mostly aligned along the minor axis and cover a
relatively small radial range, we do not expect to be sensitive to these
effects.
To test for a dependency of the dynamical quantities on radius, we analyze the single
Gaussian fits to the velocity distributions in each of the five fields which
do not show
clear evidence of substructure as described in \S\,\ref{sec:anal_ind}
(Table~\ref{table:rvfits}). The $\Delta\chi^2$
error estimates for the best-fit single Gaussian parameters in these fields are shown in
Figure~\ref{fig:errors_sg}.
The best-fit $\langle v\rangle^{\rm sph}$\ values in each field are largely consistent with
the systemic velocity of M31 ($v_{\rm sys}=-300$~\,km~s$^{-1}$) once the effect of
MW dwarf star contamination is taken into account,
with the exception of f135, whose
$\langle v\rangle^{\rm sph}$\ is significantly more negative.
Our data show no evidence for a decreasing $\sigma^{\rm sph}_{\rm v}$\ with increasing radius.
However, the fields discussed in this paper only span a range
in projected radial distance of $R_{\rm proj}\sim9$ to 30~kpc. This is small
compared to the size of the total M31 spheroid (inner spheroid and outer halo), which has been shown
to extend to 165 kpc \citep{gil06}.
\citet{bat05} use a sample of 240 Galactic halo objects to determine the
radial velocity dispersion of the Milky Way halo, and find that it has an
almost constant value of 120~\,km~s$^{-1}$\ out to 30 kpc, beyond which it decreases
with increasing radial distance, declining to 50~\,km~s$^{-1}$\ at 120 kpc.
\begin{figure}
\plotone{f7.eps}
\caption{
Results of the maximum-likelihood analysis for the
single-Gaussian fits to individual
fields that do not show clear evidence of the $\sim -300$~\,km~s$^{-1}$\ cold component in
their radial velocity distributions (\S\,\ref{sec:anal_ind}).
The panels show $\Delta\chi^2$ for the mean velocity $\langle v\rangle^{\rm sph}$\ (\textit{left}) and
velocity dispersion $\sigma^{\rm sph}_{\rm v}$\
(\textit{right}) of M31 RGB stars in fields (\textit{a}) f109, (\textit{b})
f115, (\textit{c}) f135, (\textit{d}) f130, and (\textit{e}) a0. As in
Figure~\ref{fig:errors_all}, the optimal values of $\langle v\rangle^{\rm sph}$\
and $\sigma^{\rm sph}_{\rm v}$\ are marked by arrows, and the 90\% confidence limits are shown
as dashed lines.
The horizontal arrow in each of the left panels
represents the correction (magnitude and direction) to the $\langle v\rangle^{\rm sph}$\ value necessary to offset the
bias caused by MW dwarf contamination at $v_{\rm hel}>-150$~\,km~s$^{-1}$\ (\S\,A.3).
}
\label{fig:errors_sg}
\end{figure}
There have been a few previous
measurements of the velocity dispersion based on M31 spheroid stars.
The closest analog to the present study is \citet{rei02}, who fit 80
candidate M31 RGB stars at $R_{\rm proj}=19$ kpc on the southeastern minor axis
with a combination of the
Galactic standard model and a wide Gaussian. They found $\sigma^{\rm sph}_{\rm v}$
$\sim 150$~\,km~s$^{-1}$\ for the M31 velocity dispersion, with the number of
M31 RGB stars estimated to be 43\% of the population.
\citet{guh06} studied a field on the giant
southern stream at $R_{\rm proj}=33$ kpc and found a velocity dispersion
of $\sigma^{\rm sph}_{\rm v}$$=65^{+32}_{-21}$~\,km~s$^{-1}$\ for the underlying spheroid
based on a sample of $\approx21$
stars. However, if 3 likely RGB stars with $v_{\rm hel}>-150$~\,km~s$^{-1}$\ are included,
the estimated velocity dispersion in this field increases to $\sigma^{\rm sph}_{\rm v}$$=116^{+31}_{-22}$~\,km~s$^{-1}$.
\citet{cha06} have measured a mean velocity dispersion of $\sigma^{\rm sph}_{\rm v}$$=126$~\,km~s$^{-1}$\
for the inner spheroid
of M31 using $\sim 800$ RGB stars in multiple fields
surrounding M31. They determine that
the spheroid has a central velocity dispersion of 152~\,km~s$^{-1}$\ which decreases by
$-0.9$~\,km~s$^{-1}$~kpc$^{-1}$ out to $R_{\rm proj}\sim 70$~kpc. Many of their fields are near
M31's major axis and have significant contamination from the extended
rotating component identified as the extended disk in \citet{iba05}.
They isolate a sample of
M31 spheroid stars by removing all stars within 160~\,km~s$^{-1}$\ of the disk
velocity in each field, and by
removing all stars with $v_{\rm hel} > -160$~\,km~s$^{-1}$\ (MW dwarf star contaminants).
This ``windowing'' technique leaves them with a sample of M31 spheroid stars that is
significantly incomplete, but is largely uncontaminated by M31's extended disk or MW dwarf stars.
Based on the \citet{cha06} result, we would expect to measure a velocity
dispersion of 146~\,km~s$^{-1}$\ in our innermost field, f109, which would decrease to
a velocity dispersion of 125~\,km~s$^{-1}$\ in our outermost field, a0. The predicted
velocity dispersion of 146~\,km~s$^{-1}$\ in field f109 exceeds the 90\% confidence
limits of the maximum-likelihood fit and is just within the 99\% confidence limit,
and the data presented in this paper show no evidence of a strong trend
in $\sigma^{\rm sph}_{\rm v}$\ with radius.
The previous measurements of the stellar velocity
dispersion of the M31 spheroid
relied either on samples that were chosen on the basis of radial velocity cuts
or on a statistical
fit to the combined M31 RGB and MW dwarf populations. Our measurement
of the velocity dispersion
of M31's spheroid is unique in that it is based on a sample of spectroscopically confirmed
M31 RGB stars that were chosen
\textit{without} the use of radial velocity (\S\,\ref{sec:sample}).
Our method also allows us
to quantify and correct for the effect of MW dwarf star contamination (\S\,A.2 and \S\,A.3).
\section{Properties of the Minor-Axis Substructure}\label{sec:coldpop}
\subsection{Spatial Trends: Kinematics and Structure}\label{sec:spatdist}
As discussed in \S\,\ref{sec:anal_ind}, fields H11, f116, and f123 show
evidence of the cold component at $\sim-300$~\,km~s$^{-1}$\ in their radial velocity
distributions
(Fig.~\ref{fig:velhist}), and are well fit by a sum of two Gaussians with
$G^{\rm sph}(v)$\ as the fixed wide Gaussian component (Table~\ref{table:rvfits}). The
$\Delta\chi^2$ error estimates for the free parameters
($\langle v\rangle^{\rm sub}$, $\sigma^{\rm sub}_v$, and $N_{\rm sub}/N_{\rm tot}$) are shown in
Figure~\ref{fig:errors_sub} for each field.
\begin{figure}
\plotone{f8.eps}
\caption{
Results of the maximum-likelihood analysis for the
narrow Gaussian parameters
from the constrained double-Gaussian fits to fields (\textit{a}) H11,
(\textit{b}) f116, and (\textit{c}) f123.
The panels show $\Delta\chi^2$ as a function of (\textit{left})
mean velocity $\langle v\rangle^{\rm sub}$, (\textit{middle}) velocity dispersion $\sigma^{\rm sub}_v$,
and (\textit{right}) fraction of stars in the cold component, $N_{\rm sub}/N_{\rm tot}$.
The parameters of the wide Gaussian component in these fits have been fixed at
the values of the Gaussian $G^{\rm sph}(v)$\ (\S\,\ref{sec:anal_comb}).
As in Figure~\ref{fig:errors_all}, the optimal values of each parameter
are marked by arrows, and the 90\% confidence limits are shown
as dashed lines. The velocity dispersion decreases with increasing radial
distance from the center of M31.
}
\label{fig:errors_sub}
\end{figure}
If the three fields are considered together, a pattern emerges. Both the
velocity dispersion and the fraction of stars in the cold component
decrease with increasing radial distance. The cold component in fields H11
and f116 (at $R_{\rm proj}=12$ and 13 kpc, respectively) is significantly
wider than the cold component in field f123 ($R_{\rm proj}=18$ kpc).
The cold component also appears to be more dominant by number (surface
brightness) over the hot component in fields H11 and f116
than in field f123.
Figure~\ref{fig:spatdist} presents the velocities of the M31 RGB stars as a
function of their distance along the major and minor axes of M31. The cold
component can be seen as a triangular-shaped feature that narrows to
a sharp point as the distance along the minor axis increases. The fields that
overlap the triangular-shaped feature in minor axis distance
are (in order of increasing distance along the minor axis) H11, f116, f115,
f135, and f123. Stars in fields
H11, f116 and f123 that are within the
area denoted by the dotted line are shown as red crosses in the bottom panel.
In the fields
in which it is observed, the cold component is spread evenly along the
direction of the major axis, and is centered at $v_{\rm hel}$$ \sim -285$~\,km~s$^{-1}$.
The majority of the fields overlap in position along the major axis; the
exception is field f135, which is the isolated set of points at
large major axis distances
($-0.51$$^\circ$\ to $-0.36$$^\circ$, or $-7.0$~kpc to $-4.9$~kpc) in the bottom panel of Figure~\ref{fig:spatdist}.
Field f123 extends from $-1.18$$^\circ$\
to $-1.39$$^\circ$\ (16.1~kpc to 19.0~kpc) along the minor axis; the tip of the feature is seen in this field.
This field also has the coldest substructure in the radial velocity
histograms (Fig.~\ref{fig:velhist}, Table~\ref{table:rvfits}). Field H11
brackets the other edge of the feature along the minor axis. Field H11 has a range in minor
axis distance of $-0.78$$^\circ$\ to $-0.99$$^\circ$\ (10.7~kpc to 13.5~kpc), and has a large degree of
overlap with field f116 in both
their minor and major axis distance ranges. The fits to the cold component in
these two fields return similar $\sigma^{\rm sub}_v$\ and $N_{\rm sub}/N_{\rm tot}$ estimates
(Table~\ref{table:rvfits}).
\begin{figure}
\plotone{f9.eps}
\caption{
Distribution of M31 RGB stars in velocity vs. distance along the minor
(\textit{top}) and major (\textit{bottom}) axes
of M31. The range of minor (major) axis distances
of stars in each field are shown in the top (bottom) panel. \textit{Top:} The
cold component is visible as a triangular feature that starts
at $-0.8$$^\circ$\ (10.9~kpc) and narrows to a point at $-1.35$$^\circ$\ (18.4~kpc) along the minor axis.
This feature is outlined in red (\textit{dotted line}). \textit{Bottom:}
Stars in fields H11, f116, and f123 that fall within the
triangular outline in the top panel are colored red. In the fields in
which the $\sim-300$~\,km~s$^{-1}$\ cold component is present, it is spread evenly
as a function of projected distance from the minor axis.
}
\label{fig:spatdist}
\end{figure}
Of the fields in which substructure was not clearly detected kinematically
(\S\,\ref{sec:anal_ind}), fields f135 and f115 are both within the minor axis
distance range spanned by the observed substructure (dotted triangular region
in upper panel of Fig.~\ref{fig:spatdist}).
A two-sided Kolmogorov-Smirnov (KS) test finds that the radial velocity
distribution of field f115 is consistent
with the radial velocity distribution of its closest neighbor, field f116
(Fig.~\ref{fig:cfht}), although it is inconsistent with the best
constrained double Gaussian fit to field f116. Although the radial velocity distribution of
field f135 is not well-fit by a double Gaussian (\S\,\ref{sec:anal_ind}),
there is a concentration of stars near $v_{\rm hel}\sim -300$~\,km~s$^{-1}$\
in its radial velocity distribution (Fig.~\ref{fig:velhist}). We will
discuss these fields further in the context of the physical interpretation
of the cold substructure (\S\,\ref{sec:gss}). Fields f130 and a0 are at
larger minor axis distances than the tip of the
feature, and field f109 is interior to the feature. The fact that
substructure is not detected in these fields is consistent with our favored
physical interpretation of the $-300$~\,km~s$^{-1}$\ cold component, which will be discussed in \S\,\ref{sec:gss}.
\subsection{Metallicity Distribution}\label{sec:met}
So far we have considered only the kinematic properties of the minor axis
population. The distribution of stellar metallicities, however, is also
a powerful diagnostic of the presence and origin of substructure, since
different galactic components (e.g., disk, inner spheroid) and tidal debris
have different formation histories, and therefore different chemical
abundances.
Figure~\ref{fig:vel_met}\,(\textit{a}) displays $\rm[Fe/H]$\ vs. $v_{\rm hel}$
for the M31 RGB stars. The $\rm[Fe/H]$\ values
are based on a comparison of the star's position within the ($I,\,V-I$)\
CMD to a finely spaced grid of theoretical 12 Gyr, [$\alpha$/Fe]~$=0$
stellar isochrones \citep{kal06b,vdb06} adjusted to the distance of M31
\citep[783~kpc;][]{sta98,hol98}.
Stars with $\rm[Fe/H]$$ <-1$ appear to be
evenly distributed in velocity, while there is an obvious clump of
metal-rich stars with velocities near $-300$~\,km~s$^{-1}$. The bottom panels
(\textit{b--d}) show velocity histograms for stars in three $\rm[Fe/H]$\ bins:
(\textit{b}) $\rm[Fe/H]$$\ge-0.5$, (\textit{c}) $-1.0<$$\rm[Fe/H]$$<-0.5$, and
(\textit{d}) $\rm[Fe/H]$$\le-1.0$. The strength of the cold component in each
metallicity bin is measured by performing a maximum-likelihood double-Gaussian
fit to the velocity distribution. Only the fraction of stars in the
cold component ($N_{\rm sub}/N_{\rm tot}$) is allowed to vary; the rest
of the parameters are held fixed at the best-fit values from the fit to the
full M31 RGB sample (\S\,\ref{sec:anal_comb}, Table~\ref{table:rvfits}). The fraction of stars
in the cold component is 29.2\% in (\textit{b}), 20.9\% in (\textit{c}), and
is negligible in (\textit{d}), indicating that the cold component is
metal-rich relative to the dynamically hot component.
\begin{figure}
\plotone{f10.eps}
\caption{
(\textit{a}) Metallicity vs. heliocentric velocity for the full M31 RGB sample.
The majority of the population is metal-rich, with an evenly distributed
metal-poor tail over the full range of velocities. A concentration of
metal-rich stars
near $v_{\rm hel}=-300$~\,km~s$^{-1}$\ can be seen.
The bottom three panels show velocity histograms for subsets of the data:
(\textit{b}) $\rm[Fe/H]$$\ge -0.5$, (\textit{b}) $-1.0<$$\rm[Fe/H]$$< -0.5$, and
(\textit{d}) $\rm[Fe/H]$$\le -1.0$. The $\sim-300$~\,km~s$^{-1}$\ cold component is more dominant in
the metal-rich samples. A maximum-likelihood double-Gaussian fit was
performed for each subset
of the data, with all of the parameters except $N_{\rm sub}/N_{\rm tot}$
held fixed at the best-fit values for the complete M31 RGB sample
(\S\,\ref{sec:anal_comb}, Fig.~\ref{fig:vel_all}, Table~\ref{table:rvfits}).
The cold component
comprises a negligible fraction of the population in (\textit{d}), 20.9\% of the population
in (\textit{c}), and 29.2\% of the population in (\textit{b}), indicating that
the $\sim-300$~\,km~s$^{-1}$\ cold component is relatively metal-rich.
}
\label{fig:vel_met}
\end{figure}
Figure~\ref{fig:met} compares the metallicity distributions of stars in the velocity
range of the $\sim-300$~\,km~s$^{-1}$\ cold component (\textit{solid line}) discovered in
fields H11, f116, and f123 and stars that are identified with the hot spheroidal
component (\textit{dashed line}) in those fields. The hot
spheroidal distribution ($v_{\rm outer}$) is based on stars whose velocities are greater than
$\pm 2$$\sigma^{\rm sub}_v$\ away from $\langle v\rangle^{\rm sub}$. This minimizes contamination of the
spheroidal component by stars associated with the $\sim-300$~\,km~s$^{-1}$\ cold
component. An additional
constraint on the $v_{\rm outer}$ distribution is that only stars with
$v_{\rm hel}<-150$~\,km~s$^{-1}$\ are included in order to avoid residual
MW dwarf star contaminants that lie in the range
$v_{\rm hel}>-150$~\,km~s$^{-1}$\ (\S\,A.2).
Stars within
the velocity range of the cold component can
only statistically be identified as belonging to the hot spheroid or
cold component, thus it is not possible to identify an
uncontaminated sample of the $\sim-300$~\,km~s$^{-1}$\ cold component. Stars with
velocities within
$\pm1$$\sigma^{\rm sub}_v$\ of $\langle v\rangle^{\rm sub}$\ are used for the $v_{\rm inner}$
$\rm[Fe/H]$\ distribution, in
order to maximize the number of members of the cold component while minimizing
the contribution of spheroid stars.
\begin{figure}
\epsscale{0.9}
\plotone{f11.eps}
\caption{
The $\rm[Fe/H]$\ distribution of the $v_{\rm inner}$ (\textit{solid line}) and $v_{\rm outer}$
(\textit{dashed line}) components
in fields H11, f116, and f123, in histogram (\textit{top}) and cumulative
(\textit{bottom}) form. The $v_{\rm inner}$ (cold component)
sample is defined to be stars with velocities within the range
$\langle v\rangle^{\rm sub}$~$-$~$\sigma^{\rm sub}_v$~$<v_{\rm hel}<$~$\langle v\rangle^{\rm sub}$~$+$~$\sigma^{\rm sub}_v$, where $\langle v\rangle^{\rm sub}$\ and $\sigma^{\rm sub}_v$\ represent the best-fit
narrow Gaussian components in each field. This range was chosen to maximize the
percentage of substructure stars compared to hot spheroid stars.
The $v_{\rm outer}$ (hot spheroid) sample consists of stars with
velocities $v_{\rm hel}<$~$\langle v\rangle^{\rm sub}$~$- 2$$\sigma^{\rm sub}_v$\ or ~$\langle v\rangle^{\rm sub}$~$+ 2$$\sigma^{\rm sub}_v$$<v_{\rm hel}<-150$~\,km~s$^{-1}$.
This minimizes contamination from substructure stars and MW dwarf stars (\S\,~A.2) in the hot spheroidal $\rm[Fe/H]$\
distribution.
The $v_{\rm inner}$ sample is slightly more metal rich than the $v_{\rm outer}$ sample.
Since the $v_{\rm inner}$ sample is contaminated
by spheroid stars, the true difference in $\rm[Fe/H]$\ between the substructure and spheroid
populations is somewhat greater than indicated in this plot.
The cumulative $\rm[Fe/H]$\ distribution of stars in M31's
GSS \citep[from fields at 33 and 21~kpc;][]{guh06,kal06a}
is plotted in the bottom panel for comparison (\textit{thin dotted line}),
and will be discussed in \S\,\ref{sec:gss}.
}
\label{fig:met}
\end{figure}
The $\rm[Fe/H]$\ distribution of the $v_{\rm outer}$ sample has a
peak at lower $\rm[Fe/H]$\ values and a larger metal-poor tail than the $\rm[Fe/H]$\
distribution of the $v_{\rm inner}$ sample. The mean (median) metallicity of the
$v_{\rm outer}$ sample is $\langle$$\rm[Fe/H]$$\rangle_{\rm mean}=-0.72$
($\langle$$\rm[Fe/H]$$\rangle_{\rm med}=-0.63$), while the mean (median) metallicity of
the $v_{\rm inner}$ sample is $\langle$$\rm[Fe/H]$$\rangle_{\rm mean}=-0.55$
($\langle$$\rm[Fe/H]$$\rangle_{\rm med}=-0.49$). A KS test
returns a probability of 0.7\% that the two distributions are drawn from the
same parent distribution. The $v_{\rm inner}$ distribution
is highly contaminated by spheroid stars even within $\pm1$$\sigma^{\rm sub}_v$\ of $\langle v\rangle^{\rm sub}$\
(Fig.~\ref{fig:velhist}); the estimated contamination of the $v_{\rm inner}$ sample
by spheroid stars is 32.5\%.
A statistical subtraction of the $v_{\rm outer}$ distribution (scaled by
the contamination rate) from the
$v_{\rm inner}$ distribution yields a distribution with a mean (median) metallicity of
$\langle$$\rm[Fe/H]$$\rangle_{\rm mean}=-0.52$
($\langle$$\rm[Fe/H]$$\rangle_{\rm med}=-0.45$).
An important effect on the measured $\rm[Fe/H]$\ values is the assumed age
(12 Gyr) of the stellar population. This assumption is wrong for at least
field H11, which has been shown by deep HST/ACS imaging to contain a
wide spread of stellar ages, ranging from 6\,--\,13.5 Gyr
\citep{bro03,bro06b}. Our data
show a sharp cutoff in the $\rm[Fe/H]$\ distribution at $\rm[Fe/H]$$\sim0$, with a
metal-poor tail that extends out to $\rm[Fe/H]$$\sim-1.5$ to $-2.0$
(Fig.~\ref{fig:met}). The \citet{bro06b} HST/ACS data show a metal-rich
cutoff in the $\rm[Fe/H]$\ distribution at $\rm[Fe/H]$$\sim0.3$, with a metal-poor tail
which extends to $\rm[Fe/H]$$\sim-1.5$ to $-2.0$ (Fig.~9 of \citet{bro06b}).
The star formation history derived from the HST/ACS CMDs show that the
intermediate age population (6\,--\,9~Gyr) is metal-rich ($\rm[Fe/H]$$\sim0$),
while the old (10\,--\,14 Gyr) stellar population is relatively metal-poor.
Thus, our assumed age of 12 Gyr will underestimate metallicities for stars with
$\rm[Fe/H]$\ values near solar, but is appropriate for
the more metal-poor stars ($\rm[Fe/H]$$\lesssim-0.5$).
Varying the age between 6 and 14~Gyr introduces an $\approx 0.3$--$0.4$ dex
spread in the $\rm[Fe/H]$\ values derived from the isochrone fitting \citep{kal06b}; after
accounting for this offset at the metal-rich end of the $\rm[Fe/H]$\ distribution,
our $\rm[Fe/H]$\ distribution is consistent with the \citet{bro06b} result.
The intrinsic spread in ages in the spheroid found by \citet{bro03,bro06b} can
have two possible effects on our comparison of the $\rm[Fe/H]$\ distributions
of the $v_{\rm inner}$ and $v_{\rm outer}$ samples.
If all the stars in these fields have a common spread
in ages regardless of their kinematical properties, the error in the $\rm[Fe/H]$\
measurement introduced by
assuming a uniform age for the population will cause a shift in the actual
$\rm[Fe/H]$\ values, but the relative difference between the $v_{\rm inner}$ and $v_{\rm outer}$
populations will not be greatly affected. If the stars associated with the
$\sim-300$~\,km~s$^{-1}$\ cold component
are systematically younger than the underlying smooth spheroid
population,
the measurement of $\rm[Fe/H]$\ for the $v_{\rm inner}$ sample derived from the 12 Gyr isochrones will be biased
towards low metallicities, and
thus the intrinsic difference
in metallicity between the two populations will be greater than shown.
We have also
assumed that all stars are at the same line-of-sight distance as M31's center. This is
a valid approximation for the inner spheroid: at $R_{\rm proj}=20$~kpc, the
spread in line-of-sight distances is expected to be about $\pm20$~kpc
(a spread in apparent magnitude of $\pm 0.05$~dex),
which corresponds to a spread in $\rm[Fe/H]$\ of approximately $\pm 0.03$~dex. If the
cold component is
systematically more (less) distant than M31's spheroid (\S\,\ref{sec:gss}), the intrinsic difference
in metallicity between the populations will be slightly smaller (greater).
Although fields f115 and f135 do not show clear evidence of substructure
(\S\,\ref{sec:anal_ind}), they
are both within the minor axis range of the $\sim-300$~\,km~s$^{-1}$\ cold component
(Fig.~\ref{fig:spatdist}; \S\,\ref{sec:spatdist}).
If the cold components
in fields H11, f116, and f123 have the same physical origin, it is reasonable
to postulate that there may be substructure in fields f115 and f135 that is
not detected by the fits
to the radial velocity distributions. The stars in fields f115 and f135
that fall within the triangular region marked in the upper panel of
Figure~\ref{fig:spatdist} have mean (median) $\rm[Fe/H]$\ values of $-0.67$ ($-0.57$)
and $-0.56$ ($-0.46$), respectively.
Stars from fields f115 and f135
that have velocities both outside the triangular region and
$v_{\rm hel}<-150$~\,km~s$^{-1}$\ have mean (median) $\rm[Fe/H]$\ values of $-0.61$ ($-0.53$) and $-0.64$ ($-0.59$), respectively.
The $\rm[Fe/H]$\ distributions of stars in fields f135 and f115 that are within the
triangular region of the $\sim -300$~\,km~s$^{-1}$\ cold component
are consistent with both the $v_{\rm inner}$ and $v_{\rm outer}$ distributions.
The difference between the substructure and spheroid metallicity distributions
is small, and is only statistically significant when the three
fields contributing to the $\sim-300$~\,km~s$^{-1}$\ cold component are combined (into
the $v_{\rm inner}$ and
$v_{\rm outer}$ samples). The number of stars within a restricted
velocity range in any given field is too small to support a statistically
significant comparison.
\subsection{Comparison to Previous Observations}
The H11 field has been presented in previous papers as a smooth spheroid field
that is well-described by a single, kinematically hot component
\citep{bro06,bro06b,kal06a}. There are two related factors that have caused
this field to be reinterpreted as containing substructure.
First, the data set presented in this paper represents an order of
magnitude increase in the sample size of confirmed M31 RGB stars over
previous spectroscopic samples published by our group in the SE minor
axis region of the inner spheroid ($R_{\rm proj}\lesssim 30$).
Only the H11 and a0 fields have
been previously published, and the data published in H11 contained
only a fraction of the full data set for that field, based on preliminary reductions
(\S\,\ref{sec:dataredux}). This paper presents the full H11 data set and
combines it with new data from neighboring fields. Second, the
triangular shape of the $\sim-300$~\,km~s$^{-1}$\ cold component in a plot of
velocity vs.\ position along the minor axis implies that the debris
has a relatively
large velocity dispersion in the H11 field. This makes the debris
harder to detect against the broad underlying spheroid than if it were
kinematically colder. The increase in the sample size of stars with
recovered velocities in
the H11 field, coupled with the context provided by the velocity
distributions in neighboring fields, has proved to be crucial in
detecting the
substructure in H11.
Previous observational studies have suggested the possibility of substructure
along M31's southeastern minor axis. \citet{rei02} found
a dynamically cold grouping of 4 metal-rich M31 RGB stars (out of $\sim 35$)
along a southeastern minor axis field at $R_{\rm proj}\sim 19$~kpc. This hint of
substructure was strengthened by subsequent observations at 7 and 11~kpc along
the southeastern minor axis, which increased the total M31 RGB sample to $\sim 100$ stars
\citep{guh02}. The starcount maps in \citet{fer02} (their Fig.~2) also show
hints of a population of metal-rich stars along the southeastern minor axis, and
deep HST/ACS imaging has discovered a significant intermediate-age population
in field H11 \citep[\S\,\ref{sec:intm_age},][]{bro03}.
With the spectroscopic sample of $>1000$ stars presented in this paper, we are able to confidently
identify a cold component
along the southeastern minor axis and characterize its properties.
\section{Physical Origin of the Cold Component}\label{sec:origin}
The observed M31 RGB population along the southeastern minor axis exhibits a
spatially
varying kinematically cold component, which has a higher mean
metallicity than the underlying inner spheroid population.
A cold component with these properties could be part of the
outskirts of M31's disrupted disk or debris left by disrupted satellites.
The models of F07 predict debris stripped from the progenitor of the
GSS should be present in these fields. (It is also
possible, of course, that the observed substructure is satellite debris {\em unrelated} to
the GSS.)
This section examines both the continuation of the GSS (\S\,\ref{sec:gss}) and M31's disturbed
disk (\S\,\ref{sec:disk}) as possible physical origins of the cold component.
\subsection{Relation to the Giant Southern Stream}\label{sec:gss}
\subsubsection{Model of a Recent Interaction}
Debris in the form of coherent shells has been observed in many elliptical
galaxies; these shells are believed to be formed by the tidal disruption of a satellite
galaxy on a nearly radial orbit \citep[e.g.,][]{sch80,her88,bar92}.
F07 presents the hypothesis that the Northeast and Western ``shelves''
observed in M31 (Fig.~\ref{fig:roadmap}, also see Fig.~1 of F07) are a similar phenomenon to these shell systems,
and have been created by the disruption of the progenitor of the GSS.
Shells have coherent velocities, and display a distinctive
triangular shape in the $v_{\rm los}$ vs. $R_{\rm proj}$ plane \citep[][F07]{mer98}. In
general, as $R_{\rm proj}$ approaches the boundary of the shell the spread in velocities
approaches zero, with the mean velocity at the tip of the triangle expected to
be at the systemic velocity of the system.
F07 present an $N$-body simulation of an accreting dwarf satellite within
M31's potential. The
simulations use a static bulge+disk+halo model
which is based on the
M31 mass models in \citet{gee06} combined with the
observed stellar density distribution in the halo \citep{guh05},
and assumes an
isotropic velocity distribution in the outer halo.
The simulated satellite's physical
and orbital properties have been chosen
to reproduce the observed properties of the GSS and Northeast shelf, using
the methods of \citet{far06}.
The simulations show that the orbit which reproduces these features
also reproduces a photometric feature identified in F07 as the ``Western shelf''
and an observed
stream of counter-rotating planetary nebulae \citep{mer03,mer06}.
\begin{figure}
\plotone{f12.eps}
\caption{
Projected sky position (in M31 centric coordinates $\xi$ and $\eta$)
of tidal debris in the F07 simulations of the merger of a
dwarf galaxy with M31.
Particles approaching their first pericentric passage are part of
the GSS (green). Particles approaching their second pericentric passage form the Northeast shelf
(red), and particles approaching their third pericentric passage form the Western shelf
(magenta) identified in F07. Particles in blue are approaching their fourth
pericentric passage, and form a faint shelf feature which is predicted to be
most easily visible in the southeast.
The position of our spectroscopic masks
are also shown; fields f123 and f135 straddle the edge of the Southeast shelf (Fig.~\ref{fig:cfht}).
The
two masks at $\xi= 0.3$$^\circ$, $\eta= -1.5$$^\circ$\ are in field H13s,
which is discussed in \S\,\ref{sec:f135} and \S\,\ref{sec:intm_age}.
}
\label{fig:xieta_sim}
\end{figure}
Figure~\ref{fig:xieta_sim} shows the projected sky positions
(in M31-centric coordinates) of the satellite
particles from the F07 simulation. The particles are color-coded by shell,
or equivalently, by the number of pericentric passages
they have made. Green particles represent particles approaching their first
pericentric
passage; they correspond to the observed GSS. Red particles correspond to the
observed Northeast shelf; they are between their first and second pericentric passages. Magenta
particles represent the Western shelf identified in F07, and are between their second and third pericentric passages.
The blue particles are between their third and fourth
pericentric
passages, and represent the ``Southeast shelf'' predicted by F07.
This last feature is predicted to extend out to a radius of 18 kpc and is expected to be very faint,
as it consists of particles further forward in the continuation of the stream than the
more visible Northeast and Western shelves.
This feature
actually covers $\sim 180$$^\circ$\ in position angle on the east side of M31,
although it is likely to only be
visible in
the southeast
due to its overlap with the Northeast shelf and M31's disk.
In the F07 simulations, the Northeast shelf is made up of both the leading material from the progenitor's
first pericentric passage and trailing material from its second
pericentric passage, while the Western shelf is formed by leading material.
The simulations are unable to constrain whether or not the
satellite disrupts completely, as this is dependent on the central
density of the satellite.
The bottom panel of Figure~\ref{fig:spatdist_sim} presents the distribution of particles from
the F07 simulation in the $v_{\rm los}$ vs. $R_{\rm proj}$ plane.
The figure shows particles related
to the merging satellite as well as particles associated with the static
bulge+disk+stellar-halo M31 model used in F07.
In order to carry out a precise comparison to our observational data set, the
F07 simulation particles were
selected based on their projected sky position; all particles that fall inside a
16$'$$\times$10$'$\ area (the approximate area of one DEIMOS mask is
16$'$$\times$4$'$) centered
on the position of our observed fields, and oriented at the position angle of our
observed masks, are displayed.
The satellite particles are color-coded
by shell (or, equivalently, the number of orbits they have made) as in
Figure~\ref{fig:xieta_sim}. Green particles are associated with the GSS,
red particles with the Northeast shelf, and black particles with the bulge+disk+stellar-halo model for M31. The blue particles, which
are part of the predicted ``Southeast'' shelf,
form the distinctive triangular shape expected of a shell feature in the
$R_{\rm proj}$-$v_{\rm los}$ plane.
\begin{figure}
\epsscale{0.9}
\plotone{f13.eps}
\caption{
Line-of-sight velocity vs. projected radial distance from
M31's center ($R_{\rm proj}$) of spectroscopically confirmed M31 RGB stars
(\textit{top panel}) and particles from the F07 simulations
of the orbit of the progenitor of the GSS (\textit{bottom panel}).
Particles are drawn from the locations of each of our DEIMOS
masks. The area from which particles have
been drawn has been increased relative to the size of a slitmask to increase the
number of particles. The
satellite particles are color-coded according to which shell they are in: the giant
southern stream (green), the Northeast shelf (red), and the predicted Southeast shelf (blue). Black points are particles from the bulge+disk+stellar-halo
of M31. The blue particles form a triangular shape, with an
increasingly wide kinematic profile as the minor axis distance to the
center of M31 decreases, as seen in the data.
The tip of the triangle at $R_{\rm proj}\sim 1.3$$^\circ$\ (18 kpc) in the
simulated data agrees well with the observed tip in the data in field f123.
The cold concentration of M31 particles at
$v_{\rm M31}\approx 0$~\,km~s$^{-1}$\ extending from $R_{\rm proj}\sim 0.58$$^\circ$\ to $\sim 0.73$$^\circ$\
(7.9 to 10.0 kpc) corresponds to the disk of M31 (\textit{bottom panel});
the kinematical signature of a smooth, cold disk is not seen in our data (\textit{top panel}). }
\label{fig:spatdist_sim}
\end{figure}
In the following discussion, it is important to distinguish between ``spillover''
from the GSS (material associated
with the GSS that is in roughly the same orbital phase as the material
in the GSS, i.e., green particles in Figs.~\ref{fig:xieta_sim} \&
\ref{fig:spatdist_sim}) versus wrapped around portions of the GSS
(material associated
with the GSS progenitor that is leading the GSS and has undergone one or more
additional pericentric passages, i.e., red, magenta and blue particles in
Figs.~\ref{fig:xieta_sim} \&
\ref{fig:spatdist_sim}). The latter is the main theme of this paper, although
we also briefly discuss the former in \S\S\,7.1.2\,--\,7.1.3.
\subsubsection{Comparison of Model to Data}
\subsubsection*{Sky Position}
Figure~\ref{fig:xieta_sim} shows the projected sky positions (in M31-centric coordinates) of the satellite
particles from the F07 simulation as well as the
size, position and orientation of our Keck/DEIMOS slitmasks
(\textit{rectangles}).
Fields f123 and f135 land on the edge of the predicted Southeast shelf (\textit{blue
particles}), fields f115, f116, H11 and f109 all lie within the boundary of the Southeast shelf, and fields f130 and a0 lie beyond it.
Thus, the model predicts that the edge of the shell feature should pass
directly through our CFHT/MegaCam image. Indeed, there is
an apparent edge visible in the CFHT starcount map (Fig.~\ref{fig:cfht},
passing through field f123), in the same location as that
predicted for the Southeast shelf.
A close inspection of the \citet{iba05} starcount map
(Fig.~\ref{fig:roadmap}) reveals a point of bifurcation between the edge of
the Northeast shelf and a fainter feature at $\xi\approx 1.6$$^\circ$, $\eta\approx 0.2$$^\circ$,
in rough agreement with the bifurcation of the two features in
Figure~\ref{fig:xieta_sim}. This bifurcation is more evident in the Sobel-filtered
map in F07 (their Fig. 1). The radii of the shells in the simulation are
robust (\S\,4.2 of F07); thus the agreement
between the observations and the simulations is a strong confirmation of the
validity of the F07 model.
\subsubsection*{Kinematic Trends}
The $\sim-300$~\,km~s$^{-1}$\ cold component
observed in our
minor axis fields shows the distinctive triangular velocity pattern expected
of a shell feature in the $R_{\rm proj}$-$v_{\rm los}$ plane.
Figure~\ref{fig:spatdist_sim} compares our data (top panel) to the F07 model (bottom panel).
The distribution of observed velocities narrows to
a tip at $R_{\rm proj}\approx-1.3$$^\circ$\ (18~kpc) in the simulated particle
distribution, which is similar to the position of the tip of the velocity
distribution in our observed data.
At $R_{\rm proj}\approx-1$$^\circ$\ (13.7~kpc), the velocity
distribution of the observed substructure has widened to a spread of
$\sim200$~\,km~s$^{-1}$\ (measured from the edges of the feature),
also in agreement with the velocity spread of the predicted Southeast shelf.
The ``boxy'' shape of the velocity distribution in field H11
(Fig.~\ref{fig:velhist}) is also consistent with the interpretation of
the substructure as being part of a shell system. The velocity distributions
of shells have a clearly defined minimum and maximum line-of-sight velocity at a
given $R_{\rm proj}$. Stars tend to congregate at the minimum and maximum
velocities \citep[][F07]{mer98}, although their location in the
$R_{\rm proj}$-$v_{\rm los}$ plane depends on the region they occupy
in space (cf. the discussion in F07).
A maximum-likelihood Gaussian fit to the particles identified with the
Southeast shelf and within the minor axis distance spanned by field
f123 yields parameters of
$\langle v\rangle=-280.6$~\,km~s$^{-1}$\ and $\sigma_{v}=19.4$~\,km~s$^{-1}$. A Gaussian fit to
the Southeast shelf particles within the minor axis range spanned by fields
f116 and H11 returns $\langle v\rangle=-292.6$~\,km~s$^{-1}$\ and $\sigma_{v}=60.5$~\,km~s$^{-1}$.
The
mean velocity and dispersion of the predicted shelf is in good agreement
with the properties of the observed substructure (Table~\ref{table:rvfits}).
\subsubsection*{Metallicity Distribution}
If the substructure
identified in this paper is part of the predicted Southeast shelf in
F07, it should have a similar metallicity distribution to that of the
GSS, since the two structures originated from the same progenitor.
As part of our Keck/DEIMOS survey of M31's inner spheroid and halo, we
have taken spectra in two fields located on the GSS: a field at
$R_{\rm proj}=33$~kpc \citep{guh06} and a field at $R_{\rm proj}=21$~kpc
\citep[H13s;][]{kal06a}. The cumulative $\rm[Fe/H]$\ distribution of stars identified
kinematically as belonging to the GSS in these two fields is
plotted in the bottom panel of Figure~\ref{fig:met} (thin dotted line).
It is very similar to the $\rm[Fe/H]$\ distribution of stars that are kinematically
associated with the substructure in fields H11, f116, and f123.
The mean and median $\rm[Fe/H]$\ of the stars in the GSS are 0.1 and 0.05 dex
more metal-poor, respectively, than the mean and median $\rm[Fe/H]$\ of the
substructure in fields H11, f116, and f123, after correcting for
spheroid contamination
(\S\,\ref{sec:met}). The estimated
number of inner spheroid star contaminants in the GSS sample is a few stars
\citep{guh06,kal06a}. If the 3 most metal-poor stars are removed
($\rm[Fe/H]$$<-2.25$) from the GSS distribution, the mean and median
metallicity of the GSS stars are only 0.01 dex more metal-poor than the
the minor axis substructure.
The $\rm[Fe/H]$\ values of the GSS stars
have not been corrected for the GSS' measured distance relative to M31
\citep[$\sim
50$~kpc behind M31 for these 2 fields;][]{mccon03}. Accounting for this effect
would decrease the average metallicity of the GSS by $\sim 0.1$~dex. The
distance to the minor axis substructure is not known, although the F07 simulations predict
that the Southeast shelf should be approximately at M31's distance, with a spread in
distances of $\pm 9.2$~kpc (this corresponds to $\pm 2\sigma$ in terms of the
distribution of particle distances).
\subsubsection*{Strength of the Cold Component}
The cold component comprises
44\% of the total population of observed stars in fields H11 and f116 and 31\% of
observed M31 RGB stars in field f123 (Table~\ref{table:rvfits}). This
corresponds to a lower limit for the total fraction of stars in the cold
component of 21.7\% in the fields within the predicted range of the Southeast
shelf (f109, H11, f116, f115, f123, and f135). The Southeast shelf in the
simulations is much weaker, comprising only
2.7\% of the total population in these fields (this number increases to 3.4\%
if the number of shelf particles is compared only to the number of
bulge+disk+stellar-halo M31 particles). The strength of the feature in the
simulations is highly dependent on the mass of the progenitor and the time
since the first collision (F07). Thus, the strength of the observed
substructure will place interesting constraints on future models of the stream,
but cannot be used as a reliable discriminant of the applicability of
the model at the present time.
\subsubsection*{Fields Without Clear Detection of Substructure}
We do not
find a clear detection of substructure in fields f130 and a0. In the
context of the Southeast shelf, this is not surprising as both these
fields are beyond the radial range spanned by the shelf
(Fig.~\ref{fig:xieta_sim}). Field f109 is significantly inward of
the innermost field in which we detect substructure. In the
simulation, the particles associated
with the Southeast shelf continue into the region covered by field f109 with a
spread in velocities of $\sim350$~\,km~s$^{-1}$\ (Fig.~\ref{fig:spatdist_sim}, bottom panel).
The data in field f109 is consistent with being drawn from a single Gaussian
(\S\,\ref{sec:anal_ind}) and shows no evidence of substructure.
A secondary component
with a spread in velocities as wide as predicted would be very difficult to
differentiate from the broad spheroidal component, and would require a
much larger sample of M31 RGB stars in this field than is currently available.
Field f115 is well within the boundaries of the Southeast shelf. As
discussed in \S\,\ref{sec:spatdist}, its velocity distribution is consistent
with being drawn from the same parent distribution as field f116.
The shell in field f115 may be difficult to detect in our data due to
the broad ($\sim 55$~\,km~s$^{-1}$) nature of the substructure and the smaller
number of stars available in this field ($\sim 50\%$ less than in
field f116), or the shelf may be inherently clumpy.
Field f135 is on the edge of the simulated shelf, and shows evidence
for a peak of stars near $v_{\rm hel}\sim -300$~\,km~s$^{-1}$\ in its radial velocity
histogram.
In light of the simulations, we discuss this field in detail in
\S\,\ref{sec:f135}, and show that it may contain a kinematically-cold
component whose properties are consistent with both the observations
of the $\sim-300$~\,km~s$^{-1}$\ cold component
in fields H11, f116, and
f123 and the simulations of the Southeast shelf.
\subsubsection*{Debris from the GSS}
The GSS is observed
to have an asymmetric shape, with a sharp edge on the
eastern side and a more gradual decline in density on the western side
\citep{mccon03}. However, the eastern edge is not an absolute one in
the models, and ``spillover'' material from the GSS (\S\,\ref{sec:gss})
is predicted by the F07 simulations to be present in all of our fields.
Field f135 is the closest of our fields to the GSS' eastern edge
(Figs.~\ref{fig:roadmap} and \ref{fig:xieta_sim});
\S\,\ref{sec:f135} discusses the evidence for spillover debris from the
GSS in this field.
Since the density of M31 RGB spheroid stars falls off strongly with
increasing radius in the inner spheroid,
the contrast of cold GSS debris against the dynamically hot spheroid is
expected to be greatest in our outermost field, a0.
The GSS debris in field a0 is predicted to have a mean velocity of
$\langle v\rangle_{\rm GSS}=-364$~\,km~s$^{-1}$. Although
field a0 shows hints of peaks in the radial velocity distribution
at $v_{\rm hel}<-300$~\,km~s$^{-1}$\
(Fig.~\ref{fig:velhist}), only a handful of these stars are as metal-rich
as the GSS ([Fe/H]$\gtrsim -0.75$).
This allows us to place an upper limit on the
contamination of field a0 by the GSS of $\lesssim 5$ stars
($\lesssim 6\%$).
Although the F07 simulations reproduce many of the observations
in the GSS, they predict a much larger amount of debris
on the eastern side than is observed in our fields
(Fig.~\ref{fig:spatdist_sim}). Many factors can influence the
structure of the debris
in the simulations, including the shape and rotation of the progenitor.
The current models of the stream (F07) use a spherical, non-rotating
progenitor. A more complex model of the progenitor may be required to
reproduce the observations (Fardal et al., private communication).
\subsubsection{Evidence of the Southeast Shelf and Spillover Debris from the GSS in Field f135}\label{sec:f135}
The F07 simulations predict that in addition to the Southeast shelf
some spillover debris associated with the GSS (\S\,\ref{sec:gss}) should be present in field
f135.
The sky coordinates of
field f135 are $\xi=0.7$, $\eta=-1.1$, which places it on the edge of
both the GSS and the Southeast shelf in the F07 simulations (Figure~\ref{fig:xieta_sim}).
In addition, the radial velocity distribution of stars in field f135
is not well-fit by either a single or double Gaussian
(\S\,\ref{sec:anal_ind}) and
shows evidence of a metal-rich population (\S\,\ref{sec:met}).
Motivated by the close match between the observations and simulations
of substructure
in fields H11, f116 and f123,
we carry out a constrained fit of the radial velocity
histogram of field f135 to the sum of three Gaussians to determine if the
Southeast shelf is present in this field.
The mean velocity and velocity dispersion of the simulated Southeast
shelf and GSS particles in
field f135 are $\langle v\rangle_{\rm SE}=-286$~\,km~s$^{-1}$,
$\sigma^{\rm SE}_v=19$~\,km~s$^{-1}$\ and $\langle v\rangle_{\rm GSS}=-458$~\,km~s$^{-1}$,
$\sigma^{\rm GSS}_v=40$~\,km~s$^{-1}$. These values were used as rough constraints for
the triple Gaussian fit: the means were allowed to vary within $\pm100$~\,km~s$^{-1}$\ of the
predicted values and the dispersions were allowed to vary
from 1~\,km~s$^{-1}$\ to $3\sigma^{\rm SE}_v$ and $2\sigma^{\rm GSS}_v$.
The wide Gaussian component parameters were held fixed at the values for $G^{\rm sph}(v)$:
$\langle v\rangle^{\rm sph}$$=-287.2$~\,km~s$^{-1}$\ and $\sigma^{\rm sph}_{\rm v}$$=128.9$~\,km~s$^{-1}$\
(\S\,\ref{sec:anal_comb}). The maximum-likelihood triple Gaussian fit
is displayed in Figure~\ref{fig:velhist_f135} (\textit{solid curve}).
The wide underlying inner spheroid component ($G^{\rm sph}(v)$, \textit{dot-dashed curve})
comprises 45\% of the population. The Southeast shelf component
(\textit{dashed curve}), which is the narrow peak at
$\langle v\rangle^{\rm sub}$$=-273$~\,km~s$^{-1}$, comprises 30\% of the population and has
a width of $\sigma^{\rm sub}_v$$=30$~\,km~s$^{-1}$. The ``GSS'' component (\textit{dotted curve}) at $\langle v\rangle$$=-449$~\,km~s$^{-1}$\ has
a dispersion of $\sigma_v=55$~\,km~s$^{-1}$\ and comprises 25\% of the total population.
If a more constrained fit is carried out with the $\langle v\rangle$ and $\sigma_v$ parameters for
all three Gaussian components held fixed (at the predicted values
for the simulated shelf and stream particles and at the parameters of the Gaussain $G^{\rm sph}(v)$)
and only the fractions of stars in the
various components are allowed to vary, the best-fit distribution has
$N_{\rm shelf}/N_{\rm tot}=0.18$ and $N_{\rm GSS}/N_{\rm tot}=0.17$.
\begin{figure}
\epsscale{0.85}
\plotone{f14.eps}
\caption{
Radial velocity histogram of M31 RGB stars in field f135.
A constrained triple Gaussian (\textit{solid curve}) has been
fit to the observed
data using a maximum-likelihood technique, with rough constraints imposed on
the parameters based on the properties of the simulated substructure
(\S\,\ref{sec:f135}).
The observed velocity distribution is well fit by a sum of three Gaussians:
(i) $G^{\rm sph}(v)$, the wide Gaussian which corresponds to the underlying inner
spheroid of M31 (\textit{dot-dashed curve}, \S\,\ref{sec:anal_comb}),
(ii) a component centered at $\langle v\rangle^{\rm sub}$$=-273$~\,km~s$^{-1}$\
with a width of $\sigma^{\rm sub}_v$$=30$~\,km~s$^{-1}$, which comprises 30\% of
the total population and which likely corresponds to
the Southeast shelf (\textit{dashed curve}), and (iii) a narrow component
centered at $\langle v\rangle$$=-449$~\,km~s$^{-1}$\ with a width of $\sigma_v=55$~\,km~s$^{-1}$, which
comprises 25\% of the total population
(\textit{dotted curve}).
The mean velocity and velocity dispersion ($\pm 1\sigma_v$) of the cold components in field H13s, at a
similar radial distance along the GSS as field f135, are shown as arrows and
horizontal lines \citep[\S\,\ref{sec:f135};][]{kal06a}.
}
\label{fig:velhist_f135}
\end{figure}
The kinematic properties (mean velocity and
velocity dispersion) of the Southeast shelf component in the triple
Gaussian fit to field f135 are consistent not only
with the simulations, but also with
what one would expect for the Southeast shelf in this field based on the
observations (e.g., Fig.~\ref{fig:spatdist_sim},
Table~\ref{table:rvfits}). In addition, the fraction of the population
in f135 which is in this component is consistent with the fraction of the
population which is identified with the Southeast shelf in fields
H11, f116, and f123.
As further evidence that the Southeast shelf is detected in field f135,
Figure~\ref{fig:f135_met_vel} shows $\rm[Fe/H]$\ vs. $v_{\rm hel}$ for the M31
RGB stars in field f135 (panel \textit{a}), as well as velocity histograms
for stars in two $\rm[Fe/H]$\ bins [(\textit{b}) $\rm[Fe/H]$$>-0.75$ and (\textit{c})
$\rm[Fe/H]$$<-0.75$].
As in fields H11, f116 and f123, the substructure in field f135
that is identified with the Southeast shelf ($\langle v\rangle$ $=-273$\,km~s$^{-1}$) is
relatively metal-rich.
The velocity dispersion of $\sigma_v=55$~\,km~s$^{-1}$\ for the ``GSS component''
inferred from the first
of the triple-Gaussian fits above is large compared
to previous measurements of $\sim 15$~\,km~s$^{-1}$\ for the dispersion of the GSS
\citep{iba04,guh06,kal06a}. In addition, although the best-fit $\langle v\rangle$$^{\rm sub}$
of the most negative component in f135 is similar to that of the F07 model,
the predicted GSS mean velocities in the F07 model are not as negative as the
observed velocities of the GSS. \citet{kal06a} analyzed a field centered
on a high surface brightness protion of the GSS, at approximately the
same radial distance along the stream as field f135 (field H13s, located at
$\xi=0.3$$^\circ$, $\eta=-1.5$$^\circ$\ in Figure~\ref{fig:xieta_sim})
and found a
secondary cold component, the ``H13s secondary stream,'' whose origin
and physical extent are unknown.
The observed mean velocities of the GSS and secondary stream in the
H13s field are $\langle v \rangle^{\rm GSS}=-513$~\,km~s$^{-1}$\ and
$\langle v \rangle^{\rm sec.str.}=-417$~\,km~s$^{-1}$, respectively,
with velocity dispersions of $\sigma_v^{\rm GSS}=\sigma_v^{\rm sec.str.}=16$~\,km~s$^{-1}$\ \citep{kal06a}.
The mean velocities
and velocity dispersions ($\pm 1\sigma_v$) of these two components are
shown as arrows with horizontal lines
in Figures~\ref{fig:velhist_f135} and \ref{fig:f135_met_vel}.
There appear to be two metal-rich peaks in f135 with the approximate velocities
of the GSS and secondary stream in H13s.
If both the GSS and H13s secondary stream
are present in field f135, they appear
in approximately equal proportion.
In H13s
the GSS dominates over the secondary stream by a factor of two \citep{kal06a}.
\begin{figure}
\plotone{f15.eps}
\caption{
(\textit{a}) Metallicity vs. heliocentric velocity for the M31
RGB stars in field f135, which lies to the east of the edge of
the GSS. The mean velocities and dispersions ($\pm 1\sigma_v$) of
the GSS and the secondary stream from
the GSS field H13s \citep{kal06a} are marked, as in Figure~\ref{fig:velhist_f135}.
(\textit{b}) Velocity distribution of stars with $\rm[Fe/H]$$>-0.75$. In addition
to the $\sim-300$~\,km~s$^{-1}$\ cold component, the metal-rich subset
shows evidence of concentrations of stars corresponding to the GSS and secondary
stream in the H13s field.
(\textit{c}) Velocity distribution of stars with $\rm[Fe/H]$$<-0.75$. The
velocities of stars with
metallicities lower than $-0.75$~dex appear evenly distributed.
}
\label{fig:f135_met_vel}
\end{figure}
In the simulations, the GSS dominates over the Southeast shelf in
field f135 by about a factor of 10; the observations indicate that, at best,
these two populations are roughly equal. The simulations predict that the GSS
should comprise a total of 35\% of the population in field f135, which
is somewhat larger than the fraction of the total population in the
most negative cold component from the triple-Gaussian fit shown in
Figure~\ref{fig:velhist_f135} (25\%). If the ``GSS''
component from the triple-Gaussian fit
is actually comprised of two narrower streams,
the GSS comprises a much smaller percentage of the stars in this field than
predicted by the simulations.
\subsection{Arguments Against a Disk Origin}\label{sec:disk}
The substructure discovered in fields H11, f116 and f123 is centered at close to the systemic velocity
of M31 ($v_{\rm sys}=-300$~\,km~s$^{-1}$), which is also the radial velocity expected
for an M31 disk component on the minor axis.
Recent observational evidence suggests that M31's stellar disk
extends smoothly out to $R_{\rm disk}\sim 40$ kpc and has a velocity dispersion
of $\sim30$~\,km~s$^{-1}$; isolated features with disk-like
kinematics have been observed as far out as $R_{\rm disk}\sim 70$ kpc
\citep{rei04,iba05}. We present three lines of evidence
arguing against a disk origin for the $\sim-300$~\,km~s$^{-1}$\ cold component found
on the southeastern minor axis.
\subsubsection*{Disk to Inner Spheroid Surface Brightness Ratio}
Previous
measurements of M31's
stellar disk and inner spheroid \citep{wal88,pri94} indicate that the
disk constitutes a negligible fraction of the
total light in our minor axis fields. Even in our innermost field f109
(at $R_{\rm proj}\sim 9$~kpc,
corresponding to $R_{\rm disk}\sim 38$~kpc for a disk inclination of 77$^\circ$),
the disk fraction is expected
to be only $\sim10\%$ \citep{guh05} based on disk scale radii of 5.0\,--\,6.0~kpc
\citep{iba05}. We see no evidence of a cold, disk-like feature in f109's
radial velocity distribution, although a $\lesssim10$\% component could
be difficult to detect.
The disk fraction drops sharply at larger radii: for example, at the
distance of our innermost field containing the $\sim-300$~\,km~s$^{-1}$\
cold component, H11 ($R_{\rm disk}\sim 51$~kpc), the expected smooth
disk fraction is $\sim1\%$ \citep{bro06}.
Non-uniformities in M31's stellar disk could
result in a higher disk fraction in our fields. To explain the
strength of the $\sim-300$~\,km~s$^{-1}$\
cold component in field H11 ($N_{\rm sub}/N_{\rm tot}=44$\%), the disk would have to contain a $45\times$
enhancement in this field relative to the smooth disk. Even more extreme disk
enhancements are needed to explain the cold component in fields f116 and f123.
If there is a warp such that the outer disk is more face on than the inner
disk (\citet{iba05} find a best fit disk inclination angle of 64.7$^\circ$\
from $20<R_{\rm disk}<40$~kpc), the effective disk radii of our fields will
be smaller and the smooth disk contribution larger than in the above
calculation. As an extreme example of a warp, we consider a disk of scale
length 5.7~kpc whose inclination changes from 77$^\circ$\ at small radii
($R_{\rm disk}<20$~kpc) to 60$^\circ$\ at large radii ($R_{\rm disk}>20$~kpc).
In this case, H11 is at $R_{\rm disk}\sim 35$~kpc while f123 is at
$R_{\rm disk}\sim 47$~kpc. With this disk model, taking into account
projection effects in the disk surface brightness (the surface brightness
decreases as the disk becomes more face on) but ignoring dust effects,
the expected disk fractions in fields f109, H11, f116, and f123 are 20\%,
18\%, 17\%, and 10\%, respectively. Thus, it would still require a
$\sim 2.5\times$ enhancement in the disk to explain the cold component in
H11. More importantly, the 20\% cold, smooth disk fraction predicted by
this warp model in field f109 is inconsistent with our radial velocity
data (Fig.~\ref{fig:velhist}). This disk fraction estimate for the warp is
optimistic (high) in that it does not account for the reduction in surface
brightness that would be caused by any stretching associated with the
putative warp.
\subsubsection*{Velocity Dispersion}
The measured velocity dispersion of the $\sim-300$~\,km~s$^{-1}$\
cold component in fields H11 and f116 is $\gtrsim50$~\,km~s$^{-1}$, while the velocity
dispersion of this component is only $\sim10$~\,km~s$^{-1}$\ in field f123. These
measurements are significantly above and below, respectively, the typical
velocity dispersion of $30$~\,km~s$^{-1}$\ measured for the extended, disk-like
structure by \citet{iba05}, for which they observe a range in velocity
dispersions from $\sim 20$\,--\,$40$~\,km~s$^{-1}$. If the
cold component in each of these three fields is from the disk, it would require
a warp and non-uniformities in the disk that happen to have the observed
triangular shape in the position-velocity plane (top panel of
Figure~\ref{fig:spatdist}).
\subsubsection*{Stellar Ages and Metallicities}
If the $\sim-300$~\,km~s$^{-1}$\ cold component is
debris associated with M31's extended disk, its stellar population should
reflect this. However, a comparison
of the stellar ages and metallicities of the H11 field
with deep HST/ACS and ground-based imaging of disk-dominated M31
fields yields very different star formation
histories. Obviously disturbed sections of M31's disk (e.g., the Northern
Spur and the G1 clump) show evidence of recent star formation
\citep[$\sim 3$~Gyr ago in the Northern Spur and $\sim 250$~Myr ago
in the G1 clump;][]{fer05}, while deep HST/ACS imaging of field H11
reveals that very few of the stars are younger than 4 Gyr \citep{bro03,bro06b}.
\citet{bro06b} find that their disk-dominated HST/ACS field H13d, located at 25~kpc
along the northeastern major axis, contains a significantly younger and more
metal-rich population than H11.
In addition, \citet{bro06} find very good agreement between the
stellar populations of field H11 and an HST/ACS field on the GSS (H13s, discussed
further in \S\,\ref{sec:intm_age}).
The comparison of the stellar ages and metallicities of M31 disk fields vs.\
that of field H11 is complicated by several uncertainties. A radial gradient
in disk properties could result in M31's outer disk (in field H11)
being more metal-poor and older than the inner parts of the disk. Also, the
\citet{bro06b} {\it HST/ACS\/} field H13d likely includes multiple galactic
components including spheroid and wrap-around debris from the progenitor of
the GSS, although the radial velocity distribution indicates that M31's disk
is the dominant component in this field \citep[see Fig.~9 of][]{kal06a}.
\bigskip
In conclusion, the observations disfavor an extended rotating disk model as
the physical origin of the $\sim-300$~\,km~s$^{-1}$\ cold component identified along
M31's southeastern minor axis. Although a disk origin for this substructure
cannot be ruled out, it requires simultaneous contrivance of multiple
properties
of the disk to explain the observations: patchy disk structure (to explain
the absence of an observed disk in f109), a warp and
enhancement (which cannot be due to recent star formation) of the disk
to explain the strength of the cold component,
an anomalously large velocity dispersion for fields H11 and f116
and a smaller than average velocity dispersion in field f123,
and a significant radial gradient in the metallicity and age of the stellar
disk populations. Compared to the elegance of the southeastern shelf
interpretation based on F07's simulations, which is a true prediction and
explains the observed properties of the cold component in
all the fields in which it is
detected, the disk origin clearly fails the test of Occam's razor for the most
likely physical origin of the $\sim-300$~\,km~s$^{-1}$\ cold component.
\section{Implications for the Intermediate-Age Spheroid Population}\label{sec:intm_age}
\citet{bro03,bro06,bro06b} present HST/ACS photometry of fields in M31 down to 1\,--\,1.5
magnitudes below the main-sequence turnoff. Our field H11
is coincident with the
\citet{bro03} spheroid field (Fig.~\ref{fig:cfht}). The photometry presented in \citet{bro06}
is from a field on the GSS of Andromeda at a projected radial
distance of 20 kpc, and is coincident
with the Keck/DEIMOS spectroscopy field H13s presented in \citet{kal06a}
and shown in Figure~\ref{fig:xieta_sim} ($\xi= 0.3$$^\circ$, $\eta= -1.5$$^\circ$).
In the ``smooth'' spheroid field, \citet{bro03} find that
$\sim30$\% (by mass) of the stellar population is intermediate-age (6\,--\,8 Gyr) and
metal-rich, while another 30\% of the population is old (11\,--\,13.5 Gyr) and
metal-poor. \citet{bro06} find remarkable agreement in the CMDs of the stream and
spheroid fields, indicating that the two fields have very similar age and
metallicity distributions. They query whether the similarities between the
populations could ``be explained by the stream passing through the spheroid field,'' but
note that this explanation is problematic:
the stream would have to dominate the spheroid by approximately the same
factor in both fields \citep[3:1 based on a kinematical analysis
of the stream field H13s;][]{kal06a}, yet the kinematical profiles of the
two fields are distinctly different, with the H11 field failing to show
the cold ($\sigma_{\rm v}=16$~\,km~s$^{-1}$) signature of the stream seen
in the H13s field. However, they presciently suggested that the similarity
in the two populations (spheroid and stream) implies that ``the inner
spheroid is largely polluted by material stripped from either the stream's
progenitor or similar objects.''
In light of the substructure presented in this paper, this seems to be
the correct interpretation of the similarity between the ``spheroid'' and
GSS stellar populations.
The spatial and kinematic properties of the substructure suggest that the
region of the spheroid imaged in the original HST field \citep{bro03} is
in fact contaminated by stars from the progenitor of the GSS. The
kinematical signature of the substructure at the minor axis distance of
the HST/ACS field (H11) is both predicted
and observed to be relatively wide (Fig.~\ref{fig:spatdist_sim}), and thus
less obvious against the underlying hot component. In the context of
the F07 simulations, the minor axis substructure
is not isolated, but is part of one of a series of shells caused by the
disruption of the GSS' progenitor, which collectively contaminate a
large part of the inner spheroid of M31 (Fig.~\ref{fig:xieta_sim}).
The current analysis suggests that $\sim 45$\% of the M31 RGB stars in the
H11 field are in fact part of the $\sim-300$~\,km~s$^{-1}$\ cold component, and not part of the broad
spheroid. In the H13s field, 75\% of the M31 RGB stars are part of a cold
component \citep{kal06a}.
This difference in substructure fraction agrees nicely with the difference in
the fraction of intermediate-age ($<10$ Gyr), metal-rich stars found in the stream and
spheroid fields in \citet{bro06b}: 70\% vs. 40\%, respectively. However,
recent HST/ACS observations of a field in the location of our f130 masks
at 21~kpc imply that this is not the end of the story: \citet{bro07} find that
the stellar population in H11 can \textit{not} be fit by a linear
combination of the GSS (H13s) and the 21~kpc spheroid (f130) stellar
populations, due largely to the presence of a greater number of stars
younger than 8~Gyr in H11 than in the GSS field.
Nevertheless, the observational evidence, combined with the theoretical predictions
of F07, strongly favor the explanation that the age and metallicity
distributions of the stream and spheroid HST fields are so remarkably similar
because the same progenitor polluted both fields with substructure.
\section{Summary}\label{sec:concl}
The use of the diagnostic method described in \citet{gil06} has enabled us
to isolate the first sample of spectroscopically confirmed M31 RGB stars defined \textit{without} the
use of radial velocity. We use this
sample of $\sim 1000$ M31 RGB stars to measure the velocity dispersion of the inner spheroid of M31; in
the radial range $R_{\rm proj}=9-30$~kpc the inner spheroid
has a velocity dispersion of
$\sigma^{\rm sph}_{\rm v}$=129~\,km~s$^{-1}$. Our data show no evidence of a decrease in the velocity
dispersion over this radial range.
The stellar radial velocity distribution in these fields shows evidence of a significant amount
of substructure. Compared to the large velocity dispersion seen in the
underlying hot spheroid
population, the substructure is kinematically cold, exhibiting
a decrease in velocity dispersion
with increasing projected radius. In the fields in which the $\sim-300$~\,km~s$^{-1}$\ cold component is observed,
$\approx41$\% of the stars are estimated to belong to it; the rest are
members of the hot inner spheroid of M31. The metallicity of the substructure
is higher than that of the broad
spheroidal component in the fields in which it is observed.
The physical origin
of the substructure discovered in this paper is most likely tidal debris stripped from
the progenitor of the GSS. The data agree very
well with the location and kinematical properties of the Southeast shelf
predicted by
the F07 simulations of the disruption of the GSS' progenitor,
and will add
significant observational constraints to those already existing from
the GSS,
Northeast shelf, and
Western shelf, enabling detailed modeling
of M31's dark matter distribution (F07). The minor axis fields
also place constraints on the spatial distribution of the GSS itself.
The GSS contamination in our minor axis fields is much smaller
than predicted by the current models of the stream (F07),
which suggests the stream's progenitor had a more complex structure
than the spherical, non-rotating models used so far.
The newly-discovered substructure sheds light on the discovery of a
significant intermediate-age population in the ``smooth''
spheroid field by \citet{bro03}, and the subsequent discovery of the
similarity in ages and metallicities of the stars in the spheroid field and
a field on the GSS \citep{bro06, bro06b}. The spheroid
HST/ACS field
was not in fact placed on a ``smooth'' spheroid field, and the
intermediate-age population may be part of the substructure observed in
this field. If the
substructure identified in this paper is indeed from the same progenitor as the giant southern stream,
it is not surprising that the two HST/ACS fields would have very similar age
and metallicity distributions. Given the number of observed fields in the inner
spheroid which are contaminated by substructure, both in the current work and
in the literature \citep{irw05,fer05,kal06a}, it seems likely that the inner
spheroid is highly contaminated by tidal debris. A ``smooth'' inner spheroid
field may in fact be a rarity.
\acknowledgments
We are grateful to Sandy Faber and the DEIMOS team for building an
outstanding instrument and to Mike Rich for his role in the acquisition of
many of the Keck/DEIMOS masks. We thank Peter Stetson, Jim Hesser, and James Clem for
help with the acquisition and reduction of CFHT/MegaCam images, Phil Choi,
Alison Coil, Geroge Helou, Drew Phillips, and Greg Wirth for observing some
DEIMOS masks on our behalf, Drew Phillips for help with
slitmask designs,
Jeff Lewis, Bill Mason, and Matt Radovan for fabrication of slitmasks, and
the DEEP2 team for allowing us use of the {\tt spec1d}/{\tt zspec} software.
We also thank Tom Brown for stimulating discussions and comments on the draft. The {\tt spec2d} data reduction pipeline for DEIMOS was developed at UC
Berkeley with support from NSF grant AST-0071048. This project was supported
by an NSF Graduate Fellowship (K.M.G.), NSF grants AST-0307966 and
AST-0507483 and NASA/STScI grants GO-10265.02 and GO-10134.02 (P.G., K.M.G.,
and J.S.K.), NSF grant AST-0205969 and NASA ATP grants NAGS-13308 and NNG04GK68G (M.F.), NSF grants AST-0307842 and
AST-0307851, NASA/JPL contract 1228235, the David and Lucile Packard
Foundation, and The F.~H.~Levinson Fund of the Peninsula Community Foundation
(S.R.M., J.C.O., and R.J.P.) and NSF grant AST-0307931 (D.B.R.).
J.S.K. is supported by NASA
through Hubble Fellowship grant HF-01185.01-A, awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
|
2005.10715
|
\section{Introduction}
\label{intro:1}
One of the most celebrated problem in the field of Celestial Mechanics is the restricted three-body problem (R3BP). Many researchers and scientists are attracted towards it due to its applications in various other fields (e.g. \cite{AA19a}, \cite{AGL19b}, \cite{A12}, \cite{AAG09}, \cite{AAG17}, \cite{CM87}, \cite{S07}, \cite{PAT19}, \cite{SGA19}). In addition, several modifications have been proposed by various researchers to be more realistic in the classical R3BP which make the applications of this problem in wider sense. In this proposed problem we have considered two modifications i.e., the radiation effects of both the primaries and the variation in the angular velocity (Chermnykh problem, see \cite{C87}). A generalization of the Euler's problem of two fixed gravitational centers and the restricted problem of three bodies where the third body, whose mass is negligible in comparison of the other bodies, orbits in the configuration plane of dumbbell which rotates around their center of mass with a constant angular velocity $\omega$, is always referred as Chermnykh problem. Many authors have studied this problem due to its important applications in the field of Chemistry (\cite{PFG96}), Celestial mechanics and Dynamical Astronomy.
One of the paramount issue in the dynamical system is to know the geometry of the "basins of convergence" linked with the equilibrium points of the dynamical system. The domain of the BoC unveils the fact that how the different initial conditions on the configuration plane are enticed by the particular equilibrium point when an iterative method is applied to solve the system of equations. Undoubtedly, to solve the system of simultaneous equations with two or more variables, the N-R iterative scheme is contemplated as classical one. Previously, many authors have studied the BoC by applying the N-R iterative method to reveal the numerous intrinsic properties of different dynamical system (e.g., the R3BP and the Hill's problem with oblateness and radiation effects(\cite{Z16}, \cite{Z17}, \cite{D10}), the restricted four-body problem (\cite{BP11},\cite{SAP17}, \cite{SAA17}, \cite{SUR20}), the restricted problem of five bodies (\cite{ZS18}, \cite{SAR19}, \cite{sur19}, \cite{sur19b}, \cite{Sur19d}).
Presently, we wish to analyze the effect of angular velocity on the topology of the BoC when both of the primaries are source of radiation. Moreover, the fractality of the BoC is also discussed as the function of angular velocity.
The present paper has following structure: along with the literature review regarding the R3BP presented in Sec. \ref{intro:1}, the description of the mathematical model is presented in Sec. \ref{Sec:2}. The parametric evolution of the locations of the equilibrium points is depicted in Sec. \ref{Sec:3} whereas the influences of the angular velocity on the geometry of the BoC by using the bivariate sort of the N-R iterative method are illustrated in detail in Sec. \ref{Sec:4}. The degree of fractality of the BoC is depicted in Sec. \ref{Sec:5}. The paper ends with Sec. \ref{Sec:6} where the analysis of the study and the obtained results are discussed.
\section{Mathematical descriptions and the equations of motion}
\label{Sec:2}
In the present study, we have considered the dynamical model same as in Ref.\cite{PK15} which can be reviewed as follows: the rotating, barycentric, and a dimensionless co-ordinate system with origin "O" is considered as the centre of mass of the system. The two primaries namely $m_1$ and $m_2$ rotate in circular orbit around "O" with angular velocity $\omega \geq 0$ and in addition, the primaries always lie on the $x-$axis with co-ordinate $(x_1, 0)=(-\mu, 0)$ and $(x_2, 0)=(1-\mu, 0)$ (see Fig. \ref{Fig:00}). In dimensionless unit $m_1=1-\mu$ and $m_2=\mu$ where the mass parameter $\mu=\frac{m_2}{m_1+m_2} \leq \frac{1}{2}$. The restricted problem of three bodies reduces to the Copenhagen problem when $\mu=\frac{1}{2}$. We analyse the motion of the third body $m_3$ whose mass is negligible in comparison of the primaries. In addition, it is also considered that both the primaries are source of radiation. Consequently, the motion of the infinitesimal mass is governed by two type of forces i.e., the gravitational forces of the primaries and the repulsive force of the light pressure. It is necessary to note that the radiation factors can achieve the negative value as well which means that these forces will give strength to the gravitational force.
In the dimensionless rectangular rotating co-ordinate system, the equations of motion of the third body, which also referred as test particle in the restricted three-body problem with angular velocity, are (see \cite{C87}, \cite{PR04} and \cite{PK15}):
\begin{subequations}
\begin{eqnarray}
\label{Eq:1a}
\ddot{x} -2\dot{y}&=&(\omega^2-\mathfrak{Q}^*)x-\mathfrak{M}^*\mathfrak{R}^*,\\
\label{Eq:1b}
\ddot{y}+2\dot{x} &=&(\omega^2-\mathfrak{Q}^*)y,
\end{eqnarray}
\end{subequations}
where
\begin{subequations}
\begin{eqnarray}
\label{Eq:2a}
\mathfrak{M}^*&=&\mu(1-\mu),\\
\mathfrak{Q}^* &=& \frac{q_1(1-\mu)}{r_1^3}+\frac{q_2\mu}{r_2^3},\\
\label{Eq:2b}
\mathfrak{R}^* &=&\frac{q_1}{r_1^3}-\frac{q_2}{r_2^3},
\end{eqnarray}
\end{subequations}
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{Fig_1}}
\caption{The restricted three-body problem. (colour figure online).}
\label{Fig:00}
\end{figure}
while the time independent potential function $\Omega$ is given by:
\begin{subequations}
\begin{eqnarray}
\label{Eq:3a}
\Omega&=& \frac{\omega^2}{2}(x^2+y^2)+\sum_{i=1}^{2}\frac{q_i m_i}{r_i},\\
\label{Eq:3b}
r_i^2 &=& \tilde{x_i}^2+\tilde{y_i}^2,\\
\label{Eq:3c}
\tilde{x_i}&=&(x-x_i),\\
\label{Eq:3d}
\tilde{y_i}&=&(y-y_i),
\label{Eq:2e}
\end{eqnarray}
\end{subequations}
where $r_i$ represents the distances of the test particle from the primaries $m_i$, respectively. The radiation parameters $q_i$,(see \cite{C70}) due to the radiating primaries $m_i$ are defined as:
\begin{equation*}
q_i=1-\frac{F_{p_i}}{F_{g_i}},
\end{equation*}
where, $F_{p_i}$ are the solar radiation pressure forces whereas $F_{g_i}$ are the gravitational forces due to primaries $m_i, i=1,2$.
The system admits the Jacobi integral i.e.,
\begin{equation}\label{Eq:4}
C=2\Omega -(\dot{x}^2+\dot{y}^2).
\end{equation}
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{Fig_2}}
\caption{The movement of the libration points for $q_1=0.15$, $q_2=0.25$ and consequently $\omega \in (0.03097574, 1.25144256)$. The color codes are : $L_1= $\emph{Purple}, $L_2= $\emph{Orange}, $L_3= $\emph{Green}, $L_4= $\emph{Olive}, $L_5= $\emph{Cyan}. (colour figure online).}
\label{Fig:M1}
\end{figure}
\section{The libration points: a parametric evolution}
\label{Sec:3}
The parametric evolution of the positions of libration points are presented in this section by using the same procedure given by Ref.\cite{PK15}. The collinear libration points are those points which lie on $x-$axis and we can evaluate by system of equations (\ref{Eq:1a}-\ref{Eq:1b}), by setting the velocity and acceleration components equal to zero and solving for $x$ by taking $y=0$, we get
\begin{equation}
\label{Eq: 5}
f(x)=\omega^2x-\frac{(1-\mu)q_1(x+\mu)}{|x+\mu|^3}-\frac{\mu q_2(x+\mu-1)}{|x+\mu-1|^3}=0,\\
\end{equation}
by keeping the value of the parameters $\omega, \mu$ and $q_i$, $i=1,2$, fixed. The presented problem reduces to the photo-gravitational version of the classical restricted problem when $\omega=1$. It is shown that the angular velocity $\omega$ has no affect on the existence of totality of number of collinear libration points (for detail see Ref.\cite{PK15}) and these libration points are named as $L_{i}$, $i=1,2,3$ where their positions are defined as follows:
\begin{align*}
L_{3}< & -\mu<L_{1}<1-\mu<L_{2},
\end{align*}
where $-\mu$ and $1-\mu$ are the positions of the primaries $m_1$ and $m_2$ respectively.
As far as the non-collinear triangular libration points are concerned, their positions can be described as follows:
\begin{subequations}
\begin{eqnarray}
\label{Eq:6a}
x &=& \frac{1}{2}\Big\{1+\Big(\frac{q_1}{\omega^2}\Big)^\frac{2}{3}-\Big(\frac{q_2}{\omega^2}\Big)^\frac{2}{3}\Big\}-\mu,\\
\label{Eq:6a}
y&=&\pm\Big[\Big(\frac{q_1}{\omega^2}\Big)^\frac{2}{3}-\frac{1}{4}\Big\{1+\Big(\frac{q_1}{\omega^2}\Big)^\frac{2}{3}-\Big(\frac{q_2}{\omega^2}\Big)^\frac{2}{3}\Big\}^2\Big]^\frac{1}{2},\nonumber\\
\end{eqnarray}
\end{subequations}
for detail see Ref. \cite{PK15}. In addition, it is unveiled that the planar non-collinear libration points i.e., $y\neq 0$, exist only when the following conditions are satisfied simultaneously:
\begin{align}\label{Eq:7}
q_1^\frac{1}{3} > 0, & \quad q_2^\frac{1}{3} > 0, \text{and} \quad| q_1^\frac{1}{3}-q_2^\frac{1}{3}|<\omega^\frac{2}{3}<(q_1^\frac{1}{3}+q_2^\frac{1}{3}).
\end{align}
When the effect of the radiation pressure is neglected the positions of the non-collinear equilibrium points are defined by the co-ordinates $(x , y)$ (see Ref. \cite{PK15}, \cite{PR04}) where
\begin{align}\label{Eq:8}
x=\frac{1}{2}(1-2\mu), \quad y=\pm \sqrt{\omega^{-\frac{4}{3}}-\frac{1}{4}},
\end{align}
and consequently, these points exist only when $\omega \in (0, 2\sqrt{2})$. It is necessary to mention that the non-collinear libration points exist only for the particular value of $\omega$ which depend on $q_1$ and $q_2$ (where $q_1, q_2\neq 1$), the triangular libration points exist only when $\omega\in(\omega_1, \omega_2)$ where $\omega_i=\omega_i(q_1, q_2), i=1,2$. However, the collinear libration points exist for $\omega \in (0, \infty)$ and at $\omega=2\sqrt{2}$(where $q_1, q_2=1$) the non-collinear libration points coincide with $L_{1}$. In Fig. \ref{Fig:M1}, the movements of the position of libration points (as the value of parameter $\omega \in (\omega_1,\omega_2)$) are shown for constant values of the parameters $q_i$, and $\mu$ and different increasing values of $\omega$. We can observe that the libration point $L_{3}$ move towards the primary $P_1$ whereas the libration points $L_{1,2}$ move towards the primary $P_2$ as the value of $\omega$ increases. It is also observed that the non-collinear libration points originate in the vicinity of the libration point $L_{3}$ at $\omega\approx0.03097574$ and these points annihilate in vicinity of the libration point $L_{1}$ at $\omega\approx1.25144256$. Moreover, these particular values of $\omega$ are associated to the value of $q_1=0.15$ and $q_2=0.25$.
\begin{figure*}
\centering
\resizebox{\hsize}{!}{\includegraphics{Fig_3}}
\caption{A characteristic example of the consecutive steps that are followed by the Newton-Raphson iterator and the corresponding crooked path-line that leads to an equilibrium point. (colour figure online).}
\label{Fig:C1}
\end{figure*}
\section{The Newton-Raphson basins of convergence (N-RBoC)}
\label{Sec:4}
We perform a numerical analysis of the influence of angular velocity, mass parameter, radiation parameters on the geometry of the BoC linked with the libration points of the dynamical system by using the bivariate version of the N-R iterative scheme. This iterative method can be applicable to the of system of bivariate function $\mathbf{f(x)}=0$, using the iterative method:
\begin{equation}\label{Eq:}
\mathbf{ x}_{n+1}=\mathbf{x}_n-\mathbf{J}^{-1}\mathbf{f(x_n)}.
\end{equation}
Here, $\mathbf{f(x_n)}$ denotes the system of equations, whereas $\mathbf{J}^{-1}$ is denoting the inverse Jacobian matrix.
The iterative scheme for the $x$ and $y$ co-ordinates can be decomposed as:
\begin{eqnarray*}
x_{n+1}&=&x_n-\frac{\Omega_{x_n}\Omega_{y_ny_n}-\Omega_{y_n}\Omega_{x_ny_n}}{\Omega_{x_nx_n}\Omega_{y_ny_n}-\Omega_{x_ny_n}\Omega_{y_nx_n}},\\
y_{n+1}&=&y_n+\frac{\Omega_{x_n}\Omega_{y_nx_n}-\Omega_{y_n}\Omega_{x_nx_n}}{\Omega_{x_nx_n}\Omega_{y_ny_n}-\Omega_{x_ny_n}\Omega_{y_nx_n}},
\end{eqnarray*}
where the values of the $x$ and $y$ coordinates are represented by $x_n$ and $y_n$ respectively at the $n$-th step.
The philosophy which works in the background of the N-R iterative scheme is same as described in \cite{Z16}. The collection of all those initial conditions which converge to the particular attractor (i.e., the same root of the equations) compose the so-called N-RBoC. Further, we apply color coded diagrams (CCDs), where each pixel is linked with a non-identical color, as per the concluding state of the associated initial conditions, to classify the nodes in the orbital plane. The color codes for the domain of BoC linked to the respective libration points are same in each figure and the codes are same as in Fig. \ref{Fig:M1}. In Fig. \ref{Fig:C1}, it is depicted that the successive approximation points move in a crooked path and also for different initial conditions but for same attractor the number of required iterations to converge are different.
\begin{figure*
\centering
\includegraphics[scale=0.45]{Fig_4a}
\includegraphics[scale=0.45]{Fig_4b}\\
\includegraphics[scale=0.18]{Fig_4c}
\includegraphics[scale=0.435]{Fig_4d}\\
\includegraphics[scale=0.18]{Fig_4e}
\includegraphics[scale=0.43]{Fig_4f}
\caption{The (BoC) linked with the libration points on $(x, y)$-plane for $\mu=0.5, q_1=0.15, q_2=0.25$, and then the permissible range is $0.0309757<\omega<1.25144$: (a)\emph{top left:} for $\omega=0.0309757 + 0.001,$ (b)\emph{top right:} for $\omega=0.1034199519$, (c)\emph{middle left:} for $\omega=0.25$, (d)\emph{middle right:} for $\omega=0.375$, (e)\emph{bottom left:} for $\omega=0.95$, (f)\emph{bottom right:} for $\omega=1.2332376089$. The dots show the positions of libration points. (colour figure online).}
\label{Fig:Basin_1}
\end{figure*}
The numerical analysis with the Copenhagen case where the mass ratio $\mu=0.5$ and for varying values of the angular velocity are illustrated whereas the value of radiation parameters $q_1=0.15$, $q_2=0.25$. We start our analysis with Fig. \ref{Fig:Basin_1}a, which is depicted for $\omega=0.0309757 + 0.001$, very close to the critical value. We can observe that the domain of the BoC linked to the libration points is well formed and majority of the area of the finite domains are composed of the mixtures of various types of initial conditions whose final state is unpredictable. Consequently, these areas turn into the chaotic sea. Further, it is noticed that the large number of initial conditions (i.e., $45.25\%$ of the considered initial conditions) converges to the libration point $L_1$, which has infinite extent as well. Whereas $17.12\%$ of considered initial conditions converge to the $L_{4,5}$ and $4.66\%$ of initial conditions converge to the libration point $L_3$. The majority of the area of the finite domain of the BoC is occupied by those initial conditions which either converge to one of the non-collinear libration points. In Fig. \ref{Fig:Basin_1}b, when the $\omega=0.1034199519$, there exist five equilibrium points and the domain of the BoC linked to the equilibrium points $L_{3}$ and $L_{2}$ resembles to shape of exotic bugs with many legs and antennas which are separated by the chaotic strip composed of various type of initial conditions whose final states are not same. However, the entire $xy$-plane is covered by the well formed BoC. The extent of the domain of BoC linked to the equilibrium point $L_{1}$ is infinite on the other hand for all other equilibrium points these extents are finite. The domain of the BoC linked to the libration points $L_{4}$ and $L_{5}$ looks like butterfly wings whose wings boundaries are segregated by chaotic mixture of various types of initial conditions whose final state are different. It is observed that $6.87\%$ of considered initial conditions are converging to the equilibrium point $L_{3}$, $9.6\%$ of initial conditions are converging to $L_{2}$ whereas $11.74\%$ of initial conditions are converging to each of equilibrium points $L_{4}$ and $L_{5}$ and remaining are converging to $L_{1}$ which has infinite extent. Further, when the value of $\omega$ increases, the domain of BoC associated to the equilibrium points shrinks significantly except the domain of BoC linked to the libration point $L_{1}$ which consequently increases. Moreover, with the increase in value of angular velocity, the domain of the BoC linked to $L_{4}$ and $L_{5}$ becomes more regular but decreases. Further, $3.09\%$ of considered initial conditions converge to the libration point $L_{3}$, $3.79\%$ of initial conditions converge to libration point $L_{2}$ whereas $11.22\%$ of initial conditions converge to each of libration points $L_{4}$ and $L_{5}$ and $69.1\%$ of initial conditions converge to $L_{1}$ when $\omega= 0.375$.
The finite domain of the BoC continue to decrease with the increase in value of angular velocity and consequently the infinite domain of BoC increases. It is observed that the most of the finite region of the BoC is covered by the domain of the BoC associated with the non-collinear in-plane equilibrium points whereas the BoC linked to collinear equilibrium points $L_{2,3}$ look like a very small bugs without legs and antenna (see Fig. \ref{Fig:Basin_1}f ) when $\omega= 1.233237608897815$, only $0.16\%$ of the initial conditions converge to the libration point $L_{3}$, $0.22\%$ of initial conditions converge to $L_{2}$ whereas $18.55\%$ of initial conditions converge to each of libration points $L_{4}$ and $L_{5}$ which is slightly higher than the previous cases and remaining are converging to $L_{1}$ with infinite extent. It can be seen that each of the initial conditions converge to one of the attractors sooner or later.
\begin{figure*}
\centering
\includegraphics[scale=0.22]{Fig_5a
\includegraphics[scale=0.525]{Fig_5b}\\%\omega_1.5
\includegraphics[scale=0.525]{Fig_5c}\\%\omega_3.5
\caption{The (BoC) linked with the libration points on $(x, y)$-plane for $\mu=0.5, q_1=0.15, q_2=0.25$, and (a)\emph{top left:} for $\omega=0.02$, (b)\emph{top right:} for $\omega=1.5$, (c)\emph{bottom:} for $\omega=3.5$. The dots show the positions of libration points. (colour figure online).}
\label{Fig:Basin_2}
\end{figure*}
In Fig.\ref{Fig:Basin_2}, the domain of BoC is depicted for those values of $\omega$ for which there exist only collinear libration points, i.e., when $0< \omega <0.0309757$ or $\omega >1.25144$. When $\omega=0.02$ (see Fig.\ref{Fig:Basin_2}a) it is observed that the extent of BoC corresponding to each of libration points looks infinite. We believe that this happens since the value of $\omega$ is very close to zero. It is seen that $53.61\%$ of initial conditions converge to the libration point $L_{3}$ whereas $14.1\%$ and $32.28\%$ of the investigated initial conditions converge to $L_{1}$ and $L_{2}$ respectively. It is noticed that in Fig. \ref{Fig:Basin_2}(b, c) the domain of BoC linked to libration point $L_{1}$ has infinite extent and for remaining equilibrium points the domain of BoC are finite. However, in this case when three libration points exist, it is seen that for $\omega=1.5$ (Fig.\ref{Fig:Basin_2}b) only $1.6\%$ and $2.23\%$ of initial conditions converge to collinear libration points $L_{3}$ and $L_{2}$ respectively and rest of initial conditions converge to $L_{1}$ which has infinite extent. Further, when $\omega=3.5$ (Fig.\ref{Fig:Basin_2}c) only $0.459\%$ and $0.671\%$ of total considered initial conditions finally enticed by the collinear libration points $L_{3}$ and $L_{2}$, which unveil the fact that the domain of the BoC linked to these libration points reduces as value of $\omega$ increases.
Indeed, it is very remarkable to compare the Fig.\ref{Fig:Basin_2}a and Fig.\ref{Fig:Basin_1}a, where the number of libration points changes from three to five, respectively. It can be noticed that when value of $\omega$ is just above zero, the domain of the BoC linked to the libration point $L_3$ (see Fig.\ref{Fig:Basin_2}a) looks like antennas of the exotic bugs shaped region, constitutes the domain of the BoCs linked to the libration points $L_{4,5}$ when the value of $\omega$ increases slightly from the critical value (see Fig.\ref{Fig:Basin_2}a). This happens, since the non-collinear libration points just originate in the vicinity of the $L_3$ at the critical value of $\omega$. Further, when we compare Fig.\ref{Fig:Basin_1}f and Fig.\ref{Fig:Basin_2}b, it can be noticed that the domain of the BoC linked to non-collinear libration points $L_{4,5}$ shrinks to the BoC linked to the collinear libration point $L_1$ when the value of $\omega$ crosses the critical value. The main reason for this is the libration points $L_{4,5}$ annihilate in the vicinity of $L_1$ at the critical value of the $\omega$.
\begin{figure*}
\centering
\includegraphics[scale=0.54]{Fig_6a
\includegraphics[scale=0.55]{Fig_6b}
\includegraphics[scale=0.55]{Fig_6c}\\
\caption{The (BoC) linked with the libration points on $(x, y)$-plane for $\omega=0.5, q_1=q_2=1$, and (a)\emph{top left:} for $\mu=0.5$, (b)\emph{top right:} for $\mu=0.25$, (c)\emph{bottom:} for $\mu=0.05$. The dots show the positions of libration points. (colour figure online).}
\label{Fig:Basin_3}
\end{figure*}
In Fig. \ref{Fig:Basin_3}, the BoC are depicted for three different values of the mass parameter $\mu$ in the presence of fixed angular velocity $\omega=0.5$ when the primaries are not radiating. When $\mu=0.5$, the domain of BoC is symmetrical about both the axes and also the BoC have finite extent connected to all the equilibrium points except $L_{1}$ which has infinite extent. Further, when the value of the mass parameter decreases, the domain of the BoC connected to those libration points which have finite extent, expands. The domain of the BoC connected to the equilibrium points $L_{2, 3}$ appear as exotic bugs having various legs and antennas whereas the domain of BoC linked to non-collinear libration points appear as multiple butterfly wings when the value of mass parameter decreases these wings become larger. Further, only $9.92\%$ and $6.21\%$ of initial conditions converge to the collinear libration points $L_{3}$ and $L_{2}$ respectively, whereas $11.78\%$ of those initial conditions converge to $L_{4, 5}$ and remaining $60.27\%$ of initial conditions converge to $L_{1}$ when $\mu=0.25$. In Fig. \ref{Fig:Basin_3}c, when $\mu=0.05$, it can be seen that finite domain of the BoC expand and consequently the domain of BoC linked to $L_{1}$ decreases. In addition, the domain of BoC associated to $L_{3}$, which looks like exotic bugs, increases significantly.
If we compare Fig.3f in \cite{Z16} with the Fig. \ref{Fig:Basin_3}a which is illustrated for the same value of $\mu=0.5$ but for $\omega=0.5$, we can observe that the well formed domain of the BoC associated to the libration points having finite extent increases significantly. Moreover, the domain of BoC linked to $L_{4,5}$ which was regular becomes more chaotic when $\omega\neq 1$.
Moreover, when $\omega \neq 1$(in particular $<1$), the fractal structure (in the sense explained in Section \ref{Sec:5}) is higher, as also shown in Fig. \ref{Fig:BE1}b. We rather note the same behavior when $\mu$ decreases from $0.5$ to $0$ in both the cases when $\omega=0.5$ and $\omega=1$. Furthermore, the Fig. \ref{Fig:Basin_3} is done with a more refined initial conditions grid with respect to Fig.3f in \cite{Z16}, and this allows better visualization of the fractal structure.
In Fig. \ref{Fig:Basin_4N}(a,b), the BoC are presented for two different values of radiation parameter $q_1$ and fixed value of $\omega=0.85$. The topology of the domain of BoC associated with the equilibrium points significantly changes with the change in the radiation parameter. In both the cases the domain of BoC linked to libration point $L_1$ has infinite extent. It is noticed that when the value of $q_1$ increases from $0.004$ to $0.01$, the number of initial conditions which converge to $L_1$ has infinite extent, increases from $43.29\%$ to $52.56\%$ and consequently, the area related to finite extent decreases. Moreover, the initial conditions which compose the BoC of the finite extent linked to the non-collinear libration points decrease from $23.85\%$ to $19.83\%$ and the initial conditions which compose the BoC linked to the libration point $L_3$ also decrease whereas for $L_2$, it increase.
In Fig. \ref{Fig:Basin_4N}(c, d), the BoC are presented for two different values of radiation parameter $q_1$ and fixed value of $\omega=2$. We can observe that in this case there exist only three libration points, and the domain of the BoC linked to libration points $L_{2,3}$ increase and consequently, the domain of BoC associated to $L_1$ which has infinite extent decreases with the increase in value of $q_1$.
It is observed that the values of $\omega$ depend on the value of $q_i, i=1,2$, when $q_1=0.004, q_2=1$, the value of $\omega \in (\omega_1=0.771605, \omega_2=1.24732)$ for which the non-collinear libration points exist. Moreover, for Fig. \ref{Fig:Basin_4N}, the value of $\omega=0.85$ is close to the critical value of $\omega$ and for \ref{Fig:Basin_4N}a, where $q_1=0.004$ and for Fig. \ref{Fig:Basin_4N}b, where $q_1=0.01$. Therefore, a comparison with Fig.\ref{Fig:Basin_1}a and Fig. \ref{Fig:Basin_4N}a, which are both illustrated for the very close value of $\omega$ to its critical value, the topology of the domain of BoC in Fig.\ref{Fig:Basin_1}a is very noisy whereas in Fig.\ref{Fig:Basin_4N}a it looks much regular. It is also shown in Fig.\ref{Fig:BE1}b, that as the value of the $\omega$ is small, the value of the basin entropy increases. However, in both the cases the extent of the BoC linked to the central collinear libration point is infinite and for the remaining libration points it is finite. For Fig. \ref{Fig:Basin_4N}c, where $q_1= 0.01$, we get $\omega\in (\omega_1=0.694922, \omega_2=1.33999)$ and for \ref{Fig:Basin_4N}d, where $q_1=0.2$, we get $\omega\in (\omega_1=0.267535, \omega_2=1.99509)$ which shows the interval of $\omega$ for which five libration points exist, consequently in \ref{Fig:Basin_4N}(c,d) the value of $\omega$ is set out of the range so that only three equilibrium points exist. Moreover, \ref{Fig:Basin_4N}d is illustrated for very close value of $\omega$ to its critical value.
We compare Fig. \ref{Fig:Basin_2}(b,c) with \ref{Fig:Basin_4N}(c,d) and observe that the domain of BoC linked to $L_{2,3}$ increases in both cases when $\omega$ increases as well as $q_1$ increases. Further, in all the cases the topology of the BoC are symmetrical about the $x-$axis.
If we compare Fig.9a of Ref.\cite{Z16} (when $\omega=1$) with Fig. \ref{Fig:Basin_4N}b where the value of $\omega=0.85$, we can notice that the basins boundaries are more chaotic in comparison of the previous case moreover, finite regions of the domain of the BoC also increases significantly, however, not much changes are noticed in the topology of the BoC. Also the similar behaviour has been observed for the domain of BoC linked to the libration points $L_1, L_2$, i.e., it increases with the increase in the value of $q_1$. On the contrary, when the value of $\omega\neq1$, the domain of BoC linked to $L_{4,5}$ decreases with the increase in the value of $q_1$.
\begin{figure*}
\centering
\includegraphics[scale=0.45]{Fig_7a}
\includegraphics[scale=0.45]{Fig_7b}\\
\includegraphics[scale=0.45]{Fig_7c}
\includegraphics[scale=0.45]{Fig_7d}\\
\caption{The BoC linked with the libration points on $(x, y)$-plane. When $\omega=0.85, q_2=1, \mu=0.5$, and (a)\emph{top left:} for $q_1=0.004$, (b)\emph{top right:} for $q_1=0.01$. When $\omega=2, q_2=1$,(c)\emph{bottom left:} for $q_1=0.01$ (d)\emph{bottom right:} for $q_1=0.2$. The dots show the positions of libration points.(colour figure online).}
\label{Fig:Basin_4N}
\end{figure*}
\section{The Basin Entropy}
\label{Sec:5}
In the analysis of color coded diagrams (CCDs), it is observed that the basin of convergence is highly fractal in the locality of the basins boundaries which unveil the fact that it is quite impossible to judge the final state of the initial conditions falling inside these fractal regions. The term "fractal" is simply used in the text to unveil the particular area which shows the fractal-like geometry, without evaluating the fractal dimension (see \cite{AVS01}, \cite{AVS09}). Recently, a new tool to measure the uncertainty of the basins has been presented in paper \cite{Daz16}, is named as the “basin entropy” and refers to the geometry of the basins and consequently explore the concept of unpredictability and fractality in the context of BoC.
The philosophy that works in the background of the method is to split the phase space into $N$ small cells in which every cell contains at least one of the total number of final states $N_A$. In addition, the probability to evaluate the state $j$ in the $k-$th cell is denoted by $p_{j,k}$. Using the Gibbs entropy formulae, the entropy for $j-$th cell is
\begin{equation}\label{Eq:9}
S_j=\sum_{k=1}^{N_A}p_{j,k}\log\Big(\frac{1}{p_{j,k}}\Big).
\end{equation}
The average entropy for the total number of cells $N$ is called as basin entropy, i.e., $S_b$
\begin{equation}\label{Eq:10}
S_b=\frac{1}{N}\sum_{j=1}^{N}S_j=\frac{1}{N}\sum_{j=1}^{N}\sum_{k=1}^{N_A}p_{j,k}\log\Big(\frac{1}{p_{j,k}}\Big).
\end{equation}
It is necessary to mention that the result for the basin entropy is highly influenced by the total number of cells $N$, so that a precise value of $S_b$ can be obtained for larger value of $N$. In an attempt to overcome this problem, we use Monte Carlo procedure to select randomly small cells in the phase space, and we observe that for $N>2\times 10^5$ cells, the final value of the basin entropy remain unchanged.
\begin{figure*}
\centering
\resizebox{\hsize}{!}{\includegraphics{Fig_8}}
\caption{The evolution of the basin entropy $S_b$, of the configuration $(x, y)$ space with $\mu=0.5$: (a)\emph{left:} as a function of the perturbation parameter $\omega$. The vertical, dashed, green lines referred as the value of $\omega$ where the tendency of the parametric evolution of the basin entropy changes as these are the critical value of $\omega$. (b)\emph{right:} as a function of the perturbation parameter $q_1$ when $q_2=1$. The blue line shows the basin entropy when the value of $\omega=0.85$ and gray line shows the basins entropy when $\omega=1$. (colour figure online).}
\label{Fig:BE1}
\end{figure*}
In Fig. \ref{Fig:BE1}a, we have illustrated the parametric evolution of the basin entropy for various values of the angular velocity $\omega$, with $\omega\in(0, 3.5)$ when values of radiation parameters are fixed i.e., $q_1=0.15, q_2=0.25$. The gray dashed line shows the value of $\omega\approx0.03097574$, where the value of $S_b$ is maximum. We believe that the value of $S_b$ is maximum as the value of $\omega$ is very close to the critical value. It is further observed that the unpredictability linked to the N-RBoC for the restricted three-body problem is higher when the value of the $\omega\in(0.03097574, 1.25144256)$. However, it started decreasing when $\omega$ increases and at $\omega=0.65$ the value of $S_b=0.1738758105$ is recorded lowest for the all examined value of $\omega$, and again the value $S_b$ increases almost monotonically till $\omega=1.25144256$. The value of $S_b$ decreases monotonically when $\omega\in(1.25144256, 3.5)$, which refers to the case where only three collinear libration points exist. Moreover, in the Fig. \ref{Fig:BE1}b, the parametric evolution of the basin entropy is illustrated for the increasing value of the radiation parameter $q_1$ in the both cases i.e., when $\omega=1$ and when $\omega\neq1$.
We observe that the value of the basin entropy remains always higher when the angular velocity $\omega\neq1$. However, the similar tendency in the value of basin entropy has been noticed for the increasing value of the radiation parameter $q_1$ in both the cases.
It is necessary to mention the fact that to illustrate this diagram we have used the numerical results for various additional values of the angular velocity $\omega$ which are not necessarily presented in the Figs. \ref{Fig:Basin_1},\ref{Fig:Basin_2}, \ref{Fig:Basin_3} and \ref{Fig:Basin_4N}.
The main observations can be summarized as follows:
\begin{itemize}
\item When $\omega\approx1.25144256$ the domain of the BoC and basins boundaries become complicated and consequently the increase in the basin entropy is observed, which is around 0.31. However as the value of the angular velocity increases, the number of the libration points remains three and in this case the value of basin entropy $S_b$ decreases monotonically.
\item When $\omega= 3.5$ (see Fig. \ref{Fig:Basin_2}c) the value of $S_b\approx0.002981$ which is very close to zero since the topology of the BoC appears very smooth and further increase in $\omega$ shows the same tendency i.e., the smoothness in the basins increases.
\item When $ 0.0309757<\omega < 1.25144$, there exist five libration points and in this range of $\omega$ it can be observed that the value of the basin entropy changes abruptly. Consequently, for this range of $\omega$ the unpredictability linked to the N-RBoC for the R3BP in the presence of angular velocity $\omega$ is higher.
\item When the value of radiation parameter $q_1$ increases the value of the basin entropy decreases monotonically when $q_1\in(0,0.4)$ and increases monotonically when $q_1\in(0.4,1)$ and $\omega=1$. Whereas basin entropy decreases monotonically when $q_1\in(0,0.3)$ and increases monotonically when $q_1\in(0.3,1)$ and $\omega=0.85$. It is necessary to note that, although the curves are different, their behaviour are same.
\end{itemize}
\section{Discussion and conclusion}
\label{Sec:6}
In the present paper, we numerically explored the BoC by applying the bivariate version of iterative scheme in the photo-gravitational version of restricted problem of three bodies when the angular velocity is not equal to unity. The main outcomes of the present study can be summarized as follows:
\begin{itemize}
\item[*] There exist either five or three libration points for the system. For fixed values of the $q_i$, $\mu$ and varying values of $\omega$, it can be seen that the libration point $L_{3}$ moves towards the primary $P_1$ whereas the libration points $L_{1,2}$ move toward the primary $P_2$ as the value of $\omega$ increases. It is observed that the non-collinear libration points originate in the vicinity of the libration point $L_{3}$ at $\omega\approx |q_1^\frac{1}{3}-q_2^\frac{1}{3}|^ \frac{3}{2}$ and these points annihilate in the neighbourhood of the libration point $L_{1}$ when $\omega\approx(q_1^\frac{1}{3}+q_2^\frac{1}{3})^ \frac{3}{2}$.
\item[*]The attracting domains, linked to the equilibrium point $L_{1}$, extend to infinity, in all studied cases (except for Fig. $\ref{Fig:Basin_2}a$), while the domain of BoC associated to other libration points are finite. The BoC diagrams, on the configuration $(x, y)$ plane are symmetrical in all the studied cases, with respect to the horizontal $x$-axis.
\item[*]The numerical investigations suggest that the multivariate version of Newton-Raphson iterative scheme converges very fast for those initial conditions which lie in the vicinity of the libration point and converge very slow for those initial conditions which are lying in the vicinity of the basin boundaries. However, all the initial conditions converge to one of the attractors sooner or later.
\item[*]The numerical investigations unveil that for the interval of $\omega$ where only three libration points exist, the lowest value of $S_b$ is attained near $\omega=3.5$, whereas the highest value of basin entropy was achieved near $\omega\approx1.25144256$ which is the critical value of $\omega$ where the number of libration points changes. Moreover, for those intervals of $\omega$ in which five libration points exist, the maximum value of the basins entropy $S_b$ is achieved for the value of $\omega\approx0.03097574$, i.e., the starting critical value of $\omega$ when there exit five libration points. This reveals the unpredictability, regarding the attracting regions, in the photo-gravitational restricted three-body problem with angular velocity.
\end{itemize}
In addition, we have used the latest version 12 of Mathematica$^\circledR$ for all the graphical illustrations in this paper. In future, it is worth studying problem by using different iterative schemes to analyze the similarity as well as difference on the associated basins of attraction.
\section*{Compliance with Ethical Standards}
\begin{description}
\item[-] Funding: The authors state that they have not received any research
grant.
\item[-] Conflict of interest: The authors declare that they have no conflict of
interest.
\end{description}
\section*{Acknowledgments}
\footnotesize
The authors would like to express their warmest thanks to the anonymous referee for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.
|
1711.01296
|
\section{Introduction}
In the Lagrangian formalism, gravity and Yang-Mills theories seem to be completely unrelated theories. However, if one looks at the structure of their scattering amplitudes, an unexpected relationship between them arises. This relation is called the BCJ (Bern, Carrasco, Johannson) double copy~\cite{Bern:2008qj,Bern:2010yg,Bern:2010ue}, and at tree-level it is equivalent to the KLT (Kawai, Lewellen, Tye) relations \cite{Kawai:1985xq} between open and closed string amplitudes in the limit of large string tension. More explicitly, the BCJ double copy is a procedure that relates scattering amplitudes of different theories obtained by interchanging color and kinematic factors. This replacement is motivated by color-kinematics duality---the observation that the kinematic factors satisfy the same algebra as the color factors.
The BCJ relation holds both with and without supersymmetry, and even in the presence of spontaneous symmetry breaking \cite{Chiodaroli:2015rdg}. The original and best-known form of the double copy is the so-called Gravity=(Yang-Mills)$^2$ relationship, relating Einstein Gravity (the ``double copy'') to two copies of Yang-Mills theory (each of which is usually referred to as a``single copy''). By replacing the kinematic factors in the amplitudes of a Yang-Mills theory with color factors, this procedure can be further extended to yield the amplitudes for a bi-adjoint scalar theory (the ``zeroth copy''). The bi-adjoint scalar is a real massless scalar with a cubic interaction $f^{abc}\tilde{f}^{ijk}\phi^{a\,i}\phi^{b\,j}\phi^{c\,k}$, where $f^{abc}$ and $\tilde{f}^{ijk}$ are the structure constants of the groups $G$ and $\tilde{G}$ respectively, and $\phi^{a\,i}$ is in the bi-adjoint representation of $G\times\tilde{G}$.
This procedure can be applied more generally to theories other than Einstein gravity and Yang-Mills, such as Born-Infield theory and the Special Galileon \cite{Cheung:2017ems,Cheung:2017pzi,Cheung:2016prv,Carrasco:2016ldy,Cachazo:2014xea,Chen:2013fya}, or Einstein-Maxwell and Einstein-Yang-Mills theories~\cite{Nandan:2016pya,Stieberger:2015vya,Stieberger:2016lng,Chiodaroli:2014xia,Chiodaroli:2015wal,Chiodaroli:2017ngp,Chiodaroli:2015rdg}.The extent of this remarkable relation between scalar, gauge and gravity amplitudes is the subject of ongoing research, and we refer the reader to~\cite{Carrasco:2015iwa} for a more extensive review of the literature.
The BCJ double copy is a perturbative result that has been proven at tree level \cite{Carrasco:2015iwa,Stieberger:2009hq,BjerrumBohr:2010zs,Mafra:2011kj,Cachazo:2012uq,Feng:2010my,Tye:2010dd}, but is also believed to hold at loop level~\cite{Bjerrum-Bohr:2013iza,Boels:2013bi,Carrasco:2011mn,Carrasco:2012ca,Bern:1997nh,Bern:1998ug,Bern:2013uka,Bern:2014sna,Bern:1994zx}. Recently, BCJ-like double copy relations between classical solutions have been suggested using different perturbative approaches~\cite{Saotome:2012vy,Neill:2013wsa,Luna:2016hge,Goldberger:2016iau,Goldberger:2017frp}. More broadly, this suggests the possibility of using the double copy technique to generate classical gravitational solutions from simpler, classical gauge configurations. Although this is an intrinsically perturbative procedure, it is interesting to understand the extent to which an exact relationship might persist at the classical non-perturbative level. This idea was first explored in \cite{Monteiro:2014cda}, where it was shown that a classical double copy that resembles the BCJ double copy exists for spacetimes that can be written in (single or multiple) Kerr-Schild (KS) form (see also~\cite{Luna:2015paa,Luna:2016due,Ridgway:2015fdl}). Other approaches to the classical double copy were considered in \cite{Anastasiou:2014qba,Anastasiou:2017nsz,Borsten:2015pla,Cardoso:2016amd,Cardoso:2016ngt,Chu:2016ngc}.
Most of the work thus far has been devoted to the study of solutions in asymptotically-flat space-times. Recently, however, it has been shown that the BCJ double copy of three-point scattering amplitudes is successful also in certain curved backgrounds~\cite{Adamo:2017nia}. More precisely, it was shown that graviton amplitudes on a gravitational sandwich plane wave are the double copy of gluon amplitudes on a gauge field sandwich plane wave. It is therefore natural to wonder to what degree the classical double copy procedure can also be extended to curved backgrounds. In this paper, we take a first step in this direction by focusing on curved, maximally symmetric spacetimes. The viewpoint taken here is slightly different from the one adopted in~\cite{Adamo:2017nia}, in that there the curved background was also ``copied''. At the level of the classical double copy, this approach would associate to the (A)dS background a single copy sourced by a constant charge filling all space. Here, on the other hand, we will treat the curved (A)dS background as fixed and find the single and zeroth copy solutions in de Sitter (dS) or Anti-de Sitter (AdS) spacetimes.
The rest of the paper is organized as follows. In Secs. \ref{bcj} and \ref{sec:classical double copy}, we briefly review the BCJ and classical double copies respectively, and their relation to each other. In Secs. \ref{Sch} and \ref{kerr}, we construct single and zeroth copies of the (A)dS-Schwarzschild and (A)dS-Kerr black hole solutions in $d\geqslant 4$ spacetime dimensions. We then show how to use the single copy obtained from these solutions to construct the full Einstein-Maxwell solution for the charged (A)dS-Reissner-Nordstrom and (A)dS-Kerr-Newman black holes (Sec. \ref{EM}). In Sec. \ref{others}, we consider the double copy construction for black strings and black branes in (A)dS, and in Sec. \ref{waves} we turn our attention to time-dependent solutions by studying the case of waves in (A)dS. Finally, in Sec. \ref{btz} we consider the Yang-Mills and scalar copies for the BTZ (Ba\~nados, Teitelboim, Zanelli) black hole. This last example extends the classical double copy beyond the regime of applicability of the BCJ procedure for amplitudes, since in $d=3$ there are no graviton degrees of freedom to which copies could correspond. We conclude by summarizing our results and discussing future directions in Sec. \ref{dis}.
\\
\emph{Note added:} During the completion of this paper, \cite{Bahjat-Abbas:2017htu} appeared, which also considers the classical double copy in curved spacetimes. In particular, two different kinds of double copies were considered. The so-called ``Type A'' double copy consists of taking Minkowski as a base metric and mapping both the background and perturbations. Thus, it is close in spirit to the approach followed in~\cite{Adamo:2017nia}. The ``Type B'' double copy considered in \cite{Bahjat-Abbas:2017htu} instead keeps the curved background fixed, which is the same prescription that we have adopted here. However, the examples analyzed in this paper are different from the ones in \cite{Bahjat-Abbas:2017htu}. Moreover, here we have paid particular attention to obtaining the correct localized sources for the Yang-Mills and scalar copies, and in analyzing the equations of motion that these copies satisfy in $d\geq4$ for both stationary and time-dependent cases. Contrary to the solutions analyzed in \cite{Bahjat-Abbas:2017htu}, we find that all examples we have considered lead to reasonable ``Type B'' single and zeroth copies (although time-dependent solutions warrant additional study).
\section{The BCJ double copy}\label{bcj}
To set the stage, we start with a concise review of the Gravity=(Yang-Mills)$^2$ correspondence.
The central point is that it is possible to construct a gravitational scattering amplitude from the analogous object for gluons. The gluon scattering amplitudes in the BCJ form can be expressed schematically as
\begin{equation} \label{A_YM}
A_\text{YM}=\sum_i\frac{N_i\,C_i}{D_i} \ ,
\end{equation}
where the $C_i$'s are color factors, the $N_i$'s are kinematic factors in the BCJ form, and the $D_i$'s are scalar propagators. It is convenient to expand the factors in the numerator in the half-ladder basis~\cite{DelDuca:1999rs}, so that they read
\begin{equation}
C_i=\sum_{\alpha}\gamma_i(\alpha)C(\alpha),\quad\qquad\qquad N_i=\sum_{\beta}\sigma_i(\beta)N(\beta) \ .
\end{equation}
Here, the $\gamma_i(\alpha)$'s and $\sigma_i(\beta)$'s are the expansion coefficients, $C(\alpha)$ is the color basis whose elements consist of products of structure constants, and $N(\beta)$ is the kinematic basis, the elements of which are products of polarization vectors and momenta.
The double copy procedure consists of exchanging the color factors $C_i$ in the numerator on the RHS of Eq. \eqref{A_YM} for a second instance of kinematic factors $\tilde N_i$, which in general may be taken from a different Yang-Mills theory and thus differ from the $N_i$'s. Remarkably, this replacement gives rise to a gravitational scattering amplitude,
\begin{equation}
A_\text{G}=\sum_i\frac{\tilde N_i\,N_i}{D_i} \equiv \sum_{\beta}\tilde N(\beta)A_\text{YM}(\beta) \equiv\sum_{\alpha \beta} N(\alpha) \tilde N(\beta)A_\text{S} (\alpha|\beta) \ , \label{ampl}
\end{equation}
where $A_\text{YM}(\beta)$ is the \emph{color-ordered} Yang-Mills amplitude given by
\begin{equation}
A_\text{YM}(\beta)=\sum_{\alpha} N(\alpha)A_\text{S}(\alpha|\beta) \ ,
\end{equation}
and $A (\alpha|\beta)$ is the \emph{doubly color-ordered} bi-adjoint scalar amplitude,
\begin{equation}
A_\text{S} (\alpha|\beta)=\sum_i\frac{\gamma_i(\alpha)\sigma_i(\beta)}{D_i} \ .\label{ampl3}
\end{equation}
Different choices of kinematic factors $N_i$ and $\tilde N_i$ yield gravitational amplitudes with the same number of external gravitons but different intermediate states. As we will see in the next section, the Kerr-Schild formulation of the classical double copy will be somewhat reminiscent of the relations \eqref{ampl}--\eqref{ampl3}, although to the best of our knowledge the exact connection remains to be worked out.
It is also worth mentioning that, besides the double copy procedure, other relations between scattering amplitudes have been shown to exist---see for instance~\cite{Cheung:2017ems}. One such relation corresponds to the multiple trace operation of \cite{Cheung:2017ems}, which relates a gravity amplitude to an Einstein-Maxwell one. This operation consists of applying trace operators $\tau_{ij}$ to the original amplitude. The trace operator is defined as $\tau_{ij}=\partial_{e_i\cdot e_j}$, where $e_i$ denotes the polarization vector of the particle $i$. Each trace operator reduces the spin of particles $i$ and $j$ by 1, and places them in a color trace. Applying these trace operators to a graviton amplitude exchanges some of the external gravitons for photons, which leads to an Einstein-Maxwell amplitude. In Sec. \ref{EM}, we will suggest a classical counterpart of this relation. A similar relation exists between pure Yang-Mills and Yang-Mills-scalar amplitudes, where the Yang-Mills and scalar field are coupled with the usual gauge interactions~\cite{Cheung:2017ems}.
\section{The classical double copy} \label{sec:classical double copy}
Let us now turn our attention to the classical double copy, first introduced in \cite{Monteiro:2014cda}. In its simplest implementation, one considers a space-time with a metric that admits a Kerr-Schild form with a Minkowski base metric, i.e.
\begin{equation}
g_{\mu\nu} = \eta_{\mu\nu} + \phi \, k_\mu k_\nu \ , \label{KSm}
\end{equation}
where $\phi$ is a scalar field, and $k_\mu$ is a vector that is null and geodetic with respect to both the Minkowski and the $g_{\mu\nu}$ metrics:
\begin{equation}
g^{\mu\nu}k_\mu k_\nu=\eta^{\mu\nu}k_\mu k_\nu=0, \qquad\qquad\quad k^\mu\nabla_\mu k^\nu=k^\mu\partial_\mu k^\nu=0 \ .
\end{equation}
For our purposes, the crucial property of a metric in Kerr-Schild form is that the Ricci tensor ${R^\mu}_\nu$ turns out to be linear in $\phi$ provided all indices are raised using the Minkowski metric~\cite{Stephani:2003tm}.
Starting from the metric \eqref{KSm} in Kerr-Schild form, one can define a ``single copy'' Yang-Mills field via
\begin{equation}
A_\mu^a=c^a k_\mu\phi \ , \label{A}
\end{equation}
where the $c^a$ are constant but otherwise arbitrary color factors. Then, if $g_{\mu\nu}$ is a solution to Einstein's equations, the Yang-Mills field \eqref{A} is guaranteed\footnote{This is true as long as we pick the correct splitting between the null KS vector and the KS scalar. We will discuss this further below.} to satisfy the Yang-Mills equations, provided the gravitational coupling is replaced by the Yang-Mills one,~i.e.
\begin{equation}
8\pi G\rightarrow g, \label{subs}
\end{equation}
and any gravitational source is replaced by a color source. In fact, because of the factorized nature of the ansatz \eqref{A}, this implies that the field $A_\mu \equiv k_\mu\phi$ defined without color factors satisfies Maxwell's equations, in which case the color charges can be thought of just as electric charges.\footnote{Magnetic charges are instead related to NUT charges~\cite{Luna:2015paa}, which we will not consider in this paper.} In what follows we will restrict our attention to $A_\mu$, which we will also refer to as a single copy, with a slight abuse of terminology.
We can further combine the Kerr-Schild scalar $\phi$ with two copies of the color factors to define a bi-adjoint scalar
\begin{equation}
\phi^{a\, b}= c^a c'^{b}\phi \ , \label{phi}
\end{equation}
which satisfies\footnote{Again, this is true as long as we pick the correct splitting between the KS vector and the KS scalar.} the linearized equations $\bar \nabla^2 \phi^{a\, b} = c^a c'^{b} \bar \nabla^2 \phi = 0$. As in the case of the gauge field, in the following we will restrict out attention to the field $\phi$ stripped of its color indices. It is worth emphasizing that the equations of motion for the single copy $A_\mu$ and the zeroth copy $\phi$ turn out to be linear precisely because of the Kerr-Schild ansatz~\cite{Monteiro:2014cda}.
It is interesting to notice that the expressions for the ``metric perturbation'' $k_\mu k_\nu \phi$, the single copy $A_\mu = k_\mu \phi$, and the zeroth copy $\phi$ bear a superficial and yet striking similarity with the BCJ amplitudes in Eqs. \eqref{ampl}--\eqref{ampl3}. Specifically, a comparison between the two double copy procedures would seem to suggest that the vector $k_\mu$ somehow corresponds to the kinematic factors $N(\alpha)$, while the scalar $\phi$ is the analogue of $A_S(\alpha|\beta)$. Finally, the color factors $c^a$ can be thought of as the analogue of the color factors $C(\alpha)$. Although an exact mapping between the two double copies has not yet been derived, several analyses suggest that they are indeed related~\cite{Monteiro:2014cda,Luna:2015paa,Ridgway:2015fdl,Neill:2013wsa,Luna:2016hge,Luna:2016due,Goldberger:2016iau,Goldberger:2017frp}.
\section{Extending the classical double copy to curved spacetime}
In the following sections, we extend the classical double copy procedure to curved, maximally symmetric spacetimes: AdS and dS. One example of the classical double copy in a maximally symmetric spacetime was already considered in~\cite{Luna:2015paa}, which studied the Taub-NUT-Kerr-de Sitter solution. Our goal here is to consider the double copy procedure in (A)dS more systematically, and to obtain a more complete understanding of what happens in curved backgrounds by finding additional examples in which the double copy procedure is applicable.
To this end, we will use the generalized Kerr-Schild form of the metric
\begin{equation} \label{KS g bar}
g_{\mu\nu}=\bar g_{\mu\nu}+ \phi \, k_\mu k_\nu \ ,
\end{equation}
where the base metric $\bar g_{\mu\nu}$ is now (A)dS (unless otherwise specified), while $k_\mu$ is again null and geodetic with respect to both the full and base metrics. A detailed analysis of these kinds of metrics can be found in \cite{TAUB1981326,0264-9381-4-5-005,Malek:2010mh}. Even with a more general choice for the base metric, the Ricci tensor $R^{\mu}_{\ \nu}$ is still linear in $\phi$~\cite{Stephani:2003tm}:
\begin{equation} \label{R curved}
R^\mu{}_\nu = \bar R^\mu{}_\nu -\phi k^\mu k^\lambda \bar R_{\lambda \nu} +\tfrac{1}{2} \left[ \bar \nabla^\lambda \bar \nabla^\mu (\phi k_\lambda k_\nu ) + \bar \nabla^\lambda \bar \nabla_\nu (\phi k^\mu k_\lambda ) - \bar \nabla^2 (\phi k^\mu k_\nu ) \right] \ ,
\end{equation}
and the single and zeroth copy of the metric \eqref{KS g bar} are still defined by Eqs. (\ref{A}) and (\ref{phi}) respectively.
At this point, we should discuss one aspect of the classical double copy construction that so far has not been mentioned in the literature but is nevertheless crucial to ensure that the classical double copy procedure gives rise to sensible results. For any given choice of coordinates that allows the metric to be written in the Kerr-Schild form, the null vector $k_\mu$ and the scalar $\phi$ are not uniquely determined, since Eq. \eqref{KS g bar} is invariant under the rescalings
\begin{equation}
k_\mu \to f k_\mu, \qquad \qquad \quad \phi \to \phi / f^2 \ , \label{resc}
\end{equation}
for any arbitrary function $f$. If we demand that the null vector $k_\mu$ is geodetic, this imposes restrictions on $f$, but does not fix it completely. Of course, this ambiguity is immaterial when it comes to the gravitational theory, since the Ricci tensor in Eq. \eqref{R curved} is also invariant under this redefinition. However, the single and zeroth copies defined in Eqs. \eqref{A} and \eqref{phi} are not, and neither are the equations that they satisfy. It is worth stressing that this ambiguity is not a peculiarity of curved space-time, but a general feature of any metric in Kerr-Schild form.
To further illustrate this point, it is convenient to recast Eq. \eqref{R curved} in the following form:
\begin{equation}
2(\bar R^\mu{}_\nu - R^\mu{}_\nu)= \left[ \bar \nabla_\lambda F^{\lambda\mu} + \tfrac{(d-2)}{d(d-1)} \bar R A^\mu \right] k_\nu + X^\mu{}_\nu +Y^\mu{}_\nu \ , \label{rr}
\end{equation}
where here and in what follows, $F^{\lambda\mu}$ is the usual field strength for an abelian gauge field\footnote{Our anti-symmetrization conventions are such that $B_{[\mu\nu]} \equiv B_{\mu\nu}-B_{\nu\mu}$.}, and its components have been raised using the base metric. Moreover, we have simplified our notation by introducing the following quantities:
\begin{align}
X^\mu{}_\nu\equiv& - \bar\nabla_\nu\left[A^\mu\left(\bar\nabla_\lambda k^\lambda+\frac{k^\lambda \bar\nabla_\lambda \phi}{\phi}\right)\right] \ ,\\
Y^\mu{}_\nu\equiv& \, F^{\rho\mu}\bar{\nabla}_\rho k_\nu-\bar{\nabla}_\rho\left(A^\rho \bar{\nabla}^\mu k_\nu-A^\mu \bar{\nabla}_\rho k_\nu \right) \ .
\end{align}
When the full metric solves the Einstein equations with a cosmological constant, the LHS is equal to $-16\pi G\left({T^\mu}_\nu-\delta^\mu_\nu T/(d-2)\right)$, with ${T^\mu}_\nu$ the stress-energy tensor. If we contract Eq. \eqref{rr} with a Killing vector $V^\nu$ of both the base and full metric, we obtain the equation of motion for the single copy $A^\nu$ in $d$ dimensions:
\begin{equation}
{\bar\nabla}_\lambda F^{\lambda\mu} + \tfrac{(d-2)}{d(d-1)} \bar R A^\mu + \tfrac{V^\nu}{V^\lambda k_\lambda}\left(X^\mu{}_\nu + Y^\mu{}_\nu\right) =8\pi G\, J^\mu \ , \label{Aeom}
\end{equation}
where we have defined
\begin{equation}
J^\mu\equiv-\tfrac{2 V^\nu}{V^\rho k_\rho}\left({T^\mu}_\nu-\delta^\mu_\nu \tfrac{T}{d-2}\right). \label{current}
\end{equation}
To obtain the zeroth copy equation, we can further contract Eq. \eqref{rr} with another Killing vector $V_\mu$ and find
\begin{equation}
\bar \nabla^2\phi=j-\tfrac{(d-2)}{d(d-1)} \bar R \phi -\tfrac{V_\nu}{( V^\mu k_\mu)^2}\left(V^\mu X^\nu{}_\mu + V^\mu Y^\nu{}_\mu+ Z^\nu\right) \label{phieom}
\end{equation}
where
\begin{equation}
Z^\nu\equiv( V^\rho k_\rho)\bar\nabla_\mu\left(\phi\bar\nabla^{[\mu}k^{\nu]}- k^\mu\bar\nabla_\nu\phi\right),
\end{equation}
and the source is defined as
\begin{equation}
j=\frac{V_\nu J^\nu}{V^\rho k_\rho}.
\end{equation}
In what follows, we will use the timelike Killing vector for stationary solutions, and the null Killing vector for wave solutions. The Killing vector allows us to find the correct sources for the single and zeroth copies. Clearly, Eq. \eqref{Aeom} is not invariant under the rescaling of Eq. \eqref{resc}. This freedom allows us to choose the null vector and scalar such that the copies satisfy `reasonable' equations of motion. By this, we mean that when there is a localized source on the gravitational side, we obtain a localized source in the gauge and scalar theories; when there is no source for Einstein's equations, there is no source in the Abelian Yang-Mills and scalar equations either. At this stage, we are unable to formulate a more precise criterion that selects the correct splitting between $\phi$ and $k_\mu$ based on fundamental principles. However, we believe this is an important question to address in future work and we will touch upon it again in the final section of this paper.
Before constructing explicit examples, it is worth discussing a few details of the two cases of interest in this paper: stationary spaces and waves. First, note that the terms in $X^\mu{}_\nu$ in the inner parentheses correspond to the expansion of $k^\mu$ and the derivative of $\phi$ along the direction of $k^\mu$. For stationary solutions, where $\bar\nabla_\lambda k^\lambda\neq0$ and $k^\lambda \bar\nabla_\lambda \phi\neq0$ , $X^\mu{}_\nu$ is non-zero. In these cases, we may choose $k_\mu V^\mu$ (which corresponds to choosing the scaling function $f$ between $k_\mu$ and $\phi$) such that the single copy satisfies Maxwell's equations. On the other hand, for wave solutions the expansion term is zero and the null vector is orthogonal to the gradient of $\phi$, so that $X^\mu{}_\nu=0$. In order to obtain a reasonable equation of motion for the wave solutions, we require that the terms in $ Y^\mu{}_\nu$ that contain derivatives of the gauge field cancel out. These terms can be rewritten as $ Y^\mu{}_\nu\supset \left(F^{\rho\mu}+\nabla^\rho A^\mu\right) \nabla_\rho k_\nu$; setting them to zero is equivalent to choosing $k_\mu$ such that $\bar\nabla_\rho k_\nu=0$, with $\rho\neq u$, for a wave traveling in the direction of the light-cone coordinate $u$. One can see that this choice will in fact set $Y^\mu{}_\nu=0$. This choice doesn't completely fix $k_\mu$; in fact, we can still do a rescaling as in Eq.\eqref{resc} with $f=f(u)$. The fact that we can re-scale our solution in such a way is a property of wave solutions; multiplying by $f(u)$ only changes the wave profile. Remarkably, we seem unable to choose $V^\mu k_\mu$ in such a way as to also cancel the second term in Eq. \eqref{Aeom}. Therefore, in this case the single copy satisfies an equation in which the gauge symmetry is broken by a non-minimal coupling to the background curvature. Furthermore, once we have fixed the splitting we find that
\begin{equation}
\frac{V^\nu Z^\nu}{V^\rho k^\rho} =-\frac{2(d-3)}{d(d-1)}\bar R\phi,
\end{equation}
in both cases. This means that the zeroth copy has a mass proportional to the Ricci scalar. For the special case of solutions in 4d, the stationary solutions follow the conformally invariant equation and the wave solutions the equation for a massless scalar.
\section{(A)\lowercase{d}S-Schwarzschild} \label{Sch}
The simplest example of the classical double copy procedure in maximally symmetric curved spacetimes is the (A)\lowercase{d}S-Schwarzschild black hole in $d=4$ space-time dimensions. In order to find the corresponding single and zeroth copies, we write this solution in the Kerr-Schild form, using an (A)dS base metric in global static coordinates,
\begin{equation}
\bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu = -\left(1-\frac{\Lambda \, r^2}{3} \right) \mathrm{d} t^2 + \left(1-\frac{\Lambda \, r^2}{3} \right)^{-1} \mathrm{d} r^2 + r^2 \mathrm{d} \Omega^2
\end{equation}
with $\Lambda$ the cosmological constant, and choose the null vector $k_\mu$ and scalar function $\phi$ in the following way:
\begin{equation}
k_\mu dx^\mu = \mathrm{d} t + \frac{\mathrm{d} r}{1-\Lambda \, r^2/3}\ , \qquad \qquad\qquad \phi =\frac{2 G M}{r} \ , \label{kmu phi SAdS}
\end{equation}
In this case, the full metric $g_{\mu\nu}$ defined by \eqref{KS g bar} is a solution to the Einstein equations
\begin{equation}
G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu} \ .
\end{equation}
One can remove the singularity at $r=0$ by including a localized source with stress-energy tensor
\begin{equation}
T^\mu{}_\nu =\frac{M}{2}\text{diag} (0,0,1,1) \delta^{(3)} (\vec{r}) \ , \label{TS}
\end{equation}
where the $\delta^{(3)} (\vec{r})$ should be expressed in spherical coordinates. Thus, the only two nonzero components of the stress-energy tensor in Kerr-Schild coordinates are the angular ones along the diagonal.
At first sight, this result might seem at odds with one's ``Newtonian intuition''. In fact, in the usual Newtonian limit the dominant component of Einstein's equation is the $(t,t)$ one, which reduces to a Poisson's equation for $\delta g_{tt}$ sourced by $T_{tt}$. However, this result is based on the assumption that the off-diagonal components of the metric are negligible. In Kerr-Schild coordinates, the $\delta g_{tr}$ components are of the same order in $GM/r$ as the perturbations to the diagonal components of the metric---they are all of order $\phi$. In this case, the Poisson's equation for the Newtonian potential arises from the angular components of Einstein's tensor, rather than from the $(t,t)$ component---a result in agreement with what was found in Sec. III of~\cite{Ridgway:2015fdl}.
We can now follow the procedure discussed in Sec. \ref{sec:classical double copy} to construct the single and zeroth copies. It is in fact easy to show using the replacement rule \eqref{subs} that the single copy $A_\mu=k_\mu\phi$ with $k_\mu$ and $\phi$ given by Eq. \eqref{kmu phi SAdS} satisfies Maxwell's equations on an (A)dS background,
\begin{equation}
\bar \nabla_\mu F^{\mu\nu}=g\,J^\nu \ , \label{YMeom}
\end{equation}
with a localized, static source given by
\begin{equation}
J^\mu = M\, \delta^{(3)}(\vec{r}) \, \delta^\mu_0 \ .
\end{equation}
This source can be derived from Eq. \eqref{current} using the timelike Killing vector of the Schwarzschild metric. As expected, this source describes a static point-particle with charge $Q=M$ in (A)dS, in perfect analogy with the flat case.
The zeroth copy $\phi$ instead satisfies the equation of motion
\begin{equation}
\left(\bar \nabla^2-\frac{\bar R}{6}\right)\phi= j \ , \label{seom}
\end{equation}
with a localized source $j = M\, \delta^{(3)}(\vec{r})$. Thus, moving away from a flat background it becomes apparent that the zeroth copy satisfies the equation for a conformally coupled scalar field rather than simply $\bar \nabla^2 \phi = j$. This was first noticed in \cite{Luna:2015paa}, where it was also argued that this might be tied to the conformal symmetry of the Yang-Mills equations in $d=4$. In fact, in $d \neq 4$ the non-minimal coupling between $\phi$ and the Ricci scalar does not have a conformal value~\cite{Luna:2015paa}, as we will see in the next section.
Let us now restrict our attention to the dS solution and consider the case of small dS black holes, i.e. black holes such that $0<M<M_\text{max}\equiv1/(3G\Lambda^{1/2})$. This spacetime has both a cosmological horizon and a black hole horizon. As the mass increases, the black hole horizon grows and the cosmological horizon shrinks. At the particular value $M=M_\text{max}$, both horizons have the same area but the distance between them remains finite. In this limit, the singularity disappears and the patch between the two horizons corresponds to dS$_2\times S^2$. This spacetime is known as Nariai solution~\cite{nariai1950some,nariai1951some}. At the level of the single and zeroth copies, however, there is no privileged value that the charge can take---all solutions to Eqs. \eqref{YMeom} and \eqref{seom} look qualitatively the same regardless of the value of $M$. This can be easily understood from the fact that, if there existed a special value for the charge $Q_*$, then in the spirit of the double copy procedure it should be such that $Q_* \propto M_{\rm max} / M_{\rm pl} \propto 1/\sqrt{G \Lambda}$. However, by considering a fixed background for the single and zeroth copy, we are effectively working in the limit $G \Lambda \to 0$ while keeping $M^2 /\Lambda$ constant, in which case $Q_* \to \infty$. Another point of view is that on the gravitational side, $M=M_{\rm max}$ is a special value due to the existence of horizons. This is strictly a gravitational property which does not have an analogue in Yang-Mills or scalar theories.
Finally, it is instructive to discuss how the wrong choice of the Kerr-Schild vector $k_\mu$ can give rise to an unreasonable double copy. Instead of the definitions in Eq.\eqref{kmu phi SAdS}, we will pick the splitting such that
\begin{equation}
k_\mu dx^\mu = f(\theta) \left( \mathrm{d} t + \frac{\mathrm{d} r}{1-\Lambda \, r^2/3} \right) \ , \qquad\qquad \phi =\frac{1}{f(\theta)^2}\frac{2 G M}{r} \ , \label{Alt kmu phi SAdS}
\end{equation}
where $f(\theta)$ is an arbitrary function. This choice preserves several properties of the Kerr-Schild vector, namely $k_\mu$ is null, geodetic, shear-free, and twist-free\footnote{In fact, $k_\mu$ is null, geodetic, shear-free, and twist-free for an arbitrary function $f(\theta,\phi)$, but we restrict ourselves to $f(\theta)$ for simplicity.}. As before, we can define the single copy as $A_\mu=k_\mu\phi$ and find that it satisfies Eq.\eqref{YMeom} with a source current given by
\begin{equation}
J^\mu = M\, \delta^{(3)}(\vec{r}) \, \delta^\mu_0 + \tilde{j}^\mu \ ,
\end{equation}
where
\begin{equation}
\tilde{j}^\mu=-\frac{g(\theta)}{r^2
f(\theta)^3 }\left(\frac{\delta^\mu_t}{\left(1-\Lambda r^2/3\right)}+ \delta^\mu_r\right) , \quad g(\theta)=2 f'(\theta)^2-f(\theta) \left(f''(\theta)+\cot
(\theta) f'(\theta)\right) \ .
\end{equation}
This extra term in the current is clearly non-localized and changes the total charge. Given our criteria for a reasonable single copy, this term is unacceptable. We conclude that the choice Eq.\eqref{Alt kmu phi SAdS} with an arbitrary function $f(\theta)$ is incorrect. We can see that, taking $f(\theta)=1$ sets $\tilde{j}=0$ recovering the correct result for the single copy.
\section{Kerr-(A)\lowercase{d}S} \label{kerr}
We now consider a more involved example, namely that of a rotating black hole in (A)dS. As in the previous section, we will derive the single and zeroth copies of the Kerr-(A)dS solution in $d=4$ by casting the full metric in a Kerr-Schild form. To this end, it is convenient to express the base (A)dS metric in spheroidal coordinates~\cite{Gibbons:2004uw},
\begin{align}
\bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu = -\frac{\Delta}{\Omega}\, f\, \mathrm{d} t^2+ \frac{\rho^2 \mathrm{d} r^2}{(r^2+a^2)f} + \frac{\rho^2 \mathrm{d} \theta^2}{\Delta} + \frac{(r^2+a^2)}{\Omega}\sin^2 \theta \, \mathrm{d} \varphi^2 \ ,
\end{align}
where we have defined
\begin{equation}
f \equiv 1-\frac{\Lambda \, r^2}{3}, \qquad \rho^2\equiv r^2+a^2\cos^2{\theta},\qquad \Delta\equiv 1+\frac{\Lambda}{3}a^2\cos^2{\theta},\qquad \Omega\equiv 1+\frac{\Lambda}{3}a^2 \ .
\end{equation}
The corresponding null vector and scalar function read~\cite{Gibbons:2004uw}
\begin{align}
k_\mu \mathrm{d} x^\mu = \frac{\Delta \mathrm{d} t}{\Omega} + \frac{\rho^2 \mathrm{d} r}{(r^2+a^2)f} - \frac{a\sin^2{\theta} \, \mathrm{d} \theta}{\Omega} \ , \qquad \qquad \phi =\frac{2 M G\, r}{\rho^2}.
\end{align}
It is easy to see that, when $a \to 0$, these expressions reduce to the ones used in the previous section for the (A)dS-Schwarzschild solution. Notice also that the Kerr-(A)dS black hole solution becomes singular if $|a|\geq\sqrt{3 /(-)\Lambda}$.
The source of the metric corresponds to a negative proper surface density given by a disk of radius $a$ localized at $r=0$. This can be seen by considering the induced metric at $r=0$ which corresponds to the history of the disk,
\begin{equation}
\mathrm{d} s^2=-\Delta \mathrm{d} \tilde t^2 + \Delta^{-1} \mathrm{d} R^2 + R^2 \mathrm{d} \tilde \varphi^2,
\end{equation}
where the new coordinates are given by
\begin{equation}
\tilde{t}=t/\Omega,\qquad\qquad R=a \sin{\theta},\qquad\qquad\tilde{\varphi}=\varphi/\Omega.
\end{equation}
This disk is rotating about the $z$ axis with superluminal velocity and is balanced by a radial pressure. The corresponding stress-energy tensor can be written as
\begin{equation}
T^{\mu}{}_{\nu}=-j \cos^2{\theta}\left(\xi^\mu\xi_\nu+u^\mu u_\nu\right),\qquad\qquad j=\frac{M}{4 \pi \,a^2}\sec^3{\theta}\, \delta(r) \ , \label{jkerr}
\end{equation}
where
\begin{equation}
\xi_\mu= \frac{a\cos{\theta}}{\Delta^{1/2}}\, (0,0,1,0),\qquad\qquad u_\mu= \frac{\Delta^{1/2} \tan{\theta}}{\Omega} \, \left(1,0,0,-a\right)
\end{equation}
This is the (A)dS generalization of the source for the flat Kerr solution given in \cite{Israel:1970kp}.
The single copy solution is given as usual by $A_\mu=k_\mu\phi$, with the substitution~\eqref{subs}, and it again satisfies the Maxwell equation~\eqref{YMeom}, with the source now given by
\begin{equation}
J^\mu=j\, \zeta^\mu, \qquad\qquad \zeta_\mu=(2,0,0,2/a) \ .
\end{equation}
As expected, the single copy corresponds to the field generated by a charged disk rotating around the $z$ direction in (A)dS spacetime. This field generates both an electric and a magnetic field, with the latter proportional to the angular momentum of the charged particle. Thus, the angular momentum on the gravity side is translated into a magnetic field at the level of the single copy. As we will see in Sec.~\ref{btz}, the same correspondence will hold also for the BTZ black hole. In a similar way, the scalar field satisfies Eq.~\eqref{seom} with source $2j$, where $j$ is given as in Eq.~\eqref{jkerr}.
The previous analysis can easily be extended to higher dimensions. In fact, the Myers-Perry black hole with a non-vanishing cosmological constant also admits a Kerr-Schild form~\cite{Gibbons:2004uw}. In $d=2n+1$, the null vector and scalar field read
\begin{align}
k_\mu \mathrm{d} x^\mu =W \mathrm{d} t + F \mathrm{d} r -\sum_{i=1}^{n}\frac{a_i\mu_i^2}{1+\lambda a_i^2}\mathrm{d}\varphi_i \ ,\qquad
\phi =\frac{2GM}{\sum_{i=1}^{n}\frac{\mu_i^2}{r^2+a_i^2}\prod_{j=1}^{n}(r^2+a_j^2)} \ ,
\end{align}
where $\lambda = \frac{2 \Lambda}{(d-2)(d-1)}$,
\begin{equation}
W\equiv\sum_{i=1}^{n}\frac{\mu_i^2}{1+\lambda a_i^2} \ ,\qquad\qquad F\equiv\frac{r^2}{1- \lambda r^2}\sum_{i=1}^{n}\frac{\mu_i^2}{r^2+a_i^2} \ ,
\end{equation}
and the coordinates $\mu_i$ are subject to the constraint
\begin{equation}
\sum_{i=1}^{[d/2]} \mu_i=1 \ .
\end{equation}
Meanwhile, for $d=2n$ we instead have
\begin{align}
k_\mu \mathrm{d} x^\mu =W \mathrm{d} t + F \mathrm{d} r -\sum_{i=1}^{n-1}\frac{a_i\mu_i^2}{1+\lambda a_i^2}\mathrm{d}\varphi_i \ , \qquad \phi =\frac{2GM}{r \sum_{i=1}^{n}\frac{\mu_i^2}{r^2+a_i^2}\prod_{j=1}^{n-1}(r^2+a_j^2)} \ .
\end{align}
When all the rotation parameters $a_i$ vanish, the metrics above reduce to the higher-dimensional version of the (A)dS-Schwarzschild one, and the corresponding source becomes a static charge in (A)dS$_d$.
Constructing the single and zeroth copy is very similar to the $d=4$ case. In particular, the corresponding gauge field is sourced by a charge rotating with angular momentum proportional to $a_i$ in the corresponding directions. Most interestingly, the zeroth copy satisfies the equation
\begin{equation}
\left(\bar \nabla^2- \frac{2(d-3)}{d(d-1)} \bar R\right)\phi=j \ , \label{scalar}
\end{equation}
where the non-minimal coupling to the curvature has a conformal value only in $d=4$. We have explicitly checked the coefficient of the Ricci scalar for $4 \leq d \leq 11$. The result \eqref{scalar} remains valid in the limit $a \to 0$, and as such it generalizes Eq. \eqref{seom} for a (A)dS-Schwarzschild black hole to arbitrary dimensions.\footnote{The equation for the zeroth copy of dS-Schwarzschild black holes in arbitrary dimensions already appeared in a presentation given by Andres Luna at the conference ``QCD meets Gravity'', UCLA, Dec 2016~\cite{UCLA-link}.}
\section{Charged black hole solutions} \label{EM}
It is interesting to observe that the single copy we built for the (A)dS-Schwarzschild and (A)dS-Kerr black holes is automatically a solution to the Einstein-Maxwell's equation when the metric is promoted to its charged version~\cite{Romans:1991nq}, i.e. A(dS)-Reissner-Nordstrom and A(dS)-Kerr-Newman respectively. Moreover, there is a simple procedure that yields these charged metrics starting from their neutral counterparts in Kerr-Schild coordinates.
To illustrate this, consider a neutral black hole metric in Kerr-Schild form
\begin{equation}
g_{\mu\nu}= \bar g_{\mu\nu} + k_\mu k_\nu \phi(8 \pi G M) \ ,
\end{equation}
where for later convenience we have explicitly shown the dependence of $\phi$ on the gravitational coupling and the black hole mass. As shown in previous examples, one can define the corresponding single copy as $A_\mu=k_\mu\phi(g Q)$, where we have made the substitution $M \to Q$ and have applied the replacement rule \eqref{subs}. Of course, in the case of an Abelian gauge theory the coupling $g$ in \eqref{subs} is redundant, and in what follows we will set $g=1$ by an appropriate rescaling of the charge $Q$.
Before constructing the charged black hole solutions, we analyze Kerr-Schild solutions in Einstein-Maxwell theory; for a review of these types of solutions see \cite{Griffiths:2009dfa,Stephani:2003tm}. The fact that the metric is in Kerr-Schild form imposes restrictions on the stress-energy tensor that can be translated into restrictions on the field strength $F^{\mu\nu}$ when the matter is a $U(1)$ field. When the null KS vector is geodetic and shear-free, it should also be an eigenvector of the Maxwell field strength
\begin{equation}
k_\mu F^{\mu\nu}=\lambda k^\nu \ .
\end{equation}
This requirement is a necessary but not sufficient condition for the gauge field to be a solution of the field equations. As a consistency check, we confirm that the single copy ansatz $A^{EM}_\mu=k_\mu \phi$ satisfies the above requirement with an eigenvalue
\begin{equation}
\lambda=k_\mu\nabla^\mu\phi \ .
\end{equation}
We now show how to construct charged black holes using the single copy. Using the Kerr-Schild ``building blocks'' above, we can immediately write down an electrically charged solution to the Einstein-Maxwell equations, where the metric and gauge field are given by
\begin{align}
g_{\mu\nu}^\text{EM} = \bar g_{\mu\nu} + k_\mu k_\nu\, \phi^\text{EM}(M,Q) \ , \qquad \qquad A^{EM}_\mu = A_\mu = k_\mu\phi(Q) \ ,
\end{align}
with
\begin{equation}
\phi^\text{EM}(M,Q)=\phi(8 \pi G M)- \frac{Q}{r^{d-3}} \phi(Q)\ .
\end{equation}
This construction works both in curved and flat space, but it is not applicable in $d<4$ because in that case there are no graviton degrees of freedom.
This ``recipe'' allows us to turn a solution to Einstein's equations into one that satisfies the Einstein-Maxwell equations. This is somewhat reminiscent of the transmutation operations for scattering amplitudes described in \cite{Cheung:2017ems}. In particular, the procedure we have described appears to be a classical analog of the multiple trace operation that turns gravity amplitude into Einstein-Maxwell ones. However, significantly more evidence is required to establish if there is a connection between these two procedures.
\section{Black strings and black branes} \label{others}
Black strings and black branes are black hole solutions with extended event horizons. In this section, we construct their corresponding single and zeroth copies in (A)dS in $d>4$ spacetime dimensions.
\subsection{Black strings}
In order to construct black strings in AdS$_d$, we start from the base metric
\begin{equation}
\bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu = a^2(z)\left(\bar \gamma_{ab}\mathrm{d} x^a \mathrm{d} x^b+\mathrm{d} z^2\right) \ ,
\end{equation}
where the $(d-1)$-dimensional metric $\bar \gamma_{ab}$ can be that of (A)dS$_{d-1}$ or Mink$_{d-1}$, depending on whether one chooses a (A)dS or a Minkowski slicing of AdS$_d$. The corresponding form of the scale factor is~\cite{Hirayama:2001bi}
\begin{align}
a^{-1}(z)=\begin{cases}
\ell_{d-1} / \ell_d\,\sinh{\left(z/\ell_{d-1}\right)} & \qquad\quad \text{(dS slicing)}\\
z / \ell_d & \qquad\quad \text{(Minkowski slicing)}\\
\ell_{d-1} / \ell_d\,\sin{\left(z/\ell_{d-1}\right)} & \qquad\quad \text{(AdS slicing)}
\end{cases} \ ,
\end{align}
where $\ell_d$ and $\ell_{d-1}$ are the AdS length scales in $d$ and $d-1$ dimensions, respectively. A black string solution is then obtained by replacing the $(d-1)$-dimensional metric $\bar \gamma_{ab}$ with a $(d-1)$-dimensional Schwarzschild black hole with the same cosmological constant~ \cite{Chamblin:1999by,Gregory:2000gf}.
A similar construction is possible for a de Sitter black string starting from a dS$_d$ space foliated by dS$_{d-1}$, in which case the base metric reads
\begin{equation}
\bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu = \sin^2(z/\ell_d) \, ds^2_{{\rm dS},d-1} + dz^2 \ .
\end{equation}
The metric for a black string in dS is then obtained by replacing $ds^2_{{\rm dS},d-1}$ with the line element for a dS-Schwarzschild black hole in $(d-1)$-dimensions.
In both the AdS and dS cases, if the black hole metric is in the Kerr-Schild form, the full metric automatically inherits a similar form. More precisely, writing the black hole metric as
\begin{equation}
\gamma_{\mu\nu}=\bar{\gamma}_{ab}+ \psi k_a k_b \ ,
\end{equation}
the null vector $k_\mu$ and scalar $\phi$ for the (A)dS black string can be chosen to be
\begin{equation}
k_\mu= a^{2}(z) \delta_\mu^a k_a \ , \qquad \qquad \quad \phi = \frac{\psi}{a^2(z)} \ . \label{k phi black strings}
\end{equation}
The stress-energy tensor for the AdS black string in Kerr-Schild coordinates reads
\begin{equation}
T^\mu{}_\nu =\frac{m}{2 a(z)^{2} }\, \text{diag} (0,0,1,1, \vec 0) \, \delta^{(d-2)} (\vec{r}) \ ,\label{TString}
\end{equation}
where $m$ is the mass per unit length of the string.
It is now easy to show that the single copy $A_\mu = \phi \, k_\mu$ and the zeroth copy $\phi$ satisfy the equations (\ref{YMeom}) and (\ref{seom}) respectively with sources:
\begin{equation}
J^\mu=j \, \delta^\mu_0 \ ,\qquad\qquad\quad j=\frac{m}{a^4(z)} \, \delta^{(d-2)} (\vec{r}) \ .
\end{equation}
As expected, the YM source is a charged line aligned with the $z$ direction with charge per unit length $q = m$, living in either AdS or dS. Notice that the judicious insertion of scale factors in Eq. \eqref{k phi black strings} is crucial to obtaining sensible classical copies.
\subsection{Black branes}
We now turn to the case of black branes, or planar black holes, in AdS$_{p+2}$ . The most familiar form of the metric for black branes is
\begin{equation}
\mathrm{d} s^2=\frac{r^2}{\ell^2}\left\{-\left[1-\left(\frac{r_h}{r}\right)^{p+1}\right]\mathrm{d} t^2+\eta_{ab} \mathrm{d} x^a \mathrm{d} x^b \right\} + \frac{\ell^2}{r^2} \left[1-\left(\frac{r_h}{r}\right)^{p+1}\right]^{-1}\mathrm{d} r^2 \ , \label{metric brane}
\end{equation}
where $a,b = 1,\dots,p$, and the horizon is located at $r=r_h$. This is a solution to Einstein's equations with a source
\begin{equation}
T^\mu{}_\nu = \frac{r_h^{p+1}}{2 \ell^2} \, \text{diag} (0,0,1,1, ... , 1) \, \delta(r) \ .
\end{equation}
The metric in Eq. \eqref{metric brane} can be put in a Kerr-Schild form by introducing a new time coordinate. The base metric is then just AdS in Poincar\'e coordinates, and the Kerr-Schild null vector and scalar can be chosen as follows:
\begin{align}
k_\mu \mathrm{d} x^\mu & = d\tau - \frac{\ell^2}{r^2} dr \ , \qquad \qquad \quad \phi= \frac{r^2}{\ell^2} \left(\frac{r_h}{r}\right)^{p+1} \ .
\end{align}
The single copy given by $A_\mu=k_\mu\phi$ satisfies the Abelian Yang-Mills equations of motion with a source
\begin{equation}
J^\mu = j \delta^\mu_0 \delta(r) \ , \qquad \qquad j = \frac{r_h^{p+1}}{\ell^2} \delta (r) \ .
\end{equation}
which gives rise to an electric field in the $r$ direction. Meanwhile, the scalar field satisfies Eq. \eqref{scalar} with source $j$.
\section{Wave solutions} \label {waves}
We now turn our attention to time-dependent solutions, and in particular to wave solutions. For negative values of the cosmological constant, there are three different types of wave solutions in vacuum that can be written in the Kerr-Schild form: {\it Kundt waves}, {\it generalized pp-waves}, and {\it Siklos waves} \cite{Griffiths:2003bk,Griffiths:2009dfa}. All of these solutions are Kundt spacetimes of Petrov-type N. By contrast, in the case of a positive cosmological constant, there is only one kind of wave in vacuum --- Kundt waves --- which in this case are the same as pp-waves~\cite{Griffiths:2009dfa}. Finally, we consider \emph{shock waves}, which unlike the previous solutions are generated by a non-trivial localized source. Since these spacetimes are not stationary, there is no timelike Killing vector. However, all these cases feature a null Killing vector, which we can use to construct the classical single and zeroth copies. As in previous cases, the ambiguity in choosing the form of the null KS vector and the KS scalar will play a crucial role in ensuring the existence of reasonable single and zeroth copies. Unlike the stationary cases, here we have the freedom of performing a rescaling as in Eq.\eqref{resc} with $f=f(u)$; such rescaling is a property of wave solutions and it only changes the wave profile. We will find that the single and zeroth copy satisfy the same equations in all of these cases (albeit with a source term in the case of shock waves). However, the equation for the single copy is no longer gauge invariant when the base metric is curved. For simplicity, in this section we will restrict ourselves to $d=4$ spacetime dimensions.
\subsection{Kundt waves}
We begin by analyzing the case of Kundt waves, which exist in both de Sitter and anti-de Sitter spacetimes. The Kundt waves in (A)dS can be written in Kerr-Schild form with a base metric that reads
\begin{equation}
\bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu = \frac{1}{P^2}\left[-4 x^2 \mathrm{d} u\left(\mathrm{d} v - v^2 \mathrm{d} u\right)+\mathrm{d} x^2 + \mathrm{d} y^2\right] \ ,\qquad P=1+\frac{\Lambda}{12}(x^2+y^2) \ , \label{wdS}
\end{equation}
where $u$ and $v$ are light-cone coordinates. The null vector and the scalar are given by
\begin{align}
k_\mu =\frac{x}{P} \, \delta_\mu^u \ ,\qquad \qquad \quad \phi= \frac{P}{x} H(u,x,y) \ .
\end{align}
The full metric $g_{\mu\nu} = \bar g_{\mu\nu} + \phi \, k_\mu k_\nu$ is a vacuum solution to the Einstein equations provided $H(u,x,y)$ satisfies the following partial differential equation:
\begin{equation}
\left[ \partial_x^2 + \partial_y^2 + \frac{2\Lambda}{3 P^2} \right] H(u,x,y) = 0 \ . \label{eq H Kundt}
\end{equation}
The singularity of the metric Eq.~\eqref{wdS} at $x=0$ corresponds to an expanding torus in de Sitter, and to an expanding hyperboloid in anti-de Sitter. In dS, the wavefronts are tangent to the expanding torus and correspond to hemispheres with constant area $4 \pi \ell^2$, with $\ell = \sqrt{3/\Lambda}$ the dS radius---see Fig.~\ref{dswave}. For AdS, the wave surfaces are semi-infinite hyperboloids. In both cases, the wavefronts are restricted to $x\geq 0$ to avoid caustics (except for the singularity $x=0$)~\cite{Griffiths:2003bk} and different wave surfaces are rotated relative to each other. It should also be noted that the wave surfaces in the dS and AdS cases only exist outside the expanding singular torus or hyperboloid respectively.
\begin{figure}[!t]
\includegraphics[scale=0.5]{dswave.pdf}
\caption{Kundt waves in de Sitter space. We can consider dS as a four-dimensional hyperboloid embedded in a flat 5d space with coordinates $Z^a$. This figure shows portions of de Sitter covered by the coordinates in Eq.~\eqref{wdS} at different values of the time $Z^0$ with $Z^4=0$. The portions of 2-spheres correspond to different snapshots in time and the semi-circles on them are the wavefronts of constant $u$. The gravitational, gauge, and scalar waves all have wavefronts of this shape. For more details, see \cite{Griffiths:2009dfa,Griffiths:2003bk}.}
\label{dswave}
\end{figure}
Contrary to what we have seen in the time-independent cases, in this case the gauge field $A_\mu = \phi \, k_\mu$ and scalar field $\phi$ satisfy the following equations:
\begin{eqnarray}
&\displaystyle \bar \nabla_\mu F^{\mu\nu}+\frac{\bar R}{6}A^\nu=0 \ , & \label{AeqW} \\
&\bar \nabla^2\phi=0 \ . & \label{SeqW}
\end{eqnarray}
This can be seen by using the $(\mu,u)$ component of Einstein's equations, and the equation for $H(u,x,y)$ in \eqref{eq H Kundt}. The copies correspond to waves in the gauge and scalar theory whose wavefronts are the same as the gravitational wave wavefronts. An important observation is that the single copy has broken gauge invariance due to the mass term proportional to the Ricci scalar. This fact will be discussed at length in our final section. In the following, we will see that other wave solutions give rise to single and zeroth copies that satisfy exactly the same equations.
\subsection{Generalized pp-waves}
Next, we consider the generalization of pp-waves to maximally symmetric curved spacetimes. The case of de Sitter pp-waves is identical to the Kundt waves analyzed above~\cite{Griffiths:2009dfa}; thus, here we will only consider the AdS case. The wavefronts of these AdS waves are hyperboloids that foliate the entire space. The generalized AdS pp-waves are written in Kerr-Schild form with an AdS base metric expressed as
\begin{eqnarray}
\displaystyle \bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu =\frac{1}{P^2}\left[-2Q^2 \mathrm{d} u\left(\mathrm{d} v - \frac{\Lambda}{6}v^2 \mathrm{d} u\right)+\mathrm{d} x^2 + \mathrm{d} y^2\right] \ ,
\end{eqnarray}
with
\begin{equation}
\displaystyle P=1+\frac{\Lambda}{12}(x^2+y^2),\qquad\qquad Q=1-\frac{\Lambda}{12}(x^2+y^2) \ .
\end{equation}
We choose the corresponding null vector and scalar to be
\begin{align}
k_\mu =e^{-\tanh
^{-1}\left(P-1\right)}\sqrt{\frac{Q}{P}} \delta_\mu^u \ , \qquad \qquad \quad \phi= e^{2\tanh
^{-1}\left(P-1\right)} H(u,x,y) \ .
\end{align}
The full Kerr-Schild metric is then a solution to the vacuum Einstein equations provided $H(u,x,y)$ again satisfies \eqref{eq H Kundt}. In the limit $\Lambda\rightarrow0$, this metric reduces to that for pp-waves in flat space~\cite{Griffiths:2009dfa}. We can find the classical copies corresponding to these generalized pp-waves in the same way as in the previous case, and they again turn out to satisfy Eqs. (\ref{AeqW}) and (\ref{SeqW}).
\subsection{Siklos AdS waves}
The Siklos metric in Kerr-Schild form is written with an AdS base metric that reads
\begin{equation}
\bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu =\frac{\ell^2}{x^2}\left[- 2 \, \mathrm{d} u \mathrm{d} v + \mathrm{d} x^2 + \mathrm{d} y^2 \right] \ , \label{adsP}
\end{equation}
where the Kerr-Schild null vector and scalar are chosen to be
\begin{align}
k_\mu =\frac{\ell}{x} \, \delta_\mu^u \ , \qquad \qquad \quad \phi= \frac{x}{\ell} H(u,x,y) \ .
\end{align}
The full metric satisfies the Einstein equations in vacuum provided the function $H(u,x,y)$ is such that
\begin{equation}
\left[ \partial_x^2 + \partial_y^2 - \frac{2}{x^2} \right] H(u,x,y) = 0 \ . \label{eq H Siklos}
\end{equation}
In this case, the wavefronts are planes perpendicular to the $v$ direction. This metric is the only non-trivial vacuum spacetime that is conformal to flat space pp-waves. In a similar way, one can also construct waves with spherical wavefronts~\cite{Gurses:2012db}. Once again, the single and zeroth copy turn out to satisfy Eqs. (\ref{AeqW}) and (\ref{SeqW}).
\subsection{Shock waves}
\begin{figure}[!b]
\includegraphics[scale=0.5]{shock.pdf}
\caption{Planar shock wave in AdS. The source travels on a null geodesic at fixed $u=0$, $x=x_0$, and $y=0$. The gravitational, gauge, and scalar shock waves all have this structure.} \label{shocks}
\end{figure}
Finally, we consider planar shock waves in AdS~\cite{Hotta:1992qy,Horowitz:1999gf}. (Note that the case of spherical shock waves follows analogously.) Planar shock waves have the same base metric as Siklos AdS waves---see Eq.~(\ref{adsP})---but unlike the latter they are not vacuum solutions. In this case, the null vector and the scalar field are given by
\begin{align}
k_\mu =\frac{\ell}{x} \delta_\mu^u \ ,\qquad \qquad \quad \phi= \frac{x^2}{\ell^2} H(x,y)\delta(u) \ ,
\end{align}
where we will assume that the source travels on a null geodesic at fixed $u=0$, $x=x_0$, and $y=0$ as shown in figure \ref{shocks}, i.e.
\begin{equation}
T_{\mu\nu} = E\, \frac{x_0^2}{\ell^2}\, \delta(x-x_0) \delta(y)\delta(u)\,\delta_\mu^0 \delta_\nu^0 \ ,
\end{equation}
with $E$ the total energy carried by the shock wave. Notice that we need to place the source away from $x=0$, since the base metric and the Kerr-Schild vector and scalar become singular at that point. With our ansatz, the Einstein equations reduce to
\begin{equation}
x \left[x \partial^2_x H(x,y)+2 \partial_x H(x,y)+x
\partial^2_y H(x,y)\right]-2 H(x,y)= -16 \pi G E x_0^2 \delta(x-x_0) \delta(y) \ . \label{H eq}
\end{equation}
The solution to this equation is a hypergeometric function, the exact form of which will not be needed here. Imposing Einstein's equations, the gauge and scalar copies satisfy
\begin{eqnarray}
\nabla_\mu F^{\mu\nu}+\frac{R}{6}A^\nu = g J^\nu \ , \qquad \qquad \quad \nabla^2\phi=j \ ,
\end{eqnarray}
where the sources are
\begin{equation}
J^\nu=j \frac{x}{\ell}\delta^\nu_v,\qquad\qquad \quad j=2 E \frac{x_0^2}{\ell^2}\frac{x^2}{\ell^2}\delta(u)\delta(x-x_0) \delta(y)\ .
\end{equation}
It is easy to check that the first source follows indeed from Eq. \eqref{current} using the null Killing vector $V^\mu =\delta^\mu_v$. As in the gravitational case, the sources for the shock waves in the gauge and scalar theory are localized at $u=0$, $x=x_0$, and $y=0$.
\section{An unusual example: the BTZ black hole} \label{btz}
Asymptotically flat black holes in $d=3$ space-time dimensions do not exist, but the situation changes in the presence of a negative cosmological constant. Black hole solutions in AdS$_3$ are known as BTZ black holes, and can be viewed as a quotient space of the covering of AdS$_3$ by a discrete group of isometries \cite{Banados:1992wn,Banados:1992gq}. In this section, we construct the single and zeroth copy of these solutions. Given that there are no graviton degrees of freedom in $d=3$, we can at most expect to apply the double copy procedure to the entire BTZ black hole geometry. Therefore, in the following analysis we will use a flat base metric. This approach is different from the one we have adopted in the rest of the paper, since in the previous examples we worked with a curved base metric. This is however an interesting example to consider, because it does not have an immediate counterpart at the level of scattering amplitudes.
We will write the BTZ black hole metric in Kerr-Schild form with a Minkowski base metric expressed in spheroidal coordinates,
\begin{equation}
\bar g_{\mu\nu} \mathrm{d} x^\mu \mathrm{d} x^\nu = -\mathrm{d} t^2+\frac{r^2}{r^2+a^2}\mathrm{d} r^2+(r^2+a^2) \mathrm{d} \theta^2 \ , \label{Mbtz}
\end{equation}
and a null vector and scalar field given by
\begin{align}
k_\mu =\left(1,\frac{r^2}{r^2+a^2},-a\right) \ , \qquad \qquad \quad \phi =1+8GM+\Lambda r^2\ .
\end{align}
As for the Kerr black hole, $M$ is the mass of the black hole and $a$ is the angular momentum per unit mass. The corresponding single copy field $A_\mu=k_\mu\phi$ satisfies the Abelian Yang-Mills equations of motion where, as expected, the source is a constant charge density filling all space, that is
\begin{equation}
J^\mu=4 \rho \delta^\mu_0 \ ,
\end{equation}
where we have replaced the vacuum energy density $\Lambda$ with the charge density $\rho$. By looking at the non-zero components of the field strength tensor $F^{\mu\nu}$,
\begin{equation}
F^{r\,t}=-2\rho\frac{r^2+a^2}{r} \ , \quad\quad F^{r\,\theta}=-2\rho\frac{a}{r} \ ,
\end{equation}
we can see that the non-rotating case ($a\rightarrow0$) gives rise only to an electric field, whereas the rotating case yields both electric and magnetic fields. Thus, the rotation of the BTZ black hole is translated at the level of the single copy into a non-zero magnetic field, as in the case of the Kerr solution studied in Sec. \ref{kerr}. For completeness, we mention that the equation for the zeroth copy $\phi$ also features a constant source filling all space, i.e. $\nabla^2\phi=-4\rho$.
\section{Discussion and future work} \label{dis}
We have constructed several examples of a classical double copy in curved, maximally symmetric backgrounds. Some black hole copies are straightforward extensions of the double copy in flat space, while other solutions have more involved interpretations. The (A)dS-Schwarzschild and (A)dS-Kerr single copy corresponds to a field sourced by a static and rotating electric charge in (A)dS respectively. Black strings and black branes copy to charged lines and charged planes in (A)dS. A more interesting situation occurs when we consider a black hole in AdS$_3$. The rotating BTZ black hole gives rise to a single copy which produces a magnetic field. Thus, even though there are no gravitons in $d=3$, it seems possible to consider the copy of the geometry. In principle, this relationship should be unrelated to the scattering amplitudes double copy, since there are no gravitons scattering. In this sense, this classical double copy may exhibit a deeper relationship between gravitational and Yang-Mills theories. In all these static cases, the zeroth copy satisfies an equation of motion with a coupling to the Ricci scalar. In $d=4$, it has been conjectured that this is a remnant of the conformal invariance of the Yang-Mills equations. In higher dimensions, it remains unknown if this coefficient is related to a symmetry of these theories. When we turn to time-dependent solutions, the situation seems to change. For the wave solutions, the single copy satisfies an equation of motion corresponding to Maxwell's equation in addition to a term proportional to the Ricci scalar. On the other hand, the zeroth copy equation of motion is simply that of a free scalar field. Despite this change in the equations of motion, we are able to construct the corresponding single and zeroth copies in both time-dependent and time-independent cases.
We have briefly mentioned how some properties (or special limits) of gravitational solutions have no associated mapping to Yang-Mills or scalar fields. This is expected, given that some structures are inherently gravitational, for example horizons. In this sense, when performing the classical copies, one loses information. This is similar to the observation that information of the gauge theory is lost during the BCJ double copy procedure \cite{Bjerrum-Bohr:2013bxa,Oxburgh:2012zr}. Given this, there is no reason to expect that the gravitational instabilities of black hole, black string, or black brane solutions get copied to instabilities in the gauge and scalar theory. Nevertheless, a more detailed study of this should be performed.
Some of our results, obtained by using the classical double copy procedure, are yet to find a completely satisfactory interpretation. One of these is the ambiguity in choosing $k^\mu$ and $\phi$, even after imposing the conditions that $k^\mu$ be geodetic, shear-free, and twist-free. In Sec. \ref{sec:classical double copy} we were able to track down the origin of this ambiguity by extracting Maxwell's equations from the contraction of the Ricci tensor and a Killing vector by using the Einstein equations. In all the examples we have given, we have fixed this ambiguity in a way such that the single and zeroth copies obtained were `reasonable'. Nevertheless, we have yet to identify the exact property required by the null vector and scalar to give rise to the correct copies. This could be related to the fact that, when considering the BCJ double copy, the kinematic factors need to be in BCJ form, where the kinematics factors satisfy the same algebra as the color factors. It is possible that the null vector needs to satisfy a relation that is the analogue of this, but we are not aware of such a relation. We have also found that the time-independent and the time-dependent copies satisfy different equations. For the time-independent case, the scalar copy equation of motion includes an extra factor proportional to the Ricci scalar. In the time-dependent case, this extra factor appears in the equation for the gauge field. These extra factors correspond in both cases to mass terms; this means that the Yang-Mills copy corresponds to a theory with broken gauge symmetry. The reasons for these differences between the stationary and wave solutions remains elusive.
One interesting future direction consists of finding an extension of the Kerr-Schild copy by considering metrics in a non-Kerr-Schild form. For example, not all waves in $d>4$ can be written in Kerr-Schild form \cite{Ortaggio:2008iq}, but there are examples that can be written in extended Kerr-Schild (xKS) form \cite{Vaidya:1947zz,Ett:2010by}. This xKS form considers the use of a spatial vector orthogonal to the Kerr-Schild null vector. If the Kundt-waves are of Type III, they cannot be written in Kerr-Schild form. Another example of an xKS space time is the charged Chong, Cvetic, Lu, and Pope solution in supergravity \cite{Chong:2005hr,Aliev:2008bh}. Another possible application of this classical copy in curved spacetimes may be in the context of AdS/CFT. The holographic duals to the gravitational AdS solutions that we have considered above have been largely studied in the literature, and it is possible that one could extend the copy procedure to the CFT side of the duality, although this is extremely speculative.\\
\noindent{\bf Acknowledgments}
\\
\noindent We thank Joe Davighi, Matteo Vicino and Adam Solomon for helpful discussions. This work was supported in part by US Department of Energy (HEP) Award DE-SC0013528. R.P. is also supported in part by funds provided by the Center for Particle Cosmology.
\bibliographystyle{apsrev4-1}
|
2209.00035
|
\section{Introduction}
The relative entropy is a measure of the distinguishability of two states. In quantum mechanics, for two density matrices $\rho$ and $\sigma$ it is defined as
\begin{equation}
S(\rho||\sigma)=\text{Tr}(\rho \log \rho-\rho \log \sigma)
\label{QMrelativeentropy}
\end{equation}
If the system is bi-partite, one can reduce each density matrix to one of the partitions and compute the relative entropy between the reduced density matrices. The counterpart in QFT would be to reduce the states to a spacetime region. Intuitively, the smallest this region is, the lesser operators one has at hand to characterise the states and therefore the relative entropy decreases. However, generic bounded regions of spacetime such as causal diamonds are assigned a von Neumann algebra which is a Type III factor, and these algebras have no trace-class operators and there are no density matrices \cite{haag1996local}. Nevertheless, the definition of the relative entropy \eqref{QMrelativeentropy} can be suitably extended \cite{Araki:1973hh}: given two (normal and faithful) states $\omega$ and $\omega'$ of the von Neumann algebra $A$ associated to some region, with both states represented on a Hilbert space by $\Omega$ and $\Omega'$, the relative entropy is
\begin{equation}
S_{A}(\omega||\omega')=-(\Omega,\log\Delta_{\Omega',\Omega}\,\Omega),
\label{Arakientropy}
\end{equation}
where $\Delta_{\Omega',\Omega}$ is the relative modular operator (which is defined in the next section). This is in sharp contrast with the entanglement entropy which necessarily diverges due to the generic UV behavior of correlations through the boundary of the corresponding region \cite{Witten:2018zxz}.
Araki's relative entropy \eqref{Arakientropy} can be hard to compute in general cases. Recently it has been computed in a number of specific situations for coherent states acting on the vacuum of the free scalar QFT: on a Rindler wedge \cite{Longo:2019mhx,Ciolli:2019mjo,Casini:2019qst}, on a causal diamond in spacetime dimension greater than 2 \cite{Longo:2020amm}, and later the two-dimensional case \cite{Longo:2021rag}. In addition, in the bosonic vacuum U(1)-current model (namely, a massless boson on a light ray) the relative entropy of coherent states was computed for a half-line in \cite{Longo:2018obd} and used to obtain the analogous expression for (unbounded) regions of the null plane
for a free scalar in \cite{Morinelli:2021nsx}. For similar results in free fermionic CFTs, see \cite{Longo:2017mbg}. For interacting theories, as far as we know the only results available correspond to coherent states in chiral CFTs \cite{Hollands:2019czd,Panebianco:2019plp}, while for free QFT in curved spacetimes see \cite{Ciolli:2021otw}.
All of the above computations compare the vacuum state to a coherent excitation of itself (with the exception of \cite{Ciolli:2021otw}). Recently in \cite{Bostelmann:2020srs}, among other things, the relative entropy for a half-line in the \textit{thermal} U(1)-current model was computed. That is, a KMS state of inverse temperature $\beta$ was compared to a coherent excitation of itself, when restricted to a half-line. The fact that the relative entropy was computed on the half-line is an important point, since by general arguments its modular operator had already been obtained in \cite{Borchers:1998ye}. Even in the vacuum case, one can see that to obtain the modular operator for a bounded interval it is necessary to make use of the full PSL$(2,\mathbb{R})$ group of conformal transformations of the chiral boson theory (we will discuss this later in detail).
The main goal
of this article is to compute the relative entropy of coherent states on a bounded interval $I \subset \mathbb{R}$ for the thermal U(1)-current model and the free massless boson at finite temperature in $1+1$ dimensions restricted to a causal diamond. We will find two bounds for each theory, a Bekenstein-like bound and mainly a controlled violation of the Quantum Null Energy Condition (QNEC), which was already anticipated in \cite{Bostelmann:2020srs}.
\section{Preliminaries on the modular structure of the Weyl algebra}
Given a symplectic space $(\mathcal{K},\sigma)$ we get a CCR algebra with relations
\begin{equation}
W(f)W(g)=e^{-i\sigma(f,g)}W(f+g),\quad W(f)^*=W(-f)
\end{equation}
where $f,g \in \mathcal{K}$. We will refer to this algebra as CCR$(\mathcal{K},\sigma)$.
A positive symmetric bilinear form $\tau$ such that
\begin{equation}\label{technical}
\sigma(f,g)^2\leq \tau(f,f)\tau(g,g)
\end{equation}
defines a quasi-free state by \cite{Kay:1988mu}
\begin{equation}
\omega(W(f))=e^{-\frac{1}{2}\tau(f,f)}
\end{equation}
with 2-point function $w_2(f,g)=\tau(f,g)+i\sigma(f,g)$. However the state may not be pure. It is pure if and only if $w_2$ is a complex inner product. More precisely, from \eqref{technical} it can be shown that a contraction $D$ exists such that
\begin{equation}
\sigma(f,g)=\tau(f,D g).
\label{D}
\end{equation}
Because of the non-degeneracy of $\sigma$, $ D$ is invertible and $D=C|D|$ is its polar decomposition . $C$ is a complex structure and the state $\omega$ is pure if and only if $w_2$ is a complex inner product, bi-linear with respect to the complex structure $C$. This is actually equivalent to the statement that $\omega$ is pure if and only if $|D|=1$ (see \cite{Petz:1990gb} for further details). Let us differ how to construct a purification of a non-pure state to the end of this section.
\subsection{Modular theory and relative entropy}
We assume we have a complex Hilbert space $\mathcal{H}$ with inner product $\langle f,g\rangle = w_2(f,g)$. In other words, we are assuming for now that the state is pure. Let us call its (bosonic) Fock space $\Gamma(\mathcal{H})$. We have a representation of the CCR$(\mathcal{K},\sigma)$ algebra on the Fock space $\Gamma(\mathcal{H})$. Indeed, $W(f)$ acts on $\Gamma(\mathcal{H})$ as $V(f)$:
\begin{equation}
V(f)e^0=e^{-\frac{1}{2}\langle f,f\rangle}e^f,\qquad f \in \mathcal{H}
\end{equation}
with
\begin{equation}
e^f:=1\oplus f\oplus\frac{1}{\sqrt{2!}} f \otimes f \oplus ...
\end{equation}
Calling $\Omega:=e^0$ the vacuum vector
\begin{equation}
(\Omega,V(f)\Omega)=e^{-\frac{1}{2}\langle f,f\rangle}=\omega(W(f))
\end{equation}
We can define the local algebras associated to a given real-linear subspace $H\subset \mathcal{H}$:
\begin{equation}
R(H):=\left\{V(f);\quad f\in H \right\}''
\end{equation}
It turns out that $\Omega$ is cyclic (respectively separating) for $R(H)$ if and only if $H$ is cyclic (respectively separating). $H$ is cyclic if $\overline{H+iH}=\mathcal{H}$ while separating if $H\cap iH=0$. If $H$ is also closed it is called a \textit{standard} subspace .
The Tomita operator $S$ associated to $R(H)$ (and $\Omega$) is defined by (the closure of)
\begin{equation}
S V(f) \Omega=V(f)^*\Omega,\quad V(f)\in R(H)
\end{equation}
The \textit{relative} Tomita operator $S_{\Omega',\Omega}$ associated to $R(H)$ is defined by (the closure of)
\begin{equation}
S_{\Omega',\Omega} V(f) \Omega=V(f)^*\Omega'
\end{equation}
with polar decomposition
\begin{equation}
S_{\Omega',\Omega}=J_{\Omega',\Omega}\Delta_{\Omega',\Omega}^{1/2}
\end{equation}
For some algebra $R(H)$ and cyclic and separating states $\omega$ and $\omega'$, Araki's relative entropy is defined as
\begin{equation}
S_{A}(\omega||\omega')=-(\Omega,\log\Delta_{\Omega',\Omega}\,\Omega)
\label{Arakirelativeentropy}
\end{equation}
This is hard to compute for generic cases, however it simplifies considerably for coherent states\footnote{The relative entropy between coherent states satisfies $S(\omega_f||\omega_g)=S(\omega_{f-g}||\omega)$, so there is no loss of generality in assuming that one state is not excited by a Weyl unitary.}, namely when $\Omega'=V(f)\Omega$, and we shall call the corresponding algebraic state $\omega_f$. In order to see this, we first need to introduce a modular theory for $H\subset \mathcal{H}$, with $H$ \textit{standard}, following \cite{Longo08}.
The analogous Tomita operator is defined by the closure of $S_H(f+ig)=f-ig$ and its polar decomposition is
\begin{equation}
S_H=J_H\Delta_H^{1/2}
\end{equation}
Then, the entropy of a vector $f\in H$ w.r.t. $H$ is defined by
\begin{equation}
S_H(f):=-\langle f,\log\Delta_H f\rangle
\end{equation}
Actually, in \cite{Ciolli:2019mjo} it was generalized for $f\in \mathcal{H}\cap \text{Dom}(K_H)$,
\begin{equation}
S_H(f):=-\langle f,P_H\log\Delta_H f\rangle=\sigma(f,P_H i K_H f),
\label{longoentropy}
\end{equation}
where
\begin{equation}
K_H:=-\log \Delta_H=i\frac{d}{du}\Delta^{iu}_H\bigg|_{u=0},
\end{equation}
is the 1-particle modular Hamiltonian and where $P_H:H+H'\rightarrow H$ is the cutting projector and we are assumming that $H$ is \textit{factorial}, namely $H \cap H'=0$. Here $H'$ is the symplectic complement of $H$. In that work they showed that Araki's relative entropy \eqref{Arakirelativeentropy}, between a coherent state $V(f)\Omega$ and a \textit{pure
} state $\Omega$, is nothing but the entropy of the vector $f\in \mathcal{H} \cap \text{Dom}(K_H)$:
\begin{equation}
S(\omega_f||\omega)=\sigma(f,P_H i K_H f)
\label{relativeentropy}
\end{equation}
We will take advantage of this result to compute the relative entropy by working exclusively at the level of the 1-particle Hilbert space $\mathcal{H}$.
\subsection{Purification of $\omega$}
Since we are interested in computing the relative entropy in the case that $\omega$ is a thermal state, and since \eqref{relativeentropy} is valid for pure states, we need to work with a purification of $\omega$. This can be achieved by a procedure we recall in this subsection, following \cite{Bostelmann:2020srs} (see also \cite{Petz:1990gb}).
By means of \eqref{D} we can get\footnote{We assume that $\mathcal{K}$ is complete with respect to $\tau$ and that $\sigma$ is non-degenerate.} a complex structure on $\mathcal{K}^\oplus:=\mathcal{K}\oplus \mathcal{K}$
\begin{equation}
i^\oplus=\left(
\begin{matrix}
-D& C\sqrt{1+D^2}\\
C\sqrt{1+D^2} & D
\end{matrix}\right)
\label{iplus}
\end{equation}
Let us call $\mathcal{H}^\oplus$ the complexification of $\mathcal{K}^\oplus$. The complex inner product is given by
\begin{equation}
\langle \cdot ,\cdot\rangle^\oplus=\tau^\oplus(\cdot,\cdot ) + i \sigma^\oplus(\cdot,\cdot ) ,
\end{equation}
with $\tau^\oplus:=\tau \oplus \tau$ and $\sigma^\oplus(\cdot,\cdot)=\tau^\oplus(\cdot,-i^\oplus \cdot)$. This inner product reduces to $w_2$ on $\mathcal{K}\simeq \mathcal{K}\oplus 0$:
\begin{equation}
\langle f\oplus 0,g\oplus 0\rangle^\oplus=\tau(f,g)+i\sigma(f,g)=w_2(f,g)
\end{equation}
We have
\begin{equation}
\text{CCR}(\mathcal{K},\sigma)\subset \text{CCR}(\mathcal{K}^\oplus,\sigma^\oplus)
\end{equation}
Similarly, for closed subspaces $H\subset \mathcal{K}$:
\begin{equation}
\text{CCR}(H,\sigma)\subset \text{CCR}(\mathcal{K},\sigma)
\end{equation}
And most importantly on the CCR$(\mathcal{K},\sigma)$ algebra,
\begin{equation}
\omega^\oplus(W(f\oplus 0))=e^{-\frac{1}{2}\tau^\oplus(f\oplus 0,\,f\oplus 0)}=\omega(W(f))
\end{equation}
which justifies why the pure state $\omega^\oplus$ of the CCR$(\mathcal{K}^\oplus,\sigma^\oplus)$ algebra is a purification of $\omega$.
The relative entropy for non-pure states associated to $R(H)$, with $H\simeq H \oplus 0$ the standard and factorial subspace of $\mathcal{H}^\oplus$ reads \cite{Bostelmann:2020srs},
\begin{equation}
S_{R(H)}(\omega_f||\omega)=\sigma^\oplus(f ,P_H i^\oplus K_H f),\qquad f\in \mathcal{K}\oplus 0 \cap \text{Dom}(K_H).
\label{relativeentropypurified}
\end{equation}
In \cite{Bostelmann:2020srs} a more general expression was obtained for other subspaces. We will not need this since for the $U(1)$ model the subspace associated to the bounded interval is standard and factorial as we will show. Note that in \eqref{relativeentropypurified} the modular Hamiltonian $K_H$ is associated to the modular operator that acts on the larger space $\mathcal{H}^\oplus$ while $P_H$ is the the real-linear cutting projector onto $H \oplus 0$.
\section{The chiral boson current}
We are interested in applying all the above to the case of a free chiral boson. More precisely, we consider the current usually denoted $\phi'(x)$. In the smeared version, the classical theory is defined by the symplectic space of compactly supported real functions $\mathcal{K}=C_c^\infty(\mathbb{R})$ with symplectic structure
\begin{equation}
\sigma(f,g)= \int_{\mathbb{R}} f(x)g'(x)dx
\label{symplectic}
\end{equation}
As reviewed in the previous section, we have a CCR$(\mathcal{K},\sigma)$ algebra associated to $(\mathcal{K},\sigma)$. In order to proceed, we need to define a quasi-free state by means of a positive symmetric bilinear form $\tau$. We start with the vacuum state which we will denote $\omega$ and then move on to a thermal state $\omega_\beta$.
\subsection{The vacuum case}
The vacuum state is defined by
\begin{equation}
\tau(f,g)=-\frac{1}{\pi}PV\int_{\mathbb{R}^2}dx\,dy\frac{f(x)g(y)}{(x-y)^2}=(f,\mathfrak{H}g')_{L^2}
\end{equation}
Here $PV$ denotes the principal value integral and $\mathfrak{H}$ the Hilbert transform. It is pure since $D=-\mathfrak{H}$ which is unitary or equivalently $D=-i \text{sgn}(p)$ in momentum space\footnote{We are taking the Fourier transform as $\hat{f}(p)= \int_{\mathbb{R}}dx\,e^{ixp}f(x)$. With this convention the Hilbert transform in momentum space is $i$ sgn$(p)$. More importantly, the generator of unitary translations \eqref{unitaryrep} is positive. \label{footnoteFourier}}, and therefore the complexification described in the previous section can be applied directly to $\mathcal{K}$ and we obtain the complex inner product
\begin{align}
\langle f,g \rangle&=-\frac{1}{\pi}\int_{\mathbb{R}^2}dx\,dy\frac{f(x)g(y)}{(x-y-i\epsilon)^2}=(f,\mathfrak{H}g')_{L^2}+i\sigma(f,g)\\
&=\frac{1}{\pi}\int_{0}^\infty\hat{f}(p)^*\hat{g}(p)pdp
\end{align}
which enables to establish an isomorphism with $\mathcal{H}\simeq L^2(\mathbb{R}_+,pdp)$. This is the vacuum 1-particle Hilbert space. We will mostly work in coordinate space.
An important role in this work is played by the PSL$(2,\mathbb{R})$ symmetry of the model. The unitary representation on $\mathcal{H}$ is given by,
\begin{equation}
U(g)f(x)=f(g^{-1}\cdot x)
\end{equation}
with an element of the group $g$ acting on the coordinate $x$ by linear fractional transformations:
\begin{equation}
g\cdot x=\frac{ax+b}{cx+d},\qquad g\in \text{PSL}(2,\mathbb{R}).
\end{equation}
We have three one-parametric subgroups related to the KAN decomposition of PSL$(2,\mathbb{R})$
\begin{equation}
r(\theta)=\left(
\begin{matrix}
\cos\frac{\theta}{2} & \sin\frac{\theta}{2}\\
-\sin\frac{\theta}{2} & \cos\frac{\theta}{2}
\end{matrix}\right),\quad \delta(s)=\left(
\begin{matrix}
e^{\frac{s}{2}} & 0\\
0 & e^{-\frac{s}{2}}
\end{matrix}\right),\quad \tau(t)=\left(
\begin{matrix}
1 & t\\
0 & 1
\end{matrix}\right),
\end{equation}
which we will refer to as rotatation, dilation and translation subroups, respectively. For example the dilation-translation subgroup acts as:
\begin{equation}
U(\tau(t))f(x)=f(x-t),\qquad U(\delta(s))f(x)=f(e^{-s}x)
\label{unitaryrep}
\end{equation}
Note that the generator of translations $P$ defined by $U(\tau(t))=e^{itP}$ is positive. This is most easily seen in momentum-space with the convention of footnote \ref{footnoteFourier}.
Now we can discuss the modular theory of standard subspaces of $\mathcal{H}$. Consider the subspaces $H(I)=\overline{C_c^\infty(a,b)}\in \mathcal{H}$, which are standard and factorial \cite{Longo08}. Since the above representation has positive $P$, the modular evolution on $H(\mathbb{R}_+)$ is (Theorem 3.3.1 of \cite{Longo08})
\begin{equation}
\Delta_{(0,\infty)}^{iu}=U\left(\delta\left(-2\pi u \right)\right),\quad u\in \mathbb{R}
\end{equation}
This can be seen by checking that $F(u):=\langle g,\Delta^{iu} f\rangle$ admits an analytic continuation to the strip $-1<$ Im$(u)<0$ and that the KMS property at temperature $-1$ is satisfied:
\begin{equation}
\langle\Delta_{(0,\infty)}^{iu}f,g\rangle=\langle g,\Delta_{(0,\infty)}^{i(u-i)}f\rangle
\end{equation}
By covariance $U(g)H(I)=H(g\cdot I)$ we have
\begin{equation}
\Delta_I^{iu}=U(\Bar{g})^{-1}\Delta_{(0,\infty)}^{iu}U(\Bar{g}),\quad u\in \mathbb{R},\quad \Bar{g}\cdot I= \mathbb{R}_+
\label{Imodular}
\end{equation}
Note that $\Bar{g}$ is defined modulo multiplication by a dilation on the left, but this does not affect $\Delta_I^ {iu}$.
\subsubsection*{The interval $(-\infty,t)$}
For instance, by considering $\bar{g}$ a rotation in $\pi$ followed by a translation of $t$,
\begin{equation}
\Delta^{iu}_{(-\infty,t)}f(x)=f\left(e^{-2\pi u}(x-t)+t\right).
\label{Bostelmannvacuumflow}
\end{equation}
We can compute then the modular Hamiltonian:
\begin{equation}
K_{(-\infty,t)}=i\frac{d}{du}\Delta_{(-\infty,t)}^{iu}\bigg|_{u=0}=2\pi i(t-x) \frac{d}{dx}
\end{equation}
And from here the relative entropy of a coherent state:
\begin{equation}
S_{R((-\infty,t))}(\omega||\omega_f)=S_{(-\infty,t)}(f)=2\pi\int_{-\infty}^t (t-x)f'(x)^2 dx
\end{equation}
Here $f$ need not be localized in the interval $(-\infty,t)$ \cite{Bostelmann:2020srs}.
\subsubsection*{The interval $(a,b)$}
We can repeat what we have just done for an interval $I=(a,b)$. The first step is to find $\Bar{g}$ such that $\Bar{g}\cdot I=\mathbb{R}_-$. The idea is to first project $I$ to the circle, forming an arc $(\theta_a,\theta_b)$. Then, we employ two symmetries (see Figure \ref{figure:gbar}). The first one consists of a rotation that maps $I$ to $(-\infty,b')$, which can be achieved by noticing that in the circle this is just a rotation of magnitude $-\pi-\theta_a$. Then, $(-\infty,b')$ is mapped by a translation of magnitude $b'$ to $\mathbb{R}_-$.
\begin{figure}[h]
\centering
\includegraphics[width=11cm]{geometric-transformations.eps}
\caption{In the left, a generic interval $(a,b)$ projected to the circle. It is first mapped to $(-\infty,b')$ by a rotation of magnitude $-\pi-\theta_a$, as shown in the middle diagram. Then it is mapped by a translation by $b'$ to $\mathbb{R}_-$, on the right.}
\label{figure:gbar}
\end{figure}
Having obtained such $\Bar{g}$, we can compute the modular evolution using \eqref{Imodular},
\begin{equation}
\Delta^{iu}_{(a,b)}f(x)=f\left(\frac{a e^{-2\pi u}(b-x)+b (x-a)}{e^{-2\pi u} (b-x)+x-a} \right).
\end{equation}
The modular Hamiltonian is
\begin{equation}
K_{(a,b)}=\frac{2\pi i (x-a)(b-x)}{b-a}\frac{d}{dx}.
\label{IModHam}
\end{equation}
The last ingredient to compute the relative entropy is the cutting projector $P_{I}$ associated to the interval $I=(a,b)$. Following the same lines in the proof of Proposition 4.2 in \cite{Bostelmann:2020srs} one can see that $P_{I}K_{I}f(x)=\chi_{I}(x)K_{I}f(x)$ where $\chi_{I}$ is the characteristic function of the interval.
\begin{proposition}
Let $f\in C^{\infty}_{c}(\mathbb{R})$, then it holds that $P_{I}K_{I}f(x)=\chi_{I}(x)K_{I}f(x)$
\end{proposition}
\begin{proof}
Given $f\in C^{\infty}_{c}(\mathbb{R})$ define the functions $g(x)=\chi_{I}(x)K_{I}f(x)$ and $g_{c}(x)=\chi_{I^{c}}(x)K_{I}f(x)$, which are piecewise-differentiable functions so their Fourier transforms decay at least like $p^{-2}$ for large $p$ and
\begin{equation*}
\|g\|_{\tau}^{2}=\frac{1}{\pi}\Re{\int_{0}^{\infty}|\hat{g}(p)|^2 pdp}<\infty,
\end{equation*}
then $g\in \mathcal{H}$ and also $g_{c}\in \mathcal{H}$. Moreover $\sigma(g_{c},\varphi)=0$ for all $\varphi \in C_{c}^{\infty}(I)$ because supp$(g_{c})\subseteq I_{c}$, then by continuity of $\sigma$ (with respect to the topology induced by $\tau$), $g_{c}\in H(I)'$. Similarly one can see that $g\in H(I)$ and therefore, as $H(I)$ is factorial, $P_{I}K_{I}f(x)=P_{I}(g(x)+g_{c}(x))=g(x)=\chi_{I}(x)K_{I}f(x)$, which completes the proof.
\end{proof}
Finally the relative entropy is
\begin{equation}
S_{R(I)}(\omega||\omega_f)=S_{I}(f)=2\pi\int_{a}^b \frac{ (x-a)(b-x)}{b-a}f'(x)^2 dx
\label{vacuumrelativeentropy}
\end{equation}
It is translation invariant, in the sense that $S_{I}(f)=S_{\tau\cdot I}(f\circ \tau^{-1})$. Is is also immediate to see that it is increasing with $L=(b-a)$: $\frac{d}{dL}S_{I(L)}(f)>0$. Interestingly, the relative entropy \eqref{vacuumrelativeentropy} satisfies a Bekenstein-like bound\footnote{Here $E(f):=\int_a^b f'(x)^2 dx$ is the 1-particle energy associated to the interval of $f \in \mathcal{H}$. }
\begin{equation}
S_{I}(f)\leq \pi \frac{L}{2} \int_a^b f'(x)^2\,dx =:\pi \frac{L}{2} E(f)
\label{Bekensteinvacuum}
\end{equation}
and a QNEC-like bound\footnote{This bound (for the interval centered at $0$) is the one in Proposition 3.7 of \cite{Bostelmann:2020srs}, where the appropriate $T_{f}(s,L)$ is $2\pi \int_{-\min\left\{s,\frac{L}{2}\right\}}^{\min\left\{s,\frac{L}{2}\right\}}\dfrac{(x+\frac{L}{2})(\frac{L}{2}-x)}{L}f'(x)^2\,dx$. It is straightforward to see that such $T_{f}$ satisfies the smoothness hypothesis $C^1$ of the Proposition. }:
\begin{align}
S_I'':= \frac{d^2}{dL^2}S_{I(L)}(f)&=\frac{\pi}{2}\left(f'(b)^2+f'(a)^2\right)-\frac{4\pi}{L^3}\int_a^b \left(x-\frac{a+b}{2}\right)^2f'(x)^2\,dx\nonumber\\
&\geq -\frac{4\pi}{L^3}\int_a^b \left(x-\frac{a+b}{2}\right)^2f'(x)^2\,dx. \label{QNECvacuum}
\end{align}
The QNEC, when stated in terms of the relative entropy, reads $S''(\lambda)>0$, with the understanding that $\lambda$ continuously labels nested spacetime regions. However in \eqref{QNECvacuum} we see a violation of the QNEC, which was anticipated in \cite{Bostelmann:2020srs}. What \eqref{QNECvacuum} says is that in order to have a large violation of the QNEC, a considerable amount of energy must be concentrated near the boundaries of the interval (note also that this negative bound can be saturated). Similarly, the Bekenstein-like bound \eqref{Bekensteinvacuum} implies that in order to make a coherent state largely distinguishable from the vacuum, a considerable amount of energy needs to be placed in the interval.
\subsection{The thermal case}
We now turn our attention to thermal states. The underlying symplectic space is again $(C_c^\infty(\mathbb{R}),\sigma)$ with \eqref{symplectic}. The thermal state is defined by
\begin{equation}
\tau_\beta(f,g)=-\pi PV\int_{\mathbb{R}^2}dx\,dy\frac{f(x)g(y)}{\beta^2\sinh^2\left(\frac{\pi}{\beta}(x-y)\right)}
\end{equation}
This gives the 2-point function \cite{Borchers:1998ye},
\begin{equation}
w_2^{(\beta)}( f,g)=-\pi\int_{\mathbb{R}^2}\frac{f(x)g(y)}{\beta^2\sinh^2\left(\frac{\pi}{\beta}(x-y)-i\epsilon\right)}dx\,dy.
\end{equation}
Such 2-point function satisfies being translation invariant and the KMS condition with respect to translations. The real Hilbert space $\mathcal{K}=L^2(\mathbb{R}_+,\frac{pdp}{1-e^{-\beta p}})$ is obtained after completion of $C_c^\infty(\mathbb{R})$ with $\tau_\beta$ \cite{Bostelmann:2020srs}. Note that this thermal state is the geometric KMS state of \cite{Longo:2016pci,Camassa:2011wk}\footnote{We thank Yoh Tanimoto for pointing this out.}.
Since the state is not pure, we first proceed to ``purify''. In momentum space $D=-i(1-e^ {-\beta p})$, which we use to construct $i^\oplus$ (as given by \eqref{iplus}) and then $\mathcal{H}^\oplus$, the complexification of $\mathcal{K}\oplus\mathcal{K}$.
There are operators acting as the dilation-translation group\footnote{We are not claiming these operators form a representation on $\mathcal{K}$. For instance, there are values of the parameters where the logarithms are not defined and the function is instead defined to be zero. We will not need to take this into account.} on the half-lines. For example, on $H(\mathbb{R}_-)$ \cite{Borchers:1998ye}:
\begin{align}
U_\beta(\delta(s))f(x)&=f\left( -\frac{\beta}{2\pi}\log\left(1+e^{-s}(e^{-\frac{2\pi x}{\beta}}-1)\right) \right),\nonumber\\
U_\beta(\tau(t))f(x)&=f\left(x- \frac{\beta}{2\pi}\log\left(1+\frac{2\pi t}{\beta}e^{\frac{2\pi x}{\beta}}\right) \right),
\label{BYoperators}
\end{align}
which satisfy
\[ U_\beta(\delta(s))U_\beta(\tau(t))U_\beta(\delta(-s))=U_\beta(\tau(e^st)).\]
These operators leave $w_2^{(\beta)}$ invariant, which implies that
\begin{equation}
U_\beta^\oplus (f \oplus 0 + i^\oplus g \oplus 0):=U_\beta f \oplus 0 + i^ \oplus U_\beta g \oplus 0, \qquad f,g\in\mathcal{K}, \
\label{Uplus}
\end{equation}
are unitaries of $\mathcal{H}^\oplus$ (we show this later on). Because of this, the modular operator associated to $H(\mathbb{R}_-)$ is given by\footnote{We have a sign difference with respect to \cite{Bostelmann:2020srs} in the parameter inside the dilation $\delta$. This translates into a sign difference in the modular Hamiltonian, but then the relative entropy coincides with equation (5.22) of that reference. } \cite{Bostelmann:2020srs},
\begin{equation}
\Delta^{iu}_{H(\mathbb{R}_-)}=U^\oplus_\beta(\delta(2\pi u)).
\label{BostelmannModularOp}
\end{equation}
Note that this reduces to \eqref{Bostelmannvacuumflow} for $\beta\rightarrow \infty$.
\subsubsection*{The interval $(-\infty,t)$}
From this last expression, and conjugating with the (vacuum) translation $U(t)$ as in \eqref{Imodular}, one can compute the modular Hamiltonian associated to $(-\infty,t)$,
\begin{equation}
K_{H((-\infty,t))}f(x)=\beta i^\oplus\left(1-e^{\frac{2\pi}{\beta} (x-t)}\right)f'(x) ,
\label{BostelmannModHamiltonian}
\end{equation}
and the relative entropy
\begin{equation}
S_{R((-\infty,t))}(\omega_f||\omega)=\beta \int_{-\infty}^t \left(1-e^{\frac{2\pi}{\beta}(x-t)}\right)f'(x)^2\,dx.
\label{BostelmannBetaRelEntropy}
\end{equation}
which is was first computed in \cite{Bostelmann:2020srs}. Note that in Proposition 5.6 of that reference it is shown that the subspace $H((-\infty,t))$ is both standard and factorial, so the relative entropy can be computed with \eqref{relativeentropy}.
\subsubsection*{The interval $(a,b)$}
Now we would like to approach the computation of the relative entropy for the bounded interval $I=(a,b)$. The strategy is analogous to the vacuum case of the previous subsection, but three issues are worth mentioning. First, the subspaces $H(I)$ must be shown to be standard and factorial (which we do at the end). Second the assignment $I \mapsto H(I)$ is not PSL$(2,\mathbb{R})$-covariant anymore, namely $U_\beta(g)H(I)\neq H(g\cdot I)$. However, we only need to find a $\bar{g}$ such that
\begin{equation}
U_\beta(\bar{g})H(I)=H(\mathbb{R}_-),
\end{equation}
then we conjugate with this unitary the modular operator of the negative real line \eqref{BostelmannModularOp} (in complete analogy with \eqref{Imodular}). Explicitly,
\begin{equation}
\Delta_I^{iu}=U(\Bar{g})^{-1}\Delta_{(-\infty,0)}^{iu}U(\Bar{g}),\quad u\in \mathbb{R}.
\label{Imodular2}
\end{equation}
Third, the attempt to construct $\bar{g}$ as described in the vacuum case, see Figure \ref{figure:gbar}, is not immediate to generalize, since the unitary rotation is no longer available (the vacuum rotation does not leave $w_2^{(\beta)}$ invariant). In \cite{Borchers:1998ye} the authors find the unitary dilations and translations \eqref{BYoperators}. We need to find a unitary operator that works as a rotation, meaning that \textit{it does not fix} $\infty$ (in the real-line picture). We propose that there exists $\alpha(\theta,x)$ such that
\begin{equation}
U_\beta(r(\theta))f(x)=f(\alpha(\theta,x)).
\label{alphadef}
\end{equation}
This means $\alpha(\theta,x)$ should obey the following three conditions:
\begin{enumerate}
\item Identity: $\alpha(0,x)=x$ \\
\item 1-parameter group: $\alpha(\theta_1,\alpha(\theta_2,x))=\alpha(\theta_1+\theta_2,x)$ \\
\item $w_2^{(\beta)}$-compatibility: $\frac{\partial \alpha(\theta,x)}{\partial x}\frac{\partial \alpha(\theta,y)}{\partial y}\sinh\left(\frac{\pi}{\beta}(\alpha(\theta,x)-\alpha(\theta,y))\right)^{-2}=\sinh\left(\frac{\pi}{\beta}(x-y)\right)^{-2}$
\end{enumerate}
Of course, $\alpha(\theta,x)$ depends also on $\beta$. The third condition, together with \eqref{Uplus}, assures that the operator $U^\oplus_\beta$ induced by $U_\beta(r(\theta))$ is unitary. Let us see why,
\begin{align}
\langle U^\oplus_\beta (f_1\oplus 0 + i^\oplus g_1 \oplus 0)\,&,\,U^\oplus_\beta (f_2\oplus 0 + i^\oplus g_2 \oplus 0) \rangle^\oplus\nonumber\\
&=\langle U_\beta(r(\theta)) f_1\oplus 0 + i^\oplus U_\beta(r(\theta)) g_1 \oplus 0 \,,\,U_\beta(r(\theta)) f_2\oplus 0 + i^\oplus U_\beta(r(\theta)) g_2 \oplus 0 \rangle^\oplus \nonumber \\
&= w_2^{(\beta)}(f_1\circ \alpha(\theta,\cdot),f_2\circ \alpha(\theta,\cdot))+w_2^{(\beta)}(g_1\circ \alpha(\theta,\cdot),g_2\circ \alpha(\theta,\cdot))\nonumber\\
&+ \tau_\beta(f_1\circ \alpha(\theta,\cdot),-D g_2\circ \alpha(\theta,\cdot))+i\tau_\beta(f_1\circ \alpha(\theta,\cdot), g_2\circ \alpha(\theta,\cdot))\nonumber\\
&+ \tau_\beta(-D g_1\circ \alpha(\theta,\cdot), f_2\circ \alpha(\theta,\cdot))-i\tau_\beta(g_1\circ \alpha(\theta,\cdot), f_2\circ \alpha(\theta,\cdot))\nonumber\\
&= w_2^{(\beta)}(f_1\circ \alpha(\theta,\cdot),f_2\circ \alpha(\theta,\cdot))+w_2^{(\beta)}(g_1\circ \alpha(\theta,\cdot),g_2\circ \alpha(\theta,\cdot))\nonumber\\
&- \sigma(f_1\circ \alpha(\theta,\cdot), g_2\circ \alpha(\theta,\cdot))+i\tau_\beta(f_1\circ \alpha(\theta,\cdot), g_2\circ \alpha(\theta,\cdot))\nonumber\\
&+ \sigma(g_1\circ \alpha(\theta,\cdot), f_2\circ \alpha(\theta,\cdot))-i\tau_\beta(g_1\circ \alpha(\theta,\cdot), f_2\circ \alpha(\theta,\cdot))\nonumber\\
&= w_2^{(\beta)}(f_1\circ \alpha(\theta,\cdot),f_2\circ \alpha(\theta,\cdot))+w_2^{(\beta)}(g_1\circ \alpha(\theta,\cdot),g_2\circ \alpha(\theta,\cdot))\nonumber\\
&+i w_2^{(\beta)}(f_1\circ \alpha(\theta,\cdot), g_2\circ \alpha(\theta,\cdot))-iw_2^{(\beta)}(g_1\circ \alpha(\theta,\cdot), f_2\circ \alpha(\theta,\cdot))\nonumber\\
&=w_2^{(\beta)}(f_1,f_2)+w_2^{(\beta)}(g_1,g_2)+i w_2^{(\beta)}(f_1, g_2)-iw_2^{(\beta)}(g_1, f_2)\nonumber\\
&=\langle f_1\oplus 0 + i^\oplus g_1 \oplus 0\,,\,f_2\oplus 0 + i^\oplus g_2 \oplus 0\rangle^\oplus,
\end{align}
where in the fifth equality we used the $w_2^{(\beta)}$-compatibility condition.
In order to find $\alpha(\theta,x)$, there is a hint coming
from the PSL$(2,\mathbb{R})$ product rules:
\begin{align*}
{\delta(s)r(\theta)\delta(-s)=r\left(2\arctan\left( e^{-s}\lambda\right)\right)\delta\left(\log\left[\frac{1+e^{-2s}\lambda^2}{1+\lambda^2}\right]\right) \tau\left(\frac{2\sinh(s)\lambda}{1+e^{-2s}\lambda^2}\right)}
\end{align*}
where $\lambda=\tan\frac{\theta}{2}$.
This translates, by means of \eqref{BYoperators} and \eqref{alphadef}, into a functional equation:
\begin{equation}
\left(e^s+e^{-s}\lambda^2\right)\left(e^{\frac{2\pi}{\beta}\alpha\left(\theta,\phi(s,x)\right) }-1 \right)=\left(1+\lambda^2\right)\left(e^{\frac{2\pi}{\beta} \alpha(2\arctan\left( e^{-s}\lambda\right),x)}-1\right)-\frac{4\pi}{\beta}\sinh(s)\lambda
\end{equation}
with
\begin{equation}
\phi(s,x)=\frac{\beta}{2\pi}\log\left(1+e^{-s}(e^{\frac{2\pi x}{\beta}}-1)\right)
\end{equation}
It is convenient to work with $A(\lambda,x)$ defined by:
\begin{equation}
\alpha(\theta,x)=\frac{\beta}{2\pi} \log\left[1+A(\lambda,x)\right]
\label{Adef}
\end{equation}
Differentiating w.r.t. $s$ and setting $s=0$ we get a PDE
\begin{equation}
\frac{\lambda^2-1}{\lambda^2+1}A(\lambda,x)+\frac{\beta}{2\pi}\left(1-e^{-\frac{2\pi}{\beta}x}\right)\partial_xA(\lambda,x)=\lambda \partial_\lambda A(\lambda,x)+\frac{4\pi}{\beta}\frac{\lambda}{\lambda^2+1},
\end{equation}
which has infinite solutions of the form
\begin{equation}
A(\lambda,x)=\frac{2\pi}{\beta} \left[-\lambda +\frac{1+\lambda^2}{\lambda} B\left(\lambda \left(e^{\frac{2\pi}{\beta }x}-1\right)\right)\right],
\label{Asol}
\end{equation}
for any function $B$. From $\alpha(0,x)=x$ we get
\begin{equation}
B(z)\sim\frac{\beta}{2\pi}z,\qquad z\rightarrow 0.
\end{equation}
From this and condition 2 above (group property) evaluated at $x=0$ we get
\begin{equation}
B(z)=\frac{z}{z+\frac{2\pi}{\beta}}
\end{equation}
Now plugging this form of $B$ into \eqref{Asol} and taking into account \eqref{alphadef} and \eqref{Adef},
$U_\beta\left(r(\theta)\right)f=f(\alpha(\theta,\cdot))$ can be shown to be compatible with $w_2^{(\beta)}$ (condition 3 above) where
\begin{equation}
\alpha(\theta,x)=\frac{\beta}{2\pi} \log\left[1+A(\lambda,x)\right],\qquad A(\lambda,x)=\frac{2\pi}{\beta} \frac{e^{\frac{2\pi}{\beta}x }-1-\frac{2\pi}{\beta}\lambda}{\lambda (e^{\frac{2\pi}{\beta}x}-1)+\frac{2\pi}{\beta}}.
\label{alphasol}
\end{equation}
Let us see this, first of all we rewrite the $w_2^{(\beta)}$-compatibility condition,
\begin{equation*}
\sinh^{2}\left(\frac{\pi}{\beta}(\alpha(\theta,x)-\alpha(\theta,y))\right)=\frac{\partial \alpha(\theta,x)}{\partial x}\frac{\partial \alpha(\theta,y)}{\partial y} \sinh^{2}\left(\frac{\pi}{\beta}(x-y)\right).
\end{equation*}
A straightforward computation (using $e^{\frac{\pi}{\beta}\alpha(\theta,x)}=(1+A(\lambda,x))^{\frac{1}{2}}$) of the square root of the left hand side gives
\begin{equation*}
\sinh\left(\frac{\pi}{\beta}(\alpha(\theta,x)-\alpha(\theta,y))\right)=
\frac{1}{2}\left[ \frac{(1+A(\lambda,x))^{\frac{1}{2}}}{(1+A(\lambda,y))^{\frac{1}{2}}}-\frac{(1+A(\lambda,y))^{\frac{1}{2}}}{(1+A(\lambda,x))^{\frac{1}{2}}} \right].
\end{equation*}
Squaring this expression and with \eqref{alphasol},
\begin{align}
\sinh^{2}&\left(\frac{\pi}{\beta}(\alpha(\theta,x)-\alpha(\theta,y))\right)=
\frac{1}{4}\left[ \frac{1+A(\lambda,x)}{1+A(\lambda,y)}-\frac{1+A(\lambda,y)}{1+A(\lambda,x)} -2\right] \nonumber\\
&=Y(\lambda,x)Y(\lambda,y) \sinh^2\left(\frac{\pi}{\beta}(x-y)\right)
\end{align}
where
\begin{align*}
Y(\lambda,x)&=\frac{4 \pi ^2 \beta \left(\lambda ^2+1\right) e^{\frac{2 \pi x}{\beta }}}{\left(\beta \lambda \left(e^{\frac{2 \pi x}{\beta }}-1\right)+2 \pi \right) \left(\beta (\beta \lambda +2 \pi ) e^{\frac{2 \pi x}{\beta }}-\left(\beta ^2+4 \pi ^2\right) \lambda \right)}.
\end{align*}
But it turns out that a straightforward computation gives
\[\frac{\partial \alpha}{\partial x}(\theta,x)=Y(\lambda,x)\]
which means that the $w_2^{(\beta)}$-compatibility condition holds.
Having found a unitary rotation, we can implement the first transformation of Figure \ref{figure:gbar} with $U_\beta(r(\Tilde{\theta}))$ where
\begin{equation}
\Tilde{\theta}=2\arctan\left(-\frac{2\pi}{\beta}\frac{e^{\frac{2\pi a}{\beta}} }{e^{\frac{2\pi a}{\beta}}-(\frac{2\pi}{\beta})^2-1}\right),
\label{tildetheta}
\end{equation}
and taking into account that the corresponding unitary $U_\beta^\oplus(r(\theta))$ on $\mathcal{H}^\oplus$ is defined by \eqref{Uplus}. This rotation sends $a$ to $-\infty$ and $b$ to $b'=\alpha(\Tilde{\theta},b)$, so it maps $H(I)$ to $H((-\infty,b'))$. Then, by a unitary vacuum translation $U(-b')$, $H((-\infty,b'))$ is mapped to $H(\mathbb{R}_-)$ as desired. The unitary $U(\bar{g})$ is the composition of these two unitary transformations.
From \eqref{Imodular2}, the modular evolution on $H(I)\oplus 0$ is
\begin{equation}
\Delta^{iu}_{(a,b)}f(x)\oplus 0=f\left[\frac{\beta}{2\pi}\log\left(
\frac{\sinh(\pi u)e^{\frac{\pi}{\beta}(a+b-x)}-\sinh(\frac{\pi}{\beta}(b-a)+\pi u)e^{\frac{\pi x}{\beta}}}{-\sinh(\pi u)e^{-\frac{\pi}{\beta}(a+b-x)}+\sinh(-\frac{\pi}{\beta}(b-a)+\pi u)e^{-\frac{\pi x}{\beta}}} \right)
\right]\oplus 0
\label{IBetaModularOp}
\end{equation}
By differentiating, the modular Hamiltonian is
\begin{equation}
K_{(a,
b)}f(x)\oplus 0=2 \beta i^\oplus \dfrac{ \sinh{(\frac{\pi}{\beta}(x-a))}\sinh{(\frac{\pi}{\beta}(b-x))}}{\sinh{(\frac{\pi}{\beta}(b-a))}}f'(x)\oplus 0
\label{IBetaModHamiltonian}
\end{equation}
It coincides with \eqref{BostelmannModHamiltonian} in the limit $a\rightarrow -\infty$ and with \eqref{IModHam} for $\beta\rightarrow\infty$.
Again like in the vacuum case now one can still see that $P_{I}K_{I}f(x)=\chi_{I}(x)K_{I}f(x)$. The proof is similar to the one we showed above, the only difference is that this time we have to see that $g$ has finite $\tau_{\beta}$ norm, but the same argument works. Given $f\in C^{\infty}_{c}(\mathbb{R})$ define once again the functions $g(x)=\chi_{I}(x)K_{I}f(x)$ and $g_{c}(x)=\chi_{I^{c}}(x)K_{I}f(x)$, these are piecewise-differentiable functions so its Fourier transform is bounded and decays at least like $p^{-2}$ for large $p$, then
\begin{equation*}
\|g\|_{\tau_{\beta}}^{2}=\frac{1}{\pi}\Re{\int_{0}^{\infty}\dfrac{|\hat{g}(p)|^2 p}{1-e^{-\beta p}}dp}<\infty.
\end{equation*}
Finally, the relative entropy is given by
\begin{equation}
S_{R(I)}(\omega_f||\omega)=2 \beta \int_a^b \dfrac{ \sinh{(\frac{\pi}{\beta}(x-a))}\sinh{(\frac{\pi}{\beta}(b-x))}}{\sinh{(\frac{\pi}{\beta}(b-a))}}f'(x)^2\,dx
\label{Ibetarelativeentropy}
\end{equation}
which is our main result. This relative entropy coincides, modulo some factor, with the modular Hamiltonian in the cut-off theory (equation (4.2) in \cite{hartman2015speed}). This can be formally understood by first noticing that the relative entropy can be related to a difference of mean values of the modular Hamiltonian $K$ and a difference of entanglement entropies,
\begin{equation}
S(\omega_2||\omega_1)=(\langle K_1\rangle_2-\langle K_1\rangle_1 )- (S_2-S_1).
\end{equation}
In our case the last parenthesis is zero since one state is a unitary applied to the other state (in the vector representation). This explains the connection of \eqref{Ibetarelativeentropy} to the modular Hamiltonian of \cite{hartman2015speed}. Since the arguments of \cite{hartman2015speed} are of general validity within CFTs, and taking into account the above discussion, it is reasonable to expect that in general \eqref{Ibetarelativeentropy} will hold with $f'(x)^2$ replaced by the classical energy density $T_{00}(x)$ of the theory. We will confirm this expectation in the next section for the massless scalar QFT in 1+1 dimensions.
Identically to the vacuum case \eqref{vacuumrelativeentropy}, the relative entropy \eqref{Ibetarelativeentropy} is translation invariant and with positive derivative wih respect to the length $L$ of the interval. There is also a Bekenstein-like bound
\begin{equation}
S_{(a,
b)}(f)\leq \pi \frac{L}{2}\,\left( \frac{\tanh\left(\frac{\pi}{\beta}\frac{L}{2}\right) } {\frac{\pi}{\beta}\frac{L}{2}}\right) \int_a^b f'(x)^2\,dx \leq \pi \frac{L}{2}\, \int_a^b f'(x)^2\,dx,
\label{IbetaQNECbound}
\end{equation}
and a QNEC-like bound \footnote{Again, this the same as the result of Proposition 3.7 of \cite{Bostelmann:2020srs}, this time $T_{f}(s,L)$ is $2\beta \int_{-\min\left\{s,\frac{L}{2}\right\}}^{\min\left\{s,\frac{L}{2}\right\}}\dfrac{\sinh{\frac{\pi}{\beta}(x+\frac{L}{2})}\sinh{\frac{\pi}{\beta}(\frac{L}{2}-x)}}{\sinh{\frac{\pi L}{\beta}}}f'(x)^2\,dx$. It is straightforward to see that such $T_{f}$ satisfies the smoothness hypothesis of the Proposition. }
\begin{equation}
\frac{d^2}{dL^2}S_{I(L)}(f)\geq -\frac{\pi^2}{\beta \sinh^3(L\frac{\pi}{\beta})} \int_a^b \resizebox{.5\hsize}{!}{$\left[\left(\cosh( \frac{\pi}{\beta}L)-1\right)^2+2\sinh^2\left(\frac{\pi}{\beta}(x-c)\right)\left(1+\cosh^2(\frac{\pi}{\beta}L)\right)\right]$}f'(x)^2\,dx,
\end{equation}
where $c=(a+b)/2$. We shall discuss this expression later on.
Before concluding this section we have to show that $H(I)$ is standard and factorial so the machinery we have been using, and in particular \eqref{relativeentropy}, is valid. We do this in the following Proposition.
\begin{proposition}
$H(I)$ is standard and factorial
\begin{proof}
The condition of separability $H(I)\cap i^\oplus H(I)=0$ follows exactly as in Proposition 5.6 of \cite{Bostelmann:2020srs} (or with the logic for what follows). The cyclicity, $(H(I)+i^\oplus H(I))^\perp=0$, can be shown to hold using the unitary rotation \eqref{alphadef}. Given any $H((a,b))$ there is an associated subspace $H_{b'}:=H((-\infty,b'))=U^\oplus_\beta(r(\Tilde{\theta}))H(I)$ obtained by a rotation in $\tilde{\theta}$ given by \eqref{tildetheta} and explained after that equation. The subspace $H_{b'}$ is, by Proposition 5.6 of \cite{Bostelmann:2020srs}, standard and factorial. It is immediate to show that $0=(H_{b'}+i^\oplus H_{b'})^\perp=U^\oplus_\beta(r(\Tilde{\theta}))(H(I)+i^\oplus H(I))^\perp$ which implies that $H(I)$ is cyclic. Similarly, we conclude that $H(I)$ is factorial since $0=H_{b'}\cap H_{b'}'=U^\oplus_\beta(r(\Tilde{\theta}))(H(I)\cap H(I)')$.
\end{proof}
\end{proposition}
\section{The free massless boson in $1+1$ dimensions at finite temperature}
In this section we take advantage of the quantities we have computed for the chiral boson and combine the two chiralities in order to obtain the modular flow, modular Hamiltonian and relative entropy on the interval $(a,b)$ for the massless free boson in two dimensions $\Phi$.
First of all, let us define $x^\pm=t\pm x$, and $j^\pm(x^\pm)=\partial_\pm\phi^\pm(x^\pm)$ are the (non-smeared) chiral currents of the previous section (below we give further details). In this section we will use $\pm$ symbols to denote copies of the objects of the chiral case (with the exception of the symplectic structure $\sigma$ and the bilinear form $\tau$). So for example $\mathcal{H}$ now refers to a Hilbert space of the two-dimensional model, and $\mathcal{H}_\pm$ are Hilbert spaces of the chiral case.
The symplectic space of the massless boson in two dimensions is\footnote{Here we are defining $f\in \dot{C}_c^\infty(\mathbb{R})$ if $f\in {C}_c^\infty(\mathbb{R})$ and $\hat{f}(0)=0$. This is necessary to avoid the well-known IR problem of the massless 2-dimensional field \cite{Streater:1989vi}.} \cite{Longo:2021rag}
\begin{equation}
\mathcal{K}=C_c^\infty(\mathbb{R})\oplus \dot{C}_c^\infty(\mathbb{R})
\end{equation}
with symplectic structure
\begin{equation}
\sigma_{2D}((f_1,g_1),(f_2,g_2))=\frac{1}{2}\int_\mathbb{R} dx (g_1(x) f_2(x)-f_1(x) g_2(x))
\label{symplectic2dim}
\end{equation}
Here the pair $(f,g)\in\mathcal{K}$ should be thought as the initial conditions $\Phi(0,x)=f(x)$, $\dot\Phi(0,x)=g(x)$ of a solution $\Phi(t,x)$ of the Klein-Gordon equation. In general,
\begin{equation}
\Phi(t,x)=\phi_+(x^+)+\phi_-(x^-),\qquad \phi^\pm \in C_c^\infty(\mathbb{R}).
\end{equation}
Then, the symplectic structure \eqref{symplectic2dim} can be written as
\begin{equation}
\sigma_{2D}((f_1,g_1),(f_2,g_2))=-\int_\mathbb{R}dx \left(\phi_1^+(x) \phi_2^+{}'(x)+\phi_1^-(x) \phi_2^-{}'(x) \right)
\label{symplecticstructures}
\end{equation}
The lack of mixing between the chiralities implies that there is a symplectic isomorphism\footnote{It is most easily written in Fourier space: $\hat\phi_\pm(\pm p)=\frac{1}{2}(\hat{f}( p)\pm \frac{i}{p} \hat{g}( p))$.} $\chi$ that maps $(\mathcal{K},\sigma_{2D})$ to $(\mathcal{K}_- \oplus \mathcal{K}_+,-(\sigma\oplus\sigma)) $, with inverse given by
\begin{equation}
\chi^{-1}\begin{pmatrix}\phi_+ \\ \phi_-\end{pmatrix}=\begin{pmatrix}\phi_+(x)+\phi_-(-x) \\ {\phi'_+}(x)+{\phi'_-}(-x)\end{pmatrix}=\begin{pmatrix} f(x) \\ g(x) \end{pmatrix}.
\label{symplectomorphism}
\end{equation}
In turn, this implies that the CCR$(\mathcal{K},\sigma_{2D})$ algebra is equivalent to the tensor product \[\text{CCR}(\mathcal{K}_-,-\sigma)\otimes
\text{CCR}(\mathcal{K}_+,-\sigma),\]
with $\sigma$ as in \eqref{symplectic}. More precisely, we identify these CCR-algebras by
\begin{equation}
W(\phi_-(x))\otimes W(\phi_+(x))\mapsto W((\phi_+(x)+\phi_-(-x),\,\,{\phi_+}'(x)+{\phi_-}'(-x)))
\end{equation}
This is in fact a $*-$isomorphism of the algebras. The change in sign in the symplectic structure $\sigma$ w.r.t to the previous section requires a change in sign in the complex structure\footnote{In order to see this, note that $\tau$ is independent of this change in sign, since it must be positive. Therefore from the defining equation of the complex structure $\tau(\cdot,D\cdot)=\sigma(\cdot,\cdot)$ it is seen that a change in sign in $\sigma$ translates into a change in sign in $D$ and therefore in the complex structure.}, and these two signs end up compensating each other in the relative entropy\footnote{The 1-particle modular Hamiltonian $K$ is not affected by this sign change, since $S$ is not affected as seen by its definition and neither is $S^*$, therefore $\Delta=S^*S$ is not affected.} \eqref{relativeentropy}.
Given a positive symmetric bilinear form $\tau_{2D}$ on $\mathcal{K}$ and its corresponding quasi-free state on CCR$(\mathcal{K},\sigma_{2D})$, by the isomorphisms mentioned above we get a quasi-free product state on CCR$(\mathcal{K}_+,\sigma)\otimes \text{CCR}(\mathcal{K}_-,\sigma)$ with the same $\tau$ for each chiral copy. Therefore the vacuum one-particle Hilbert space is
\begin{equation}
\mathcal{H}\simeq \mathcal{H}_-\oplus\mathcal{H}_+
\end{equation}
where $\mathcal{H}_\pm$ are copies of the chiral boson Hilbert space $L^2(\mathbb{R}_+,pdp)$. The isomorphism \eqref{symplectomorphism} is anti-linear, since in momentum space (or coordinate space, using properties of the Hilbert transform $\mathfrak{H}$) it is direct to show that
\begin{equation}
\chi^{-1}i_1=-i_2\chi^{-1},
\end{equation}
where $i_1$ is the complex structure of the chiral boson and $i_2$ is the complex structure in \cite{Longo:2020amm,Longo:2021rag}
\begin{equation}
i_2:=\left(\begin{matrix}
0& |p|^{-1}\\
-|p| &0
\end{matrix}\right)
\end{equation}
Therefore,
\begin{align}
\tau_{2D}(\Phi,\Psi)&= \sigma_{2D}(\Phi,i_2\Psi)\nonumber\\
&=-\sigma(\phi_+,(\chi i_2\Psi)_+)-\sigma(\phi_-,(\chi i_2\Psi)_-)\nonumber\\
&=-\tau(\phi_+,-i_1(\chi i_2\Psi)_+)-\tau(\phi_-,-i_1(\chi i_2\Psi)_-)\nonumber\\
&=\tau(\phi_+,i_1(\chi i_2\Psi)_+)+\tau(\phi_-,i_1(\chi i_2\Psi)_-)\\
&=\tau(\phi_+,\psi_+)+\tau(\phi_-,\psi_-)
\label{taus}
\end{align}
Analogously, for the thermal state we have
\begin{equation}
\mathcal{H}^\oplus\simeq\mathcal{H}_-^\oplus \oplus\mathcal{H}_+^\oplus
\end{equation}
with $\mathcal{H}_\pm^\oplus$ two copies of the purified Hilbert space that we constructed in the previous section (which was called $\mathcal{H}^\oplus$, we hope there is no confusion). The Fock spaces are related as
\begin{equation}
\Gamma(\mathcal{H}^\oplus)\simeq \Gamma(\mathcal{H}_-^\oplus)\otimes \Gamma(\mathcal{H}_+^\oplus)
\end{equation}
From now on we identify all these spaces with the appropriate isomorphisms.
\subsection{Modular flow and modular Hamiltonian}
Let us consider a causal diamond with base $(a,b)$ on the time-zero surface. Its corresponding standard subspace is $H(\Diamond)$ of pairs $(f,g)\in \mathcal{K}$ supported on the interval $(a,b)$ or equivalently Klein-Gordon fields $\Phi$ with initial conditions given by $(f,g)$. Note that such diamond is described in null coordinates as $(x^-,x^+)\in ((-b,-a), (a,b))$. Right wedges are obtained in limit $b\rightarrow \infty$ and similarly $a\rightarrow -\infty$ for left wedges.
At the one-particle level, we have
\begin{equation}
K_{H(\Diamond)}\simeq K_{H((-b,-a))} \oplus K_{H((a,b))}.
\label{Ks}
\end{equation}
This follows from the fact that for $\Phi, \Psi \in H (\Diamond)$
\begin{align}
S_{H(\Diamond)} (\Phi + i \Psi) &=\Phi - i \Psi \nonumber \\
&= \phi_+ -i \psi^+ + \phi_- -i \psi^- \nonumber\\
&=S_{H((-b,-a))}(\phi_- +i \psi^- ) + S_{H((a,b))} (\phi_+ +i \psi^+)
\end{align}
implying that $S_{H(\Diamond)}\simeq S_{H((-b,-a))} \oplus S_{H((a,b))}$ and then $\Delta_{H(\Diamond)}\simeq \Delta_{H((-b,-a))} \oplus \Delta_{H((a,b))}$. The modular evolution in the diamond,
\begin{equation}
\Delta_{H(\Diamond)}^{iu}\simeq \Delta_{H((-b,-a))}^{iu} \oplus \Delta_{H((a,b))}^{iu}
\end{equation}
which explicilty reads,
\begin{equation}
\left[\Delta_{H(\Diamond)}^{iu} \left( \begin{matrix}
f\\
g
\end{matrix}\right)\oplus \left( \begin{matrix}
0\\
0
\end{matrix}\right)\right](x)=\left(\begin{matrix}
[\Delta_{H((-b,-a))}^{iu} \phi_-](-x)+[\Delta_{H((a,b))}^{iu} \phi_+](x)\\ [\Delta_{H((-b,-a))}^{iu} \phi_-]'(-x)+[\Delta_{H((a,b))}^{iu} \phi_+]'(x)
\end{matrix}\right)\oplus \left( \begin{matrix}
0\\
0
\end{matrix}\right)
\end{equation}
where $\phi_\pm$ should be thought as given in terms of $(f,g)$ using the isomorphism $\chi$ and the evolution of each chirality is given in \eqref{IBetaModularOp}. A more intuitive presentation of the modular flow is to show the geometric transformation of the coordinates $(t,x)$ inside the diamond, as in
Figure \ref{modularevolution}.
\begin{figure}[h]
\centering
\includegraphics[width=11cm]{diamonds.eps}
\caption{Modular flow for low temperature (left) and high temperature (right)}
\label{modularevolution}
\end{figure}
\subsection{Relative entropies and bounds}
The relative entropy in two dimensions is the sum of the relative entropies of the chiral copies, which follows from \eqref{symplecticstructures} and \eqref{Ks}. Before arriving to an explicit expression of the relative entropies for different cases, we first find the modular Hamiltonians.
\subsubsection*{The wedge}
On a right wedge $W_R(a)$ with base $(a,\infty)$, given \eqref{symplectomorphism} and \eqref{Ks}, we have the corresponding vacuum modular Hamiltonian acting on the initial conditions
\begin{align}
K_{H((W_R(a))}\left(\begin{matrix}
f\\
g
\end{matrix}\right)
&=\left( \begin{matrix}
(K_{H((-\infty,-a))}\phi_-)(-x) +(K_{H((a,\infty))}\phi_+)(x)\\
(K_{H((-\infty,-a))}\phi_-)'(-x) +(K_{H((a,\infty))}\phi_+)'(x)
\end{matrix}
\right)\nonumber\\
&=-2\pi i\left( \begin{matrix}
(-a+x)\phi_-'(-x) +(x-a)\phi_+'(x)\\
((-a-x)\phi'_-)'(-x) +((x-a)\phi'_+)'(x)
\end{matrix}
\right)\nonumber\\
&=-2\pi i\left( \begin{matrix}
(-a+x)\phi_-'(-x) +(x-a)\phi_+'(x)\\
((a-x)\phi'_-(-x) +(x-a)\phi'_+(x))'
\end{matrix}
\right)\nonumber\\
&=-2\pi i\left( \begin{matrix}
(x-a) g(x)\\
((x-a) f'(x))'
\end{matrix}
\right)
\end{align}
Where in the second line we made use of the antilinearity between the chiral spaces $\mathcal{H}_\pm$ and $\mathcal{H}$. Then,
\begin{equation}
K_{H(W_R(a))}=-2\pi i\left( \begin{matrix}
0& x-a\\
\frac{d}{dx}(x-a)\frac{d}{dx} & 0
\end{matrix}
\right)
\end{equation}
Plugging this modular Hamiltonian in \eqref{relativeentropy},
\begin{equation}
S_{H(W_R(a))}((f,g))=2\pi\int_a^\infty (x-a)T_{00}(x)dx,
\end{equation}
with
\begin{equation}
T_{00}(x)=\frac{1}{2}(f'(x)^2+g(x)^2)
\end{equation}
the classical energy density at $t=0$ of the KG field $\Phi$. This is the same result as that of \cite{Longo:2019mhx}, with a translation by $a$.
Similarly, for the thermal state we have
\begin{align}
K_{H(W_R(a))}^{(\beta)}\left(\begin{matrix}
f\\
g
\end{matrix}\right)\oplus \left(\begin{matrix}
0\\
0
\end{matrix}\right)
&=-\beta i^\oplus\left( \begin{matrix}
\left(1-e^{-\frac{2\pi}{\beta}(x-a)}\right) g(x)\\
\left[\left(1-e^{-\frac{2\pi}{\beta}(x-a)}\right) f'(x)\right]'
\end{matrix}
\right)\oplus \left(\begin{matrix}
0\\
0
\end{matrix}\right)
\label{wedgebetamodularhamiltonian}
\end{align}
Where we have used \eqref{BostelmannModHamiltonian} and \eqref{IBetaModHamiltonian} in the limit $b\rightarrow\infty$. The relative entropy on the wedge at finite temperature is then,
\begin{equation}
S_{H(W_R(a))}^{(\beta)}((f,g))=\beta\int_a^\infty \left(1-e^{-\frac{2\pi}{\beta}(x-a)}\right) T_{00}(x)dx
\label{wedgebetarelativeentropy}
\end{equation}
Note that this expression is valid even for initial conditions supported outside $x>a$, since the cutting projector in \eqref{relativeentropy} restricts the integral to the wedge \cite{Longo:2020amm}. The only restriction on the initial conditions $(f,g)$ is that they belong to the domain of the modular Hamiltonian \eqref{wedgebetamodularhamiltonian}.
\subsubsection*{The interval}
Repeating the previous computations for the time-zero interval $(a,b)$, we obtain for the vacuum,
\begin{equation}
K_{H((a,b))}\left(\begin{matrix}
f\\
g
\end{matrix}\right)
=-2\pi i\left( \begin{matrix}
\frac{(b-x)(x-a)}{b-a} g(x)\\
\left[\frac{(b-x)(x-a)}{b-a} f'(x)\right]'
\end{matrix}
\right) .
\label{Imodularhamiltonian2D}
\end{equation}
The vacuum relative entropy of a coherent state is
\begin{equation}
S_{H((a,b))}((f,g))=2\pi\int_a^b \frac{(b-x)(x-a)}{b-a} T_{00}(x)dx
\label{Irelativeentropy2D}
\end{equation}
On the other hand, at finite temperature we have,
\begin{align}
K_{H((a,b))}^{\beta}\left(\begin{matrix}
f\\
g
\end{matrix}\right)\oplus \left(\begin{matrix}
0\\
0
\end{matrix}\right)
&=-2\beta i^\oplus\left( \begin{matrix}
\dfrac{ \sinh{(\frac{\pi}{\beta}(x-a))}\sinh{(\frac{\pi}{\beta}(b-x))}}{\sinh{(\frac{\pi}{\beta}(b-a))}} g(x)\\
\left[ \dfrac{ \sinh{(\frac{\pi}{\beta}(x-a))}\sinh{(\frac{\pi}{\beta}(b-x))}}{\sinh{(\frac{\pi}{\beta}(b-a))}} f'(x)\right]'
\end{matrix}
\right)\oplus \left(\begin{matrix}
0\\
0
\end{matrix}\right)
\label{Ibetamodularhamiltonian2D}
\end{align}
The relative entropy of a coherente state in the thermal state representation is,
\begin{equation}
S_{H((a,b))}^{(\beta)}((f,g))=2\beta\int_a^b \dfrac{ \sinh{(\frac{\pi}{\beta}(x-a))}\sinh{(\frac{\pi}{\beta}(b-x))}}{\sinh{(\frac{\pi}{\beta}(b-a))}} T_{00}(x)dx
\label{Ibetarelativeentropy2D}
\end{equation}
This expression confirms, at least for this model, the expectation that in a CFT the relative entropy of coherent states on a finite interval has this form, where the dependence on the model enters only in $T_{00}$. Because of this, the bounds obtained earlier for the chiral model also hold in this case. The Bekenstein-like bound reads,
\begin{equation}
S_{(a,
b)}((f,g))\leq \pi \frac{L}{2}\,\left( \frac{\tanh\left(\frac{\pi}{\beta}\frac{L}{2}\right) } {\frac{\pi}{\beta}\frac{L}{2}}\right) \int_a^b T_{00}(x)\,dx \leq \pi \frac{L}{2}\, \int_a^b T_{00}(x)\,dx.
\end{equation}
While the QNEC-like bound is,
\begin{equation}
\frac{d^2}{dL^2}S_{I(L)}((f,g))\geq -\frac{\pi^2}{\beta \sinh^3(L\frac{\pi}{\beta})} \int_a^b \resizebox{.5\hsize}{!}{$\left[\left(\cosh( \frac{\pi}{\beta}L)-1\right)^2+2\sinh^2\left(\frac{\pi}{\beta}(x-c)\right)\left(1+\cosh^2(\frac{\pi}{\beta}L)\right)\right]$}T_{00}(x)\,dx.
\end{equation}
\section{Conclusions}
We have extended the relative entropy on $\mathbb{R}_-$ with $T\geq 0$ of {\footnotesize\color{blue} [BCD '22]} to a bounded interval (see \eqref{Ibetarelativeentropy}). In order to achieve this, we found a unitary in the thermal Hilbert space implementing a rotation. Such unitary may turn out to be useful for other related computations.
From the relative entropy \eqref{Ibetarelativeentropy} a Bekenstein-like bound and a QNEC-like bound can be observed. There is however a violation of the QNEC $S''>0$, and all of this is in agreement with \cite{Bostelmann:2020srs}. For the vacuum case, given an energy $E$ we can find a family of functions $f_{n}\in H(I)$ such that $S_{I}''(f_{n})$ given in
\eqref{QNECvacuum} goes to zero (just concentrating the energy density closer and closer around the center of the interval), thus making the QNEC violation as small as desired. On the contrary, in the thermal case this is not possible because there is always a bound for the violation of the QNEC given by
\[ S''_{I}(f_n)\rightarrow -\dfrac{\pi^2}{\beta \sinh^3(L\frac{\pi}{\beta})}\left(\cosh( \frac{\pi}{\beta}L)-1\right)^2 E<0\] despite how the energy density is distributed (see \eqref{IbetaQNECbound}).
The computations in the context of a thermal $U(1)$ current left a clear path to analyse the case of a thermal state of the free massless boson in $1+1$ dimensions restricted to a causal diamond. In the last Section we obtained the modular Hamiltonian \eqref{Ibetamodularhamiltonian2D} and relative entropy \eqref{Ibetarelativeentropy2D} at finite temperature in 1+1 dimensions, with analogous bounds as in the chiral case. In principle most of these techniques could be used for the massless boson in higher dimensions and also the free massive boson in $d+1$ dimensions with $T>0$ \cite{garbarzpalau}. In addition, it would be very interesting to extend the formalism to include non-coherent states, although this seems a much more complicated affair.
\section*{Acknowledgements}
We would like to thank David Blanco and Guillem Pérez-Nadal and specially both Henning Bostelmann for correspondance regarding \cite{Bostelmann:2020srs} and Yoh Tanimoto for reading a preliminary version of the manuscript and providing valuable feedback. This work was partially supported by grants PIP and PICT from CONICET and ANPCyT. The work of G.P. is supported by an UBACYT scholarship from the University of Buenos Aires.
\bibliographystyle{toine}
|
1111.1045
|
\section{\label{}}
\section{Introduction}
Radio and X-ray observations have revealed about 140 millisecond
pulsars (MSPs) in 26 globular clusters~\cite[GCs;][]{Freire_web}. However, the presence
of much stronger X-ray emitters can contaminate the X-ray observations
of MSPs. Because MSPs are the only known steady $\gamma$-ray~sources in GCs~\cite{lat_msp}, $\gamma$-ray~observations of GCs serve as an alternative channel in studying
the underlying MSP populations in GCs.
Using the Large Area Telescope (LAT), $\gamma$-rays~from 8 GCs~\cite{lat_8GCs} have been
discovered, including 47 Tucanae~\cite{lat_47Tuc} and Terzan 5~\cite{Kong_Terzan5}.
\begin{figure*}
\includegraphics[width=75mm]{combine0p5_20jpg.eps}%
\includegraphics[width=75mm]{combine10-20jpg.eps}\\%
\caption{The count maps of the $5\ensuremath{^\circ}\times5\ensuremath{^\circ}$ region centered on Terzan 5. The insets show the test-statistic maps~\cite{Kong_Terzan5}.}
\end{figure*}
\section{Models of $\gamma$-rays~from globular clusters}
The radiation mechanism of $\gamma$-rays~is unclear. In the pulsar
magnetosphere model, e.g.~\cite{Venter08}, $\gamma$-rays~up to a few GeV come from the
MSPs through curvature radiation. On the other hand, inverse Compton
(IC) processes resulted from energetic particles up-scattering low-energy
photons, such as starlight and infrared light, may give rise to $\gamma$-rays~of
MeV to TeV energies, e.g.~\cite{Cheng_ic_10}. In either model, it is expected that the $\gamma$-ray~luminosity of a GC is proportional to the stellar encounter rate, a
measure of the number of MSPs in a GC.
\section{New $\gamma$-ray~globular clusters uncovered}
Terzan 5 contains the largest number of known MSPs among all GCs. It
was discovered as the second known $\gamma$-ray~emitting GC after 47~Tucanae~\cite{Kong_Terzan5} (see Figure~1). We note that 47~Tucanae was discovered in the bright source list~\cite{bsl_lat}, while the discovery of Terzan~5 in $\gamma$-rays~was announced~\cite{Kong_Terzan5} before the release of the first Fermi/LAT catalog~\cite{lat_1st_cat} and the report of the 8 GCs~\cite{lat_8GCs}.
Like 47~Tucanae, the $\gamma$-ray~spectrum of Terzan~5 also shows a cutoff at $\sim$3~GeV~\cite{lat_8GCs,Kong_Terzan5}.
After the discovery of other six $\gamma$-ray~emitting GCs~\cite{lat_8GCs}, we also identified
a group of GCs with high encounter rate. Using more than two years of data
taken from LAT, we found $\gamma$-ray~emission from the directions of Liller~1,
NGC~6624, and NGC~6752~\cite{Tam_gc_11}. The test-statistic maps of the regions around
these 3~GCs are shown in Figures~2 and 3. For M80, NGC~6139, and NGC~6541, the detection is marginal ($4-5\sigma$) when it was first reported~\cite{Tam_gc_11}.
For the cases where the $\gamma$-ray~emission is offset from the core (i.e. Liller 1
and NGC 6624), the $\gamma$-ray~spectra in the energy range of 200~MeV to 100~GeV are presented in Figure 4. The photons above $\sim$20~GeV are detected at significance levels of 3--4. Once the existence of these high-energy photons is established, it will be easier to be reconciled in the IC models than in the pulsar magnetosphere model. In the latter case, spectral cut-offs at several GeV are expected.
\begin{figure*}
\includegraphics[width=85mm]{Liller1.eps}
\caption{The test-statistics map of Liller 1~\cite{Tam_gc_11}}
\end{figure*}
\begin{figure*}
\includegraphics[width=160mm]{fg5_6.eps}
\caption{The test-statistics maps of NGC~6624 (left) and NGC~6752 (right)~\cite{Tam_gc_11}}
\end{figure*}
\begin{figure*}
\includegraphics[width=80mm]{fg8.eps}
\includegraphics[width=80mm]{fg9.eps}
\caption{Spectra of Liller 1 (left)
and NGC 6624 (right).
The solid and dashed lines
represent the best-fit
power law and power law
with exponential cutoff,
respectively~\cite{Tam_gc_11}.}
\end{figure*}
\begin{figure*}
\includegraphics[width=150mm]{Fp_edgeon.eps}
\caption{The edge-on views of the fundamental plane relations of $\gamma$-ray~GCs. The straight
lines in the plots represent the projected best-fits~\cite{Hui11_correlation}.}
\end{figure*}
\section{The fundamental planes of $\gamma$-ray~globular clusters}
We have investigated the properties of the $\gamma$-ray~emitting globular
clusters~\cite{Hui11_correlation}. By correlating the observed $\gamma$-ray~luminosities with
various cluster properties, we probe the origin of the high energy
photons from these GCs. We found that the $\gamma$-ray~luminosity is
positively correlated with the encounter rate and the metalicity
[Fe/H] which places an intimate link between the $\gamma$-ray~emission
and the MSP population. We also found that the $\gamma$-ray~luminosity
increases with the energy densities of the soft photons at the
cluster location. When combining two parameters at the same
time, the correlation is even stronger. The edge-on fundamental
plane relations of $\gamma$-ray~GCs are depicted in Figure~5.
This finding strongly suggests that models that incorporate optical
or infrared photons should be taken into considerations in
explaining the $\gamma$-ray~emission from GCs, e.g. the IC models~\cite{Cheng_ic_10}.
\bigskip
\begin{acknowledgments}
P. Tam acknowledges the support of the Formosa Program of Taiwan, NSC100-2923-M-007-001-MY3, and
the NSC grant, NSC100-2628-M-007-002-MY3. AK is supported by a Kenda Foundation Golden Jade Fellowship.
\end{acknowledgments}
\bigskip
|
1111.1281
|
\section{Introduction}
The desire to endow important classes of $C^*$-algebras generated by partial isometries with a structure of a more general crossed product led to the concept of a partial group action, introduced in \cite{E-1}, \cite{Mc}, \cite{E0}, \cite{E1}. The new structure permitted to obtain relevant results on $K$-theory, ideal structure and representations of the algebras under consideration, as well as to treat amenability questions, especially amenability of $C^*$-algebraic bundles (also called Fell bundles), using both partial actions and the related concept of a partial representation. Amongst prominent classes of $C^*$-algebras endowed with the structure of non-trivial crossed products by partial actions one may list the Bunce-Deddens and the Bunce-Deddens-Toeplitz algebras \cite{E-2}, the approximately finite dimensional algebras \cite{E-3}, the Toeplitz algebras of quasi-ordered groups, as well as the Cuntz-Krieger algebras \cite{ELQ}, \cite{QR}.\\
The algebraic study of partial actions and partial representations was initiated in \cite{E1}, \cite{DEP} and \cite{DE}, motivating investigations in diverse directions. In particular, the Galois theory of partial group actions developed in \cite{DFP} inspired further Galois theoretic results in \cite{CaenDGr}, as well as the introduction and study of partial Hopf actions and coactions in \cite{CJ}. The latter paper became in turn the starting point for further investigation of partial Hopf (co)actions in \cite{AB}, \cite{AB2} and \cite{AB3}. The Galois theoretic treatment in \cite{CaenDGr} was based on a coring ${\mathcal C} $ constructed for an idempotent partial action of a finite group. The coring ${\mathcal C} $ was shown to fit the general theory of cleft bicomodules in \cite{bohmverc}, and, in addition, in \cite{Brz} descent theory for corings was applied, using ${\mathcal C} ,$ to define non-Abelian Galois cohomology ($i=0, 1$) for idempotent partial Galois actions of finite groups.\\
The general notion of a (continuous) twisted partial action of a locally compact group on a
$C^*$-algebra (a twisted partial $C^*$-dynamical system) and the cor\-res\-pon\-ding crossed pro\-ducts were given by R. Exel in \cite{E0}. The new construction permitted to show that any se\-cond countable $C^*$-algebraic bundle, which satisfies a certain regularity condition (automatically verified if the unit fiber algebra is stable), is a $C^*$-crossed product of the unit fiber algebra by a continuous partial action of the base group. The algebraic version of the latter fact was established in \cite{DES1}. The
importance of partial actions and partial representations was reinforced
by R. Exel in \cite{E3} where, among other results, it was proved that given a field $K$
of characteristic $0,$ a group $G$ and subgroups $H, N \subseteq G$ with $N$ normal in
$G$ and $H$ normal in $N,$ there is a twisted partial action $\theta $ of $G/N$ on the group
algebra $K(N/H)$ such that the Hecke algebra ${\mathcal H}(G,H)$ is isomorphic to the
crossed product $K(N/H) \ast _{\theta} G/N.$ More recent algebraic results on twisted partial
actions and corresponding crossed products were obtained in \cite{BLP}, \cite{DES2} and \cite{PSantA}. The algebraic concept of twisted partial actions also motivated the study of projective partial group representations, the corresponding partial Schur Multiplier and the relation to partial group actions with $K$-valued twistings in \cite{DN} and \cite{DN2}, contributing towards the elaboration of a background for a general cohomological theory based on partial actions. Further information around partial actions may be consulted in the survey \cite{D}.\\
The aim of this article is to introduce and study twisted partial Hopf actions on rings. The general definitions are given in Section 2, including that of a partial crossed product. The cocycle and normalization conditions are needed in order to make the partial crossed product to be both associative and unital. As expected, restrictions of usual (global) twisted Hopf actions naturally result in twisted partial Hopf actions. Idempotent twisted partial actions of groups give natural examples of twisted partial actions of Hopf group algebras. Less evident examples may be obtained using algebraic groups, as it is shown in Section 3. Actions of an affine algebraic group on affine varieties give rise to coactions of the corresponding commutative Hopf algebra $H$ on the coordinate algebras of the varieties, restrictions of which produce concrete examples of partial Hopf coactions. Then one may dualize in order to obtain partial Hopf actions. This works theoretically, but the elaboration of a concrete example needs some work. One possibility is to try to identify the finite dual $H^0$ for a specific $H$ obtained this way. A more flexible possibility is to find a concrete Hopf algebra $H_1$ such that $H$ and $H_1$ form a dual pairing. Then Proposition 8 from \cite{AB} produces a partial action of $H_1.$ One still wishes to transform it into a twisted one, which in the setting specified in Section 3 is not difficult. A concrete example is elaborated in Proposition \ref{Ex:AlgGrHopf}.\\
In order to treat the convolution invertibility of the partial cocycle in a manageable way, we introduce symmetric twisted partial Hopf actions in Section 4 and establish some useful technical formulas. Our definition is inspired by the case of twisted partial group actions. We also show that a restriction of a global twisted Hopf action with convolution invertible cocycle gives a symmetric twisted partial Hopf action. Theorem \ref{the41} relates isomorphisms of crossed products by symmetric twisted partial actions with a kind of ``partial coboundaries,'' establishing an analogue of a corresponding result known in the global case.\\
The last Section 5 is dedicated to the notion of partial cleft extensions and its relation with partial crossed products, in a quite similar fashion as it is done in classical Hopf algebra theory. The definition of a partial cleft extension reflects the ``partiality'' in more than one ways, incorporating, in particular, some equalities already proved to be significant in the study of partial group actions and partial representations (see Remark \ref{interaction}).
Then the main result Theorem \ref{the51} states that the partial cleft extensions over the coinvariants $A$ are exactly the crossed products by symmetric twisted partial Hopf actions on $A.$
\section{Twisted partial actions and partial crossed
products}
In this paper, except Section~\ref{ExViaAlGroups}, $\kappa $ will denote an arbitrary (associative)
unital commutative ring and unadorned $\otimes$ will stand for $\otimes_{\kappa }$, as well as $\text{Hom}(V,W)$ will mean
$\text{Hom}_\kappa(V,W)$ for any $\kappa$-modules $V$ and $W$.
\begin{defi} \label{defi:twisted} Let $H$ be a Hopf $\kappa $-algebra, $A$ a unital $\kappa $-algebra
with unity element $\um .$ Let furthermore $\alpha:H\otimes A\to A$ and $\omega:H\otimes H\to A$ be two $\kappa $-linear
maps. We will write $\alpha(h\otimes a):=h\cdot a$, and $\omega (h\otimes l) : =\omega (h,l)$, where $a\in A$ and $h, l\in H$.
The pair $(\alpha,\omega)$ is called a \underline{\it twisted partial action}
of $H$ on $A$ if the fol\-lo\-wing conditions hold:
\begin{align}
1_H\cdot a&=a \label{unitpartial},\\
h\cdot (ab)&=\sum(h_{(1)}\cdot a)(h_{(2)}\cdot b) \label{productpartial},\\
\sum(h_{(1)}\cdot(l_{(1)}\cdot a))\omega(h_{(2)} ,l_{(2)})&=\sum\omega(h_{(1)} ,
l_{(1)})(h_{(2)}l_{(2)}\cdot a)\label{torcao},\\
\omega (h ,l) &= \sum \omega (h_{(1)} , l_{(1)})(h_{(2)}l_{(2)}\cdot\um )\label{cociclo},
\end{align}
for all $a,b\in A$ and $h,l\in H$.
\end{defi}
If $H$, $A$ and $(\alpha,\omega)$ satisfy Definition~\ref{defi:twisted}, then we shall also say that $(A, \cdot, \omega)$
is a \underline{twisted partial $H$-module algebra}.
\begin{prop} \label{FirstProp} If $(\alpha , \omega )$ is a twisted partial action, then the following identities hold:
\begin{eqnarray}
\omega (h,l)& =& \sum(h_{(1)}\cdot (l_{(1)} \cdot \um ))\omega (h_{(2)} ,l_{(2)})= \sum(h_{(1)}\cdot\um)\omega(h_{(2)} , l).\label{firstprop}\
\end{eqnarray}
\end{prop}
{\bf Proof:} The first identity is obtained from (\ref{torcao}) by taking $a=1:$
\[
\sum(h_{(1)}\cdot (l_{(1)} \cdot \um)\omega(h_{(2)} ,l_{(2)}) = \sum\omega(h_{(1)} , l_{(1)})(h_{(2)}l_{(2)}\cdot\um)
\overset{\text{(\ref{cociclo})}}{=}\omega (h,l).
\]
For the second identity notice that
\begin{eqnarray}
& \, & \sum(h_{(1)}\cdot\um)\omega(h_{(2)} , l) =\sum(h_{(1)}\cdot\um)(h_{(2)}\cdot (l_{(1)} \cdot \um ))\omega (h_{(3)} , l_{(2)}) =\nonumber\\
&\, & =\sum(h_{(1)}\cdot (l_{(1)} \cdot \um ))\omega (h_{(2)} ,l_{(2)})
= \omega (h,l), \nonumber
\end{eqnarray} is obtained by using the first identity and (\ref{productpartial}). \nolinebreak\hfill$\Box$\par\medbreak
\vspace{.3cm}
We say that the map $\omega$ is {\it trivial}, if the following condition holds
\begin{align}
h\cdot(l\cdot \um)&=\omega (h ,l)=\sum(h_{(1)}\cdot\um)(h_{(2)}l\cdot\um)\
\label{cociclotrivial}\end{align}
for all $h,l\in H$. In this case, the twisted partial action $(\alpha,\omega)$ turns out to be a partial
action of $H$ on $A$, as introduced in \cite{CJ}. Indeed, if (\ref{cociclotrivial}) holds then the
condition (\ref{cociclo}) is superfluous, and for all $h,l\in H$ and
$a\in A$ we have:
\[
\begin{array}{ccl}
h\cdot(l\cdot a)&\overset{\text{(\ref{productpartial})}}{=}&\sum(h_{(1)}\cdot(l_{(1)}\cdot a))(h_{(2)}\cdot(l_{(2)}\cdot \um))
\overset{(\ref{cociclotrivial})}{=}\sum(h_{(1)}\cdot(l_{(1)}\cdot a))\omega(h_{(2)} ,l_{(2)})\\
&\overset{\text{(\ref{torcao})}}{=}&\sum\omega(h_{(1)} , l_{(1)})(h_{(2)}l_{(2)}\cdot a)
\overset{(\ref{cociclotrivial})}{=} \sum(h_{(1)}\cdot \um)(h_{(2)}l_{(1)}\cdot \um)(h_{(3)}l_{(2)}\cdot a)\\
&\overset{\text{(\ref{productpartial})}}{=}&(h_{(1)}\cdot \um)(h_{(2)}l\cdot a).\
\end{array}
\]
Observe also that if $h\cdot \um =\varepsilon(h)\um$, for all $h\in H$, then the condition (\ref{cociclo}) is a
trivial consequence of the counit's properties and the $\kappa $-linearity of $\omega$, and so
we recover the classical notion of a twisted (global) action of $H$ on $A$ (see, for instance, \cite{Mont}).
\begin{ex}\label{ex:kG} This example is inspired by
\cite[Proposition 4.9]{CJ} and suits\footnote{If one assumes in Definition 2.1 of \cite{DES1} that each $D_g$
is generated by a central idempotent, then the definition below is more general, as neither the invertibility
in $D_g D_{gh}$ of each $w_{g,h}$ is required, nor the $2$-cocycle equality.} Definition 2.1 of \cite{DES1}.
An idempotent twisted partial action of a group $G$ on a $\kappa $-algebra $A$ is
a triple
$$\left(\{D_g\}_{g\in G}, \{\alpha_g\}_{g\in G},
\{w_{g,h}\}_{(g,h)\in G\times G}\right),$$ where for each $g\in G$,
$D_g$ is an ideal of $A$ generated by a central idempotent $1_g$ of
$A$, $\alpha_g:D_{g^{-1}}\to D_g$ is an isomorphism of unital
${\kappa} $-algebras, and for each $(g,h)\in G\times G$, $w_{g,h}$ is an element of $ D_g
D_{gh}$, and the following statements are satisfied:
\begin{align}
1_e=\um \quad\text{and}\quad \alpha_e=I_A \label{pga1},\\
\alpha_g(\alpha_h(a1_{h^{-1}})1_{g^{-1}})w_{g,h}=w_{g,h}\alpha_{gh}(a1_{(gh)^{-1}}) \label{pga2},
\end{align}
for all $a\in A$ and $g,h\in G$, where $e$ denotes the identity element of $G$
and $I_A$ the identity map of $A$.
Let $\alpha:{\kappa} G\otimes A\to A$ and $\omega:{\kappa} G\otimes {\kappa} G\to A$ be the
$\kappa $-linear maps given respectively by $\alpha(g\otimes
a)=\alpha_g(a1_{g^{-1}})$ and $\omega(g, h)=w_{g,h}$, for all
$a\in A$ and $g,h\in G$. This pair $(\alpha,\omega)$ is a twisted
partial action of ${\kappa} G$ on $A$. Indeed, conditions (\ref{productpartial}) and (\ref{cociclo}) are
obvious since each $\alpha_g$ is multiplicative and each $w_{g,h}$
lives in $D_gD_{gh}$. Conditions (\ref{unitpartial}) and (\ref{torcao}) follow
easily from (\ref{pga1}) and (\ref{pga2}) respectively. Notice in addition that $g \cdot \um=1_g$
is central, for all $g\in G$.
Conversely, consider a twisted partial action $(\alpha,\omega)$
of ${\kappa} G$ on A and set $\alpha(g\otimes a)=g\cdot a$ and $\omega_{g,h}=
\omega(g,h)$, for all $g, h\in G$ and $a\in A$. By (\ref{productpartial}) we have that
$1_g:=g\cdot\um$ is an idempotent of $A$ and by (\ref{cociclo}) and Proposition~\ref{FirstProp}
$\omega_{g,h}\in (1_gA)\cap(A1_{gh})$, for all $g,h\in G$. Since
by (\ref{unitpartial}) $1_e=\um$, we have $\omega_{g,g^{-1}}\in 1_gA$ for all $g\in G$.
Now assume, in addition, that $1_g$ is central in $A$ and $\omega_{g,g^{-1}}$
is invertible in $1_gA$, for all $g\in G$. Thus, $1_gA$ is a unital $\kappa $-algebra,
$\omega_{g,h}\in (1_gA)(1_{gh}A)$, $$g\cdot 1_{g^{-1}}=g\cdot(g^{-1}\cdot\um)
\overset{(\ref{torcao})}{=}\omega_{g,g^{-1}}\um
\omega_{g,g^{-1}}^{-1}=1_g$$ and $$g\cdot(1_{g^{-1}}a)\overset{(\ref{productpartial})}{=}
(g\p1_{g^{-1}})(g\cdot a)=1_g(g.a)=(g\cdot\um)(g\cdot a)\overset{(\ref{productpartial})}{=}g\cdot a,$$
for all $g,h\in G$ and $a\in A$.
Hence, $\alpha$ induces by restriction a map of $\kappa $-algebras $\alpha_g:1_{g^{-1}}A\to 1_gA$,
given by $\alpha_g(1_{g^{-1}}a)=g\cdot(1_{g^{-1}}a)=g\cdot a$, for all $g\in G$ and $a\in A$.
Furthermore, it follows from (\ref{torcao}) that $\alpha_g\circ\alpha_{g^{-1}}(1_g a)=
\omega_{g,g^{-1}}(1_ga)\omega_{g,g^{-1}}^{-1}$, that is, $\alpha_g\circ\alpha_{g^{-1}}$
is an inner automorphism of $1_gA$. In particular, $\alpha_{g^{-1}}$
is injective and $\alpha_g$ is surjective, for all $g\in G$. Consequently,
$\alpha_g$ is an isomorphism and $1_gA=g\cdot A$, for all $g\in G$. Finally, (\ref{unitpartial}) and (\ref{torcao})
imply the above conditions (\ref{pga1}) and (\ref{pga2}) respectively. Therefore,
$$\left(\{ g\cdot A \}_{g\in G}, \{ g\cdot \underline{\,}\}_{g\in G},
\{\omega(g,h)\}_{(g,h)\in G\times G}\right)$$ is a twisted partial action of $G$ on $A,$ as defined above.\nolinebreak\hfill$\Box$\par\medbreak
\end{ex}
\begin{ex}{(Induced twisted partial action.)}\label{induced}
Let $B$ be a unital $\kappa $-algebra measured by an action $\beta: H \otimes B \rightarrow B$, denoted by
$\beta (h,b) = h \rhd b$, which is twisted by a map $u: H \otimes H \rightarrow B$, i.e.,
\begin{align}
h \rhd (ab) & = \sum (h_{1} \rhd a)(h_{2} \rhd b),\\
h \rhd 1_B &= \varepsilon (h) 1_B,\\
\sum (h_{(1)} \rhd (k_{(1)} \rhd a )) u(h_{(2)},k_{(2)}) &= \sum u(h_{(1)},k_{(1)}) (h_{(2)}k_{(2)} \rhd a), \label{twistedglobal}
\end{align} for all $h, k \in H$ and $a, b\in B.$ Assume furthermore that
\begin{equation}\label{1 cutuca a}
1_H \rhd a = a
\end{equation} for all $a\in A.$ Here $u$ is neither supposed to be convolution invertible, nor to satisfy the $2$-cocycle equality.
Suppose that $\um$ is a non-trivial central idempotent of $B$, and let $A$ be the ideal generated by $\um$. Given $a \in A, h \in H$,
define a map $\cdot : H \otimes A \rightarrow A$ by
\begin{equation}\label{cotucapartial}
h \cdot a = \um (h \rhd a).
\end{equation}
It is clear that (\ref{1 cutuca a}) implies (\ref{unitpartial}), and (\ref{productpartial}) follows from the fact that $\um b = b \um$
for all $b \in B$. We still have to define a map $\omega : H \otimes H \rightarrow A$.
From equation (\ref{twistedglobal}) we obtain
\begin{eqnarray*}
\sum (h_{(1)} \cdot (k_{(1)} \cdot a)) u(h_{(2)} , k_{(2)}) & = & \sum \um (h_{(1)} \rhd \um (k_{(1)} \rhd a)) u(h_{(2)} ,k_{(2)})\\
& = & \sum (h_{(1)} \cdot \um)(h_{(2)} \rhd (k_{(1)} \rhd a)) u(h_{(3)} , k_{(2)}) \\
& = & \sum (h_{(1)} \cdot \um)u(h_{(2)} , k_{(1)}) (h_{(3)}k_{(2)} \rhd a) \\
& = & \sum (h_{(1)} \cdot \um)u(h_{(2)} , k_{(1)}) (h_{(3)}k_{(2)} \cdot a),
\end{eqnarray*}
where the last equality follows from the fact that $ \sum (h_{(1)} \cdot \um)u(h_{(2)} , k)$ lies in $A$. In particular, for $a = \um$ we obtain
\begin{equation}\label{passagem}
\sum (h_{(1)} \cdot (k_{(1)} \cdot \um)) u(h_{(2)} , k_{(2)}) = \sum (h_{(1)} \cdot \um)u(h_{(2)} , k_{(1)}) (h_{(3)}k_{(2)} \cdot \um),
\end{equation}
which, in view of conditions (\ref{torcao}) and (\ref{cociclo}), suggests to define $\omega$ by
\begin{equation}
\omega (h , k) = \sum (h_{(1)} \cdot \um) u(h_{(2)} ,k_{(1)}) (h_{(3)}k_{(2)} \cdot \um) \label{def.omega}.
\end{equation}
\noindent With $\omega$ thus defined, (\ref{torcao}) and (\ref{cociclo}) are clearly satisfied, and $(A, \cdot, \omega)$ is a twisted partial $H$-module algebra.
\vspace{.3cm}
In particular, when $B$ is an $H$-module algebra, i.e., when $u$ is the trivial cocycle $u(h,k) = \varepsilon(h) \varepsilon(k) 1_B$,
it follows from (\ref{passagem}) that $\omega$ is also trivial, i.e. $\omega $ satisfies (\ref{cociclotrivial}).
Therefore, in this case $A$ becomes a partial $H$-module algebra as defined in \cite{CJ}. \nolinebreak\hfill$\Box$\par\medbreak
\end{ex}
Given any two $\kappa$-linear maps $\alpha:H\otimes A\to A$, $h\otimes
a\mapsto h\cdot a$, and $\omega:H\otimes H\to A$, we can define on the
$\kappa$-module $A\otimes H$ a product, given by the
multiplication
$$(a\otimes h)(b\otimes l)=\sum a(h_{(1)}\cdot b)\omega(h_{(2)} , l_{(1)})\otimes
h_{(3)}l_{(2)},$$ for all $a,b\in A$ and $h,l\in H$. Write $A\#_{(\alpha,\omega)} H =(A\otimes H) (\um \otimes 1_H ).$
It is readily seen that this corresponds to the $\kappa $-submodule of
$A\otimes H$ generated by the elements of the form $a\# h:=\sum
a(h_{(1)}\cdot\um)\otimes h_{(2)}$, for all $a\in A$ and $h\in H$.
\vspace{.1cm}
In general $A\otimes H$, with this above defined product, is neither associative nor unital. The following proposition
gives necessary and sufficient conditions
under which $A\otimes H$ (so, also $A\#_{(\alpha,\omega)} H$) is
associative and $A\#_{(\alpha,\omega)}H$ is unital with $\um\#
1_H=\um\otimes 1_H$ as the identity element. This proposition is a
generalization of \cite[Lemmas 4.4 and 4.5]{BCM} to the setting of
twisted partial Hopf algebra actions.
\vspace{.4cm}
\begin{prop}
Let $A$ be a unital $\kappa $-algebra, $H$ a Hopf ${\kappa} $-algebra, $\omega:H\otimes H\to A$ and
$\alpha:H\otimes A\to A$, $h\otimes a\mapsto h\cdot a$, two $\kappa $-linear
maps satisfying the conditions (\ref{unitpartial}) , (\ref{productpartial}) and (\ref{cociclo}).
\begin{enumerate}
\item[(i)] $\um\# 1_H$ is the unity of $A\#_{(\alpha,\omega)}H$ if and
only if, for all $h\in H$,
\begin{align}
\omega(h, 1_H)=\omega(1_H , h)=h\cdot\um. \label{8}
\end{align}
\item[(ii)] Suppose that $\omega(h, 1_H)=h\cdot\um$, for all $h\in
H$. Then $A\otimes H$ is associative if and only if the condition
(\ref{torcao}) holds and, for all $h,l,m\in H$,
\begin{align}
\sum(h_{(1)}\cdot\omega(l_{(1)} , m_{(1)}))\omega(h_{(2)} , l_{(2)}m_{(2)})&=
\sum\omega(h_{(1)}, l_{(1)})\omega(h_{(2)} l_{(2)} , m).\label{9}
\end{align}
\end{enumerate}
\end{prop}
\noindent{\bf Proof.}\,\, The proof is quite similar to that of
\cite[Lemmas 4.4 and 4.5]{BCM}.
\vspace{.1cm}
(i) Assume that $\omega(h,1_H)=\omega(1_H,h)=h\cdot\um$. Then
\[
\begin{array}{ccl}
(\um\# 1_H)(a\# h)&=& (\um \otimes 1_H) \left( \sum a (h_{(1)} \cdot \um ) \otimes h_{(2)} \right) \\
&=& \sum \um(1_H\cdot(a(h_{(1)}\cdot \um))\omega(1_H ,h_{(2)})\otimes 1_H h_{(3)}\\
&=&\sum a(h_{(1)}\cdot\um)(h_{(2)}\cdot\um)\otimes h_{(3)}\\
& \overset{(\ref{productpartial})}{=} & \sum
a(h_{(1)}\cdot\um)\otimes h_{(2)} = a\# h
\end{array}
\]
and
\[
\begin{array}{ccl}
(a\# h)(\um\# 1_H)&=& \left( \sum a(h_{(1)} \cdot \um ) \otimes h_{(2)} \right)(\um \otimes 1_H ) \\
&=& \sum a(h_{(1)}\cdot\um)(h_{(2)}\cdot\um)\omega(h_{(3)} ,1_H)\otimes h_{(4)}1_H\\
&=&\sum a(h_{(1)}\cdot\um)(h_{(2)}\cdot\um)(h_{(3)}\cdot 1_A)\otimes h_{(4)}\\
&\overset{(\ref{productpartial})}{=}&\sum a(h_{(1)}\cdot\um)\otimes h_{(2)}= a\# h\
\end{array}
\]
for every $a\in A$ and $h\in H$.
\vspace{.1cm}
Conversely, if $\um\# 1_H$ is the unity of $A\#_{(\alpha,\omega)} H$
then applying $I_A\otimes\varepsilon$ to the equalities
\[
\sum(h_{(1)}\cdot\um)\otimes h_{(2)}=\um\# h=(\um\# 1_H)(\um\# h)=
\sum\omega(1_H , h_{(1)})\otimes h_{(2)}
\]
and
\[
\begin{array}{ccl}
\sum(h_{(1)}\cdot\um)\otimes h_{(2)}&=&\um\# h=(\um\#
h)(\um\#1_H)\\
&=& \left( \sum (h_{(1)} \cdot \um )\otimes h_{(2)} \right) (\um \otimes 1_H ) \\
&=&\sum(h_{(1)}\cdot\um)(h_{(2)}\cdot\um)\omega(h_{(3)} , 1_H)\otimes
h_{(4)}\\
&\overset{(\ref{productpartial})}{=}&\sum(h_{(1)}\cdot\um)\omega( h_{(2)} ,1_H)\otimes h_{(3)}\\
&\overset{(\ref{cociclo})}{=}&\sum\omega(h_{(1)} ,1_H)\otimes h_{(2)},
\end{array}
\]
we obtain
\[
h\cdot\um=\sum (h_{(1)}\cdot\um)\varepsilon(h_{(2)})=\sum\omega(1_H,h_{(1)})\varepsilon(h_{(2)})=\omega(1_H,h)
\]
and
\[
h\cdot\um=\sum (h_{(1)}\cdot\um)\varepsilon(h_{(2)})=\sum\omega(h_{(1)} ,1_H)\varepsilon(h_{(2)})=\omega(h ,1_H).
\]
\vspace{.2cm}
(ii) Assume that (\ref{torcao}) and (\ref{9}) hold. Then, for all $a,b,c\in A$ and
$h,l,m\in H$ we have:
\vspace{.1cm}
$ (a\otimes h)[(b\otimes l)(c\otimes m)]$
\vspace{-.6cm}
\[
\begin{array}{ccl}
&=& \sum a[h_{(1)}\cdot(b(l_{(1)}\cdot
c)\omega(l_{(2)},m_{(1)}))]\omega(h_{(2)},l_{(3)}m_{(2)})\otimes
h_{(3)}l_{(4)}m_{(3)}\\
&\overset{(\ref{productpartial})}{=}&\sum a(h_{(1)}\cdot b)[(h_{(2)}\cdot(l_{(1)}\cdot
c))(h_{(3)}\cdot\omega(l_{(2)},m_{(1)}))\omega(h_{(4)},l_{(3)}m_{(2)})] \\
& \, & \otimes h_{(5)}l_{(4)}m_{(3)}\\
&\overset{(\ref{9})}{=}&\sum a(h_{(1)}\cdot b)(h_{(2)}\cdot(l_{(1)}\cdot
c))\omega(h_{(3)},l_{(2)})\omega(h_{(4)}l_{(3)},m_{(1)})\otimes h_{(5)}l_{(4)}m_{(2)}\\
&\overset{(\ref{torcao})}{=}&\sum a(h_{(1)}\cdot b)\omega(h_{(2)},l_{(1)})(h_{(3)}l_{(2)}\cdot
c)\omega(h_{(4)}l_{(3)},m_{(1)})\otimes h_{(5)}l_{(4)}m_{(2)}\\
&=&[(a\otimes h)(b\otimes l)](c\otimes m).\
\end{array}
\]
Conversely, we have by assumption that
\[
(\um\otimes h)[(\um\otimes
l)(a\otimes 1_H)]=[(\um\otimes h)(\um\otimes l)](a\otimes 1_H)
\]
and
\[
(\um\otimes h)[(\um\otimes l)(\um\otimes m)]=[(\um\otimes
h)(\um\otimes l)](\um\otimes m),
\]
for all $a\in A$ and $h,l,m\in H$.
Using mainly condition (\ref{cociclo}) (and the hypothesis on $\omega$ in the
first case only) one easily obtains, by a straightforward calculation,
from the first equality:
\[
(h_{(1)}\cdot(l_{(1)}\cdot a))\omega(h_{(2)} ,l_{(2)})\otimes
h_{(3)}l_{(3)}=\omega(h_{(1)} , l_{(1)})(h_{(2)}l_{(2)}\cdot a)\otimes h_{(3)}l_{(3)},
\]
and from the second:
\begin{eqnarray} &\, & (h_{(1)}\cdot\omega(l_{(1)} , m_{(1)}))\omega(h_{(2)} , l_{(2)}m_{(2)})\otimes h_{(3)}l_{(3)}m_{(3)} \nonumber \\
& \, & = \omega(h_{(1)} , l_{(1)})\omega(h_{(2)}l_{(2)} , m_{(1)})\otimes
h_{(3)}l_{(3)}m_{(2)}.\nonumber
\end{eqnarray}
Now, applying $I_A\otimes\varepsilon$ in both sides of these
equalities the conditions (\ref{torcao}) and (\ref{9}) follow respectively. \nolinebreak\hfill$\Box$\par\medbreak
\vspace{.3cm}
Given a twisted partial action $(\alpha,\omega)$ of a Hopf
${\kappa} $-algebra $H$ on a ${\kappa} $-algebra $A$, the ${\kappa} $-algebra
$A\#_{(\alpha,\omega)}H$ is called a {\it crossed product by a
twisted partial action} (shortly, a {\it partial crossed product})
if the additional conditions (\ref{8}) and (\ref{9}) hold.
\vspace{.2cm}
In order to establish some notation, we give the following lemma.
\begin{lemma} In $A\#_{(\alpha,\omega)} H$ we have the following identities:
\begin{enumerate}
\item[(i)] $a\# h=\sum a(h_{(1)} \cdot \um )\# h_{(2)}$.
\item[(ii)] $(a\# h)(b\# k)=\sum a(h_{(1)} \cdot b )\omega (h_{(2)} ,k_{(1)}) \# h_{(3)} k_{(2)}$.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof.}\,\, Item (i) is straightforward,
\begin{eqnarray*}
\sum a(h_{(1)} \cdot \um )\# h_{(2)} &=& \sum a(h_{(1)} \cdot \um ) (h_{(2)}\cdot \um ) \otimes h_{(3)}\\
&=& \sum a(h_{(1)} \cdot \um )\otimes h_{(2)} =a\# h.
\end{eqnarray*}
For item (ii), we have
\begin{eqnarray*}
(a\# h)(b\# k) & = & \left( \sum a(h_{(1)} \cdot \um )\otimes h_{(2)} \right) \left( \sum b(k_{(1)} \cdot \um )\otimes k_{(2)} \right) \\
& =& \sum a(h_{(1)} \cdot \um ) (h_{(2)} \cdot (b(k_{(1)}\cdot \um )))\omega ( h_{(3)},k_{(2)}) \otimes h_{(4)}k_{(3)} \\
& =& \sum a(h_{(1)} \cdot (b(k_{(1)}\cdot \um )))\omega ( h_{(2)},k_{(2)}) \otimes h_{(3)}k_{(3)} \\
&=& \sum a(h_{(1)} \cdot b)(h_{(2)}\cdot (k_{(1)}\cdot \um ))\omega ( h_{(3)},k_{(2)}) \otimes h_{(4)}k_{(3)} \\
&\overset{(\ref{firstprop})}{=}& \sum a(h_{(1)} \cdot b)\omega ( h_{(2)},k_{(1)}) \otimes h_{(3)}k_{(2)} \\
&\overset{(\ref{cociclo})}{=}& \sum a(h_{(1)} \cdot b)\omega ( h_{(2)},k_{(1)}) (h_{(3)}k_{(2)} \cdot \um ) \otimes h_{(4)}k_{(3)} \\
&=& \sum a(h_{(1)} \cdot b)\omega ( h_{(2)},k_{(1)}) \# h_{(3)}k_{(2)}.
\end{eqnarray*}\nolinebreak\hfill$\Box$\par\medbreak
If, in particular, $\omega$ is trivial then the multiplication in
$A\#_{(\alpha,\omega)}H$ becomes
\[
\begin{array}{ccl}
(a\# h)(b\# l)&=&(\sum a(h_{(1)}\cdot\um)\otimes h_{(2)})(\sum
b(l_{(1)}\cdot\um)\otimes l_{(2)})\\
&=&\sum a(h_{(1)}\cdot\um)(h_{(2)}\cdot(b(l_{(1)}\cdot\um)))\omega(h_{(3)} , l_{(2)})\otimes
h_{(4)}l_{(3)}\\
&\overset{(\ref{cociclotrivial})}{=}&\sum a(h_{(1)}\cdot\um)(h_{(2)}\cdot(b(l_{(1)}\cdot\um)))(h_{(3)}\cdot(l_{(2)}\cdot\um))\otimes
h_{(4)}l_{(3)}\\
&\overset{(\ref{productpartial})}{=}&\sum a(h_{(1)}\cdot(b(l_{(1)}\cdot\um))\otimes h_{(2)}l_{(2)}\
\end{array}
\]
for all $a,b\in A$ and $h,l\in H$. Consequently, in this case
we recover the partial smash product introduced in \cite{CJ}.
\vspace{.3cm}
\begin{remark} \label{firstremark}\,\ If $(\alpha,\omega)$ is a twisted partial
action of ${\kappa} G$ on a ${\kappa} $-algebra $A$ arisen from a twisted partial
action of a group $G$ on $A$, as defined in Example~\ref{ex:kG}, and the
conditions (\ref{8}) and (\ref{9}) also hold in this case, then
$A\#_{(\alpha,\omega)}{\kappa} G$ is an slight generalization of the partial
crossed product introduced in \cite{DES1}.
\end{remark}
\begin{ex}
Consider an induced partial twisted $H$-module structure as given in Example~\ref{induced}, and suppose that
the map $u: H \rightarrow A$ is a \emph{normalized cocycle}, i.e., assume that
\begin{eqnarray}
\sum (h_{(1)} \rhd u(k_{(1)},l_{(1)}))u(h_{(2)},k_{(2)}l_{(2)}) & = & u(h_{(1)},k_{(1)}) u(h_{(2)}k_{(2)},l),
\label{leidoscociclos}
\\ u(h , 1_H) & = &u(1_H , h) = \varepsilon(h) 1_B
\end{eqnarray}
for all $h,k,l \in H$. It is clear that the induced map
$\omega$ (see equality (\ref{def.omega}))
satisfies condition (\ref{8}), and we will show that (\ref{9}) is also satisfied. In what follows, note that
$\sum(h_{(1)} \cdot \um)(h_{(2)} \cdot x) = \sum(h_{(1)} \cdot x)(h_{(2)} \cdot \um)$, and that if $a,b \in A$, then $a(h \cdot b)
= a\um(h \rhd b) = a(h \rhd b)$. Using equality (\ref{leidoscociclos}),
\begin{eqnarray*}
& & \sum (h_{(1)} \cdot \omega(l_{(1)} , m_{(1)}) ) \omega(h_{(2)} , l_{(2)}m_{(2)}) = \\
& = & \sum(h_{(1)} \cdot [(l_{(1)} \cdot \um ) u(l_{(2)} , m_{(1)}) (l_{(3)}m_{(2)}\cdot \um)](h_{(2)} \cdot \um) \times \\
& & \times u(h_{(3)} , l_{(4)}m_{(3)}) (h_{(4)}l_{(5)}m_{(4)} \cdot \um) \\
& = & \sum(h_{(1)} \cdot(l_{(1)}\cdot \um))( (h_{(2)} \cdot u(l_{(2)} , m_{(1)})) (h_{(3)} \cdot (l_{(3)}m_{(2)} \cdot \um)) \times \\
& & \times (h_{(4)} \cdot \um) u(h_{(5)} , l_{(4)}m_{(3)})(h_{(6)}l_{(5)}m_{(4)} \cdot \um) \\
& = & \sum\underbrace{(h_{(1)} \cdot(l_{(1)} \cdot \um)(h_{(2)} \cdot \um)}( (h_{(3)} \cdot u(l_{(2)} , m_{(1)})) \times \\
& & \times (h_{(4)} \cdot (l_{(3)}m_{(2)} \cdot \um))
u(h_{(5)} , l_{(4)}m_{(3)})(h_{(6)}l_{(5)}m_{(4)} \cdot \um) \\
& = & \sum(h_{(1)} \cdot(l_{(1)} \cdot \um)) (h_{(2)} \cdot u(l_{(2)} , m_{(1)})) \times \\
& & \times \underbrace{(h_{(3)} \cdot (l_{(3)}m_{(2)} \cdot \um)) u(h_{(4)} , l_{(4)}m_{(3)})} (h_{(5)}l_{(5)}m_{(4)} \cdot \um)
\end{eqnarray*}
\begin{eqnarray*}
& \overset{\text{(\ref{passagem})}}{=} & \sum(h_{(1)} \cdot(l_{(1)} \cdot \um)) (h_{(2)} \cdot u(l_{(2)} , m_{(1)})) (h_{(3)} \cdot \um) u(h_{(4)} , l_{(3)}m_{(2)}) \times \\
& & \times (h_{(5)}l_{(4)}m_{(3)} \cdot \um) \\
& = & \sum(h_{(1)} \cdot(l_{(1)} \cdot \um)) (h_{(2)} \cdot u(l_{(2)} , m_{(1)})) u(h_{(3)}, l_{(3)}m_{(2)})
(h_{(4)}l_{(4)}m_{(3)} \cdot \um) \\
& = & \sum(h_{(1)} \cdot(l_{(1)} \cdot \um)) \underbrace{(h_{(2)} \rhd u(l_{(3)} , m_{(1)})) u(h_{(3)} , l_{(4)}m_{(2)})}
(h_{(4)}l_{(5)}m_{(4)} \cdot \um) \\
& \overset{\text{(\ref{leidoscociclos})} }{=} & \sum\underbrace{(h_{(1)} \cdot(l_{(1)} \cdot \um)) u(h_{(2)} , l_{(2)})} u(h_{(3)}l_{(3)} , m_{(1)})(h_{(4)}l_{(4)}m_{(2)} \cdot \um) \\
& \overset{\text{(\ref{passagem})}}{=} & \sum(h_{(1)} \cdot \um)u(h_{(2)} , l_{(1)})(h_{(3)}l_{(2)} \cdot \um) u(h_{(4)}l_{(3)} , m_{(1)})
(h_{(5)}l_{(4)}m_{(2)} \cdot \um) \\
& = & \sum\omega(h_{(1)}, l_{(1)})\omega(h_{(2)}, l_{(2)}m).
\end{eqnarray*}
\end{ex}
If one forms the usual crossed product $B\#_{u} H$, then it is easy to see that $\um \# 1_H$ is an idempotent of this algebra, that
\[
(\um \# 1_H) (B\#_{u} H) = A \otimes H
\]
and that
\[
(\um \# 1_H)(B\#_{u} H) (\um \# 1_H) = A\#_{(\alpha,\omega)}H.
\] \nolinebreak\hfill$\Box$\par\medbreak
\section{Examples of (twisted) partial actions via algebraic groups}\label{ExViaAlGroups}
In this section we use the relation between algebraic groups and commutative Hopf algebras (see \cite{Abe}, \cite{Water})
to explain a way of producing examples of partial Hopf (co)actions. A concrete example is elaborated to
which we attach a twisting resulting in a twisted
partial Hopf action.\\
We shall extract our example from central notions of the theory of algebraic groups such as maximal tori, Cartan
subgroups and the Weyl group. Let ${\kappa}$ be an algebraically closed field and let ${\bf G}$ be a linear algebraic
group over ${\kappa},$ by which we mean a subgroup of ${\rm GL}_n({\kappa} )$ (for some positive integer $n$), which is
closed in the Zariski topology of ${\rm GL}_n({\kappa} ).$ Let $T$ be a maximal torus in $G.$ We recall that a torus
is a connected diagonalizable linear algebraic group. Let $C$ be the centralizer of $T$ in ${\bf G}$ and $N$ be
the normalizer of $T$ in ${\bf G}.$ Then $C$ is the Cartan subgroup of ${\bf G}$ and $W=N/C$ is the corresponding
Weyl group (which is finite). Then evidently $N$ acts on itself by left multiplication and this permutes the left
cosets of $N$ by $C.$ Then taking the Hopf algebra $H$ which corresponds to $N,$ its comultiplication
$\Delta: H \otimes H \to H$ is the right coaction of $H$ on itself, which corresponds to the left action of $N$ on itself.
Let $Y\subsetneq N$ by a union of some left cosets of $N$ by $C.$ Then $N$ acts (globally) on $N$ and only partially on $Y.$
Then one may take a two-sided ideal $A$ in $H$ determined by $Y$ (see the concrete example below) so that one comes to
a partial coaction $\rho : A \to A \otimes H$ which is obtained by the restriction $\rho = (1_{A} \otimes 1_{H}) \Delta .$
Then taking a Hopf algebra $H_1$ such that there exist a pairing $\langle , \rangle \: H_1\otimes H \to {\kappa} ,$ one
can dualize to obtain a partial action of $H_1$ on $A,$ as given in \cite[Prop. 8]{AB}. In particular, $H_1$ can be
the finite dual of $H.$
For a concrete example we take one of the most classical cases, in which ${\bf G}= {\rm GL}_n({\kappa} )$ and
$T\subseteq {\rm GL}_n({\kappa} )$ is the group of all diagonal matrices of $GL_n(k).$ Then $T \cong ({\kappa} ^*)^n ,$
where ${\kappa} ^*$ is the multiplicative group of the field ${\kappa} $. It is directly verified that in this case
$C=T$ and $N $ is formed by the monomial matrices, that is, the matrices whose rows and columns have only
one nonvanishing entry. The Weyl group $W$ can be identified with the group of $n\times n$ permutation
matrices, which is isomorphic to the symmetric group $S_n$.\\
The group $N$ is an algebraic group and is isomorphic to the semidirect product of $T$ by the action of the Weyl group
\[
N\cong T\rtimes W= T\rtimes S_n .
\]
Here, the left action of $S_n$ on $T$ is given by conjugation, whose net effect is the permutation of the diagonal
matrix entries. By the fact that all these groups are algebraic groups, one can associate to the action
\[
\begin{array}{rccl} \alpha : & S_n \times T & \rightarrow & T, \\
\, & (g,x) & \mapsto & g\cdot x=gxg^{-1}, \end{array}
\] a left coaction of the corresponding Hopf algebras. It is a basic fact that the Hopf algebra which corresponds
to a finite group is the dual of the group algebra, i.e. in our case it is $({\kappa} S_n)^*.$ It is also basic that the
algebra corresponding to ${\kappa} ^*$ is the Hopf algebra of the Laurent polynomials ${\kappa} [t,t^{-1} ].$ Since tensor products
of Hopf algebras correspond to direct product of algebraic groups, it follows that the Hopf algebra corresponding to
$T$ is ${\kappa} [t,t^{-1} ]^{\otimes n}.$ Consequently there is a left coaction of $({\kappa} S_n)^*$ on
${\kappa} [t,t^{-1} ]^{\otimes n},$ which corresponds to the above action of $S_n$ on $T,$ i.e.
${\kappa} [t,t^{-1} ]^{\otimes n}$ turns out to be a left $({\kappa} S_n )^*$-comodule coalgebra. Since $N\cong T\rtimes S_n , $
it follows by \cite[p. 143, p. 208]{Abe} that the Hopf algebra associated to the group $N$ is the co-semidirect product
\[
{\kappa} [t,t^{-1}]^{\otimes n} >\!\!\blacktriangleleft ({\kappa} S_n)^*.\\
\]
A typical element of ${\kappa} [t,t^{-1}]^{\otimes n}$ is a tensor polynomial of the form
\[
\sum_{N\in \mathbb{Z}} \sum_{k_1 +\cdots +k_n =N} \lambda_N t^{k_1} \otimes \ldots \otimes t^{k_n} .
\]
In order to simplify the notation, write
\begin{equation}\label{notation1}
t_i = 1 \otimes \ldots \otimes 1 \otimes t \otimes 1\otimes \ldots \otimes 1,
\end{equation} where $t$ belongs to the $i$-copy of ${\kappa} [ t, t{}^{-1}].$ Then we have $t_1^{k_1}\ldots t_n^{k_n} = t^{k_1} \otimes \ldots \otimes t^{k_n}$.
Since $S_n$ operates on $T$ by permuting the entries, it follows by a direct verification that the left $({\kappa} S_n)^*$-coaction on ${\kappa} [t,t^{-1}]^{\otimes n}$ is given by
\[
\delta (t_1^{k_1}\ldots t_n^{k_n}) =\sum_{g\in S_n} p_g \otimes t_{g{}^{-1} (1)}^{k_1} \ldots t_{g{}^{-1} (n)}^{k_n} .
\]
With this coaction, ${\kappa} [t,t^{-1}]^{\otimes n}$ is a left $({\kappa} S_n)^*$-comodule coalgebra, and the comultipication of the cosemidirect product
\[
H = {\kappa} [t,t^{-1}]^{\otimes n} >\!\!\blacktriangleleft ({\kappa} S_n)^*
\] is given explicitly by
\begin{eqnarray}
\Delta (t_1^{k_1}\ldots t_n^{k_n} \otimes p_g ) &=& \sum_{s,f\in S_n} t_1^{k_1}\ldots t_n^{k_n} \otimes p_s p_f \otimes
t_{s{}^{-1} (1)}^{k_1}\ldots t_{s{}^{-1} (n)}^{k_n} \otimes p_{f^{-1}g} \nonumber \\
&=& \sum_{s\in S_n} t_1^{k_1}\ldots t_n^{k_n} \otimes p_s \otimes
t_{s{}^{-1} (1)}^{k_1}\ldots t_{s{}^{-1} (n)}^{k_n} \otimes p_{s^{-1}g} .\nonumber
\end{eqnarray}
The cosemidirect product acts on the right on itself by the comultiplication. In order to construct a partial coaction one can simply project over a two-sided ideal.\\
Let $X$ be a subset of $S_n$ which is not a subgroup. Write $L = {\kappa} [t,t^{-1}]^{\otimes n}.$ Then evidently
\begin{equation}\label{idempotent}
e_X = 1_L \otimes (\sum _{g \in X} p_g)
\end{equation} is a central idempotent in $H= L >\!\!\blacktriangleleft ({\kappa} S_n)^*,$ and the algebra $A = e_X H $ is a two-sided ideal. Write
$e_X \cdot $ for the map $ H \to A$ given by multiplication by $e_X .$ Then the restriction $\rho : A \to A\otimes H,$ $\rho = (e_X \cdot \otimes I ) \circ \Delta , $
of $\Delta _H : H \to H \otimes H$ is a right partial coaction of $H$ given by
\begin{equation}\label{rho}
{\rho } (t_1^{k_1}\ldots t_n^{k_n} \otimes p_g)= \sum_{s\in X} t_1^{k_1}\ldots t_n^{k_n} \otimes p_s \otimes
t_{s{}^{-1} (1)}^{k_1}\ldots t_{s{}^{-1} (n)}^{k_n} \otimes p_{s^{-1}g},
\end{equation} where $g \in X.$ Since $X \subseteq S_n$ is not a subgroup, it is readily seen that $\rho$ is not a (global) coaction
(if $X$ was a subgroup then one would have $\rho : A \to A \otimes A$).\\
We shall obtain a partial action from a partial coaction using Proposition 8 from \cite{AB}, which we recall for reader's convenience:
\begin{prop}\label{prop:pairing} Let $H_1 $ and $H_2 $ be two Hopf algebras with a pairing between them:
\[
\begin{array}{rccl} \langle , \rangle : & H_1 \otimes H_2 & \rightarrow & {\kappa} \\
\, & h\otimes p & \mapsto & \langle h ,p \rangle .
\end{array}
\]
Then a partial right $H_2$-comodule algebra $A$, acquires a structure of partial left $H_1$-module algebra by the partial action
\[
h\cdot a = \sum a^{[0]} \langle h,a^{[1]} \rangle ,
\]
where ${\rho}(a) =\sum a^{[0]} \otimes a^{[1]}$ is the partial right coaction of $H_2$ on $A$.
\end{prop}
Assume now that ${\kappa} $ is an isomorphic copy of the complex numbers ${\mathbb C} ,$ and let $\mathbb{S}^1 $
be the unit circle group. The elements of $\mathbb{S}^1 $ can be viewed as the complex roots of $1,$
however we assume that ${\kappa} $ and $\mathbb{S}^1 \subseteq {\mathbb C}$
are disjoint and consider the group algebra $ {\kappa} \mathbb{S}^1$ of $\mathbb{S}^1$ over ${\kappa} ,$
so that the roots of unity $\chi \in \mathbb{S}^1 $ are linearly independent over ${\kappa} .$ Then $S_n$ acts on
$ ({\kappa} \mathbb{S}^1 )^{\otimes n} $ by permutation of roots, which gives an action of the group Hopf algebra
${\kappa} S_n $ on $ ({\kappa} \mathbb{S}^1 )^{\otimes n} .$ Note that ${\kappa} S_n $ and $ ({\kappa} \mathbb{S}^1 )^{\otimes n} $
are both cocommutative. Then we may consider the smash product Hopf algebra
\[
H_1 = ({\kappa} \mathbb{S}^1 )^{\otimes n} \rtimes {\kappa} S_n.
\] Write
\begin{equation}\label{notation2}
\chi_{\theta_1 , \ldots \theta_n } = \chi_{\theta_1} \otimes \ldots \otimes \chi_{\theta_n } \in ({\kappa} \mathbb{S}^1 )^{\otimes n},
\end{equation} where $\chi_{\theta_i} \in \mathbb{S}^1 $ is the root of $1$ whose angular coordinate is $\theta _i $ and which
belongs to the $i$-factor of $({\kappa} \mathbb{S}^1 )^{\otimes n}.$ Then evidently the elements
$ \chi_{\theta_1 , \ldots \theta_n } \otimes u_g $ $(g\in S_n)$ form a ${\kappa} $-basis of $H_1.$
With this notation define the map $\langle , \rangle : H_1 \otimes H \to {\kappa} ,$ by setting
\[
\langle \chi_{\theta_1 , \ldots \theta_n } \otimes u_g , t_1^{k_1} \ldots t_n^{k_n} \otimes p_s \rangle =\delta_{g,s}
\exp\{ik_1 \theta_1 \} \ldots \exp \{ ik_n \theta_n \},
\] where $ \delta_{g,s} $ is the Kronecker delta and $i^2=-1.$ It is an easy straightforward verification that this defines a
pairing of Hopf algebras. Observe that it is non-degenerate, however we do not need to use this property.\\
Now using the coaction $\rho : A \to A\times H$ we obtain by Proposition~\ref{prop:pairing} a partial action
$H_1 \times A \to A.$ To specify it, take $h = \chi_{\theta_1 , \ldots \theta_n } \otimes u_g \in H_1$ and
$ a= t_1^{k_1} \ldots t_n^{k_n} \otimes p_s \in A$ and check by the formula in Proposition~\ref{prop:pairing} that the partial action is explicitly given by
\begin{align*} & h \cdot a = \sum _{f \in X} t_1^{k_1} \ldots t_n^{k_n} \otimes p_f \;
\langle \chi_{\theta_1 , \ldots \theta_n } \otimes u_g , t_{f{}^{-1}(1)}^{k_ 1} \ldots t_{f{}^{-1}(n)}^{k_n} \otimes p_{f{}^{-1} s} \rangle = \\
& \sum_{f\in X} t_1^{k_1} \ldots t_n^{k_n} \otimes p_f \; (\delta_{g,f{}^{-1} s} \exp\{ik_1 \theta_{f{}^{-1}(1)} \} \ldots \exp \{ ik_n \theta_{f{}^{-1}(n)} \})
=\\ & \exp\{ik_1 \theta_{ g{}^{-1} s(1)} \} \ldots \exp \{ ik_n \theta_{ g{}^{-1} s (n)} \} \; t_1^{k_1} \ldots t_n^{k_n} \otimes p_{ s{}^{-1} g},
\end{align*} where $g\in S_n$ and $x\in X.$
Since any finite group $G$ can be seen as a subgroup of $S_n$ for some $n,$ we may replace in the above considerations $S_n$ by an
arbitrary finite group $G$ as follows. Fix a monomorphism $G \to S_n$ so that $G$ will be considered as a subgroup of $S_n.$ Then the formula
\[
\delta (t_1^{k_1}\ldots t_n^{k_n}) =\sum_{g\in G} p_g \otimes t_{g{}^{-1} (1)}^{k_1} \ldots t_{g{}^{-1} (n)}^{k_n}
\] gives a structure of a left $({\kappa} G)^*$-comodule coalgebra on $L= {\kappa} [t,t^{-1}]^{\otimes n},$ and one can take the cosemidirect product
$$H_2 = L >\!\!\blacktriangleleft ({\kappa} G)^* $$ with comultiplication given by
\begin{eqnarray}
\Delta (t_1^{k_1}\ldots t_n^{k_n} \otimes p_g )
&=& \sum_{s\in G} t_1^{k_1}\ldots t_n^{k_n} \otimes p_s \otimes
t_{s{}^{-1} (1)}^{k_1}\ldots t_{s{}^{-1} (n)}^{k_n} \otimes p_{s^{-1}g} \;\; (g\in G). \nonumber
\end{eqnarray} Clearly, $H_2 = e H,$ where $e= 1_L \otimes (\sum _{g \in G} p_g).$\\
Let now $X$ be an arbitrary subset of $G$ which is not a subgroup. The element $e_X$ defined by the formula (\ref{idempotent}) is obviously a central idempotent in $H_2$ and the algebra $A' = e_X H_2 $ is a two-sided ideal. The restriction
${\rho}' : A' \to A' \otimes H,$ ${\rho }' = e_X \Delta _{H_2} $ of $\Delta _{H_2} : H_2 \to H_2 \otimes H_2$ is a right partial coaction of $H_2$ given by exactly the same formula (\ref{rho}) which was used for $\rho.$ The elements of $G \subseteq S_n$ act on $ ({\kappa} \mathbb{S}^1 )^{\otimes n}, $ as above, by permutation of roots, and we have the smash product
$$H'_1 = ({\kappa} \mathbb{S}^1 )^{\otimes n} \rtimes {\kappa} G.$$ Then the formula above which defined the left partial action of $H_1$ on $A$ gives a left partial action $H'_1 \times A' \to A' :$
\begin{equation}\label{ex:ParAc}
(\chi_{\theta_1 , \ldots \theta_n } \otimes u_g) \cdot (t_1^{k_1} \ldots t_n^{k_n} \otimes p_s ) =
\exp\{ i(k_1 \theta_{ g{}^{-1} s(1)} + \ldots + k_n \theta_{ g{}^{-1} s (n)}) \} \; t_1^{k_1} \ldots t_n^{k_n} \otimes p_{ s{}^{-1} g},
\end{equation} where $g \in G$ and $s \in X \subseteq G.$\\
In order to turn the partial action (\ref{ex:ParAc}) into a twisted one take a finite group $G$ whose Schur Multiplier over ${\kappa} ={\mathbb C}$ is not trivial. Then there exists a $2$-cocycle $\gamma : G \times G \to {{\mathbb C}}^{\ast}$ which is not a coboundary. The $2$-cocycle equality means that
\begin{equation}\label{GroupCocycle}
\gamma (x,y) \gamma (xy, z) = \gamma (x, yz) \gamma (y,z) \;\;\;\; \;\;\;\; \forall x,y,z \in G.
\end{equation} Assume also that $\gamma$ is normalized, i.e.
\begin{equation}\label{normalized}
\gamma(g,1) = \gamma (1,g) = 1.
\end{equation} For arbitrary
$h= \chi_{\theta_1 , \ldots \theta_n } \otimes u_g$ and $l= \chi_{{\theta}'_1 , \ldots {\theta }'_n } \otimes u_s$ in $H'_1$ set
\begin{equation}\label{ex:omega}
\omega ( h, l ) = \gamma (g,s)\; ( h \cdot l \cdot {\mathbf{1}_{A'} } ).
\end{equation} The fact that (\ref{ex:ParAc}) and (\ref{ex:omega}) define a twisted partial action of $H'_1 = ({\kappa} \mathbb{S}^1 )^{\otimes n} \rtimes {\kappa} G$ on $A',$ which satisfies (\ref{8}) and (\ref{9}), will follow from the next easy:
\begin{prop}\label{ex:omega2} Let $G$ be a finite group and $L$ be a cocommutative Hopf algebra over a field ${\kappa} ,$ such that $L$ is a left ${\kappa} G$-module algebra. Suppose that there is a left partial action of the smash product $H= L \rtimes {\kappa} G$ on a ${\kappa} $-algebra $A:$
$$H \otimes A \ni h\otimes a \mapsto h\cdot a \in A.$$ If $\gamma : G \times G \to {\kappa} ^{\ast}$ is a normalized $2$-cocycle, then the map
\begin{equation}\label{ex:omega3}
\omega(h , m)= \sum \gamma(g,s) \; (h \cdot m \cdot 1_A),
\end{equation} where $h = \sum l\otimes g, m=\sum l'\otimes s \in L \rtimes {\kappa} G,$ turns the partial action $H \otimes A \to A$ into a twisted one such that (\ref{8}) and (\ref{9}) are satisfied.
\end{prop}
\begin{proof} One needs to check (\ref{torcao}), (\ref{cociclo}), (\ref{8}) and (\ref{9}). It is obviously enough to verify these properties for the elements $h, m, k \in H$ of the form $h = l\otimes g,$ $m = l'\otimes s,$ and $ k = l''\otimes f,$ with $l. l', l''\in L, g, s, f, \in G.$ Recall from \cite[p. 142]{Abe} that
\begin{equation*}
\Delta _H (l \otimes g) = \sum ( l_{(1)} \otimes g) \otimes ( l_{(2)} \otimes g)
\end{equation*} Then using (\ref{normalized}) and (\ref{GroupCocycle}) it is readily seen that the properties (\ref{torcao}), (\ref{cociclo}), (\ref{8}) and (\ref{9}) are resumed respectively to the following equalities:
\begin{align*}
& \sum ( h _ {(1)} \cdot m_{(1)} \cdot a ) ( h _ {(2)} \cdot m_{(2)} \cdot \um ) = \sum ( h_{(1)} \cdot m_{(1)} \cdot \um ) ( h_{(2)} m_{(2)} \cdot a )\;\;\; \;\;\;\;\;\;\; (\forall a\in A),\\
& h \cdot m \cdot \um = \sum ( h_{(1)} \cdot m_{(1)} \cdot \um ) ( h_{(2)} m_{(2)} \cdot \um ),\\
& h \cdot 1_H \cdot \um = 1_H \cdot h \cdot \um = h \cdot \um ,\\
&\sum( h_{(1)} \cdot m_{(1)} \cdot k_{(1)} \cdot \um ) ( h_{(2)} \cdot m_{(2)}k_{(2)} \cdot \um ) = \sum ( h_{(1)} \cdot m_{(1)} \cdot \um ) ( h_{(2)} m_{(2)} \cdot k _{(2)} \cdot \um ).
\end{align*} The first three equalities are immediate consequences of the definition of a (non-twisted) partial action. As to the last one, write
\begin{align*} & \sum( h_{(1)} \cdot m_{(1)} \cdot k_{(1)} \cdot \um ) ( h_{(2)} \cdot m_{(2)}k_{(2)} \cdot \um ) =\\
&\sum h \cdot [(m_{(1)} \cdot k_{(1)} \cdot \um ) ( m_{(2)}k_{(2)} \cdot \um ) ]=\\
&\sum h \cdot [(m_{(1)}\cdot \um )( m_{(2)} k_{(1)} \cdot \um ) ( m_{(3)}k_{(2)} \cdot \um ) ]=\\
&\sum h \cdot [(m_{(1)}\cdot \um )( m_{(2)} k_{(1)} \cdot \um ) ]=
\end{align*}
\begin{align*}
& h \cdot m \cdot k \cdot \um =\sum (h_{(1)} \cdot \um) ( h_{(2)} m \cdot k \cdot \um ) =\\
&\sum (h_{(1)} \cdot \um) ( h_{(2)} m_{(1)} \cdot \um ) ( h_{(3)} m_{(2)} \cdot k \cdot \um ) =\\
&\sum ( h_{(1)} \cdot m_{(1)} \cdot \um ) ( h_{(2)} m_{(2)} \cdot k _{(2)} \cdot \um ),
\end{align*} which completes the proof. \end{proof}
\vspace{.3cm}
Note that if in the proposition above we do not assume (\ref{GroupCocycle}) and (\ref{normalized}), i.e. we take an arbitrary map
$\gamma : G \times G \to {{\kappa}}^{\ast},$ then we obtain a twisted partial action which in general does not satisfy (\ref{8}) and (\ref{9}).\\
The above example can be made more specific by taking a concrete group $G.$ The smallest finite group with non-trivial Schur Multiplier is the Klein-four group $G = \langle a \rangle \times \langle b \rangle ,$ $a^2 = b^2=1.$
In this case the Schur Multiplier $M(G)$ has order $2,$ and a $2$-cocycle $\gamma $, which is not a coboundary, can be easily obtained by considering the covering group $G^*$ of $G,$ which is the quaternion group of order $8.$
In order to obtain $\gamma $ one takes a function $\phi : G \to G^*,$ which is a choice of representatives of cosets of $G^*$ by $G,$ and defines
$\tilde{\gamma }(g,s) = \phi( g) \phi(s) (\phi (gs)){}^{-1} ,$ $g,s \in G.$ Denote by $\varphi : {\mathcal Z}(G^*) \to \langle -1\rangle $ the isomorphism between the center of $G^*$ (which has order $2$) and $\langle -1\rangle \subseteq {\mathbb C} .$ Then $\gamma = \varphi \circ \tilde{\gamma} $ is a $2$-cocycle which is not a coboundary. One readily checks that this gives the cocycle $\gamma : G \times G \to {\kappa} ^*$ with $\gamma (g,1) = \gamma (1,g) =1,$ for all $g\in G$ and
$\gamma (a,a) = \gamma (a,ab) = \gamma (b,a) =\gamma (b,b) =\gamma (ab,b) =\gamma (ab,ab)=-1,$
$\gamma (a,b) = \gamma (b,ab) =\gamma (ab,a)=1.$\\
We resume the example of this section in the next:
\begin{prop}\label{Ex:AlgGrHopf} Let ${\kappa} $ be an isomorphic copy of the complex numbers ${\mathbb C} $ and let $\mathbb{S}^1 \subseteq {\mathbb C}$ be the circle group, i. e the group of all complex roots of $1.$ Let, furthermore, $G$ be an arbitrary finite group seen as a subgroup of $S_n$ for some $n.$ Taking the action of $G \subseteq S_n$ on $ ({\kappa} \mathbb{S}^1 )^{\otimes n} $ by permutation of roots, consider the smash product Hopf algebra $$H'_1 = ({\kappa} \mathbb{S}^1 )^{\otimes n} \rtimes {\kappa} G.$$ Let $X\subseteq G$ be an arbitrary subset which is not a subgroup, and consider the subalgebra $\tilde {A} = (\sum _{g\in X} p_g) ({\kappa} G)^* \subseteq ({\kappa} G)^*,$ and write $A'= {\kappa} [t,t^{-1}]^{\otimes n} \otimes \tilde{A}.$ Then with the notation established in (\ref{notation1}) and (\ref{notation2}), the formula
\begin{equation*}
(\chi_{\theta_1 , \ldots \theta_n } \otimes u_g) \cdot (t_1^{k_1} \ldots t_n^{k_n} \otimes p_s ) =
\exp\{ i(k_1 \theta_{ g{}^{-1} s(1)} + \ldots + k_n \theta_{ g{}^{-1} s (n)}) \} \; t_1^{k_1} \ldots t_n^{k_n} \otimes p_{ s{}^{-1} g},
\end{equation*} where $g \in G$ and $s \in X \subseteq G$ gives a left partial action $\alpha : H'_1 \times A' \to A' .$ Assume now that the Schur Multiplier of $G$ is non-trivial and take a normalized (see (\ref{normalized}) ) $2$-cocycle $\gamma : G \times G \to {{\kappa} }^{\ast}$ which is not a coboundary. For arbitrary
$h= \chi_{\theta_1 , \ldots \theta_n } \otimes u_g$ and $l= \chi_{{\theta}'_1 , \ldots {\theta }'_n } \otimes u_s$ in $H'_1$ set
\begin{equation*}
\omega ( h, l ) = \gamma (g,s)\; ( h \cdot l \cdot {\mathbf{1}_{A'} } ).
\end{equation*} Then the pair $(\alpha , \omega )$ forms a twisted partial action of
$H'_1 = ({\kappa} \mathbb{S}^1 )^{\otimes n} \rtimes {\kappa} G$ on $A',$ which satisfies (\ref{8}) and (\ref{9}).
\end{prop}
\section{Symmetric Twisted Partial Actions}
In \cite{DES1} a twisted partial action of a group $G$ over a unital ${\kappa} $-algebra $A$ was defined as a triple
\[
\left(\{D_g\}_{g\in G}, \{\alpha_g\}_{g\in G},
\{w_{g,h}\}_{(g,h)\in G\times G}\right),
\]
where for each $g, h \in G$,
$D_g$ is an ideal of $A$ and $w_{g,h}$ is a multiplier of $D_g D_{gh}$ with some properties. If each $D_g$ is generated by a central idempotent $1_g,$ then, as we have seen in Example~\ref{ex:kG}, this matches our concept of a partial action of the group Hopf algebra ${\kappa} G$ over $A$, and in this case, $1_g =g\cdot 1_A$. Then, from now on, unless explicitly stated, we are going to consider only partial actions of a Hopf algebra $H$ over some unital algebra $A$ such that the map ${\bf e}\in \mbox{Hom} (H,A)$, given by ${\bf e}(h)=(h\cdot \um )$, is central with respect to the convolution product. These partial actions are, in some sense, more akin to partial group actions.
\vspace{.1cm}
The second point of interest in twisted partial group actions is the case where the cocycles $\omega_{g,h}$ are invertible in $D_g D_{gh}$, for all $g,h\in G$. If the group action is global, then every element $\omega_{g,h}$ is an invertible element in $A$, this is automatically translated into the Hopf algebra setting by saying that the cocycle $\omega \in \mbox{Hom}(H\otimes H ,A)$ is convolution invertible. In the partial case, we have to search more suitable conditions to replace the convolution invertibility for the cocycle.
Let $A = (A, \cdot, \omega)$ be a twisted partial $H$-module algebra. From the definition it follows that $f_1(h,k) = (h \cdot \um)\varepsilon(k) $ and $f_2(h,k) = (hk \cdot \um) $ are both (convolution) idempotents in $Hom(H \otimes H,A)$. We also have that ${\bf e}$ is an idempotent in $\operatorname{Hom}(H,A)$ (and $f_1(h,k) = {\bf e}(h) \varepsilon(k)$).
Let us assume that both $f_1$ and $f_2$ are central in $\operatorname{Hom}(H \otimes H, A)$. In this case condition (\ref{cociclo}) of the definition of a twisted partial action reads as
\begin{eqnarray*}
\sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot \um)& = & \sum (h_{(1)}k_{(1)} \cdot \um) \omega(h_{(2)} , k_{(2)})= \omega(h , k) ,
\end{eqnarray*} and by Proposition~\ref{FirstProp} one also has:
\begin{eqnarray*}
\sum \omega(h_{(1)} , k)(h_{(2)} \cdot \um) & = & \sum (h_{(1)} \cdot \um)\omega(h_{(2)} , k) = \omega(h , k). \\
\end{eqnarray*} Notice that this actually says that $\omega$ is an element of the ideal $\langle f_1 * f_2 \rangle \subset \operatorname{Hom}(H \otimes H,A )$ generated by $ f_1 * f_2$. Clearly $f_1 * f_2 $ is the unity element of $\langle f_1 * f_2 \rangle .$ Observe also that the centrality of $f _1$ evidently implies that of ${\bf e} \in \operatorname{Hom}(H,A).$
\begin{defi}\label{symm} Let $A = (A, \cdot, \omega)$ be a twisted partial $H$-module algebra.
We will say that the partial action
is \emph{symmetric} if
\begin{enumerate}[\rm (i)]
\item $f_1$ and $f_2$ are central in $\operatorname{Hom}(H \otimes H, A);$
\item $\omega$ is a normalized cocycle which is an \emph{invertible} element of the ideal $\langle f_1 * f_2 \rangle \subset \operatorname{Hom}(H \otimes H,A)$, i.e., $\omega$ satisfies conditions (\ref{8}) and (\ref{9}) and has a convolution inverse $\omega'$ in $\langle f_1 * f_2 \rangle ;$
\item $\sum (h \cdot (k \cdot \um)) = \sum (h_1 \cdot \um)(h_2 k \cdot \um)$, for every $h,k \in H .$
\end{enumerate}
\end{defi}
We remark once more that to say that $\omega': H \otimes H \rightarrow A$ lies in $\langle f_1 * f_2 \rangle$ is equivalent to require the equalities:
\begin{equation}
\sum \omega'(h_{(1)} , k_{(1)})(h_{(2)} \cdot \um) = \omega'(h , k) = \sum
\omega'(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot \um), \label{abs.omegalinha}
\end{equation}
and that $\omega'$ is the inverse of $\omega $ in $\langle f_1 * f_2 \rangle$ if and only if
\begin{equation}
(\omega * \omega' )(h , k) = (\omega' * \omega) (h , k) = \sum (h_{(1)} \cdot \um)(h_{(2)} k \cdot \um) . \label{omega.omegalinha}
\end{equation}
It readily follows from (\ref{abs.omegalinha}) and (\ref{omega.omegalinha}) that $\omega '$ is also normalized, i.e. ${\omega}'(1_H,h) = {\omega}'(h,1_H) = h\cdot \um $ for all $h \in H.$
\vspace{.3cm}
Multiplying equality (\ref{torcao}) on the right by $\omega'$ and using (iii) of Definition~\ref{symm}, we obtain
\begin{equation}
h \cdot (k \cdot a) = \sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot a)\omega'(h_{(3)} , k_{(3)}) \label{h.k.a}
\end{equation}
for all $h,k \in H$ and $a \in A$, which is an expression analogous to that of global twisted actions of Hopf algebras and also of partial twisted actions of groups. It is easy to prove that if one assumes that the two first items of Definition~\ref{symm} and equality (\ref{h.k.a}) hold, then item (iii) of Definition~\ref{symm} follows.
Formula (\ref{h.k.a}) also provides another equality for $\omega'$ which is similar to (\ref{torcao}). Multiplying (\ref{h.k.a}) by
$\omega'$ on the left and using the centrality of ${\bf e},$ we obtain
\begin{eqnarray*}
& & \sum \omega'(h_{(1)} , k_{(1)}) (h_{(2)} \cdot (k_{(2)} \cdot a)) =\nonumber \\
& & = (\omega' * \omega)(h_{(1)} , k_{(1)}) (h_{(2)}k_{(2)} \cdot a) \omega'(h_{(3)} , k_{(3)}) \\
& & = (h_{(1)} \cdot \um)(h_{(2)}k_{(1)} \cdot \um ) (h_{(3)}k_{(2)} \cdot a) \omega'(h_{(4)} , k_{(3)}) \\
& & = \sum(h_{(1)}k_{(1)} \cdot a) (h_{(2)} \cdot \um)\omega'(h_{(3)} , k_{(2)})\\
& & \overset{\text{(\ref{abs.omegalinha})}}{=} \sum(h_{(1)}k_{(1)} \cdot a) \omega'(h_{(2)} , k_{(2)}).
\end{eqnarray*} Therefore, $\omega'$ satisfies
\begin{equation}
\sum \omega'(h_{(1)} , k_{(1)}) (h_{(2)} \cdot (k_{(2)} \cdot a)) = \sum(h_{(1)}k_{(1)} \cdot a) \omega'(h_{(2)} , k_{(2)})
\end{equation}
for all $h,k \in H$ and $a \in A$.
We shall need expressions for $h \cdot \omega(h , k)$ and $h \cdot \omega'(h, k)$, and for this we prove first an intermediate result, which is interesting on its own.
\begin{lemma}\label{lema.semigrupo}
Let $\mathcal{S}$ be a semigroup and let $v,e,e'$ be elements of $\mathcal{S}$. If there is an element $v' \in \mathcal{S}$ such that
\begin{equation}\label{3.igualdades.semigrupo}
vv' = e, \ \ v'v = e' \text{ and } \ \ v'e = v',
\end{equation}
then $v' \in \mathcal{S} $ satisfying (\ref{3.igualdades.semigrupo}) is unique.
\end{lemma}
\begin{proof}
In fact, assume that $v'$ is a solution of (\ref{3.igualdades.semigrupo}). Then $v'$ also satisfies
\begin{equation}
e'v' = v' \label{4a.igualdade.semigrupo},
\end{equation}
because
\[
e'v' = (v'v)v'=v'(vv')=v'e=v'.
\]
Suppose that $v''$ is another solution. It follows from (\ref{3.igualdades.semigrupo}) and
(\ref{4a.igualdade.semigrupo}) that
\[
v'' = v''e = v''(vv') = (v''v)v' = e'v' = v'.
\]
\end{proof}
\begin{prop}
Let $(A,\cdot, (\omega,\omega'))$ be a symmetric twisted partial $H$-module algebra. Then
\begin{eqnarray}
h \cdot \omega(k , m) & = & \sum \omega(h_{(1)} , k_{(1)}) \omega(h_{(2)}k_{(2)} , m_{(1)}) \omega'(h_{(3)} , k_{(3)}m_{(2)}), \label{h.em.omega}\\
h \cdot \omega'(k , m) & = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \omega'(h_{(3)} , k_{(3)}). \label{h.em.omega.linha}
\end{eqnarray}
\end{prop}
\begin{proof}
To prove (\ref{h.em.omega}), multiply (\ref{9}) by $\omega'$ on the right, obtaining
\begin{eqnarray}
& & \sum (h_{(1)} \cdot \omega (k_{(1)} , m_{(1)})) (\omega * \omega' )(h_{(2)} , k_{(2)}m_{(2)}) = \nonumber \\
& & = \sum
\omega (h_{(1)} , k_{(1)}) \omega (h_{(2)}k_{(2)} , m_{(1)}) \omega' (h_{(2)} , k_{(2)}m_{(2)})
\nonumber
\end{eqnarray}
Since the left hand side equals
\begin{eqnarray*}
& & \sum (h_{(1)} \cdot ( \omega(k_{(1)} , m_{(1)}))(h_{(2)} \cdot (k_{(2)}m_{(2)} \cdot \um)) = \\
& = & h \cdot (\sum \omega(k_{(1)} , m_{(1)}) (k_{(2)}m_{(2)} \cdot \um)) = h \cdot \omega(k,m) , \\
\end{eqnarray*} equation (\ref{h.em.omega}) follows.
The proof of (\ref{h.em.omega.linha}) is a bit more involved and uses Lemma~\ref{lema.semigrupo}. Consider $\operatorname{Hom}(H^{\otimes^3},A)$ as a multiplicative semigroup, and take the elements
\begin{eqnarray*}
v(h , k , m) & = & h \cdot \omega (k , m), \\
e(h , k , m) & = & e'(h \otimes k \otimes m) = (h \cdot (k \cdot (m \cdot \um))) =\nonumber\\ & = & \sum h \cdot [(k_{(1)} \cdot \um)(k_{(2)}m \cdot \um)].
\end{eqnarray*}
We will show that
\begin{eqnarray*}
v'(h , k , m) & = & h \cdot \omega' (k , m), \\
v''(h , k , m) & = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \omega'(h_{(3)} , k_{(3)})
\end{eqnarray*} satisfy
\begin{equation*}
v * v' = e, \ \ v' * v = e, \ \ v' * e = v'
\end{equation*}
and
\begin{equation*}
v * v'' = e, \ \ v'' * v = e, \ \ v'' * e = v'',
\end{equation*}
thus proving, via Lemma~\ref{lema.semigrupo}, that $v'=v''$.
Keep in mind that $e$ can be written also as
\begin{eqnarray*}
e(h , k , m) & = & \sum h \cdot [(k_{(1)} \cdot \um)(k_{(2)}m \cdot \um)] \\
& = & \sum (h_{(1)} \cdot \um)(h_{(2)}k_{(1)} \cdot \um)(h_{(3)}k_{(2)}m \cdot \um).
\end{eqnarray*}
The equalities involving $v'$ are straightforward. For instance,
\[
(v'*e)(h, k, m) =
h [\cdot (\omega' (k , m))(k \cdot (m \cdot \um))]
= h \cdot (\omega' (k , m)) = v'(h \otimes k \otimes m).
\]
As for $v''$, we first compute $v * v'',$ using (\ref{h.em.omega}), the centrality of ${\bf e}$ and (3) of Definition~\ref{defi:twisted}:
\begin{eqnarray*}
&& (v * v'')(h , k , m) = \\
& = & \sum \underbrace{(h_{(1)} \cdot \omega(k_{(1)} , m_{(1)})) \omega(h_{(2)} , k_{(2)}m_{(2)})} \omega'(h_{(3)}k_{(3)} , m_{(3)}) \omega'(h_{(4)} , k_{(4)}) \\
& \overset{\text{(\ref{9})}}{=} &
\sum \omega(h_{(1)} , k_{(1)})\omega(h_{(2)}k_{(2)} , m_{(1)})
\omega'(h_{(3)} , k_{(3)}m_{(2)}) \omega'(h_{(4)} , k_{(4)})\\
& \overset{\text{\ref{symm}.iii,(\ref{omega.omegalinha})}}{=} &
\sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot ( m \cdot \um))
\omega'(h_{(3)} , k_{(3)})\\
& = & (h \cdot (k \cdot (m \cdot \um))) = e(h , k , m).
\end{eqnarray*}
Observe next that given $m \in H$, the linear function $\nu_{m}: H \otimes H \rightarrow A$ given by $h \otimes k \mapsto \omega (h, k m) $ lies in $\operatorname{Hom} (H \otimes H ,A)$ and therefore commutes with $f_2$, i.e.
\begin{equation}\label{commutef2}
\sum \omega(h_{(1)}, k_{(1)} m) (h_{(2)} k_{(2)} \cdot \um ) = \sum (h_{(1)} k_{(1)} \cdot \um ) \omega(h_{(2)}, k_{(2)} m) ,
\end{equation} for all $h,k,m\in H.$ Then we calculate $v'' \ast v$:
\begin{eqnarray*}
&& (v'' * v)(h \otimes k \otimes m) =\\
& = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \omega'(h_{(3)} , k_{(3)}) \, (h_{(4)} \cdot \omega(k_{(4)} , m_{(3)})) \\
& \overset{\text{(\ref{h.em.omega})}}{=} & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \underbrace{\omega'(h_{(3)} , k_{(3)}) \omega (h_{(4)} , k_{(4)})} \times \\
&& \times \omega (h_{(5)} k_{(5)} , m_{(3)}) \omega'(h_{(6)} , k_{(6)}m_{(4)})\\
& = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) (h_{(3)} \cdot \um)(h_{(4)}k_{(3)} \cdot \um)
\times \\ && \times \omega (h_{(5)} k_{(4)} , m_{(3)}) \omega'(h_{(6)} , k_{(5)}m_{(4)})\\
& = & \sum \underbrace{\omega(h_{(1)} , k_{(1)}m_{(1)}) (h_{(2)} \cdot \um)}
\overbrace{\omega'(h_{(3)}k_{(2)} , m_{(2)}) (h_{(4)}k_{(3)} \cdot \um)}
\times \\
&& \times \omega (h_{(5)} k_{(4)} , m_{(3)})\omega'(h_{(6)} , k_{(5)}m_{(4)})\\
& = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \underbrace{\omega'(h_{(2)}k_{(2)} , m_{(2)}) \omega (h_{(3)} k_{(3)} , m_{(3)})} \omega'(h_{(4)} , k_{(4)}m_{(4)})\\
& = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) (h_{(2)}k_{(2)} \cdot \um) \underbrace{(h_{(3)} k_{(3)} m_{(2)} \cdot \um)
\omega' (h_{(4)} , k_{(4)}m_{(3)})}\\
& = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) (h_{(2)}k_{(2)} \cdot \um)\omega'(h_{(3)} , k_{(3)}m_{(2)})\\
& \overset{\text{(\ref{commutef2})}}{=} & \sum (h_{(1)}k_{(1)} \cdot \um)\omega(h_{(2)} , k_{(2)}m_{(1)}) \omega'(h_{(3)} , k_{(3)}m_{(2)})\\
& = & \sum (h_{(1)} \cdot \um)(h_{(2)}k_{(1)} \cdot \um)(h_{(3)}k_{(2)}m \cdot \um) = e(h , k , m).
\end{eqnarray*}
And finally, the expression for $v'' * e$ can be obtained as follows
\begin{eqnarray*}
&&v'' * e (h , k , m) = \\
& = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \underbrace{\omega'(h_{(3)} , k_{(3)}) ( h_{(4)} \cdot \um) }
\times \\
&& \times \underbrace{ (h_{(5)}k_{(4)} \cdot \um) (h_{(6)}k_{(5)}m_{(3)} \cdot \um) } \\
& =& \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \omega'(h_{(3)} , k_{(3)}) (h_{(4)}k_{(4)} \cdot (m_{(3)} \cdot \um))
\\
&\overset{\text{(\ref{h.k.a})}}{=}& \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \underbrace{ \omega'(h_{(3)} , k_{(3)}) \omega(h_{(4)} , k_{(4)})} \times \\
&& \times (h_{(5)} k_{(5)} m_{(3)} \cdot \um)\omega'(h_{(6)} , k_{(6)})\\
& =& \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) (h_{(3)} \cdot \um)
(h_{(4)}k_{(3)} \cdot \um)\times \\
&& \times(h_{(5)} k_{(4)} m_{(3)} \cdot \um)\omega'(h_{(6)} , k_{(5)})
\end{eqnarray*}
\begin{eqnarray*}
&=&\sum \underbrace{ \omega(h_{(1)} , k_{(1)}m_{(1)}) (h_{(2)} \cdot \um)} \overbrace {\omega'(h_{(3)}k_{(2)} , m_{(2)})
(h_{(4)}k_{(3)} \cdot \um) (h_{(5)} k_{(4)} m_{(3)} \cdot \um) } \times \\
&& \times \omega'(h_{(6)} , k_{(5)})\\
& = & \sum \omega(h_{(1)} , k_{(1)}m_{(1)}) \omega'(h_{(2)}k_{(2)} , m_{(2)}) \omega'(h_{(3)} , k_{(3)}) = v''(h , k , m).
\end{eqnarray*}
Therefore, Lemma~\ref{lema.semigrupo} implies (\ref{h.em.omega.linha}). \end{proof}
\vspace{.1cm}
\begin{ex}
Consider a twisted $H$-module algebra $B$ as in Example~\ref{induced}, and assume that the map $u: H \otimes H \rightarrow B$, which twists the action, is a normalized \emph{invertible} cocycle with convolution inverse $u^{-1}$. Suppose furthermore, that $B$ has a nontrivial central idempotent $\um$, and consider the twisted partial $H$-module structure on the ideal $A =\um B$ as it was done in Example~\ref{induced}: the partial action and the cocycle $\omega$ are defined by
\begin{eqnarray*}
h \cdot a & = & \um (h \rhd a) \\
\omega(h , k) & = & \sum (h_{(1)} \cdot \um) u(h_{(2)} , k_{(1)}) (h_{(3)}k_{(2)} \cdot \um).
\end{eqnarray*}
Suppose also that $f_1(h \otimes k)=(h \cdot \um)\varepsilon(k)$ and $f_2(h \otimes k) = (hk \cdot \um)$ are central in $\operatorname{Hom}(H \otimes H,A)$. Under this hypothesis, the functions $h \otimes k \mapsto \um u(h,k) $ and $h \otimes k \mapsto \um u{}^{-1} (h,k) $ commute with ${\bf e}$ and $f_2,$ and it is obvious that
\begin{equation*}
\omega'(h , k) = \sum (h_{(1)}k_{(1)} \cdot \um)u^{-1}(h_{(2)} , k_{(2)}) (h_{(3)} \cdot \um)
\end{equation*}
is the inverse of $\omega$ in $\langle f_1 * f_2 \rangle$. Note also that
\begin{eqnarray*}
h \cdot (k \cdot a) & = & \um (h \rhd (\um (k \rhd a))) = \um (h \rhd (\um (k \rhd a) \um))\\
& = & \um \sum (h_{(1)} \rhd \um) (h_{(2)} \rhd (k \rhd a)) (h_{(3)} \rhd \um)\\
& = & \um \sum (h_{(1)} \rhd \um) u(h_{(2)} , k_{(1)}) (h_{(3)}k_{(2)} \rhd a) u^{-1} (h_{(4)},k_{(3)})(h_{(5)} \rhd \um)
\\
& = & \um \sum (h_{(1)} \rhd \um) u(h_{(2)} , k_{(1)}) (h_{(3)}k_{(2)} \rhd \um)(h_{(4)}k_{(3)} \rhd a) \times \\
&& \times (h_{(5)}k_{(3)} \rhd \um) u^{-1} (h_{(6)} , k_{(4)})(h_{(7)} \rhd \um)\\
& = & \sum \underbrace{(h_{(1)} \cdot \um) u(h_{(2)} , k_{(1)}) (h_{(3)}k_{(2)} \cdot \um)}(h_{(4)}k_{(3)} \cdot a) \times \\
&& \underbrace{(h_{(5)}k_{(3)} \cdot \um) u^{-1} (h_{(6)} , k_{(4)})(h_{(7)} \cdot \um)}\\
& =& \sum \omega(h_{(1)} , k_{(1)})(h_{(2)} k_{(2)} \cdot a)\omega'(h_{(3)} , k_{(3)}),
\end{eqnarray*}
and it follows that
\begin{eqnarray*}
h \cdot (k \cdot \um)
& = & \sum \omega(h_{(1)} , k_{(1)})(h_{(2)} k_{(2)} \cdot \um)\omega'(h_{(3)} , k_{(3)}) \\
& = & \sum \omega(h_{(1)} , k_{(1)})\omega'(h_{(2)} , k_{(2)}) = \sum (h_{(1)} \cdot \um)(h_{(2)}k \cdot \um),\\
\end{eqnarray*} proving that we have a symmetric twisted partial $H$-module algebra.\nolinebreak\hfill$\Box$\par\medbreak
\end{ex}
Another important point arising in the context of symmetric twisted partial Hopf actions is to give criteria in order to decide whether two twisted partial actions give rise to the same crossed product. In the classical case, two crossed products are isomorphic if, and only if the associated twisted (global) actions can be transformed one into another by some kind of coboundary (see, for instance \cite{Mont} for the main results of the classical case). In the case of abelian groups, there is, indeed, a cohomology theory involved, and the cocycles performing the twisted actions are related by coboundaries. What we shall see now is an analogue of Theorem~7.3.4 of \cite{Mont} for twisted partial Hopf actions, this result opens a window for a cohomological point of view of the twisted cocycles presented above.
\begin{thm} \label{the41}Let $A$ be a unital algebra and $H$ a Hopf algebra with two symmetric twisted partial actions on $A$, $h\otimes a \mapsto h\cdot a$, and $h\otimes a \mapsto h\bullet a$, with cocycles $\omega$ and $\sigma$, respectively. Suppose that there is an algebra isomorphism
\[
\Phi :\, A\#_{\omega} H \, \rightarrow \, A\#_{\sigma} H
\]
which is also a left $A$-module and right $H$-comodule map. Then there exists linear maps $u,v\in \operatorname{Hom}(H,A)$ such that, for all $h,k\in H$, $a\in A$,
\begin{enumerate}
\item[(i)] $u*v(h)=h\cdot \um $,
\item[(ii)] $u(h)=\sum u(h_{(1)})(h_{(2)}\cdot \um )=\sum (h_{(1)} \cdot \um )u(h_{(2)})$,
\item[(iii)] $h\bullet a=\sum v(h_{(1)}) (h_{(2)}\cdot a)u(h_{(3)})$,
\item[(iv)] $\sigma (h,k)=\sum v(h_{(1)})(h_{(2)}\cdot v(k_{(1)}))\omega (h_{(3)},k_{(2)}) u(h_{(4)}k_{(3)})$,
\item[(v)] $\Phi (a\#_{\omega} h) =\sum au(h_{(1)})\#_{\sigma} h_{(2)}$.
\end{enumerate}
Conversely, given maps $u,v\in \operatorname{Hom}(H,A)$ satisfying (i),(ii),(iii) and (iv), and in addition $u(1_H )=v(1_H )=\um$, then the map $\Phi$, as presented in (v), is an isomorphism of algebras.
\end{thm}
\begin{proof} ($\Rightarrow$) The left $A$-module structure on the crossed products is given by the left multiplication:
\[
a\blacktriangleright (b\# h)= (a\# 1_H )(b\# h)=ab\# h ,
\]
and the right $H$-comodule structure is given by $\rho =\mbox{I}_A \otimes \Delta$. Let $\Phi :\, A\#_{\omega} H \, \rightarrow \, A\#_{\sigma} H$ be the algebra isomorphism which also is a left $A$-module and right $H$-comodule map. Define $u,v\in \mbox{Hom}(H,A)$ as
\[
u(h)= (\mbox{I}_A \otimes \varepsilon )\Phi (1_A \#_{\omega} h) , \qquad \mbox{ and } \qquad v(h)= (\mbox{I}_A \otimes \varepsilon )\Phi^{-1} (1_A \#_{\sigma} h) .
\]
Let us verify that the maps $u,v$, as defined above, satisfy the items (i) to (v). For the item (v) we have, for all $a\in A$ and $h\in H$
\begin{eqnarray*}
\Phi (a\#_{\omega} h) & =& a\blacktriangleright ((\Phi (\um \#_{\omega} h)
) ) \\
&=& a\blacktriangleright \{ (\mbox{I}_A \otimes \varepsilon \otimes \mbox{I}_H) (\mbox{I}_A\otimes \Delta )\Phi (\um \#_{\omega} h) \}\\
&=& a\blacktriangleright \{ (\mbox{I}_A \otimes \varepsilon \otimes \mbox{I}_H) \Phi \otimes \mbox{I}_H )(\sum \um \#_{\omega} h_{(1)}) \otimes h_{(2)} ) \} \\
&=&a\blacktriangleright \{ (\mbox{I}_A \otimes \varepsilon ) \Phi(\sum \um \#_{\omega} h_{(1)}) \otimes h_{(2)} \}\\
&=& a\blacktriangleright ( \sum u(h_{(1)}) \#_{\sigma} h_{(2)}) =\sum au(h_{(1)}) \#_{\sigma} h_{(2)} .
\end{eqnarray*}
With a totally similar reasoning, we can conclude that
\[
\Phi^{-1} (a\#_{\sigma} h) =\sum av(h_{(1)}) \#_{\omega} h_{(2)} .
\] Notice that we readily obtain from the above that $u(1_H )= v(1_H) = \um.$
\vspace{.1cm}
For item (i) consider the expression
\begin{eqnarray*}
\sum (h_{(1)}\cdot \um )\#_{\omega} h_{(2)} &=& \um \#_{\omega} h =\Phi^{-1} (\Phi (\um
\#_{\omega} h))\\
&=& \Phi^{-1} (\sum u(h_{(1)} )\#_{\sigma} h_{(2)} )\\
&=& \sum u(h_{(1)}) v(h_{(2)}) \#_{\omega} h_{(3)} .
\end{eqnarray*}
Applying $(\mbox{Id}\otimes \varepsilon )$ on both sides, we obtain
\[
\sum u(h_{(1)} ) v(h_{(2)} ) =h\cdot \um .
\]
Analogously, we can conclude that
\[
\sum v(h_{(1)} )u( h_{(2)} ) =h\bullet \um .
\]
Item (ii) is easily obtained by applying $\mbox{I}_A\otimes \varepsilon$ on both sides of the equality
\[
\sum u(h_{(1)})\#_{\sigma} h_{(2)}= \Phi (\um\#_{\omega} h)=\Phi (\sum (h_{(1)}\cdot \um )\#_{\omega} h_{(2)})=
\sum (h_{(1)}\cdot \um )u(h_{(2)}) \#_{\sigma} h_{(3)} .
\] The absorption of $h \cdot \um$ on the other side in (ii) comes from the fact that the twisted partial action is symmetric.
In order to prove items (iii) and (iv), we use the fact that $\Phi^{-1}$ is an algebra morphism, as so is $\Phi $ either. Therefore
\[
\Phi^{-1} ((a\#_{\sigma }h )(b\# _{\sigma} k)) =\Phi^{-1}(a\#_{\sigma} h) \Phi^{-1}
(b\#_{\sigma} k) ,
\]
which gives
\begin{eqnarray*}
& &\sum a(h_{(1)}\bullet b )\sigma (h_{(2)}, k_{(1)}) v(h_{(3)}k_{(2)})\#_{\omega} h_{(4)} k_{(3)} =\\
&=& \sum av(h_{(1)})(h_{(2)}\cdot (bv(k_{(1)}))) \omega (h_{(3)}, k_{(2)})
\#_{\omega} h_{(4)}k_{(3)} .
\end{eqnarray*}
Applying $\mbox{I}_A\otimes \varepsilon$ on both sides, we get
\begin{equation}{\label{cobordos}}
\sum a(h_{(1)}\bullet b )\sigma (h_{(2)}, k_{(1)}) v(h_{(3)}k_{(2)}) =\sum av(h_{(1)})(h_{(2)}\cdot (bv(k_{(1)}))) \omega (h_{(3)}, k_{(2)}) .
\end{equation}
Using this formula for $a=\um $ and $k=1_H$ we obtain
\[
\sum (h_{(1)}\bullet b ) v(h_{(2)}) =\sum v(h_{(1)})(h_{(2)}\cdot b) .
\]
The expression (iii) is finally obtained multiplying convolutively on the right by $u$:
\[
\sum (h_{(1)}\bullet b ) v(h_{(2)}) u(h_{(3)})=\sum v(h_{(1)})(h_{(2)}\cdot b) u(h_{(3)}),
\]
and using the fact that $v*u (h)=h\bullet \um$. This gives
\[
h\bullet b=\sum v(h_{(1)}) (h_{(2)}\cdot b)u(h_{(3)}) .
\]
On the other hand, putting $a=b=\um$ in (\ref{cobordos}) we get
\[
\sum \sigma (h_{(1)}, k_{(1)}) v(h_{(2)}k_{(2)}) =\sum v(h_{(1)})(h_{(2)}\cdot v(k_{(1)})) \omega (h_{(3)}, k_{(2)}) .
\]
Therefore
\[
\sum \sigma (h_{(1)}, k_{(1)}) v(h_{(2)}k_{(2)})u(h_{(3)} k_{(3)}) =\sum v(h_{(1)})(h_{(2)}\cdot v(k_{(1)})) \omega (h_{(3)}, k_{(2)}) u(h_{(4)} k_{(3)}).
\]
Remembering that the cocycle $\sigma$ has the absorption property
\[
\sigma (h,k) =\sum \sigma (h_{(1)} ,k_{(1)})(h_{(2)}k_{(2)} \bullet \um ),
\]
we obtain
\[
\sigma (h,k)=\sum v(h_{(1)})(h_{(2)}\cdot v(k_{(1)}))\omega (h_{(3)},k_{(2)}) u(h_{(4)}k_{(3)}) .
\]
($\Leftarrow$) Conversely, let us consider unit preserving maps $u,v\in \mbox{Hom}(H,A)$, satisfying the items (i) to (iv) in the statement. We shall verify that $\Phi :A\#_{\omega} H \rightarrow A\#_{\sigma} H$ given by
\[
\Phi (a\#_{\omega } h)=\sum au(h_{(1)}) \#_{\sigma} h_{(2)}
\]
is indeed an algebra morphism.
We see immediately that $\Phi (\um \#_{\omega} 1_H)=\um \#_{\sigma} 1_H$. For the multiplicativity, we have
\begin{eqnarray*}
& &\Phi (a\#_{\omega} h)\Phi (b\#_{\omega} k) =\\
&=& \sum (au(h_{(1)})\#_{\sigma}h_{(2)} )
(bu(k_{(1)})\#_{\sigma} k_{(2)}) \\
&=& \sum au(h_{(1)}) (h_{(2)} \bullet (bu(k_{(1)}))) \sigma (h_{(3)} ,k_{(2)})
\#_{\sigma} h_{(4)} k_{(3)} \\
&=& \sum au(h_{(1)}) v(h_{(2)})(h_{(3)} \cdot (bu(k_{(1)}))) u(h_{(4)}) v(h_{(5)})(h_{(6)} \cdot v(k_{(2)})) \times \\
& & \times \omega (h_{(7)},k_{(3)}) u(h_{(8)}k_{(4)}) \#_{\sigma} h_{(9)}k_{(5)} \\
&=& \sum a(h_{(1)}\cdot b) (h_{(2)}\cdot u(k_{(1)}))(h_{(3)}\cdot v(k_{(2)})) \omega (h_{(4)} , k_{(3)}) u(h_{(5)} k_{(4)})\#_{\sigma} h_{(6)}k_{(5)} \\
&=& \sum a(h_{(1)}\cdot b) (h_{(2)}\cdot (u(k_{(1)}) v(k_{(2)}))) \omega (h_{(3)} , k_{(3)}) u(h_{(4)} k_{(4)})\#_{\sigma} h_{(5)}k_{(5)} \\
&=& \sum a(h_{(1)}\cdot b) (h_{(2)}\cdot (k_{(1)}\cdot \um )) \omega (h_{(3)} , k_{(2)}) u(h_{(4)} k_{(3)})\#_{\sigma} h_{(5)}k_{(4)} \\
&=& \sum a(h_{(1)}\cdot b) \omega (h_{(2)} , k_{(1)}) u(h_{(3)} k_{(2)})\#_{\sigma} h_{(4)}k_{(3)} \\
&=& \Phi (\sum a(h_{(1)}\cdot b) \omega (h_{(2)} , k_{(1)})\#_{\omega} h_{(3)}k_{(2)} )\\
&=& \Phi ((a\#_{\omega} h)(b\#_{\omega} k)).
\end{eqnarray*}
Now, it remains to show that $\Phi$ is invertible. Consider the map $\Psi :A\#_{\sigma} H\rightarrow A\#_{\omega} H$ given by
\[
\Psi (a\#_{\sigma} h)=\sum av(h_{(1)})\#_{\omega} h_{(2)}.
\]
Then, we have
\[
\Psi (\Phi (a\#_{\omega} h))=\sum au(h_{(1)})v(h_{(2)})\#_{\omega} h_{(3)} =\sum a(h_{(1)}\cdot \um ) \#_{\omega} h_{(2)} =a\#_{\omega} h .
\]
From (ii) and (iii), we easily conclude that $v*u (h)=h\bullet \um$, and then
\[
\Phi (\Psi (a\#_{\sigma} h))=\sum av(h_{(1)})u(h_{(2)})\#_{\sigma} h_{(3)} =\sum a(h_{(1)}\bullet \um ) \#_{\sigma} h_{(2)} =a\#_{\sigma} h .
\]
Therefore, $\Psi =\Phi^{-1}$ as we wanted to prove.
\end{proof}
\section{Partial Cleft Extensions}
It is a well-known simple fact that a group graded algebra ${\mathcal B}=\oplus_{g\in G} {\mathcal B}_g$ is isomorphic to a crossed product $\mathcal A \ast G,$ where ${\mathcal A}= {\mathcal B}_e $ and $e\in G$ is the neutral element of the group $G,$ exactly when each ${\mathcal B}_g$ contains an element $u_g$ which is invertible in $\mathcal{B}.$ Evidently, the inverse $v_g$ of $u_g$ belongs to ${B}_{g{}^{-1}}.$ Thus we have the maps $\gamma : G \to \mathcal B,$
$g \mapsto u_g \in {\mathcal B}_g$ and ${\gamma}' : G \to { \mathcal B},$ $g \mapsto v_g\in {\mathcal B}_{g{}^{-1}},$
and ${\gamma }'$ is in some sense inverse to $\gamma .$ This becomes precise if we recall that ${ \mathcal B}$ is a ${\kappa} G$-module algebra, and a more general result for a Hopf algebra $H$ says that an $H$-comodule algebra $B$ is isomorphic to a smash product $A \# H,$ $A = B^{co H},$ if and only if $A \subseteq B$ is a Cleft extension, which means that there exists a $ {\kappa} $-linear map $\gamma : H \to B$ which fits into an appropriate commutative diagram and possesses a convolution inverse ${\gamma } ': H \to B.$
The partial case is essentially more complicated. One of the results in \cite{DES1} gives a criteria for a non-degenerate $G$-graded algebra ${\mathcal B}=\oplus_{g\in G} {\mathcal B}_g$ to have the structure of a crossed product $\mathcal A \ast G$ by a twisted partial action of $G$ on $\mathcal A= {\mathcal B}_e .$ More specifically, if ${\mathcal B} $ satisfies
\begin{equation}
\label{g-graduada}
{\mathcal B}_g {\mathcal B}_{g^{-1}} {\mathcal B}_g ={\mathcal B}_g, \;\;\;\;\;\;\;\;\;\;\;\; (\forall g \in G),
\end{equation}
then using the multiplication in ${\mathcal B}$ it is possible to define, for each $g \in G$, idempotent ideals $\mathcal{D}_g =\mathcal{B}_g \mathcal{B}_{g^{-1}},$ $\mathcal{D}_{g^{-1}} =\mathcal{B}_{g^{-1}} \mathcal{B}_g$ of $\mathcal{B}_e$, a unital $\mathcal{D}_g$ - $\mathcal{D}_{g^{-1}}$ bimodule $\mathcal{B}_g$ and a unital $\mathcal{D}_{g^{-1}}$ - $\mathcal{D}_g$ bimodule $\mathcal{B}_{g^{-1}},$ such that they constitute a Morita context. The main ingredients used to the construction of the crossed product are operators $u_g$ and $v_g$ in the multiplier algebra of the context algebra
\[
\mathcal{C}_g =\left( \begin{array}{cc} \mathcal{D}_g & \mathcal{B}_g \\
\mathcal{B}_{g^{-1}} & \mathcal{D}_{g^{-1}} \end{array} \right)
\]
such that
\begin{equation}
\label{g-graduada2}
u_g v_g =e_{11}= \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \qquad
\mbox{and} \qquad
v_g u_g =e_{22}= \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right).
\end{equation} Then it turns out that a non-degenerate $G$-graded algebra $\mathcal{B}$ is isomorphic as a graded algebra to the crossed product $\mathcal A \ast G$ by a twisted partial action exactly when (\ref{g-graduada}) is satisfied and for each $g\in G$ there exist multipliers $u_g$ and $v_g$ of $\mathcal{C}_g$ such that (\ref{g-graduada2}) holds.\\
Now it becomes natural to treat this topic in the context of twisted partial Hopf actions, which is the purpose of the present section. The ``partiality'' is reflected now on the properties of $\gamma .$ Instead of assuming that $\gamma$ is convolution invertible, one declares the existence of a map ${\gamma }'$ which is related to $\gamma $ by conditions which are weaker than that of the convolution invertibility. Some of them match equalities which already played a crucial role in the study of partial actions and partial representations (see Remark~\ref{interaction}).
\begin{defi}
\label{cleft}Let $B$ be a right $H$-comodule unital algebra with coaction
given by $\rho: B \rightarrow B \otimes H$ and let $A$ be a subalgebra of $B$. We will say that $A \subset B$ is an $H$-extension if $A = B^{co H}$. An $H$-extension $A \subset B$ is \emph{partially cleft} if there is a pair of $k$-linear maps $\gamma, \gamma^\prime : H \rightarrow B$ such that
\begin{enumerate}[(i)]
\item $\gamma (1_H) = 1_B ,$
\item the diagrams below are commutative:
\begin{equation} \label{diagrams}
\xymatrix{
H \ar[r]^{\gamma} \ar[d]^{\Delta}& B \ar[d]^{\rho}
& & H \ar[r]^{\gamma'} \ar[d]^{\Delta^{cop}}& B \ar[d]^{\rho} \\
H \otimes H \ar[r]_{\gamma \otimes I_H} & B \otimes H
& & H \otimes H \ar[r]_{\gamma' \otimes S} & B \otimes H
}
\end{equation}
\item $(\gamma*\gamma')\circ M$ is a central element in the convolution algebra $\operatorname{Hom}(H \otimes H,A),$ where $M: H \otimes H \to H$ is the multiplication in $H ,$ and $(\gamma' * \gamma ) (h)$ commutes with every element of $A$ for each $h\in H,$
\end{enumerate}
and, for all $b \in B$ and $h,k \in H$, if we write $e_h= ({\gamma } \ast {\gamma}')(h)$ and $\tilde{e}_h = ({\gamma}' \ast \gamma ) (h)$, then
\begin{enumerate}
\item[(iv)] $\sum b_{(0)} \gamma^\prime (b_{(1)}) \gamma (b_{(2)}) = b ,$ \label{expansao}
\item[(v)] $ {\gamma}(h) e_k = \sum e_{h_{(1)}k} {\gamma} (h_{(2)}), $
\item[(vi)] $ {\gamma}'(k) \tilde{e}_h = \sum \tilde{e}_{hk_{(1)}}{\gamma}'(k_{(2)}),$
\item[(vii)] $ \sum \gamma (h k_{(1)}) \tilde{e}_{k_{(2)}} = \sum {e}_{h_{(1)}} \gamma (h_{(2)} k),$
\end{enumerate}
\end{defi}
Note that item (iii) makes sense because item (ii) implies that $(\gamma * \gamma')(h) \in A$, for all $h \in H$, and therefore $\gamma * \gamma' \in \operatorname{Hom}(H,A)$.
With respect to items (v), (vi) and (vii) we make the following:
\vspace{.1cm}
\begin{remark}\label{interaction} Let $\gamma : G \to \mathcal B$ be a partial representation of a group $G$ into a ${\kappa} $-algebra $\mathcal B,$ i.e. a ${\kappa} $-linear map such that $\gamma (1_G) = 1_B ,$ ${\gamma}(g) {\gamma}(s) \gamma (s{}^{-1})= {\gamma}(g s) \gamma (s{}^{-1})$
and ${\gamma}(g{}^{-1} ) {\gamma}(g) \gamma (s) = {\gamma}(g{}^{-1} ) {\gamma}(gs),$ for all $g,s \in G .$ Then by (2) of \cite{DEP} the following equality holds
$$\gamma (g) e_r = e_{gr} \gamma (g) ,$$ where $e_g =\gamma (g) {\gamma}(g{}^{-1} ).$ This corresponds to item (v) if we take $H= {\kappa} G.$ The above equality plays a crucial role for the interaction between partial actions and partial representations (see \cite{DE}), as well as for an analogous interaction in the context of partial projective representations (see \cite{DN}, \cite{DN2}). Now writing ${\gamma}'(g) = \gamma (g{}^{-1} )$ and $\tilde{e}_g = e_{g{}^{-1}},$ we readily obtain from the above equality that
$$ {\gamma}'(g) \tilde{e}_s = \tilde{e}_{sg}{\gamma}'(g)\quad \text{and} \quad \gamma (g s) \tilde{e}_{s} = {e}_{g} \gamma (gs),$$ for all $g,s \in G,$ which are exactly items (vi) and (vii) above with $H = {\kappa} G.$
\end{remark}
Note that in the case of a cleft extension, with a convolution invertible map $\gamma$, the axioms for partial cleft extensions are automatically satisfied if we take ${\gamma}'$ to be the convolution inverse of $\gamma .$
Observe furthermore that given a partial cleft extension, we also have
\begin{equation}\label{gammalinhaUm}
\gamma'(1_H) = 1_B ,
\end{equation} since by (iv) of Definition \ref{cleft} we see that
\[
1_B = \sum (1_B)_{(0)} \; \gamma'((1_B ) _{(1)}) \; \gamma((1_B)_{(2)}) = (1_B) \gamma'(1_H) \gamma(1_H) = \gamma'(1_H).
\] Moreover, since by (\ref{diagrams}) $\gamma$ is a morphism of comodules, we have that $\rho^2 (\gamma (h)) = (\text{I}_A\otimes\Delta)\rho(\gamma(h))=\sum \gamma (h_{(1)} ) \otimes h_{(2)} \otimes h_{(3)} .$ Then applying (iv) of Definition \ref{cleft} to $b = \gamma(h),$ we conclude that
\begin{equation}
\label{produtogama}
\gamma * \gamma' * \gamma = \gamma .
\end{equation}
The latter will be quite important in what follows. In particular, multiplying this equality by ${\gamma }'$ on the right we obtain that $\gamma \ast {\gamma}'$ is an idempotent, and, moreover, multiplying (\ref{produtogama}) by $\gamma'$ on the left, we see that $\gamma' * \gamma$ is also idempotent. Furthermore, since any linear function $\tau \in \operatorname{Hom} (H,A)$ can be seen as a function $ h \otimes k \mapsto \tau (h)$ in $\operatorname{Hom} (H\otimes H,A), $ item (iii) of Definition~\ref{cleft} implies that
$(\gamma \ast {\gamma}') \ast \tau = \tau \ast (\gamma \ast {\gamma}'),$ so that we have:
\begin{remark}\label{rem:central} Given a partially cleft extension, $ \gamma*\gamma' $ is a central idempotent in the convolution algebra $\operatorname{Hom}(H ,A).$
\end{remark}
The map $\gamma'$ may not satisfy an equality similar to (\ref{produtogama}), but it always can be replaced by another map $\overline{\gamma}$ that does, and the pair $(\gamma,\overline{\gamma})$ still satisfies properties (i)--(vii), as seen in the following:
\begin{lemma} We may assume that ${\gamma }'$ in Definition~\ref{cleft} satisfies the equality
\begin{equation}
\label{produtogamalinha}
\gamma' * \gamma * \gamma' = \gamma'.
\end{equation}
\end{lemma}
\begin{proof} Consider the map $\overline{\gamma} = \gamma' * \gamma * \gamma'$. Since $\gamma'*\gamma$ is an idempotent,
\begin{eqnarray*}
\overline{\gamma} * \gamma * \overline{\gamma} & = & (\gamma' * \gamma * \gamma') * \gamma * (\gamma' * \gamma * \gamma') \\
& = & (\gamma' * \gamma) * (\gamma' * \gamma) * (\gamma' * \gamma) * \gamma' \\
& = & \gamma' * \gamma * \gamma' = \overline{\gamma}.
\end{eqnarray*}
We will show that the pair $(\gamma,\overline{\gamma})$ satisfies the properties (i)--(vii). Item (i) is immediate in view of (\ref{gammalinhaUm}). Item (ii) holds, since
\begin{eqnarray*}
\rho(\overline{\gamma}(h)) & = & \sum \rho (\gamma'(h_{(1)})) \gamma(h_{(2)}) \gamma'(h_{(3)})) = \sum \rho (\gamma'(h_{(1)}) ) \rho ( \gamma(h_{(2)}) ) \rho ( \gamma' (h_{(3)}) ) \\
& = & \sum (\gamma'(h_{(2)}) \otimes S(h_{(1)}))(\gamma(h_{(3)}) \otimes h_{(4)}) ( \gamma'(h_{(6)}) \otimes S(h_{(5)}))
\end{eqnarray*}
\begin{eqnarray*}
& = & \sum \gamma'(h_{(2)})\gamma(h_{(3)})\gamma'(h_{(6)}) \otimes S(h_{(1)})h_{(4)} S(h_{(5)}) \\
& = & \sum \gamma'(h_{(2)})\gamma(h_{(3)})\gamma'(h_{(4)}) \otimes S(h_{(1)}) \\
& = & \sum \overline{\gamma}(h_{(2)})\otimes S(h_{(1)}) = (\overline{\gamma} \otimes S)\Delta^{cop} (h).
\end{eqnarray*} Item (iii) immediately follows from
\begin{equation}\label{barra}
\gamma * \overline{\gamma} = \gamma * \gamma', \ \ \overline{\gamma} * \gamma = \gamma' * \gamma.
\end{equation} Item (iv) holds because, given $b \in B$,
\begin{eqnarray}
& & \sum b_{(0)} \overline{\gamma}(b_{(1)}) \gamma(b_{(2)}) = \sum b_{(0)} \gamma'(b_{(1)}) (\gamma(b_{(2)}) \gamma'(b_{(3)}) \gamma(b_{(4)})) \nonumber \\
& & = \sum b_{(0)} \gamma'(b_{(1)}) \gamma(b_{(2)}) = b. \nonumber
\end{eqnarray} It remains to check (v)--(vii). Notice that ${e}_h= (\gamma \ast \bar{\gamma}) (h)$ and ${\tilde{e}}_h = (\bar{\gamma} \ast \gamma) (h),$ thanks to (\ref{barra}). Thus for $\bar{\gamma}$ we need to verify only (vi). For compute
\begin{align*} \bar{\gamma} (k) \tilde{e}_h = {\gamma}'(k_{(1)}) e_{k_{(2)}} \tilde{e}_h \overset{\text{(iii)}}{=} {\gamma}'(k_{(1)}) \tilde{e}_h e_{k_{(2)}} \overset{\text{(vi)}}{=} \tilde{e}_{ hk_{(1)}} \, {\gamma}'(k_{(2)}) e_{k_{(3)}} =
\tilde{e}_{hk_{(1)} } \, \bar{\gamma}(k_{(2)}),
\end{align*} taking into account that $e_{k} \in A .$ \end{proof}
Since $\rho$ is an algebra morphism, applying (iv) of Definition~\ref{cleft} to $b=\gamma(h) \gamma (k)$ and also to $b=\gamma(h) \gamma (k)a ,$ we obtain for any $a \in A = B^{coH}$ and $h,k \in H$ the following equalities:
\begin{eqnarray}
\gamma(h) \gamma (k) & = & \sum \gamma(h_{(1)}) \gamma (k_{(1)}) \gamma'(h_{(2)}k_{(2)}) \gamma (h_{(3)}k_{(3)}), \label{gammagamma}\\
\gamma(h) \gamma (k) a & = & \sum \gamma(h_{(1)}) \gamma (k_{(1)}) a \gamma'(h_{(2)}k_{(2)}) \gamma (h_{(3)}k_{(3)}) . \label{gammadir}
\end{eqnarray}
Then taking $k=1_H$ in (\ref{gammadir}) we have
\begin{eqnarray}
\gamma(h) a &=& \sum \gamma(h_{(1)}) a \gamma'(h_{(2)}) \gamma (h_{(3)}). \label{gammavezesa}
\end{eqnarray}
\begin{prop}\label{CleftProp} If $(A,\cdot, (\omega,\omega'))$ is a symmetric partial twisted $H$-module algebra, then $A \subset A\#_{(\alpha,\omega)}H$ is a partially cleft $H$-extension.
\end{prop}
\begin{proof} We see that $A\#_{(\alpha,\omega)}H$ is a right comodule algebra via the mapping $\rho = (I \otimes \Delta): A\#_{(\alpha,\omega)}H \rightarrow ( A\#_{(\alpha,\omega)}H) \otimes H$. It is easy to see that $(A\#_{(\alpha,\omega)}H)^{coH} = A \otimes 1_H$, which we will identify with $A$ via the canonical monomorphism $A \rightarrow A \otimes 1_H$.
Consider the maps $\gamma, \gamma': H \rightarrow A\#_{(\alpha,\omega)}H$ given by
\begin{align}
\gamma(h) & = \um \# h = (\um \otimes h)(\um \otimes 1_H), \\
\gamma'(h) & = \sum \omega'(S(h_{(2)}) , h_{(3)}) \# S(h_{(1)}).
\end{align}
From the definition of $\gamma$ we have $\gamma(1_H) = \um \# 1_H = 1_{A\#_{(\alpha,\omega)}H },$ which gives (i) of Definition~\ref{cleft}. With respect to item (ii), the equality $\rho \gamma = (\gamma \otimes I) \Delta$ follows directly by the definition of $\rho$. As for the second diagram in (ii), we have:
\begin{eqnarray*}
\rho \gamma' (h) & = &\sum \rho (\omega'(S(h_{(2)}) , h_{(3)}) \# S(h_{(1)})) = \sum (\omega'(S(h_{(3)}) , h_{(4)}) \# S(h_{(2)}) \otimes S(h_{(1)}) \\
& = & \sum \gamma'(h_{(2)}) \otimes S(h_{(1)}) = ( \gamma' \otimes S) \Delta^{cop}(h),
\end{eqnarray*} which completes the proof of (ii) of the definition of partial cleft extension. Now,
\begin{eqnarray*}
(\gamma * \gamma')(h)
& = & \sum (\um \# h_{(1)})(\omega'(S(h_{(3)}) , h_{(4)}) \# S(h_{(2)})) \\
& = & \sum (h_{(1)} \cdot \omega'(S(h_{(6)}) , h_{(7)})) \omega(h_{(2)} , S(h_{(5)})) \# \underbrace{ h_{(3)}S(h_{(4)}) } \\
& = & \sum (h_{(1)} \cdot \omega'(S(h_{(4)}) , h_{(5)})) \omega(h_{(2)} , S(h_{(3)})) \# 1_H \\
& \overset{\text{(\ref{h.em.omega.linha})}}{=} & \sum \omega(h_{(1)} , \underbrace{S(h_{(8)})h_{(9)}}) \omega'(h_{(2)} S(h_{(7)}) , h_{(10)})
\times \\
&& \times \underbrace{\omega'(h_{(3)} , S(h_{(6)})) \omega(h_{(4)} , S(h_{(5)}))} \# 1_H\\
&\overset{\text{(\ref{omega.omegalinha}),(\ref{8})}}{=} & \sum
(h_{(1)} \cdot \um) \underbrace{ \omega'(h_{(2)} S(h_{(6)}) , h_{(7)})(h_{(3)} \cdot \um) } \overbrace{ (h_{(4)}S(h_{(5)}) } \cdot \um) \# 1_H \\
& \overset{\text{(\ref{abs.omegalinha})}}{=} & \sum (h_{(1)} \cdot \um) \omega'( \underbrace{h_{(2)} S(h_{(3)} ) } , h_{(4)}) \# 1_H = (h \cdot \um) \# 1_H.
\end{eqnarray*} Hence $(\gamma * \gamma')(hk) = f_2(h,k) \# 1_H, $ and this implies that $(\gamma * \gamma')\circ M$ is central in $\operatorname{Hom} (H\otimes H, A)$ thanks to the convolution
centrality of $f_2 .$ Observe also that $(\gamma * \gamma')(h) $ commutes with every element of $A ,$ since each $a \in A$ gives rise to a linear map $\tau_a: H \rightarrow A$ defined by $ \tau_a(h) = \varepsilon(h) a$, and ${\bf e}(h) = (h \cdot \um)$ is central in $\operatorname{Hom}(H,A)$ by assumption. Hence
\[
(h \cdot \um)a = \sum (h_{(1)} \cdot \um)\varepsilon(h_{(2)})a = ({\bf e}*\tau_a)(h) = (\tau_a * {\bf e})(h) = a (h \cdot \um).
\]
With respect to $\gamma'*\gamma$,
\begin{eqnarray*}
\gamma' * \gamma (h ) &=&(\omega'(S(h_{(2)}) , h_{(3)}) \# S(h_{(1)}))(\um \# h_{(4)}) \\
&=&\omega' (S(h_{(4)}) , h_{(5)}) (S(h_{(3)}) \cdot \um)\omega (S(h_{(2)}) , h_{(6)})\# S(h_{(1)})h_{(7)} \\
&=&\omega' (S(h_{(3)}) , h_{(4)})\omega (S(h_{(2)}) , h_{(5)})\# S(h_{(1)})h_{(6)} \\
&=&(S(h_{(3)})h_{(4)} \cdot \um )(S(h_{(2)}) \cdot \um)\# S(h_{(1)})h_{(5)} \\
&=&(S(h_{(2)}) \cdot \um)\# S(h_{(1)})h_{(3)},
\end{eqnarray*}
and this expression implies
\begin{eqnarray*}
(\gamma' * \gamma )(h )(a \# 1_H) &= & \sum ((S(h_{(2)}) \cdot \um)\# S(h_{(1)})h_{(3)})(a \# 1_H) \\
& = & \sum (S(h_{(4)}) \cdot \um)(S(h_{(3)})h_{(5)} \cdot a) \omega (S(h_{(2)})h_{(6)} , 1_H) \# S(h_{(1)})h_{(7)} \\
& = & \sum\underbrace{ (S(h_{(4)})h_{(5)} \cdot a)(S(h_{(3)}) \cdot \um)} \omega (S(h_{(2)})h_{(6)} , 1_H) \# S(h_{(1)})h_{(7)} \\
& = & \sum a (S(h_{(3)}) \cdot \um) \omega(S(h_{(2)})h_{(4)}, 1_H) \# S(h_{(1)})h_{(5)}\\
& = & \sum a \omega(\underbrace{S(h_{(3)})h_{(4)}}, 1_H)(S(h_{(2)}) \cdot \um) \# S(h_{(1)})h_{(5)}\\
& = & \sum a (S(h_{(2)}) \cdot \um) \# S(h_{(1)})h_{(3)}\\
& = & \sum (a \# 1_H)( (S(h_{(2)}) \cdot \um) \# S(h_{(1)})h_{(3)}) = (a \# 1_H)(\gamma' * \gamma )(h ),
\end{eqnarray*} proving item (iii).
For item (iv), consider $b = a \# h$ in $A\#_{(\alpha,\omega)}H$. Applying $\rho^2=(\text{I}_A\otimes\Delta)\rho$ to $b$ we obtain
\[
\sum b_{(0)} \otimes b_{(1)} \otimes b_{(2)} = \sum (a \# h_{(1)}) \otimes h_{(2)} \otimes h_{(3)}, \] and therefore
\begin{eqnarray*}
& & \sum b_{(0)} \gamma'(b_{(1)})\gamma(b_{(2)}) = \sum (a \# h_{(1)}) \gamma'(h_{(2)}) \gamma(h_{(3)}) \\
& = & \sum (a \# h_{(1)}) (\omega'(S(h_{(3)}) \otimes h_{(4)}) \# S(h_{(2)}))(\um \# h_{(5)}) \\
& = & \sum (a \# 1_H)\underbrace{(\um\#h_{(1)}) (\omega'(S(h_{(3)}) \otimes h_{(4)}) \# S(h_{(2)}))} (\um \# h_{(5)}) \\
& \overset{(\ref{gammadir}),(\ref{gammavezesa})}{=} & \sum (a \# 1_H)(\gamma * \gamma')(h_{(1)})(\um \# h_{(2)})\\
&=& \sum (a \# 1_H)((h_{(1)} \cdot \um) \# 1_H)(\um \# h_{(2)}) = a \# h = b.
\end{eqnarray*}
Next we check (v), using (iii) of Definition~\ref{symm}, as follows:
\begin{align*} \sum e_{h_{(1)}k} {\gamma} (h_{(2)}) &= \sum ( h_{(1)}k \cdot \um \# 1_H ) (\um \# h_{(2)})
\overset{(\ref{gammadir})}{=} \sum (h_{(1)} k \cdot \um ) (h_{(2)} \cdot \um ) \# h_{(3)}\\
&= \sum (h_{(1)} \cdot (k \cdot \um )) \# h_{(2)} = \sum (h_{(1)} \cdot (k \cdot \um )) (h_{(2)}\cdot \um ) \# h_{(3)}\\
& = (\um \# h) (k \cdot \um \# 1_H) = \gamma(h)(\gamma\ast\gamma')(k)={\gamma}(h) e_{k}.
\end{align*}
In order to establish (vi) we compute, using again (iii) of Definition~\ref{symm}, that
\begin{align*} & {\gamma}'(h) \tilde{e} _k = (\sum {\omega}' (S(h_{(2)}), h_{(3)}) \# S(h_{(1)}) \; (\sum S(k_{(2)}) \cdot \um \# S(k_{(1)})k_{(3)}) =\\
&= \sum {\omega}'(S(h_{(4)}), h_{(5)}) \underbrace{(S(h_{(3)}) \cdot ( S(k_{(3)}) \cdot \um ))} \omega ( S(h_{(2)}), S(k_{(2)}) k_{(4)}) \# S(h_{(1)}) S(k_{(1)}) k_{(5)}\\
& =\sum \underbrace{{\omega}'(S(h_{(5)}), h_{(6)}) (S(h_{(4)}) \cdot \um) }(S(h_{(3)}) S(k_{(3)}) \cdot \um )\omega ( S(h_{(2)}), S(k_{(2)}) k_{(4)}) \#\\ & \# S(h_{(1)}) S(k_{(1)}) k_{(5)}\overset{(\ref{abs.omegalinha})}{=} \\
\overset{(\ref{abs.omegalinha})}{=} &\sum {\omega}'(S(h_{(4)}), h_{(5)}) \underbrace{(S(h_{(3)}) S(k_{(3)}) \cdot \um )\omega ( S(h_{(2)}), S(k_{(2)}) k_{(4)}) } \# S(h_{(1)}) S(k_{(1)}) k_{(5)}.
\end{align*}
With respect to the underbraced product, for a fixed $m\in H$ consider the function $\tau _m :H\otimes H \to A$ given by
$h\otimes k \mapsto \omega ( h, k m) .$ Since $f_2$ is central, we have
\begin{align*}
& \sum (S(h_{(2)}) S(k_{(2)}) \cdot \um )\omega ( S(h_{(1)}), S(k_{(1)}) m) = ( f_2 \ast \tau _m)(S(h)\otimes S(k ) ) =\\
& = (\tau _m \ast f_2)(S(h)\otimes S(k ) ) =
\sum \omega ( S(h_{(2)}), S(k_{(2)}) m) (S(h_{(1)}) S(k_{(1)}) \cdot \um ),
\end{align*} and consequently we obtain
\begin{align*} & {\gamma}'(h) \tilde{e} _k =\\
& = \sum {\omega}'(S(h_{(4)}), h_{(5)}) \omega ( S(h_{(3)}), \underbrace{ S(k_{(3)}) k_{(4)} }) (S(h_{(2)}) S(k_{(2)}) \cdot \um ) \# S(h_{(1)}) S(k_{(1)}) k_{(5)} \\
& = \sum \underbrace{ {\omega}'(S(h_{(4)}), h_{(5)}) ( S(h_{(3)}) \cdot \um )} \; (S(h_{(2)}) S(k_{(2)}) \cdot \um ) \# S(h_{(1)}) S(k_{(1)}) k_{(3)}\\
&\overset{(\ref{abs.omegalinha})}{=} \sum {\omega}'(S(h_{(3)}), h_{(4)}) \; (S(h_{(2)}) S(k_{(2)}) \cdot \um ) \# S(h_{(1)}) S(k_{(1)}) k_{(3)}.
\end{align*} To compute $\sum \tilde{e}_{kh_{(1)}} {\gamma}'(h_{(2)})$ consider first the function
$\mu _{l, m, n} : H\to A$ defined by $h \mapsto hl \cdot {\omega}'(m,n),$ where $l, m, n \in H$ are fixed. Then
${\bf e} \ast \mu _{l, m, n} = \mu _{l, m, n} \ast {\bf e},$ and applying both sides of this equality to $S( kh ),$ we obtain
\begin{equation}\label{commuting1} (S(k_{(2)} h_{(2)}) \cdot \um ) \; [(S(k_{(1)} h_{(1)})l) \cdot {\omega }'(m , n)] = [ (S(k_{(2)}h_{(2)})l) \cdot {\omega }'(m, n) ]\; (S(k_{(1)}h_{(1)}) \cdot \um ).
\end{equation} Similarly, taking the function $\nu _{m,n} : H \to A,$ given by $h \mapsto \omega (hm, n),$ and using
${\bf e} \ast \nu _{m,n} = \nu _{m,n} \ast {\bf e}$ applied also to $S(hk),$ we also obtain
\begin{equation}\label{commuting2} (S(k_{(2)}h_{(2)} ) \cdot \um ) \; {\omega }( S(k_{(1)}h_{(1)} ) m , n) =
\omega ( S(k_{(2)}h_{(2)} ) m, n) \; ( S(k_{(1)}h_{(1)} ) \cdot \um ).
\end{equation} Then we have:
\begin{align*} & \sum \tilde{e}_{kh_{(1)}} {\gamma}'(h_{(2)}) =\\
& [ \sum ( S(k_{(2)}h_{(2)}) \cdot \um ) \# S(k _{(1)}h _{(2)}) k_{(3)}h_{(3)} ] \; \; [ \sum {\omega}'(S(h_{(5)}), h_{(6)}) ) \# S(h_{(4)}) ]=\\
& \sum ( S(k_{(4)}h_{(4)}) \cdot \um ) \;\; [ S(k _{(3)}h _{(3)}) k_{(5)}h_{(5)} \cdot {\omega}'(S(h_{(10)}), h_{(11 )}) ) ] \times \\
& \times \omega ( S(k _{(2)}h _{(2)}) k_{(6)}h_{(6)}, S(h_{(9)}) ) \# S(k _{(1)}h _{(1)}) k_{(7)} \underbrace{ h_{(7)} S(h_{(8)}) }=\\
& \sum ( S(k_{(4)}h_{(4)}) \cdot \um ) \;\; [ S(k _{(3)}h _{(3)}) k_{(5)}h_{(5)} \cdot {\omega}'(S(h_{(8)}), h_{(9 )}) )] \times \\
&\times \omega ( S(k _{(2)}h _{(2)}) k_{(6)}h_{(6)}, S(h_{(7)}) ) S(k _{(1)}h _{(1)}) k_{(7)} \overset{\text{(\ref{commuting1})}}{=} \\
& \sum [ \underbrace{ S(k _{(4)}h _{(4)}) k_{(5)}h_{(5)} } \cdot {\omega}'(S(h_{(8)}), h_{(9 )}) ) ] \;\;
( S(k_{(3)}h_{(3)}) \cdot \um ) \times \\
& \times \omega ( S(k _{(2)}h _{(2)}) k_{(6)}h_{(6)}, S(h_{(7)}) ) \# S(k _{(1)}h _{(1)}) k_{(7)} =\\
& \sum {\omega}' (S(h_{(6)}), h_{(7 )}) ) \underbrace{ ( S(k_{(3)}h_{(3)}) \cdot \um ) \omega ( S(k _{(2)}h _{(2)}) k_{(4)}h_{(4)}, S(h_{(5)}) ) } \# S(k _{(1)}h _{(1)}) k_{(5)} =\\
& \sum {\omega}' (S(h_{(6)}), h_{(7 )}) ) \omega ( \underbrace{ S(k _{(3)}h _{(3)}) k_{(4)}h_{(4)} }, S(h_{(5)}) ) ( S(k_{(2)}h_{(2)}) \cdot \um ) \# S(k _{(1)}h _{(1)}) k_{(5)} =\\
& \sum \underbrace{ {\omega}' (S(h_{(4)}), h_{(5 )}) ) (S(h_{(3)}) \cdot \um ) } ( S(k_{(2)}h_{(2)}) \cdot \um ) \#
S(k _{(1)}h _{(1)}) k_{(3)} =\\
& \sum {\omega}' (S(h_{(3)}), h_{(4 )}) ) ( S(h_{(2)}) S(k_{(2)}) \cdot \um ) \# S(h _{(1)}) S(k _{(1)}) k_{(3)},\\
\end{align*}which coincides with the expression obtained above for $ {\gamma}'(h) \tilde{e} _k ,$ proving thus (vi).
\vspace{.2cm}
Finally, item (vii) follows from the next calculation, in which we use again the convolution centrality of $f_2$ and ${\bf e}$:
\begin{align*}
& \sum \gamma (hk_{(1)} ) \tilde{e}_{k_{(2)}} = \sum (\um \# h k_{(1)} ) \; [ ( S(k_3) \cdot \um ) \# S(k_{(2)} k_{(4)} )] =\\
& \sum (h_{(1)} k_{(1)} \cdot ( S(k_{(6)}) \cdot \um )) \; \omega ( h_{(2)} k_{(2)} , S(k_{(5)}) k_{(7)} ) \#
h_{(3)} \underbrace{ k_{(3)} S(k_{(4)}) } k_{(8)} = \\
& \sum (h_{(1)} k_{(1)} \cdot ( S(k_{(4)}) \cdot \um )) \; \omega ( h_{(2)} k_{(2)} , S(k_{(3)}) k_{(5)} ) \#
h_{(3)} k_{(6)} =
\\
& \sum \underbrace{ (h_{(1)} k_{(1)} \cdot \um) ( (h_{(2)} k_{(2)} S(k_{(5)}) \cdot \um )} \; \omega ( h_{(3)} k_{(3)} , S(k_{(4)}) k_{(6)} ) \#
h_{(4)} k_{(7)} = \\
& \sum (h_{(1)} k_{(1)} S(k_{(5)}) \cdot \um ) (h_{(2)} k_{(2)} \cdot \um) \; \omega ( h_{(3)} k_{(3)} , S(k_{(4)}) k_{(6)} ) \#
h_{(4)} k_{(7)} = \\
& \sum (h_{(1)} k_{(1)} S(k_{(5)}) \cdot \um ) \underbrace{ (h_{(2)} k_{(2)} \cdot \um) \omega ( h_{(3)} k_{(3)} , S(k_{(4)}) k_{(6)} )} \#
h_{(4)} k_{(7)} =
\end{align*}
\begin{align*}
& \sum (\underbrace{ h_{(1)} k_{(1)} S(k_{(4)}) \cdot \um ) \omega ( h_{(2)} k_{(2)} , S(k_{(3)}) k_{(5)} ) } \#
h_{(3)} k_{(6)} =\\
& \sum \omega ( h_{(1)} k_{(1)} , \underbrace{ S(k_{(4)}) k_{(5)} } ) \; (h_{(2)} \underbrace{ k_{(2)} S(k_{(3)} } ) \cdot \um ) \#
h_{(3)} k_{(6)} =\\
& \sum ( h_{(1)} k_{(1)} \cdot \um ) (h_{(2)} \cdot \um ) \# h_{(3)} k_{(2)} =
\sum (h_{(1)} \cdot \um ) \; ( h_{(2)} k_{(1)} \cdot \um ) \# h_{(3)} k_{(2)} =\\
& \sum (h_{(1)} \cdot \um ) \; \omega ( 1_H, h_{(2)} k_{(1)} ) \# h_{(3)} k_{(2)}
\overset{(\ref{firstprop})}{=}
\sum ( (h_{(1)} \cdot \um ) \# 1_H ) \; ( \um \# h_{(2)} k ) =\\
& \sum e_{h_{(1)}} \gamma (h_{(2)} k ).
\end{align*} \end{proof}
\begin{thm}\label{the51}
Let $B$ be an $H$-comodule algebra and let $A = B^{coH}$. Then the $H$-extension $A \subset B$ is partially cleft if and only if $B$ is isomorphic to a partial crossed product $A\#_{(\alpha,\omega)}H$ with respect to a symmetric twisted partial $H$-module structure on $A.$
\end{thm}
\begin{proof}
We have already proved half of this statement in Proposition~\ref{CleftProp}. So, assume that $B$ is partially cleft by the pair of maps $\gamma, \gamma' : H \rightarrow B$.
The pair $(\gamma, \gamma')$ allows us to define a twisted partial action of $H$ on $A = B^{coH}$ as follows. Given $h,k \in H$ and $a \in A$, set
\begin{eqnarray*}
h \cdot a & = & \sum \gamma(h_{(1)}) a \gamma'(h_{(2)}),\\
\omega(h ,k) & = & \sum \gamma(h_{(1)}) \gamma(k_{(1)}) \gamma'(h_{(2)}k_{(2)}),\\
\omega'(h ,k) & = & \sum \gamma(h_{(1)}k_{(1)}) \gamma'(k_{(2)}) \gamma'(h_{(2)}).\\
\end{eqnarray*}
Before anything else, we must check that these elements lie in $A$, but this is quite simple.
\begin{eqnarray*}
\rho(h \cdot a) & = & \sum \rho(\gamma(h_{(1)}))\rho( a) \rho( \gamma'(h_{(2)})) \\
& = & \sum (\gamma(h_{(1)}) \otimes h_{(2)})( a \otimes 1_H) (\gamma'(h_{(4)}) \otimes S(h_{(3)})) \\
& = & \sum (\gamma(h_{(1)})a\gamma'(h_{(4)}) \otimes h_{(2)}S(h_{(3)}) \\
& = & \sum (\gamma(h_{(1)})a\gamma'(h_{(2)}) \otimes 1_H \\
&= &(h \cdot a) \otimes 1_H, \\
\end{eqnarray*} and thus $a\in B^{coH}= A.$
In an analogous fashion, one may check that both $\omega(h ,k)$ and $\omega'(h , k)$ lie in $A$ for every $h,k$ in $H$. For instance,
\begin{eqnarray*}
\rho(\omega(h,k)) & = & \sum (\gamma(h_{(1)}) \otimes h_{(2)}) (\gamma(k_{(1)}) \otimes k_{(2)}) (\gamma'(h_{(4)}k_{(4)}) \otimes S(h_{(3)}k_{(3)}))\\
& = & \sum \gamma(h_{(1)}) \gamma(k_{(1)}) \gamma'(h_{(2)}k_{(2)}) \otimes 1_H\\
& = & \omega(h,k) \otimes 1_H,
\end{eqnarray*} and similarly for $\omega'(h , k).$
Note that, since $A=B^{coH}$, then $1_B =\um $, and since $\gamma(1_H) = 1_B =\um = \gamma'(1_H)$, we have $1_H \cdot a = a$ for all $a \in A$.
Next, given $h \in H$ and $a,b \in A$, we see that
\begin{eqnarray*}
h \cdot ab & = & \sum \underbrace{\gamma(h_{(1)}) a} b\gamma'(h_{(2)}) \\
& \overset{\text{(\ref{gammavezesa})}}{=} & \sum \gamma(h_{(1)}) a \gamma'(h_{(2)}) \gamma(h_{(3)}) b\gamma'(h_{(4)})\\
& =& \sum (h_{(1)} \cdot a) (h_{(2)} \cdot b).
\end{eqnarray*}
The partial action is twisted by $(\omega, \omega')$, since
\begin{eqnarray*}
h \cdot (k \cdot a)
& =& \sum \underbrace{\gamma(h_{(1)}) \gamma(k_{(1)}) a} \gamma'(k_{(2)}) \gamma'(h_{(2)}) \\
& \overset{\text{(\ref{gammadir})}}{=}& \sum \underbrace{\gamma(h_{(1)})\gamma(k_{(1)})} a \gamma'(h_{(2)}k_{(2)}) \gamma(h_{(3)}k_{(3)}) \gamma'(k_{(4)})\gamma'(h_{(4)})\\
& \overset{\text{(\ref{gammagamma})}}{=}& \sum [\gamma(h_{(1)})\gamma(k_{(1)}) \gamma'(h_{(2)}k_{(2)})] \gamma(h_{(3)}k_{(3)}) a \gamma'(h_{(4)}k_{(4)}) \times \\
& & \times [\gamma(h_{(5)}k_{(5)}) \gamma'(k_{(6)})\gamma'(h_{(6)})]\\
& =& \sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot a)\omega'(h_{(3)} , k_{(3)})
\end{eqnarray*}
for every $a \in A$ and $h,k \in H$.
\vspace{.2cm}
With respect to $\omega$ and $\omega'$, first we observe that
\[
\omega(h,1_H) = \sum \gamma(h_{(1)}) \gamma(1_H) \gamma'(h_{(2)}) = \gamma(h_{(1)})\gamma'(h_{(2)}) = h \cdot \um ,
\]
and also $\omega(1_H,h) = h \cdot \um ,$ showing that $\omega$ is normalized. Note, furthermore, that
\begin{equation}\label{absorbs hDotUm}
\sum \omega (h_{(1)} , k) (h_{(2)} \cdot \um) = \sum (h_{(1)} \cdot \um) \omega (h_{(2)} , k) = \omega(h , k ) ,
\end{equation} because $h \cdot \um = (\gamma * \gamma')(h)$ and $ {\bf e} = \gamma * \gamma'$ is central in $\operatorname{Hom}(H,A)$ by Remark~\ref{rem:central}. Therefore:
\begin{eqnarray*}
\sum \omega (h_{(1)} , k) (h_{(2)} \cdot \um)
& = & \sum (h_{(1)} \cdot \um)\omega (h_{(2)} , k)\\
& = & \sum \underbrace{\gamma(h_{(1)}) \gamma'(h_{(2)})\gamma(h_{(3)})} \gamma(k_{(1)}) \gamma'(h_{(4)}k_{(2)}) \\
& \overset{\text{(\ref{produtogama}) }}{=} & \sum \gamma(h_{(1)}) \gamma(k_{(1)}) \gamma'(h_{(2)}k_{(2)}) = \omega (h , k).
\end{eqnarray*} Analogously, using (\ref{produtogamalinha}), one shows that
\begin{equation}\label{omegaLinhaAbsorbs hDotUm}
\sum {\omega}' (h_{(1)} , k) (h_{(2)} \cdot \um) = \sum (h_{(1)} \cdot \um) {\omega }' (h_{(2)} , k) = \omega(h , k ).
\end{equation} For $\omega * \omega'$ we have
\begin{eqnarray*}
(\omega * \omega') (h \otimes k)
& = & \sum \underbrace{\gamma(h_{(1)}) \gamma (k_{(1)}) \gamma'(h_{(2)}k_{(2)}) \gamma(h_{(3)}k_{(3)})} \gamma'(k_{(4)}) \gamma'(h_{(4)}) \\
& \overset{\text{(\ref{gammagamma}) }}{=} & \sum \gamma(h_{(1)}) \gamma (k_{(1)}) \gamma'(k_{(2)}) \gamma'(h_{(2)}) = h \cdot (k \cdot \um).
\end{eqnarray*}
For the evaluation of $\omega' * \omega$ we use (vi) and (vii) of Definition~\ref{cleft} to compute
\begin{eqnarray*}
(\omega' * \omega) (h \otimes k)
& = & \sum \gamma(h_{(1)}k_{(1)}) \gamma'(k_{(2)}) \underbrace{ \gamma'(h_{(2)})\gamma(h_{(3)}) } \gamma (k_{(3)}) \gamma'(h_{(4)}k_{(4)}) \\
& = & \sum \gamma(h_{(1)}k_{(1)}) \underbrace { \gamma'(k_{(2)}) \tilde{e}_{h_{(2)}} } \gamma (k_{(3)}) \gamma'(h_{(3)}k_{(4)}) \\
& \overset{\text{ (vi) }}{=} & \sum \underbrace{ \gamma(h_{(1)}k_{(1)}) \tilde{e}_{h_{(2)}k_{(2)}} } \gamma'(k_{(3)}) \gamma (k_{(4)}) \gamma'(h_{(3)}k_{(5)}) \\
& \overset{\text{ (\ref{produtogamalinha}) }}{=} & \sum \gamma(h_{(1)}k_{(1)}) \gamma'(k_{(2)}) \gamma (k_{(3)}) \gamma'(h_{(2)}k_{(4)}) \\
& = & \sum \underbrace { \gamma(h_{(1)}k_{(1)}) \tilde{e}_{ k_{(2)} } } \gamma'(h_{(2)}k_{(3)}) \overset{\text{ (vi) }}{=}
\sum {e}_{ h_{(1)}} \gamma(h_{(2)}k_{(1)}) \gamma'(h_{(3)}k_{(2)}) \\
&= & \sum \gamma(h_{(1)}) \gamma'(h_{(2)}) \gamma (h_{(3)}k_{(1)})\gamma'(h_{(4)}k_{(2)}) =
\sum (h_{(1)} \cdot \um ) (h_{(2)}k \cdot \um ).
\end{eqnarray*}
We use this to obtain the initial form of the twisting condition given in (\ref{torcao}) of Definition~\ref{defi:twisted}:
\begin{eqnarray*}
& & \sum (h_{(1)} \cdot (k_{(1)} \cdot a ))\omega(h_{(2)} , k_{(2)}) = \\
& = & \sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot a)\omega'(h_{(3)} , k_{(3)})\omega(h_{(4)} , k_{(4)}) \\
& = & \sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot a)(h_{(3)} \cdot \um)(h_{(4)} k_{(3)} \cdot \um) \\
& = & \sum \underbrace{\omega(h_{(1)} , k_{(1)})(h_{(2)} \cdot \um)}\underbrace{(h_{(3)}k_{(2)} \cdot a)(h_{(4)} k_{(3)} \cdot \um)} \\
& = & \sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot a).
\end{eqnarray*} Using again $\omega' * \omega,$ we obtain the similar twisting equality for ${\omega}':$
\begin{equation}\label{OmegaLinhaTorcao}
\sum \omega'(h_{(1)} , k_{(1)}) (h_{(2)} \cdot (k_{(2)} \cdot a)) = \sum(h_{(1)}k_{(1)} \cdot a) \omega'(h_{(2)} , k_{(2)}).
\end{equation} Indeed, multiplying the above obtained equality
$$ h \cdot (k \cdot a) = \sum \omega(h_{(1)} , k_{(1)})(h_{(2)}k_{(2)} \cdot a)\omega'(h_{(3)} , k_{(3)})$$ by ${\omega}'$ on the left, and using the convolution centrality of ${\bf e}$ and (\ref{omegaLinhaAbsorbs hDotUm}), we have:
\begin{align*}
& \sum {\omega}' (h_{(1)} , k_{(1)} ) (h \cdot (k \cdot a)) =\\
&\sum ( {\omega}' \ast \omega ) (h_{(1)} , k_{(1)}) (h_{(2)}k_{(2)} \cdot a)\omega'(h_{(3)} , k_{(3)})=\\
&\sum (h_{(1)} \cdot \um ) \underbrace{ (h_{(2)}k_{(1)} \cdot \um ) (h_{(3)}k_{(2)} \cdot a) } \omega'(h_{(4)} , k_{(3)})=\\
&\sum (h_{(1)} \cdot \um ) (h_{(2)}k_{(1)} \cdot a) \omega'(h_{(3)} , k_{(2)})=\\
&\sum (h_{(1)}k_{(1)} \cdot a)\underbrace{ (h_{(2)} \cdot \um ) \omega'(h_{(3)} , k_{(2)})} =\\
&\sum(h_{(1)}k_{(1)} \cdot a) \omega'(h_{(2)} , k_{(2)}),
\end{align*} as desired.
Note now that by (v) and (iii) of Definition~\ref{cleft} we have
\begin{align*} h \cdot ( k \cdot \um) & = \sum \gamma (h_{(1)}) e_k {\gamma }'(h_{(2)}) =
\sum e_{h_{(1)} k } \gamma (h_{(2)}) {\gamma }'(h_{(3)}) \\
& = \sum (h_{(1)} k \cdot \um ) ( h_{(2)} \cdot \um ) = \sum ( h_{(1)} \cdot \um ) (h_{(2)} k \cdot \um ),
\end{align*} which gives (iii) of Definition~\ref{symm}. Next we see that $\omega$ absorbs $hk \cdot \um$ on the right:
\begin{align*} \sum \omega (h_{(1)}, k_{(1)}) (h_{(2)} k_{(2)} \cdot \um) & = \sum \gamma( h_{(1)} ) \gamma (k _{(1)})
\underbrace{ {\gamma}' (h_{(2)} k_{(2)}) {\gamma} (h _{(3)} k_{(3)}) {\gamma} '( h _{(4)} k_{(4)})} \\
&\overset{\text{(\ref{produtogamalinha}) }}{=} \sum \gamma( h_{(1)} ) \gamma (k _{(1)}) {\gamma} '( h _{(4)} k_{(4)}) =\omega (h,k),
\end{align*} showing that (\ref{cociclo}) holds. Then using the twisting condition (\ref{torcao}) we see that
\[
\sum (h_{(1)} \cdot (k_{(1)} \cdot \um)) \omega (h_{(2)} , k_{(2)}) = \sum \omega(h_{(1)},k_{(1)})(h_{(2)}k_{(2)} \cdot \um) = \omega(h , k).
\] Thus we have that $\omega(h,k)$ absorbs the elements $h \cdot k \cdot \um,$ $h\cdot \um $ and $hk\cdot \um$ from both sides for any
$h,k \in H.$ In particular, $\omega $ is contained in the ideal $\langle f_1 \ast f_2 \rangle .$
Now, $\omega'$ absorbs $(h \cdot (k \cdot \um))$ on the right, which we see by using (vi) of Definition~\ref{cleft}:
\begin{eqnarray*}
& & \sum \omega'(h_{(1)} , k_{(1)})(h_{(2)} \cdot (k_{(2)} \cdot \um)) = \\
& = & \sum \gamma(h_{(1)}k_{(1)}) \gamma'(k_{(2)}) \gamma'(h_{(2)}) \gamma(h_{(3)}) \gamma(k_{(3)}) \gamma'(k_{(4)}) \gamma'(h_{(4)}) \\
& = & \sum \gamma(h_{(1)}k_{(1)}) \underbrace{ \gamma'(k_{(2)}) \tilde{e}_{h_{(2)}} } \gamma(k_{(3)}) \gamma'(k_{(4)}) \gamma'(h_{(3)}) \\
&\overset{\text{Def.~\ref{cleft}.(vi)}}{=} & \sum \underbrace{ \gamma(h_{(1)}k_{(1)}) \tilde{e}_{h_{(2)} k_{(2)} } } \underbrace{ \gamma'(k_{(3)}) \gamma(k_{(4)}) \gamma'(k_{(5)}) } \gamma'(h_{(3)})\\
&\overset{\text{(\ref{produtogama}), (\ref{produtogamalinha}) }}{=} & \sum \gamma(h_{(1)}k_{(1)}) \gamma'(k_{(2)}) \gamma'(h_{(2)}) = \omega'(h , k),
\end{eqnarray*} By (\ref{omegaLinhaAbsorbs hDotUm}) and the convolution centrality of ${\bf e},$ this implies
$$\sum {\omega}' (h_{(1)}), k_{(1)}) ( h_{(2)} k_{(2)} \cdot \um ) = {\omega}' (h , k ) ,$$ and, moreover, it follows using (\ref{OmegaLinhaTorcao}) that
$$\omega '(h,k) = \sum \omega'(h_{(1)} , k_{(1)}) (h_{(2)} \cdot (k_{(2)} \cdot \um )) = \sum(h_{(1)}k_{(1)} \cdot \um ) \omega'(h_{(2)} , k_{(2)}).$$ Consequently, ${\omega}'$ also absorbs the elements $h \cdot k \cdot \um,$ $h\cdot \um $ and $hk\cdot \um$ from both sides. In particular, (\ref{abs.omegalinha}) is satisfied, i.e. ${\omega }'$ belongs to $\langle f_1 \ast f_2 \rangle .$
We check the cocycle equality (\ref{9}) for $\omega ,$ taking into account that ${\gamma}' \ast {\gamma}$ commutes with each element of $A,$ as follows:
\begin{align*}
& \sum [h_{(1)} \cdot \omega(k_{(1)} , l_{(1)}) ] \; \omega(h_{(2)} , k_{(2)} l_{(2)}) \\
& = \sum \gamma(h_{(1)}) \omega(k_{(1)} , l_{(1)}) [ (\gamma ' \ast \gamma ) (h_{(2)}) ]
\gamma(k_{(3)} l_{(3)}) \gamma'(h_{(3)} k_{(2)} l_{(2)} ) \\
& = \sum \underbrace{ \gamma(h_{(1)}) [ (\gamma ' \ast \gamma ) (h_{(2)}) ] } \omega(k_{(1)} , l_{(1)}) \gamma(k_{(3)} l_{(3)}) \gamma'(h_{(3)} k_{(2)} l_{(2)} ) \\
& \overset{\text{(\ref{produtogama}) }}{=} \sum \gamma(h_{(1)}) \underbrace{ \; \gamma(k_{(1)} ) \gamma(l_{(1)} ) {\gamma }' (k_{(2)} l_{(2)}) \gamma(k_{(3)} l_{(3)}) } \gamma'(h_{(2)} k_{(4)} l_{(4)} ) \\
& \overset{\text{(\ref{gammagamma}) }}{=} \sum \underbrace{ \gamma(h_{(1)}) \; \gamma(k_{(1)} ) } \gamma(l_{(1)} ) \gamma'(h_{(2)} k_{(2)} l_{(2)} ) \\
& \overset{\text{(\ref{gammagamma}) }}{=} \sum [ \gamma(h_{(1)}) \; \gamma(k_{(1)} ) \gamma'(h_{(2)} k_{(2)} ) ] [ \gamma (h_{(3)} k_{(3)} ) \gamma(l_{(1)} ) \gamma'(h_{(4)} k_{(4)} l_{(2)} ) ] \\
& = \sum \omega(h_{(1)} , k_{(1)}) \; \omega(h_{(2)} k_{(2)} , l ).
\end{align*} This completes the proof of the fact that $A = (A, \cdot, \omega, \omega')$ is a symmetric twisted partial $H$-module algebra.
\vspace{.2cm}
Finally, we claim that
\begin{eqnarray*}
\Phi: A\#_{(\alpha,\omega)}H & \rightarrow & B \\
a \# h &\mapsto & a \gamma(h)
\end{eqnarray*}
is an algebra isomorphism, with inverse given by
\begin{eqnarray*}
\Psi: B & \rightarrow & A\#_{(\alpha,\omega)}H \\
b &\mapsto & \sum b_{(0)} \gamma'(b_{(1)}) \# b_{(2)}
\end{eqnarray*}
In fact, $\Phi$ is an algebra map since it obviously takes unity to unity and
\begin{eqnarray*}
\Phi(a \# h) \Phi(b \# k) & = & a \gamma(h) b \gamma(k) \\
& \overset{\text{(\ref{gammavezesa})}}{=} & \sum a \gamma(h_{(1)})b \gamma'(h_{(2)}) \gamma(h_{(3)}) \gamma(k) \\
& \overset{\text{(\ref{gammagamma})}}{=} & \sum a [\gamma(h_{(1)})b \gamma'(h_{(2)})] [\gamma(h_{(3)}) \gamma(k_{(1)}) \gamma'(h_{(4)}k_{(2)})] \times \\
& & \gamma(h_{(5)}k_{(3)}) \\
&=& \sum a (h_{(1)} \cdot b) \omega(h_{(2)} \otimes k_{(1)}))\gamma(h_{(3)}k_{(2)}) \\
&=& \Phi(\sum a (h_{(1)} \cdot b) \omega(h_{(2)} \otimes k_{(1)})) \# h_{(3)}k_{(2)}) \\
&=& \Phi((a \# h) \Phi(b \# k)) \\
\end{eqnarray*}
In order to prove that $\Psi = \Phi^{-1}$, first note that $\sum b_{(0)} \gamma'(b_{(1)})$ lies in $A$, because \ref{cleft}.ii implies that
\begin{eqnarray*}
\rho(\sum b_{(0)}\gamma'(b_{(1)})) & = & \sum \rho(b_{(0)})
(\rho \circ \gamma')(b_{(1)}) = \\
= \sum b_{(0)} (\gamma'(b_{(3)})\otimes b_{(1)} S(b_{(2)}) & = & \sum b_{(0)} \gamma(b_{(1)}) \otimes 1_H.
\end{eqnarray*}
\vspace{.1cm}
Now, $\Phi \Psi = Id_B$ is just condition (iv) of definition \ref{cleft}. For the other composition,
given $a \#h \in A\#_{(\alpha,\omega)}H$, since $\gamma$ is a comodule morphism and $A = B^{coH}$ it follows that
\begin{eqnarray*}
\Psi (\Phi (a \# h)) & = & \Psi (a \gamma(h)) = \sum a\gamma(h_{(1)}) \gamma'(h_{(2)}) \# h_{(3)} \\
& = & \sum a (h_{(1)} \cdot \um) \# h_{(2)} = a \# h.
\end{eqnarray*}
\end{proof}
|
2011.14723
|
\subsection{Non-rigid shape correspondence}
Non-rigid shape matching is built out of aligning points with similar features, geometric and/or photometric, and a smoothness term, making sure a point can not be mapped farther from its neighbor.
Under this umbrella, we had seen various axiomatic methods focused on distances, angles, and areas \cite{gps,shot} where a large leap forward was made when deep learning was applied on top of geometric data.
We can split deep models into two categories - spatial and spectral. Under the spatial approach, we usually see a flow mechanism where the models' changes are minor ~\cite{wu2020pointpwcnet,liu2019flownet3d} or an all-to-all correlation approach \cite{puy2020flot} that can cope with large displacements. When the domain is well defined, then template matching showed great results, as seen in \cite{3dcoded} and in \cite{kanazawaHMR18}.
On the spectral side, various methods based on functional maps \cite{fmnet,dvir,halimi} showed superb results on meshes and points and even excelled on partial alignment \cite{fmnet}. Those papers' goal was to construct deep local features such that the spectra of the shapes would align following a point-to-point soft correspondence matrix.
One of the challenges in non-rigid alignment is the lack of labeled data. That is mainly because there is no feasible way to own the exact dense correspondence of bendable and stretchable domains on real scanned sets. To overcome this obstacle and remain within the learnable regime, we consider a self-supervised approach.
In the spatial domain, we have seen several useful cost functions that use templates while forcing smoothness on the structures \cite{3dcoded}. More sophisticated assumptions on the domain, such as isometry, were able to learn a mapping by minimizing the Goromov Hausdorff metric as it only needed to compare distances between pairs \cite{halimi}, but failed to converge once stretching appeared. A recent mapping with a cyclic loss measuring the error only on the source showed superior results even under local stretching \cite{dvir}.
All those methods break once there isn't enough data to train. As reported by the authors, either the system can not converge, or we witness a high number of outliers.
\emph{To overcome this limitation, we present a zero-shot alignment architecture between two-shapes, where we rethink the alignment process as denoising a soft correspondence matrix.} By that, we quickly converge into a clean outlier-free model and can cope with inter-class alignments even under large deformations.
\subsubsection{Feature matching}
One popular method of minimizing \ref{eq:min_dist_corr}without computing all $|\mathcal{X}|^{|\mathcal{Y}|}$ matches is to consider only matches among points with similar descriptors; In doing so, the key problem now turns into creating descriptive deformation-invariant local point features instead of solving the non-linear objective function.
This has been a key research goal in shape analysis with a great impact on the community. Some examples include GPS \cite{gps}, SHOT\cite{shot} and spectral signatures \cite{hks,wks}. Rotation-invariant descriptors have also been suggested\cite{shot,rotinv}, As this is an important attribute for 3d shape analysis.
These handcrafted descriptors usually require a significant amount of manual tuning and do not work well when applied in different settings.
Lately, a tremendous amount of research has been devoted to \textbf{learning} point descriptors, with methods ranging from random forests \cite{randomforest} to metric learning \cite{meta}.
In more recent years, the field of deep-learning showed remarkable results for 3d point-descriptors as well \cite{pointnet,dgcnn,fmnet,3dcoded}, Providing state-of-the-art solutions for 3d registration.
As we will prove, local learnable descriptors are not sufficient as a holistic solution due to data-variations, noise, and sensitivity for complex deformations.
\subsubsection{Unsupervised and zero-shot dense shape correspondence}
Solving machine learning problems in an unsupervised fashion is marked as one of our community's most significant future goals.
For the dense shape correspondence problem, only a limited set of methods proposed such unsupervised learning mechanisms, among them are \cite{3dcoded,halimi,dvir,surfmnet}.
The first work \cite{3dcoded} used a template that closely resembles all shapes in the dataset and tried to map each vertex of a shape to a vertex in the template using an auto-encoder; at inference, the pair correspondence is $\pi^{-1}_{\mathcal{Y}}(\pi_{\mathcal{X}}(x_i))$
where $\pi_{\mathcal{Y}}$ is the mapping from shape $\mathcal{Y}$ to the template. This method requires a great amount of training data and fail on datasets where no "natural" template can be defined.
Other works such as Ginzburg \& Raviv \cite{dvir} optimize for a smooth cyclic mapper, where the objective of the bi-directional mapper is to map each vertex to itself after a full cyclic.
A groundbreaking subdomain in unsupervised learning is zero-shot learning, where a model can complete a task by only examining the test-sample itself \cite{zeroshot1,zeroshot2}.
One of the first methods to suggest such a framework for zero-shot alignment \cite{halimi} worked in the spectral domain and tried to minimize the distortion function on the inference pair directly using gradient descent on the point descriptors until convergence. Halimi \textit{et al.} \cite{halimi} and works that followed \cite{dvir} only worked with nearly-isometric shapes, heavily influenced by the shapes' topology and data's modality.
All mentioned works have opened the door to a new era of shape registration but still shows some gap from their supervised counterparts.
Here, we show how modeling the correspondence itself as an input to the learning pipeline allows the network to overcome problems that previous unsupervised methods encountered.
\section{Background}
This work is focused on alignment in-between non-rigid models, motivated by denoising concepts of graph neural networks.
Let us elaborate on each one of those elements.
\begin{figure*}[!th]
\centering
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=3.5cm]{Sections/Experiments/images/self_sup/refinment_faust/human_reference.png}
\label{fig:refine0}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=3.5cm]{Sections/Experiments/images/self_sup/refinment_faust/human0.png}
\label{fig:refine1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=3.5cm]{Sections/Experiments/images/self_sup/refinment_faust/human3.png}
\label{fig:refine3}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=3.5cm]{Sections/Experiments/images/self_sup/refinment_faust/human5.png}
\label{fig:refine5}
\end{subfigure}\hfil
\medskip
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=4.1cm]{Sections/Experiments/images/self_sup/refinment_smal/horseref.PNG}
\caption{Reference}
\label{fig:refine00}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=3.5cm]{Sections/Experiments/images/self_sup/refinment_smal/cat0.PNG}
\caption{Step 1}
\label{fig:refine11}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=3.7cm]{Sections/Experiments/images/self_sup/refinment_smal/cat3.PNG}
\caption{Step 3}
\label{fig:refine33}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=3.5cm]{Sections/Experiments/images/self_sup/refinment_smal/cat5.PNG}
\caption{Step 5}
\label{fig:refine55}
\end{subfigure}\hfil
\medskip
\caption{Refinement illustration - self supervised setting. Given two input shapes we train NaiveNet in a zero-shot manner, and present here the refinement steps, leftmost - correspondence output from NaiveNet, rightmost - correspondence after 5 refinement steps of DG2N.\\ Mean geodesic error over time FAUST - Naivenet 18.5, First iteration 13.7, Third iteration 8.8, Fifth iteration 3.7.\\
Mean geodesic error over time SMAL - Naivenet 14.3, First iteration 9.5, Third iteration 6.2, Fifth iteration 4.1.}
\label{fig:refinment}
\end{figure*}
\input{Sections/Background/gnns/GNNs}
\input{Sections/Background/3d_shape_corr/3dcorr}
\input{Sections/Background/graphdenoise/Graphdenosing}
\subsection{Graph neural networks}
While deep learning effectively captures hidden patterns in grid sampled data, we witness an increasing number of applications where the information is better represented in graphs or manifolds \cite{appnp,chen2015signal,dgcnn}. New challenges arise from a non-Euclidean structures due to the variable size of neighbors and unordered nodes.
Graph neural networks go back to 1997, working on acyclic graphs \cite{sperduti1997supervised}, but the notion of graph neural network was officially introduced by Gori \textit{et al.} in 2005 \cite{gori2005new}. Within the idea of graph neural networks, the most relevant to this work are convolutional graph neural networks, also known as ConvGNN. Under this umbrella, we can find two main streams; spectral and spatial. The first prominent research on spectral networks was presented by Bruna \textit{et al.} \cite{bruna2013spectral}. On the other hand, a spatial convolutional structure was addressed more than a decade ago by Michei \cite{micheli}, which has recently been resurfacing, showing its usefulness for multiple tasks in geometry and computer vision.
A variety of modern graph learning algorithms \cite{defferrard2017convolutional,gat,appnp,GCN,monti2016geometric,Fey/Lenssen/2019} replaced the traditional Euclidean convolution with a general concept of pulling that can be implemented on a graph.
Among popular modern graph neural network architectures for computer vision tasks, we can find PointNet \cite{pointnet}, its successor PointNet++ \cite{qi2017pointnet} and DGCNN ~\cite{dgcnn}, which provides useful tools to convolve over a set of points.
In this paper, the unit blocks we use are based on top of the graph convolution network \cite{GCN}, the vertices are points in space, and the features are alignment probabilities in between the source and the target.
\subsection{Graph denoising}
Denoising graph signals is a ubiquitous problem that plays an important role in many areas of machine learning \cite{appnp,graphdenapp1,graphdenapp2}, and was proven to improve results on a wide range of problems \cite{graphden1,graphden2}.
The two main approaches for analyzing graph signals are graph regularization-based optimization and graph dictionary design \cite{graphdict1,graphdict2};
The optimization approach applies a regularization term that promotes certain characteristics on the model, such as smoothness or sparsity \cite{chung1997spectral,graphden1}.
The optimization function itself usually takes the form of
$$
\argmin_{\bm{x}} ||t-x||_2^2 + \lambda Q(x)
$$
where $t$ is the noisy graph signal, and $Q$ is the regularization term.
When smoothness of the graph signal is assumed, one popular choice is the quadratic
form of the graph Laplacian, which captures the second-order difference of a graph signal \cite{moon2000mathematical}. For sparsity of the graph signals, a graph total variation term that captures the first-order difference of the graph signals was proven effective \cite{graphdenapp1,chen2015signal}. Recently, denoising graphs using deep architecture showed superior results by unrolling the $L_1$ regularization term into several layers, converging iteratively into the desired cost function \cite{chen2020graph}.
\subsection{Contributions}
We present three key contributions:
\begin{itemize}
\item Build a new architecture for self-supervised non-rigid alignment based on a residual pipeline that converges into a clean, soft mapping matrix for each pair of models (zero-shot).
\item Present a novel new concept for graph features derived from the target points' pull-back probabilities.
\item Report state of the art results in a wide range of benchmarks, including FAUST, TOSCA, SURREAL, SMAL, and SHAPENET.
\end{itemize}
\section{Experiments}
The following section presents multiple scenarios in which our self-supervised architecture surpasses current state-of-the-art algorithms for non-rigid alignment.
In addition, we will present our zero-shot pipeline that achieves near-perfect results for non-isometric deformable shape matching.
\begin{table*}[]
\begin{tabular}{l|cccccc}
Method\textbackslash{}textbackslash Dataset & FAUST \cite{faust} & SURREAL \cite{surreal} & F on S & S on F & SMAL \cite{smal} & SMAL on TOSCA \cite{tosca} \\ \hline
FMNet \cite{fmnet} & 12.1 & 18.7 & 35.3 & 33.4 & * & * \\
3D-CODED \cite{3dcoded} & 8.5 & 15.5 & 28.5 & 26.0 & 8.8 & 22.4 \\
Deep GeoFM \cite{deepfm} & 3.8 & 4.2 & 7.8 & 14.2 & * & * \\ \hline
SURFMNet(Unsup) \cite{surfmnet} & 7.1 & 11.3 & 31.5 & 42.3 & * & * \\
Unsup FMNet(Unsup) \cite{halimi} & 13.1 & 14.6 & 33.2 & 38.5 & * & * \\ \hline
NaiveNet(Unsup) & 12.2 & 11.8 & 20.3 & 21.7 & 11.2 & 26.5 \\
Ours(Unsup) on NaiveNet & 6.5 & 8.3 & 15.1 & 16.2 & \textbf{7.3} & \textbf{19.2} \\
Ours(Unsup) on Deep GeoFM & \textbf{3.4} & \textbf{4.1} & \textbf{6.2} & \textbf{8.1} & * & * \\ \hline
\end{tabular}
\caption{Mean geodesic error comparison by different methods on FAUST(F), SURREAL(S), SMAL and TOSCA datasets. No post processing filters are used for any of the methods. We remark that due to the numerical instabilities of the laplacian decomposition we were not able to run the spectral methods with the code published by the authors (results marked with *) on our re-sampled smal dataset.}
\label{tb:fausursmal}
\end{table*}
\input{Sections/Experiments/shape_corr/shapecorr}
\subsection{Animals datasets - SMAL and TOSCA}\label{subsec:animals}
To better understand the different models' generalization capabilities and ensure the models are not hand-crafted for human-like structures, we also assess the network's performance on animal datasets.
SMAL \cite{smal} dataset provides a generative model
for synthetic animals creation in different categories as cats, horses, etc.; SMAL is extracted from a continuous parametric space with a fixed number of vertices and same triangulation for all shapes, with the possibility of generating "infinitely many" training samples. Unlike SMAL, TOSCA \cite{tosca} contains a fixed selection of shapes, including 9 cats, 11 dogs, 3 wolves, etc. Which is both dramatically smaller and has no topological guarantees, meaning no two shapes have the same triangulation.
The animal's datasets experiment was conducted as follows: For each SMAL category, we create 80 shapes for training and 20 for the test, resulting in 500 samples. We must emphasize that previous methods \cite{3dcoded} that worked with SMAL used two orders of magnitude more training samples in their experiments.
Table \ref{tb:fausursmal} expresses the advantages of DG2N over previous works that are considered state-of-the-art in this regime. The tested spectral based methods (FMnet variant \cite{fmnet,halimi}) failed to converge on the remeshed datasets, probably due to the unstable and noise process of the decomposition of the Laplacians.
\subsection{Humans datasets - FAUST and SURREAL}\label{subsec:faust_scape}
We follow the suggested setting \cite{deepfm} for these human datasets and split both datasets into training sets (80 shapes) and test sets (20 shapes). The specific shape splits are identical for all tested methods for a fair comparison. We test two scenarios, one in which we train and evaluate on the same dataset and one in which we test on the other dataset (e.g., training on FAUST evaluating on SURREAL). This experiment aims
attesting the generalization power of all methods to small
re-meshed datasets, as well as its ability to adapt to a different dataset at test time.
Table \ref{tb:fausursmal} stresses some of the key advantages of DG2N compared to other self-supervision methods measuring robustness, and generalization.
In addition, using our proposed simple correlation NaiveNet we converge up to X10 faster than other methods that appear in the table; running time comparison appears in the supplementary. While almost all learnable methods perform reasonably well on the same-dataset benchmark, we see the tremendous margin compared to other methods when conducting the cross-dataset test; this is due to the fact we are self-supervised and shape-pair specific, thus are almost invariant to noise added to the system by changing the statistical attributes of the data. The fact we are self-supervised will be even more substantial in the zero-shot experiments
where most of the other methods fail.
\subsection{Mesh Error Evaluation}
The measure of error for the correspondence mapping between two shapes will be according to the Princeton benchmark \cite{geodesic_error_metric}, that is, given a mapping $\pi_{\to}(\mathcal{X, Y})$ and the ground truth $\pi^*_{\to}(\mathcal{X, Y})$, the error of the correspondence matrix is the sum of geodesic distances between the mappings for each point in the source figure, divided by the area of the target figure.
\begin{equation} \label{eq:geodesic_error}
\epsilon(\pi_{\to}) = \sum_{x \in \mathcal{X}} \frac{\mathcal{D_Y}(\pi_{\to}(x),\pi^*_{\to}(x))}{\sqrt{area(\mathcal{Y})}},
\end{equation}
where the approximation of $area(\mathcal{\bullet})$ for a triangular mesh is the sum of its triangles area.
\subsection{Deformable irregular correspondences - point clouds registration}\label{subsec:point_clouds}
Point cloud registration is undoubtedly one of the hardest registration tasks for 3D shapes, while it is the most common scenario in real-world cases. We evaluate the different methods of chosen classes from SHAPENET \cite{shapenet}, namely chairs, cars, and plains; Each category contains multiple subjects, where no pair is isometric, nor has the same number of points. Unlike meshes, point clouds suffer from noise and topology ambiguity due to the sampling process involved in generating them and the surface-approximation heuristics needed to define each point's neighborhood.
Spectral based methods undergo significantly
degradation in the results, since spectral decomposition of point clouds is inaccurate and unstable ~\cite{pclap1,pclap2}. The authors of SURFMNet \cite{surfmnet} did not evaluate on point-clouds nor offered the tools for such evaluation. For fairness, we used \cite{Sharp:2020:LNT} which is a new tool for Laplacian decomposition on point clouds.
No dataset currently exists with ground-truth correspondences between deformable point clouds, so we turn to evaluate the performance of the different methods visually, in terms of smoothness, coherence\footnote{A good alignment will map a guitar neck of one shape to the other.}, and robustness to deformations.
\begin{figure}[t]
\centering
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/cars/1_5/origcar1.PNG}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/cars/1_5/surfcar5.PNG}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/cars/1_5/pisi.PNG}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/cars/1_5/car5ours.PNG}
\label{fig:1}
\end{subfigure}\hfil
\medskip
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/planes/1_16/source.PNG}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/planes/1_16/pisi.PNG}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/planes/1_16/surf.PNG}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.5cm]{Sections/Experiments/images/pc/planes/1_16/ours.PNG}
\label{fig:1}
\end{subfigure}\hfil
\medskip
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.20cm]{Sections/Experiments/images/pc/chairs/0_2/source.PNG}
\caption{Source\\\ }
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.0cm]{Sections/Experiments/images/pc/chairs/0_2/surf.PNG}
\caption{SURFMNet\\*}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=1.0cm]{Sections/Experiments/images/pc/chairs/0_2/pisI.PNG}
\caption{Elementary\\ structures}
\label{fig:1}
\end{subfigure}\hfil
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[width=0.90cm]{Sections/Experiments/images/pc/chairs/0_2/ours.PNG}
\caption{Ours\\\ }
\label{fig:1}
\end{subfigure}\hfil
\medskip
\caption{Dense correspondence on point clouds - spectral solutions fail to create smooth or coherent results due to the noisy nature of the Laplacian for point clouds, Elementary structures \cite{deprelle2019learning} and other reconstruction based methods do not enforce smoothness resulting in noisy maps.}
\label{fig:pc_reg}
\end{figure}
\subsection*{Shape correspondence}\label{subsec:exp_shapecorr}
We evaluate DG2N on a wide range of popular datasets for dense shape correspondence. To assess the network's robustness, we test it on multiple datasets with different statistical and topological attributes as humans datasets (FAUST\cite{faust} and SURREAL \cite{surreal}), animals (SMAL \cite{smal} and TOSCA \cite{tosca}) or chairs and plains (SHAPENET\cite{shapenet}). We use a remeshed and down-sampled version of FAUST, SURREAL, and SMAL, as suggested by \cite{ren}, in the generated datasets, each shape has approximately 1000 vertices. These re-meshed datasets offer significantly more variability in terms of shape structures and connectivity than the original datasets \cite{deepfm}.
\input{Sections/Experiments/shape_corr/mesh_error_eval}
\input{Sections/Experiments/shape_corr/faust_scape}
\input{Sections/Experiments/shape_corr/animals}
\input{Sections/Experiments/shape_corr/point_clouds}
\input{Sections/Experiments/shape_corr/zero_shot}
\subsection{Zero-shot correspondence}\label{subsec:zeroshot}
"Zero-shot" self-supervised methods are essential and brought great achievements and new capabilities in other domains as super-resolution and image generation \cite{zeroshot2,zeroshot1}.
Having a zero-shot registration method for 3D shapes is considered exceptionally difficult, with only a few \cite{dvir,halimi} that tried to tackle the problem. Unfortunately, as seen in previous experiments \ref{subsec:animals} \ref{subsec:point_clouds}, spectral methods are sensitive and limited in terms of the input domain. To present our self-supervision capabilities, we chose randomly 10 inter-class shape pairs\footnote{Indexes appear in the supplementary} for the FAUST-remeshed dataset. For each pair, we trained NaiveNet only on the inference pair until convergence and ran the inference. Naturally, only unsupervised methods are relevant for comparison. We present a comparison to other zero-shot methods in Table \ref{tb:self_sup} and a numerical convergence of our refinement process in the supplementary.
\begin{table}[]
\begin{tabular}{l|c}
\multicolumn{1}{c|}{Method} & Mean geodesic error \\ \hline
SURFMNet(Unsup) \cite{surfmnet} & 36.2 \\
Unsup FMNet(Unsup)\cite{halimi} & 16.5 \\
Cyclic-FMnet(Unsup) \cite{dvir} & 14.1 \\ \hline
NaiveNet(Unsup) & 15.2 \\
Ours(Unsup) on NaiveNet & \textbf{8.5} \\ \hline
\end{tabular}
\caption{Mean geodesic error in a zero-shot setting on FAUST-remeshed. We present best results among all unsupervised methods that are relevant to this experiment setting.}
\label{tb:self_sup}
\end{table}
\section{Introduction}
The alignment of non-rigid shapes is a fundamental problem in computer vision. It plays an important role in multiple applications such as pose transfer \cite{rad2018feature}, cross-shape texture mapping \cite{zhang2019image}, 3D body scanning \cite{faust}, and simultaneous localization and mapping (SLAM) \cite{wolter2004shape}.
The task of finding dense correspondence is especially challenging for non-rigid shapes, as the number of variables needed to define the mapping is vast, and local deformations might occur.
To this end, a variety of solutions were offered to solve this problem, using axiomatic and learnable methods. From defining unique key-points or local descriptors and matching such descriptors between the shapes~\cite{hks,wks,gps,shot}, spectral-based methods that try to align the spectra of the shapes \cite{fmnet,surfmnet,halimi,dvir}, or template-based approaches that assume a known pre-defined structure closely resemble all shapes and find the correspondence from each shape to that template \cite{3dcoded}.
Many algorithms for non-rigid alignment relax the problem to matching probabilities. That allowed us to consider noise and variability in the pipeline. The transition from soft mapping to vertices alignment, or directly matching points, requires a post-processing step to remove outliers and smooth the results. Unfortunately, this is a slow process and is not performed in a network, as those algorithms are resource-demanding and require a large number of repetitions \cite{vestner2017product,kuhn1955hungarian}.
In this work, we focus on the refinement of a non-rigid alignment task. We unroll the refinement process into a multi-block graph neural network that performs the map denoising. To denoise the alignment's in a learnable manner, we construct a dual graph structure, one for the forward map and one for the backward map, where we claim that the features are the actual probabilities for mapping pulled back from the target.
In simple words, what best describes a point in the source, is not just its local features but how do all the points in the target resemble it. We call that structure a \emph{Dual Geometric Graph Network (DG2N)}.
We report state-of-the-art results on multiple benchmarks and succeed in providing a stable solution even under large non-isometric deformations.
\section{Summary}
\label{sec:summary_future_work}
We presented a novel new line of thought for aligning non-rigid domains using a learnable iterative pipeline. Motivated by graph denoising and presenting a dual graph structure built on top of soft correspondences, we rapidly converge into an accurate and free of outliers mapping even under severe non-isometric deformations. We report state-of-the-art results on multiple benchmarks and different scenarios, where other methods suffer poor outcomes or fail altogether.
\subsection{Dual Geometric Graph Network} \label{subsec:dg2n}
In the heart of the proposed architecture, is our understating that the soft correspondence matrix $\mathcal{P}$ induces a graph. The nodes of the \emph{primal} graph are the source points, and the features are the correspondence measure for all target points, i.e., the rows of $\mathcal{P}$ are the features.
The \emph{dual} graph has the same structure only based on $\mathcal{P}^T$. Here the nodes are the vertices of the target shape, and the columns of $\mathcal{P}$ are the features.
We provide a visualization of the architecture in
Figure \ref{fig:architecture_scheme}.
Each primal-dual pair ($\mathcal{P},\mathcal{P}^T$) is passed through $k$ layers of $DGAT$ in a res-net structure; i.e. the output of each $DGAT$ layer is $DGAT(\mathcal{P}) + \mathcal{P}$ for an input soft correspondence matrix $\mathcal{P}$, and similar for $\mathcal{P}^T$, the dual graph pipeline.
In each iteration, we fuse $\mathcal{P}$ and $\mathcal{P}^T$ into one aligned soft correspondence matrix\footnote{There are several reasonable options for the fusion, as element-wise max or mean. In practice, no consistent improvement was noted by one option over the other.}.\\ The output of each iteration of DG2N dual graph refinement is also a soft correspondence matrix. As the correspondence statistics vary between one iteration to the next, we use different weights per iteration. As we continue to iterate, the soft correspondence matrix improves and converges to a clean, outlier-free soft mapping. The number of iteration varies between datasets and depends on the quality of the initialization.
\subsection{Differentiable GAT}\label{subsec:DGAT}
\begin{figure}[H]
\includegraphics[width=\linewidth]{Sections/Method/dgat.png}
\caption{Single DGAT layer. Phase I: Create the \textit{difference guiding feature vector} $(\vec{h_i}|\vec{h_j}|\vec{h_i}-\vec{h_j})\in \mathbb{R}^{3M}$ and pass it through a first learnable architecture. Phase II: stack Phase I features $\tilde{H_i}\in\mathbb{R}^{M\times K}$ and regress each feature individually through the second module forming the output feature per point $\tilde{h_i}\in\mathbb{R}^{M}$.}
\label{fig:gdcarch}
\end{figure}
Our DG2N network is composed of two GNN units we note as Differential Graph Attention or DGAT in short. Inspired by GAT \cite{gat}, we perform weighted local pooling, only here we stack the output feature vector from the per-pair module and apply a per-feature network to learn the best refinement step.
The most generalized structure associated with GNNs is
$$
f_{i}^{'}=\gamma_{\Theta}(f_i,\mathord{\scalerel*{\Box}{gX}}_{j\in \mathcal{N}(i)}\phi_{\Theta}(f_i,f_j,e_{j,i}))
$$
where $f$ and $f^{'}$ represent the input an output data channels respectively, $\mathord{\scalerel*{\Box}{gX}}$ is some differentiable aggregation function and $\gamma_{\Theta},\phi_{\Theta}$ denote non-linear transmission functions.
In $DGAT$ we define $\mathord{\scalerel*{\Box}{gX}}$ to be stacking of the per-neighbor output feature vector, while the features $f_i$ are the guiding vectors described earlier, and the graph's edges determine the neighborhood. $\gamma_{\Theta}$, and $\phi_{\Theta}$ are variants of a multi-layer perception (MLP) with normalization and non-linear activation function layers.
In practice, DGAT takes the form
\begin{equation}\label{eq:GDC}
f_{i}^{'}=DNN_2(\underset{{j\in \mathcal{N}(i)}}{\mathbin\Vert}(DNN_1(f_i,f_j,f_i-f_j))
\end{equation}
An illustration can be found in Figure \ref{fig:gdcarch}.\\ \\
One crucial emphasis here is the rule of $DNN_2$ and the distinction to other suggested aggregation functions. One optional aggregation would be to concatenate the output difference features resulting in a feature vector of dimension $\tilde{H_i}\in\mathbb{R}^{KM}$, while we stack the features resulting in $\tilde{H_i}\in\mathbb{R}^{M\times K}$.
Our construction not only produces a learnable module with a factor of $M^2$ fewer parameters as $DNN2$ is applied on the $k$ dimensional vectors but, more significantly, acts as a learnable weighting function that incorporates the various per-neighbor refinements into a refinement step per-node.
$$
$$
\subsection{Graph denoising}\label{subsec:denoise}
We assume $\mathcal{P}$ and $\mathcal{P}^T$ are noisy or approximate solutions to the dense shape correspondence and present a method of deriving the underlying mapping.
The maximum likelihood solution for the mapping from $\mathcal{P}$ is $$MLE(\pi(x_i))=\argmax_{j^*}(P_{i,\star}),$$ Meaning that any added noise s.t $p_{ij}+e_j>p_{ij^*}+e_{j^*}$ where $y_{j^*}$ is the true corresponding point, yield a wrong solution, regardless to a proximity or relation between $y_j$ and $y_{j^*}$.
We notate $\mathcal{P}$ as the soft correspondence mapping at the initial state, while $\mathcal{P}^*$ is the refined soft alignment we wish to extract at the end of the denoising phase. We assume there is a noise matrix $E\in \mathbb{R}^{N\times M}$ of the graph signals such that
\begin{equation}
\mathcal{P} = \mathcal{P}^* + E
\end{equation}
Without prior information on $\mathcal{P}^*$, it is impossible to restore the underlying signals. Fortunately, the most known priors for the graph denoising problem fit our situation; those priors are feature space sparsity and node pair-wise smoothness.\\
We follow Chen \textit{et al.} \cite{chen2020graph} notation of graph filters for graph denosing:
\begin{equation}\label{eq:graphobj}
\mathcal{P}^{*} = \mathbb{H} * \bm{s} = \sum_{l=1}^{L}(\mathbb{H}_lA^ls)
\end{equation}
Where $\bm{s}$ is the initial graph signal, $*$ is the graph convolution operator, and $\mathbb{H}_{l}$ are $L$ layers of Graph filter coefficients.
For stability reasons one might decide to use the residual objective as in our case:
\begin{equation}\label{eq:graphobj}
\mathcal{P}^{*} = \bm{s} + \mathbb{H} * \bm{s} = \bm{s} + \sum_{l=1}^{L}(\mathbb{H}_lA^ls)
\end{equation}
Meaning we try to infer $-E$ explicitly and subtract it from the initial noisy graph input.
In our work $\mathbb{H}$ will be the $GDC$ convolution layer \ref{subsec:GDC}, and $\bm{s}\coloneqq\mathcal{P}$.\\
Solving \ref{eq:graphobj} requires formulating the objective as an optimization function. Using our priors on $\mathcal{P}^{*}$ we have:
\begin{equation}\label{eq:graphlossv1}
\begin{aligned}
\mathcal{L}_L = {\mathcal{P}^{*}}^T&L\mathcal{P}^{*} \quad
\mathcal{L}_{l1} = ||\mathcal{P}^{*}\mathbbm{1}||_1 \\
\mathcal{P}^{*} = \min_{\mathcal{P}^{*}} ||&\mathcal{P}^{i}-\mathcal{P}^{*}||_2^2 + \alpha\mathcal{L}_L + \beta \mathcal{L}_{l1} \\
s.t \quad &\mathcal{P}^{*} = \mathcal{P} + \mathbb{H} * \mathcal{P}.
\end{aligned}
\end{equation}
The first term enforces $\mathcal{P}^{*}$ not to diverge from the source signal, while terms 2,3 act as the graph denoisers. $L$ is the graph Laplacian matrix $L=D-A\in\mathbb{R}^{N\times N}$ with diagonal degree matrix $D_{i,i}=|N(i)|$ and $A$ as the adjacency matrix.
\subsection{Losses}\label{subsec:losses}
We combine four different losses in this pipeline. $\mathcal{L}_{L}$ Laplacian loss, $\mathcal{L}_{l1}$ Sparsity loss, $\mathcal{L}_{AG}$ Anchors guidance loss
and $\mathcal{L}_{l2}$ Denoising regularization.
These constraints form together the loss objective of a single refinement step of DG2N, which is:
\begin{equation}\label{eq:graphlosswithssar}
\begin{aligned}
\mathcal{L}=\mathcal{L}_{L} + \mathcal{L}_{l1} + \mathcal{L}_{AG} + \mathcal{L}_{l2}
\end{aligned}
\end{equation}
All four losses are evaluated separately and summed together both for the primal and dual graphs and executed for every iteration output.
Let us elaborate on each term in the loss.
\subsubsection{Laplacian loss}
Laplacian regularization term pushes toward graph smoothness.
It takes the form of:
$$\mathcal{L}_L= \lambda_{L} \mathcal{P}^T L_\mathcal{P} \mathcal{P} = \lambda_{L} \sum_{(i,j) \in E_{\mathcal{X}}}w_{i,j}||\mathcal{P}_{i,\star}-\mathcal{P}_{j,\star}||_2^2
$$
where $L_\mathcal{P}=D-A$ is the graph Laplacian of the source shape $\mathcal{X}$, $D$ is the degree of each node, $A$ is its adjacency matrix, and $\mathcal{P}_{i,\star}$ is the $i$'th row of $\mathcal{P}$.
Two important items to note here, the first is that this term can be used on any structure inducing a graph. Second, while all other methods use Laplacians on the coordinates mapping in space, we claim that smoothness should apply directly to the soft correspondence matrix. Since the features are the mapping probabilities, we claim the smoothness on $\mathcal{P}$ is a better goal.
\subsubsection{Sparsity regularization}
We add the $\mathcal{L}_{1}$ regularization on the rows of $\mathcal{P}$. Specifically,
\begin{equation}
\mathcal{L}_{1} = \lambda_{l1} \sum_{i=1}^N|\mathcal{P}_{i,\star}|_1.
\end{equation}
As the rows of $\mathcal{P}$ represent the alignment probabilities, we wish to promote sparsity. Each source point corresponds to a single target point, meaning one element should hold most of the energy, and the rest should decline rapidly.
Note that we are not normalizing the rows or columns in each iteration.
\subsubsection{Anchors guidance loss}\label{subsec:anchor_guidence}
One of the caveats with the Laplacian regularization is its tendency for over-smoothing and thus hurt the overall performance \cite{wu2019simplifying}.
In our case, this phenomenon takes the shape of pushing all correspondence probabilities of $\mathcal{P}$ towards the average. While this decreases the Laplacian loss, it results in significant degradation of the results, as shown in the supplementary.
To solve the mentioned problem, we present a self-supervised anchor guidance mechanism.
Motivated by node classification tasks \cite{giles1998citeseer,sen2008collective},
a few anchor points are sampled from the initial soft correspondence map, and we seek to use their initial mapping as guidance through the refinement.
Bear in mind that those points are not fixed and are part of the learnable pipeline, only we provide extra attention to points we believe in their mapping.
Analyzing soft correspondence mappings generated from different pipelines (FMnet \cite{fmnet}, SURFMNet \cite{surfmnet}, NaiveNet), we observed two attributes that reoccur by all algorithms:
\begin{enumerate}
\item Source nodes where the highest correspondence probability of $\mathcal{P}$ is two orders of magnitude larger than the average probability ($\frac{1}{M}$) usually point to the true correspondence.
\item High probability correspondences reside in clusters, that is, if a source node corresponds to some target node with high probability, it is usually the case its neighbors will also have high probability correspondence to some neighbor of this corresponding point.
\end{enumerate}
We utilize the above observations to attend the over-smoothing caused by the Laplacian.
For each $\mathcal{X},\mathcal{P},\mathcal{Y}$ we first sample $k\leq|V_{\mathcal{X}}|$ disconnected nodes using FPS \cite{FPS}
noted as $V_{K_{\mathcal{X}}}$, and assign their soft label by defining
\begin{equation}\label{eq:pernodeSSAG}
\begin{aligned}
\hat{y}_{i}=\argmax_j\mathcal{P}&_{i,*} \quad \forall v_i \in V_{K_{\mathcal{X}}} \\
C(\hat{y}_{i}) = \mathcal{P}&_{i,\hat{y}_{i}}
\end{aligned}
\end{equation}
where $C(\cdot)$ is the confidence $x_i$ corresponds to $\hat{y}_{i}$.
Using the above formulation, we define a soft-classification problem. We constrain the network to label the anchor points similarly to how they were classified before the dual iterative layer. The penalty for each wrong classification is directly proportional to the confidence $C(\cdot)$.
We use the anchors' notations and define the anchor loss as the cross-entropy between the presumed label, and the output features of DG2N Layer. Specifically,
\begin{align}
\mathcal{L}_{AG}=\sum_{x_{i}\in V_{K_\mathcal{X}}} &C(\hat{y}_{i})^2 \\ \nonumber &\Big(-x_i[\hat{y}_{i}] + \log\big(\sum_{j=0}^{|V_{K_\mathcal{X}|-1}}\exp{(x_i[j])}\big)\Big), \label{eq:AG_los}
\end{align}
where $x_i[m]$ is the $m$'th element in the row of $\mathcal{P}$ corresponding to vertex $x_i$.
\subsubsection{Denoising regularization}
For each iteration, we assume the output is similar to the input. By that we force the network to penalize for large gaps, and de-facto promote minor updates, usually referred to as noise or outliers in this paper. That is a standard approach to perform denoising.
Denoting the previous layer matrix as $\mathcal{P}$ and the output of a single DG2N iteration as $\mathcal{P}^*$ we have
\begin{equation}\label{loss:l2}
\mathcal{L}_{l2}=\lambda_{l2} ||\mathcal{P}^{*}-\mathcal{P}||_2^2.
\end{equation}
\section{Method}
\begin{figure*}[t!]
\centerin
\includegraphics[width=\linewidth]{Sections/Implementation/DualGraphGeometricNetwork.png}
\caption{Our dual graph geometric network. Given two input graphs we first pass them through an initiator to have the initial soft correspondence matrix $\mathcal{P}$, representing correspondence probabilities between all vertex-pairs in the mapping. We then pass the graphs induced by $\mathcal{P}$ and $\mathcal{P}^T$ through stacked layers of $DGAT$ with residual connections, where the output is a refined soft correspondence matrix. The 4 loss objectives \ref{subsec:losses} allows iterative refinement over
$\mathcal{P}$, where the input $\mathcal{P}$ for the next iteration is the output of the previous one.}
\label{fig:architecture_scheme}
\end{figure*}
The proposed method is a self-supervised zero-shot pipeline. To align two non-rigid models, we first generate a naive all-to-all soft mapping based on local deep features \ref{subsec:softcorr} and iteratively learn how to update the soft correspondence matrix improving the results and removing the outliers \ref{subsec:dg2n}.
Our cost function is based on the understanding that each point's best features are the inverse probabilities for a match measured by each point in the target domain.
Specifically, if $P$ is a soft correspondence matrix, i.e., if $P \in \mathbb{R}^{N \times M}$, and $P_{ij} \in \mathbb{R}$ is the probability that point $i$ matches point $j$, then in the primal graph, the features of point $i$ is the $i$'th row of P, and the features of the dual graph are represented by the columns (or rows of $P^T$). We denote this primal-dual structure as the Dual Graph Geometric Network (DG2N).
The initial soft correspondence matrix can come from any method, such as functional maps based architectures \cite{fmnet,dvir}, which we call "the initiator". In the scenarios where there is no relevant initiator (e.g. as in the cases of point-clouds ~\ref{subsec:point_clouds}), we use graph convolutions to aggregate features from neighboring points, and use an all-to-all correlation between those deep-features using cosine-similarity as presented in detail in ~\ref{subsec:softcorr}. This is an extremely weak learner that produces outliers and inaccurate alignments, but it is sufficient to train the proposed DG2N architecture and converge to a very good mapping.
DG2N is composed of our new graph attention mechanism, activated on the two graphs (primal and dual) simultaneously. We refer to the new convolution blocks by differential-GAT, or DGAT. Inspired by \cite{gat}, we consider a pulling strategy in-between points and their neighbors, where we concatenate the differences between the node features for a fixed number of neighbors.
To keep improving the outcome and not collapsing during the denoising process, we present four cost functions. We require the alignment to be injective, smooth, not too far from the previous iteration, and to keep the most valuate points in place.
In what follows, we elaborate on the main three components of the architecture's pipeline and the four cost functions \ref{subsec:losses}.
\input{Sections/Method/soft_cor}
\input{Sections/Method/GDC}
\input{Sections/Method/DG2N}
\input{Sections/Method/losses}
\subsection{All-to-All mapping}\label{subsec:softcorr}
To achieve a coherent and smooth correspondence map between two shapes, our dual graph unit (DG2N) uses the soft correspondence mapping $\mathcal{P}$ as an input.
We can use any known method which has a soft correspondence matrix in the pipeline as an initiator, for example \cite{dvir,fmnet,halimi}.\\
In detail, \cite{fmnet} showed that using the functional mapping $\mathcal{C}$, with the graphs laplacian eigendecomposition of the shapes $\Phi,\Psi$ the soft correspondence is constructed by
\begin{equation}\label{eq:PfromFM}
\mathcal{P} \propto |\Psi \mathcal{C} \Phi^T|.
\end{equation}
In the scenarios where spectral methods are unstable or fail to create reasonable results (as in experiments \ref{subsec:animals},~\ref{subsec:point_clouds}) we show here that an elementary all-to-all correlation matrix can be constructed out of several convolutions and cosine product. We found that to be good enough as an initiator for the refinement.
Specifically, we have built on top of DGCNN \cite{dgcnn} pipeline and used the last hidden layer as a point descriptor.
See Figure \ref{fig:init_corr_arch} for visual aid.
The soft correspondence is constructed by the cosine similarity between the descriptors:
$$
\mathcal{P}_{i,j}=\frac{ h_{x_i} \cdot h_{x_j}} {|| h_{x_i} ||_2 \cdot ||h_{x_j}||_2},
$$
where $h_{x_i}$ and $h_{x_j}$ represent two feature vectors of points $x_i$ and $x_j$.
In order to train this naive correspondence pipeline we generate spatial deformations for the same model during training. The advantage of the given pipeline is that the true correspondence is known and is $T(i)=i$, yielding the unsupervised loss:
\begin{equation}\label{eq:naivenetloss}
\mathcal{L}=||P-I||_\mathcal{F}^2.
\end{equation}
In Section ~\ref{subsec:exp_shapecorr} we show such a simple solution finds a noisy correspondence between non-isometric pairs but provides sufficient initialization for our architecture.
\begin{figure}[hbt!]
\centerin
\includegraphics[width=\linewidth]{Sections/Implementation/dgcnnshapecorr.png}
\caption{Self supervised baseline for shape correspondence. Each source shape (up) is being augmented (down) by a linear transformation, than passed through a backbone layer to generate high dimensional descriptors per-point. Using cosine-similarity we evaluate the distance between the points, where the underlying correspondence is known (the identity mapping).}
\label{fig:init_corr_arch}
\end{figure}
\subsection{Ablation}
DG2N training is constructed of 4 different loss functions, each plays an important and substantial rule in the refinement process.
We provide table ~\ref{tab:Ablation} as numerical evidence to the importance of each loss, as well as the effect of disabling combinations of loss functions. The ablation was done on FAUST resampled from the main transcript, with the initial correspondence generated by NaiveNet.
\begin{table}
\begin{tabular}{cc}
Loss & MGE \\ \hline
\multicolumn{1}{c|}{NaiveNet} & 12.2 \\
\multicolumn{1}{c|}{All} & \textbf{6.5} \\
\multicolumn{1}{c|}{$\mathcal{L}_L+\mathcal{L}_{l2}+\mathcal{L}_{AG}$} & 8.5 \\
\multicolumn{1}{c|}{$\mathcal{L}_L+\mathcal{L}_{l1}+\mathcal{L}_{l2}$} & 9.3 \\
$\mathcal{L}_{l2}+\mathcal{L}_{AG}$ & 12.2 \\
$\mathcal{L}_L+\mathcal{L}_{l1}+\mathcal{L}_{AG}$ & 42.3 \\
$\mathcal{L}_L+\mathcal{L}_{l1}$ & 55.3
\end{tabular}
\caption{Ablation study evaluating the significance of the different cost functions. While some objectives improve the refinement effect, some, as $\mathcal{L}_{AG}$ or $\mathcal{L}_{l2}$ are indispensable, with substantial degradation to the results without their regularization effect to the denoising process.}
\label{tab:Ablation}
\end{table}
In addition to the numerical evaluation of the importance of the different objectives, we include a visual example of the effect of disabling the anchor guidance in figure {\color{red}8}.
\begin{center}
\begin{table*}
\begin{tabular}{cccc}
\noalign{\smallskip}
SMAL &
\adjustimage{width=.30\textwidth,valign=c}{Sections/supplementary/images/naivenet0.PNG}&
\adjustimage{width=.30\textwidth,valign=c}{Sections/supplementary/images/naivenet6.PNG}&
\adjustimage{width=.30\textwidth,valign=c}{Sections/supplementary/images/naivenet5.PNG}\\
\noalign{\smallskip}
FAUST &
\adjustimage{width=.30\textwidth,valign=c}{Sections/supplementary/images/naivenet1.PNG}&
\adjustimage{width=.30\textwidth,valign=c}{Sections/supplementary/images/naivenet2.PNG}&
\adjustimage{width=.30\textwidth,valign=c}{Sections/supplementary/images/naivenet3.PNG}\\
\end{tabular}
\caption*{\textbf{Figure 7}: Naivenet correspondence examples, the network was only trained on same-shape different-augmentations pairs, resulting with reasonable correspondences even for non-isometric shapes.} \label{fig:naivenet_corr}
\end{table*}
\end{center}
\begin{center}
\begin{table*}[btp]
\begin{tabular}{ccccccc}
\noalign{\smallskip}
No $\mathcal{L}_{AG}$ &
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/reference.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/1noag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/2noag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/3noag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/4noag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/5noag.PNG}\\
\noalign{\smallskip}
With $\mathcal{L}_{AG}$ &
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/reference.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/1ag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/2ag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/3ag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/4ag.PNG}&
\adjustimage{width=.10\textwidth,valign=c}{Sections/supplementary/images/ag/5ag.PNG}\\
\noalign{\smallskip}
\ &
NaiveNet&
Step 1&
Step 2&
Step 3&
Step 4&
Step 5\\
\end{tabular}
\caption*{\textbf{Figure 8}: Correspondence refinement without and with $\mathcal{L}_{AG}$. Without the anchor guidance mechanism the iterative refinment "forgets" the initial correspondences that had high probability by the initiator, resulting in degregation in the results within several steps of the refinment.}
\label{fig:anchor_guidence}
\end{table*}
\end{center}
\subsubsection{DGAT and DG2N}
In tables \ref{tab:dgat},\ref{tab:dg2n} we specify the different layers and the structure of DGAT, as well as the full pipeline of DG2N. We build DGAT atop \cite{Fey/Lenssen/2019}, a PyTorch framework designed for geometric NN's.
We perform 4-10 consecutive refinement steps of $DG2N$, where each input $\mathcal{P}$ is the output of the previous refinement step.
\begin{table}[!h]
\begin{tabular}{clll}
Step & \multicolumn{1}{c}{Layer} & \multicolumn{1}{c}{Input size} & \multicolumn{1}{c}{Output size} \\ \hline
\multicolumn{1}{c|}{1} & Neighbor pooling & BxNxM & BxNxKxM \\
\multicolumn{1}{c|}{2} & Difference vector & BxNxKxM & BxNxKx3M \\
\multicolumn{1}{c|}{3} & MLP+LN+LR & BxNxKx3M & BxNxKxM \\
\multicolumn{1}{c|}{4} & MLP+LN+LR+MLP & BxNxKxM & BxNxM
\end{tabular}
\caption{DGAT block, N,M are in number of vertices in the source,target shapes respectively. LN and LR stands for LayerNorm \cite{layernorm}
and LeakyRelu respectively}\label{tab:dgat}
\end{table}
\begin{table}[!h]
\begin{tabular}{cccl}
\multicolumn{1}{l}{Step} & Layer & \multicolumn{1}{l}{Topology} & Output \\ \hline
\multicolumn{1}{c|}{1.source} & DGAT block x 3 & Source & $\mathcal{P}\in \mathbb{R}^{N\times M}$ \\
\multicolumn{1}{c|}{1.target} & DGAT block x 3 & Target & $\mathcal{P}_t\in \mathbb{R}^{M\times N}$ \\
\multicolumn{1}{c|}{2} & $P_t = P_t^T$ & - & $\mathcal{P}_t\in \mathbb{R}^{N\times M}$ \\
\multicolumn{1}{c|}{3} & $\tilde{P} = \frac{P_s + P_t}{2}$ & - & $\mathcal{P}\in \mathbb{R}^{N\times M}$
\end{tabular}
\caption{DG2N block, we apply 3 DGAT layers on each soft correspondence mapping ($\mathcal{P},\mathcal{P}^T$) followed by a fusion of the refined statistics.}
\label{tab:dg2n}
\end{table}
\subsection{Architecture details}
\input{Sections/supplementary/naivenet}
\input{Sections/supplementary/dgat}
\subsubsection{NaiveNet}
In tables \ref{tab:dgcnnblock}, \ref{tab:NaiveNet_architecture} we provide specific network configurations to construct NaiveNet. Each input shape $\mathcal{X}$ is being transformed by random augmentations woth the parameters depicted in table \ref{tab:NaiveNet_augmentations} resulting in $\grave{\mathcal{X}}$. Both shapes enter into a Siamese structure of NaiveNet resulting in deep features $h_{\mathcal{X}}\in \mathbb{R}^{N\times F},h_{\grave{\mathcal{X}}}\in \mathbb{R}^{N\times F}$. Using the deep features we compute $\mathcal{P}$ according the cosine-similarity between the features and evaluate the self-supervised loss of NaiveNet \ref{eq:naivenetloss}.
\begin{table}[h]
\begin{tabular}{clll}
Step & \multicolumn{1}{c}{Layer} & \multicolumn{1}{c}{Input size} & \multicolumn{1}{c}{Output size} \\ \hline
\multicolumn{1}{c|}{1} & Neighbor pooling & BxNxI & BxNxKxI \\
\multicolumn{1}{c|}{2} & Conv 2D 1x1 & BxNxKxI & BxNxKxO \\
\multicolumn{1}{c|}{3} & BatchNorm 2D & BxNxKxO & BxNxKxO \\
\multicolumn{1}{c|}{4} & LeakyRelu & BxNxKxO & BxNxKxO \\
\multicolumn{1}{c|}{5} & NeighborMax & BxNxKxO & BxNxKxO
\end{tabular}
\caption{DGCNN block, I,O are the Input/Output feature vectors dimensions.}
\label{tab:dgcnnblock}
\end{table}
\begin{table}[h]
\begin{tabular}{cccc}
\multicolumn{1}{l}{Step} & Layer & \multicolumn{1}{l}{Input size} & \multicolumn{1}{l}{Output size} \\ \hline
\multicolumn{1}{c|}{1} & DGCNN block & 3 & 64 \\
\multicolumn{1}{c|}{2} & DGCNN block & 64 & 128 \\
\multicolumn{1}{c|}{3} & DGCNN block & 128 & 256 \\
\multicolumn{1}{c|}{4} & DGCNN block & 256 & 512 \\
\multicolumn{1}{c|}{5} & Concatenate (1,2,3,4) & * & 960 \\
\multicolumn{1}{c|}{6} & Linear layer & 960 & 512 \\
\multicolumn{1}{c|}{7} & BatchNorm1D & 512 & 512
\end{tabular}
\caption{NaiveNet architecture parameters.}
\label{tab:NaiveNet_architecture}
\end{table}
\begin{table}[h]
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccc}
Augmentation & Value & \# Parameters per shape \\ \hline
Rotation X,Y,Z (degrees) & -180 - 180 & 3 (per axis) \\
Scale (multiplicative) & 0.2-5 & 1 (per shape) \\
Random gaussian noise X,Y,Z (additive) & 0.01* area(shape) & Nx3 (x,y,z per point)
\end{tabular}%
}
\caption{NaiveNet augmentations. We treat all datasets with the same augmentation parameters.}
\label{tab:NaiveNet_augmentations}
\end{table}
Figure {\color{red}7} provides a visualizations of correspondences generated by NaiveNet.
NaiveNet is a good choice as an initiator due to its
modality robustness (meshes or
points clouds) and stable outcomes on a variety of datasets. Equally important is its superior convergence time compared to all other initiators. See numerical support in table \ref{tab:convergence_times}.
\begin{table}[]
\begin{tabular}{cc}
Architecture & Convergence time (Hours) \\ \hline
\multicolumn{1}{c|}{\textbf{NaiveNet}} & \textbf{0.65} \\
\multicolumn{1}{c|}{FMnet} & 1.92 \\
\multicolumn{1}{c|}{SURFMnet} & 1.93 \\
\multicolumn{1}{c|}{Cyclic FM} & 2.11 \\
\multicolumn{1}{c|}{Unsup FM} & 2.53 \\
\multicolumn{1}{c|}{GeoFM} & 1.66 \\
\multicolumn{1}{c|}{3Dcoded} & 4.32
\end{tabular}
\caption{Convergence times for different initiators on the FAUST-resampled dataset. NaiveNet is a robust and stable initiator (fair correspondence results as an initial soft correspondence map before DG2N denosing operation), while converging up to 10X times faster than other correspondence methods.}
\label{tab:convergence_times}
\end{table}
\subsection{Results}
\begin{figure}[!h]
\includegraphics[width=1.\linewidth]{Sections/Experiments/images/graphs/cum_error_Zero_shot_on_Scap_remeshed.pdf}
\caption{Geodesic error on FAUST-remeshed in a self-supervised setting. This figure presents a quantitative evaluation of our network's ability to refine the results in this setting. Both the backbone and DG2N witnessed only augmented versions of the test-pair at training time.}
\label{fig:self_sup_curve}
\end{figure}
\section{Supplementary}
\input{Sections/supplementary/implementation}
\input{Sections/supplementary/ablation}
|
2011.14651
|
\section{Introduction}
Recent growth of the quantum volume in noisy intermediate-scale quantum (NISQ) devices has stimulated rapid development in circuit-based quantum algorithms.
In particular, quantum machine learning (QML)~\cite{schuld2018supervised,biamonte2017quantum,dunjko2018machine} using variational quantum circuits (VQC) shows great promise in surpassing the performance of classical machine learning (ML).
A VQC is a quantum circuit with adjustable parameters that are optimized according to a predefined metric, such as an objective function.
One of the major advantages of QML compared to its classical counterpart is the drastic reduction in the number of required parameters, potentially mitigating the problem of overfitting common in ML.
A QML architecture in modern setting typically includes a classical part and a quantum part.
Prominent examples in this hybrid genre include quantum approximate optimization algorithm (QAOA)~\cite{farhi2014quantum}, and quantum circuit learning (QCL)~\cite{mitarai2018quantum} where the VQC plays an important role as an quantum component.
Various architectures and geometries of VQC have been suggested for tasks ranging from binary classification to reinforcement learning.
One major issue in QML is how to encode classical data, typically presented in the form of high-dimensional vectors or arrays, efficiently into a quantum circuit with limited number of gate operations.
The deep circuit depth required in either the basis or amplitude encoding makes them less desirable for the NISQ devices.
Straightforward approaches such as single qubit rotations promises a shallow circuit, but suffers from the lack of representation power.
This can be mitigated by preprocessing the input data with classical means to perform dimension reduction.
Principal component analysis (PCA) is a simple dimension reduction method and widely used in the QML research.
More advanced methods using neural networks, though more powerful, are less commonly utilized due to the requirement of pre-training and the significant number of parameters involved.
Therefore, it is necessary to devise a data compression scheme which can be naturally integrated with VQC.
In this work, we propose a hybrid framework where a tensor network (TN)~\cite{Orus:2014um}, in particular a matrix product state (MPS)~\cite{Ostlund:1995iz,Schollwock:2011kt}, is used as a feature extractor to produce a low dimensional feature vector, which is subsequently fed into a VQC for classification.
Unlike other QML schemes where the classical neural network has to be pre-trained, our framework is trained end-to-end, i.e., the MPS-VQC is trained as a whole.
This end-to-end training indicates the quantum-classical boundary can be moved based on the available quantum resource at the training stage.
Furthermore, since the MPS can always be realized precisely by a quantum circuit~\cite{Huggins:2019kh}, the scheme is highly adaptable and can be easily modified when more quantum resource is provided.
The main contributions of this paper are
\begin{itemize}
\item We propose a hybrid quantum-classical model based on TN and VQC which allows for an end-to-end training.
\item We perform a binary classification task of the MNIST dataset and show the MPS-VQC scheme is superior than a PCA-VQC scheme even at very low bond dimensions.
\item We show that the VQC serves as regularization for the MPS to avoid over-fitting.
\end{itemize}
\section{Methods}
\subsection{Tensor Network}
Tensor networks are efficient representation of data residing in high-dimensional space.
Originally developed to simulate quantum many-body systems, recently TN has been applied to solve problems in classical ML~\cite{Cohen:2016mi,Stoudenmire:2016ve} and showed encouraging success in both discriminative~\cite{Levine:2018qp,Stoudenmire:2018wk,Liu:2019ty,Reyes:2020fd} and generative learning tasks~\cite{Han:2018rt}.
It is common to use graphical notation to express tensor networks.
A tensor is represented as a closed shape, typically a circle, with emanating lines representing tensor indices (Fig.~\ref{TN}).
The joined line indicates the corresponding index is contracted, as in the Einstein convention where repeated indices are summed over.
The simplest TN is a MPS also known as tensor train, where tensors are contracted through the ``virtual'' indices ($\alpha$'s in Fig.~\ref{TN}(d)).
The dimension of these virtual indices are called bond dimension and is indicated by $\chi$.
In the MPS representation of a quantum wave function, the bond dimension indicates the amount of quantum entanglement the MPS can represent in the bond.
In the context of ML, this corresponds to the representation power of the MPS.
The connection between the quantum entanglement and deep learning architectures has been first explored in Ref.~\cite{Levine:2018qp,Levine:2019xt}.
In the current study, we choose the MPS as our TN for simplicity; there are other examples of TN with distinct entanglement structures such as the tree tensor network (TTN), multi-scale entanglement renormalization ansatz (MERA) and projected entangled pair state (PEPS).
The successful application of a specific TN can also give insights into the hidden correlations in the data.
The quantumness inherent in the TN gives it great advantage over other architectures in the application of QML.
In particular, since each TN can be mapped to a quantum circuit, it means that although in the current scheme, the TN is treated classically, it is possible to replace the whole or part of the TN component by an equivalent quantum circuit when more qubits are available.
This gives the current scheme the flexibility to move the quantum-classical boundary based on the available resources.
\subsection{Variational Quantum Circuit}
Variational quantum circuits are quantum circuits that have {adjustable} parameters subject to classical iterative optimizations.
The term {variational} means that certain parts of the circuit can be updated according to some predefined metric, the so-called \emph{loss}.
We describe the general structure of a VQC in \figureautorefname{\ref{Fig:GeneralVQC}}.
The $U(\mathbf{x})$ represents the data encoding block which is predefined and is not optimized.
The encoding method should be designed with respect to the problem of interest and is a crucial part in the overall architecture.
The $\Phi(\boldsymbol{\theta})$ represents the variational block which is the \emph{learnable} part and will be optimized, usually with the gradient-based methods.
These circuit parameters are similar to the \emph{weights} in the classical neural networks.
It has been shown that such circuits are potentially resilient to quantum noises~\cite{kandala2017hardware,farhi2014quantum,mcclean2016theory} and therefore are suitable for building applications on NISQ devices.
Several results have also shown that VQCs are more expressive than conventional neural networks~\cite{sim2019expressibility,lanting2014entanglement,du2018expressive} with respect to the number of parameters.
Architectures based on VQCs have successfully demonstrated its capability in function approximation~\cite{mitarai2018quantum}, classification~\cite{schuld2018circuit,havlivcek2019supervised,Farhi2018ClassificationProcessors,benedetti2019parameterized}, generative modeling~\cite{dallaire2018quantum}, deep reinforcement learning~\cite{chen19} and transfer learning~\cite{mari2019transfer}.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.9\textwidth]{figure/TN.pdf}
\caption[Graphical Notation for Tensors and Tensor Networks]{{\bfseries Graphical Notation for Tensors and Tensor Networks.} (a) Graphical tensor notation for (a) a vector, (b) a matrix, (c) a rank-3 tensor and (d) a MPS. Here we follow the Einstein convention that repeated indices, represented by internal lines in the diagram, are summed over.
}
\label{TN}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\begin{minipage}{10cm}
\Qcircuit @C=1em @R=1em {
\lstick{\ket{0}} & \multigate{3}{U(\mathbf{x})} & \qw & \multigate{3}{\Phi(\boldsymbol{\theta})} & \qw & \meter \qw \\
\lstick{\ket{0}} & \ghost{U(\mathbf{x})} & \qw & \ghost{\Phi(\boldsymbol{\theta})} & \qw & \meter \qw \\
\lstick{\ket{0}} & \ghost{U(\mathbf{x})} & \qw & \ghost{\Phi(\boldsymbol{\theta})} & \qw & \meter \qw \\
\lstick{\ket{0}} & \ghost{U(\mathbf{x})} & \qw & \ghost{\Phi(\boldsymbol{\theta})} & \qw & \meter \qw \\
}
\end{minipage}
\end{center}
\caption[Generic circuit architecture for the variational quantum classifier.]{{\bfseries Generic circuit architecture for the variational quantum classifier.}
Here $U(\mathbf{x})$ is the quantum routine for encoding classical data and $\Phi(\boldsymbol{\theta})$ is the variational circuit block with the adjustable parameters $\boldsymbol{\theta}$.
}
\label{Fig:GeneralVQC}
\end{figure}
\subsection{Hybrid TN-VQC model}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.85\textwidth]{figure/OverallArchitecture.pdf}
\caption[Hybrid TN-VQC framework.]{{\bfseries Hybrid TN-VQC model.}
The TN part is an MPS. The number of input legs matching the dimension of input data. The output leg is a four-dimensional vector that will be subsequently encoded into a VQC. Red circles: feature-mapped input. Blue circles: matrices with trainable parameters.}
\label{OverallArchitecture}
\end{figure}
\figureautorefname{\ref{OverallArchitecture}} shows our TN-VQC hybrid model, where the TN(MPS) serves as a feature extractor to compress the input data into a low-dimensional representation to be fed into a VQC.
Here we follow the architecture proposed in Ref.~\cite{Stoudenmire:2016ve} for the MPS feature extractor.
Each image in the MNIST dataset is packed as an $N$-dimensional vector $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{N}\right)$, with $N=28\times 28=784$ and each component is normalized such that $x_i\in[0,1]$.
The vector is mapped to a product state using the feature map~\cite{Stoudenmire:2016ve}
\begin{equation}
\mathbf{x} \rightarrow|\Phi(\mathbf{x})\rangle=\left[\begin{array}{c}
\cos \left(\frac{\pi}{2} x_{1}\right) \\
\sin \left(\frac{x}{2} x_{1}\right)
\end{array}\right] \otimes\left[\begin{array}{c}
\cos \left(\frac{\pi}{2} x_{2}\right)\\
\sin \left(\frac{x}{2} x_{2}\right)
\end{array}\right] \otimes \cdots \otimes\left[\begin{array}{c}
\cos \left(\frac{\pi}{2} x_{N}\right) \\
\sin \left(\frac{\pi}{2} x_{N}\right)
\end{array}\right],
\end{equation}
and then fed into an MPS.
Unlike in Refs.~\cite{Stoudenmire:2016ve,Efthymiou:2019wy} where the MPS is used as a classifier, here we use the MPS as a feature extractor.
Contracting the feature-mapped input and the MPS yields a four-dimensional feature vector, which corresponds to a compressed representation to be used as an input for the VQC to perform classification.
The dimension of the feature vector can be adjusted to fit the number of qubits available for the VQC.
With the MPS performing dimension reduction, the rest of our hybrid framework is a VQC similar to the one proposed in Ref.~\cite{chen19}, originally designed for reinforcement learning.
For the binary classification, we use four qubits in our VQC (Fig.~\ref{Fig:Basic_VQC_Hadamard}).
The encoding of the classical data is through single-qubit gates $R_y(\arctan(x_i))$ and $R_z(\arctan(x_i^2))$, which represent $y$- and $z$-rotations by the given angle $\arctan(x_i)$ and $\arctan(x_i^2)$, respectively.
The choice of arctangent function is that in general the input values may not be in the interval of $[-1, 1]$.
%
The CNOT gates are used to entangle quantum states from each qubit and $R(\alpha,\beta,\gamma)$ represents the general single qubit unitary gate with three learnable parameters $\alpha_i$, $\beta_i$ and $\gamma_i$. %
The first two qubits are measured for classification labels.
The current architecture can be trained end-to-end, i.e. the parameters within the MPS are updated together with those within the VQC at each iteration.
This is a drastic contrast to other QML architectures where the classical part has to be pre-trained.
This framework can thus be viewed as a QML architecture where the TN part, with its own quantum circuit equivalence, is now temporarily treated classically.
\begin{figure}
\begin{center}
\begin{minipage}{10cm}
\Qcircuit @C=1em @R=1em {
\lstick{\ket{0}} & \gate{H} & \gate{R_y(\arctan(x_1))} & \gate{R_z(\arctan(x_1^2))} & \ctrl{1} & \qw & \qw & \targ & \gate{R(\alpha_1, \beta_1, \gamma_1)} & \meter \qw \\
\lstick{\ket{0}} & \gate{H} & \gate{R_y(\arctan(x_2))} & \gate{R_z(\arctan(x_2^2))} & \targ & \ctrl{1} & \qw & \qw & \gate{R(\alpha_2, \beta_2, \gamma_2)} & \meter \qw \\
\lstick{\ket{0}} & \gate{H} & \gate{R_y(\arctan(x_3))} & \gate{R_z(\arctan(x_3^2))} & \qw & \targ & \ctrl{1} & \qw & \gate{R(\alpha_3, \beta_3, \gamma_3)} & \qw \\
\lstick{\ket{0}} & \gate{H} & \gate{R_y(\arctan(x_4))} & \gate{R_z(\arctan(x_4^2))} & \qw & \qw & \targ & \ctrl{-3} & \gate{R(\alpha_4, \beta_4, \gamma_4)} & \qw \gategroup{1}{5}{4}{9}{.7em}{--}\qw
}
\end{minipage}
\end{center}
\caption[Circuit architecture for the variational quantum classifier.]{{\bfseries Variational quantum circuit architecture.}
The encoding part consists of single qubit gates and parameters labeled $R_y(\arctan(x_i))$ and $R_z(\arctan(x_i^2))$ are for the state preparation.
%
The dashed square indicates the learnable part where the CNOT gates are used to entangle quantum states from each qubit and $R(\alpha,\beta,\gamma)$ represents the general single qubit unitary gate with three learnable parameters $\alpha_i$, $\beta_i$ and $\gamma_i$.}
\label{Fig:Basic_VQC_Hadamard}
\end{figure}
\section{Experiments and Results}
To demonstrate the capabilities of the proposed TN-VQC hybrid model, we perform the binary classification task on the standard MNIST dataset, specifically digits 3 and 6.
\subsection{PCA-VQC model} \label{PCA-VQC}
To develop a baseline, we study the architecture with PCA as the feature extractor and the VQC as the discriminator.
We use PCA to reduce the input dimension of $28 \times 28 = 784$ into a four-dimensional vector, which is then fed into the VQC for training.
In this experiment, we use RMSProp \cite{Tieleman2012} as the optimizer with the hyperparameters: learning rate $ = 0.01$, $\alpha = 0.99$ and $\epsilon = 10^{-8}$.
The PCA is performed with the Python package scikit-learn \cite{scikit-learn}.
We can see from the result shown in~\figureautorefname{\ref{EndToEnd_PCA}} that both the accuracy and loss saturates within the first few epochs.
As PCA has been a standard dimensionality reduction method in ML, the result validates the capability of our VQC to perform classification task.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.6\linewidth]{figure/PCAVQC.pdf}
\caption[Result: PCA preprocessing and VQC training]{{\bfseries Result: PCA preprocessing and VQC training.}
Results of the binary classification task using PCA to preprocess the input data. The input data is first reduced from dimension $28 \times 28$ to $4$ via PCA, and subsequently fed into the VQC for training.
This serves as our baseline.}
\label{EndToEnd_PCA}
\end{figure}
\subsection{MPS classifier}
Next, we demonstrate the case where the MPS part alone is fully responsible for the classification task to study its representation power.
In this experiment, the optimizer is Adam~\cite{kingma2014adam} with a learning rate of $ 0.001$ and batch size of $100$.
Figure~\ref{MPS} shows the results of the MPS classifier for $\chi=1$ and 2.
For $\chi=1$, the accuracy of both the training and testing datasets remain around $68-70\%$.
When we increase the bond dimension to $\chi=2$, we observe the accuracy reaching close to $99\%$, indicating that the classifier becomes powerful enough to be highly confident in the result.
However, we also observe that although the training loss remains low, the testing loss starts to rise.
This implies that the classifier is getting more/less confident in the wrong/correct labels over training epochs, which could be a sign of overfitting.
Such increasing test loss behavior is also seen in Ref.~\cite{Efthymiou:2019wy}.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.95\linewidth]{figure/MPS.pdf}
\caption[Result: MPS classifier.]{{\bfseries Result: MPS classifier.}
Results of the binary classification task using an MPS as a classifier. Bond dimension of the MPS is (a) $\chi = 1$ (b) $\chi=2$.}
\label{MPS}
\end{figure}
\subsection{ MPS-VQC hybrid model }
Finally, we study the capability of MPS as a feature extractor for VQC.
Here the MPS is a \emph{learnable} model with parameters subject to iterative optimization, in contrast to the PCA method presented in Section \ref{PCA-VQC}.
In this experiment, the optimizer is Adam \cite{kingma2014adam} with a learning rate of $10^{-4}$. The results are shown in~\figureautorefname{\ref{EndToEnd_Bond_1}}, where we can see that an MPS with $\chi=1$ is enough for our hybrid classifier to reach a test accuracy above $99 \%$, which is significantly better than that of the PCA-VQC model.
See \tableautorefname{\ref{tab:results_comparison}} for the performance comparison between the two methods.
For $\chi=2$ where the MPS classifier shows signs of overfitting, we find that the training of the MPS-VQC model still remains stable without the rising testing loss emerging in the MPS case, indicating that the VQC can also serve as a regularizer for the MPS part.
We note that for $\chi>2$, the training becomes unstable and we ascribe this instability to the representation power of the model being excessive for the binary classification. Such behavior, however, still requires further investigation.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\linewidth]{figure/MPSVQC.pdf}
\caption[Result: End-to-End training of MPS and VQC]{{\bfseries Result: End-to-End training of MPS-VQC model.}
In this experiment, we perform the end-to-end training of the hybrid MPS-VQC model. Bond dimension of the MPS is (a) $\chi = 1$ (b) $\chi=2$. }
\label{EndToEnd_Bond_1}
\end{figure}
\begin{table}[tbp]
\centering
\caption{Performance comparison of PCA-VQC and MPS-VQC ($\chi=1$).}
\label{tab:results_comparison}
\begin{tabular}{|l|l|l|l|l|}
\hline
& Training Acc. & Testing Acc. & Training Loss & Testing Loss \\ \hline
PCA-VQC & $87.29\%$ & $87.34\%$ & $0.3979$ & $0.4006$ \\ \hline
MPS-VQC & $99.91\%$ & $99.44\%$ & $0.3154$ & $0.3183$ \\ \hline
\end{tabular}
\end{table}
\section{Discussion}
We present a hybrid quantum-classical classifier based on the quantum-inspired tensor network and the variational quantum circuit.
Such MPS-VQC framework allows for QML to deal with sizable data with limited qubits and a shallow circuit depth.
We further demonstrate the superiority of this framework by comparing it with the baseline study of a PCA-VQC model on a binary classification task of the MNIST dataset.
One clear advantage is that the representation power of the trainable MPS feature extractor is tunable with bond dimension.
It is expected that this framework can readily adapt to more difficult tasks such as the ternary classification of the Fashion-MNIST dataset.
Our preliminary results show great promise in this direction.
\begin{ack}
This work is supported (in part) by the U.S. DOE under grant No. DE-SC-0012704 and the BNL LDRD No.20-024 and Ministry of Science and Technology (MOST) of Taiwan under grants No. 108-2112-M-002-020-MY3 and No. 107-2112-M-002-016-MY3.
\end{ack}
\newpage
\medskip
\small
\bibliographystyle{ieeetr}
|
hep-ph/9812262
|
\section{I. Quark Mass Hierarchy}
The experimental measurement a few years ago of the top quark mass
\cite{prd96} eliminated the last unknown among
the physical masses of {\it all} three generations of up--, down--,
and electron--like particles.
The masses of these particles are displayed in Table \ref{masstable1},
where I have expressed all masses in top--quark mass units.
To first order, the inter--generational mass ratio for up--type quarks is
$10^{-5}:10^{-3}:1$,
and for down--type is $10^{-5}:10^{-3}:10^{-2}$.
In minimal supersymmetric standard model (MSSM) physics,
quarks (and their supersymmetric partners)
gain mass through superpotential couplings
to Higgs bosons $H_1$ and $H_2$,\footnote{$\hat{X}$
denotes a generic superfield and $X$ its bosonic component.}
\begin{equation}
W_{u_i}\sim \lambda_{u_i} \hat{H}_2 \hat{Q_i} \hat{U}^c_i\, ;\quad
W_{d_i}\sim \lambda_{d_i} \hat{H}_1 \hat{Q_i} \hat{D}^c_i\, ;\quad
\label{nonrenql}
\end{equation}
where $i$ is the generation number. Effective mass terms appear
when the Higgs acquire a typical soft supersymmetry breaking scale
vacuum expectation value (VEV)
$\vev{H_{1,2}} \sim m_{soft}= {\cal O} (M_Z)$.\footnote{Similarly,
the three generations of electron--type leptons gain mass via superpotential
terms $W_{e_i}\sim \lambda_{e_i} \hat{H}_1 \hat{L_i} \hat{E}^c_i$.}
Inter--generational mass ratios can be induced when the associated
first and second generation superpotential terms contain
effective couplings $\lambda$ that include non--renormalizable
suppression factors,
\begin{equation}
\lambda_{(u,d,e)_i} \sim (\frac{\vev{S}}{M_{\rm Pl} })^{P'_i},
\quad {\rm ~for~} i=1,2,
\label{pval}
\end{equation}
where
$S$ is a non--Abelian singlet, $M_{\rm Pl} $ is the Planck scale (which is replaced
by the $M_{\rm string}$ for string models), and $P$ is a positive integer.
VEVs only slightly below the Planck/string scale
(often resulting from a $U(1)$ anomaly cancellation) imply large values
of $P'_{1,2}$ for $10^{-5}$ and $10^{-3}$ suppression
factors. In contrast, intermediate scale VEVs
(between $M_{\rm Z}$ and $M_{\rm string}$) require far lower values for $P'_i$.
In a series of recent papers
intermediate scales have been explored \cite{CCEEL1,CCEEL2}
and their realization in actual models has been
investigated \cite{CCEELW,GC1}.
This investigation has been the product of a
fruitful collaboration with M. Cveti\v c, J. Espinosa, L. Everett,
P. Langacker, \& J. Wang at the University of
Pennsylvania.\footnote{In addition to this mass ratio study
from the perturbative string theory perspective,
Katsumi Tanaka of the Ohio State University
and I have been investigating
quark mass ratios from the non--perturbative approach of
Seiberg--Witten duality \cite{gckt}.}
In the following section, I show how intermediate scales can occur and
how, in theory, they could produce
an inter--generational $10^{-5} : 10^{-3}: 1$ up--quark mass ratio,
and a corresponding $10^{-5}:10^{-3}:10^{-2}$ down--quark mass ratio.
\hfill\vfill\eject
\section{II. Theoretical Quark Mass Hierarchy from String Models}
One method by which an intermediate scale VEV $\vev{S}$
can generate non--renormaliz-able suppression factors
involves extending the SM gauge group by an additional non--anomalous
$U(1)'$. This approach requires (at least)
two SM singlets $S_1$ and $S_2$,
carrying respective $U(1)'$ charges $Q'_1$ and $Q'_2$.
$D$--flatness for the non--anomalous $U(1)'$,
\begin{equation}
\vev{D}_{U(1)'} \equiv Q'_{1}|\vev{S_1}|^2 + Q'_{2}|\vev{S_2}|^2 = 0,
\label{dif}
\end{equation}
necessitates that $Q'_{1}$ and $Q'_{2}$ be of opposite sign.
Together the VEVs of $S_1$ and $S_2$ form a $D$--flat scalar field
direction $S$ defined by,
\begin{equation}
\langle S_1\rangle=\cos\alpha_Q\langle S\rangle,\;\;\;\;
\langle S_2\rangle=\sin\alpha_Q\langle S\rangle,\quad
{\rm where}\quad
\tan^2\alpha_Q\equiv \frac{|Q_1|}{|Q_2|}.
\label{flatdir}
\end{equation}
The $F$--flatness constraints
\begin{equation}
\vev{F_{S_p}} \equiv \vev{\frac{\partial W}{\partial S_p }} = 0,\,\, p= 1,2;
\quad {\rm and}\quad \vev{W}=0,
\label{ff}
\end{equation}
imply that the $D$--flat direction
$S= S_1 \cos\alpha_Q + S_2 \sin\alpha_Q$ is also a
{\it renormalizable} $F$--flat direction if (as I assume hereon)
$\hat{S}_1$ and $\hat{S}_2$ do not couple among themselves
in the renormalizable superpotential.
Consider the real component of this flat direction,
$s=\sqrt{2} {\mathrm Re} S = s_1 \cos\alpha_Q + s_2 \sin\alpha_Q$.
This scalar's renormalization group equation (RGE) running mass is,
\begin{equation}
m^2=m_1^2(\mu) \cos^2\alpha_Q+m_2^2(\mu)\sin^2\alpha_Q=\left(
\frac{m_1^2}{|Q_1|}+\frac{m_2^2}{|Q_2|}\right)
\frac{|Q_1Q_2|}{|Q_1|+|Q_2|},
\label{msum}
\end{equation}
which generates a potential
\begin{equation}
V(s)=\frac{1}{2 }m(\mu= s)^2 s^2.
\label{mpot}
\end{equation}
I will assume that $m^2$ is positive at the string scale
and of order $m^2_{\rm soft} \sim {\cal O}({M_{\rm Z}}^2)$
($m^2_o$ if universality is assumed).
However, through RGE running, $m^2$ can
be driven negative (with electroweak (EW) scale magnitude)
by large Yukawa couplings
(i) of $S_1$ to exotic triplets, $W=h\hat{D}_1\hat{D}_2\hat{S}_1$;
(ii) of $S_1$ to exotic doublets and of $S_2$ to exotic triplets,
$W=h_D\hat{D}_1\hat{D}_2\hat{S}_1+h_L\hat{L}_1\hat{L}_2\hat{S}_2$; or
(iii) of $S_1$ to varying numbers of additional SM singlets
$W=h\sum_{i=1}^{N_p}\hat{S}_{ai}\hat{S}_{bi}\hat{S}_1$ \cite{CCEEL1}.
$m(\mu= s)^2$ can turn negative anywhere between a scale of
$\mu_{rad}= 10^4$ GeV and $\mu_{rad}= 10^{17}$ GeV (slightly below the string scale)
for various choices of the supersymmetry breaking parameters
$A^0$ (the universal Planck scale soft trilinear coupling) and
$M_{1/2}$ (the universal Planck scale gaugino mass).\footnote{The
standard universal scalar EW soft mass--squared parameter $m_0$
has a simple normalizing effect, with $A^0/m_0$ and $M_{1/2}/m_0$
being the actual relevant parameters.}
When $m^2$ runs negative,
a minimum of the potential develops along the flat direction
and $S$ gains a non--zero VEV.
In the case of only a mass term and no Yukawa contribution to $V(s)$,
minimizing the potential
\begin{equation}
\frac{dV}{ds}=\left.\left(m^2+\frac{1}{2}\beta_{m^2}\right)\right|_{\mu=s}s=0,
\label{vdeq}
\end{equation}
(where $\beta_{m^2}=\mu\frac{dm^2}{d\mu}$)
shows that the VEV $\langle s\rangle$ is determined by
\begin{equation}
m^2(\mu=\langle s\rangle)=-\frac{1}{2}\beta_{m^2}.
\label{m2eq}
\end{equation}
Eqs.\ (\ref{vdeq},\ref{m2eq})
are satisfied very close to the scale $\mu_{RAD}$ at which $m^2$ crosses
zero. $\mu_{RAD}$ is fixed by the renormalization group evolution of
parameters from $M_{\rm string}$ down to the EW scale and will lie
at some intermediate scale.
Location of the potential minimum can also be effected by non--renormalizable
self--interaction terms,
\begin{equation}
W_{\rm NR}=\left(\frac{\alpha_{K}}{M_{\rm Pl} }\right)^{K}\hat{S}^{3+K},
\label{nrsup}
\end{equation}
where $K=1,2...$ and $\alpha_{K}$ are coefficients.
Such non--renormalizable operators (NRO's)
lift the flat direction
(by breaking $F$--flatness)
for sufficiently large values of $s$.
The general form of the potential, $V(X_p)$, for the scalar components
$X_p$ of corresponding supermultiplets $\hat{X_p}$ is
\begin{eqnarray}
V(X_p) &=& V_{soft\,\, susy} + \sum_p \mid \frac{\partial W}{\partial \hat{X}_p} \mid^2
+\frac{1}{2} g_{\alpha}^2\sum_\alpha
\mid \sum_p Q_p^{\alpha} | X_p \mid^2 \mid^2
\label{vsnroa}\\
&=& V_{soft\,\,susy} + \sum_p \mid F_p \mid^2 \phantom{0}\phantom{0}
+\frac{1}{2} g_{\alpha}^2\sum_\alpha \mid D_{\alpha}\mid^2
\label{vsnrob}
\end{eqnarray}
Thus, NRO contributions transform $V(s)$ in eq.\ (\ref{mpot}) into
\begin{equation}
V(s)= \frac{1}{2}m^2s^2+\frac{1}{2(K+2)}
\left(\frac{s^{2+K}}{{\cal M}^K}\right)^2,
\label{vsnro}
\end{equation}
where ${\cal M} = {\cal C}_K M_{\rm Pl} / \alpha_K$, with
${\cal C}_K = [ 2^{K+1}/((K+2)(K+3)^2)]^{1/(2K)}$.
Even when an NRO is present,
the running mass effect still dominates in determining $\vev{s}$
if $\mu_{RAD}\ll 10^{12}$ GeV.
However, an NRO is the controlling factor when
$\mu_{RAD}\gg 10^{12}$ GeV. In the latter case, we find that
\begin{equation}
\label{veveq}
\langle s\rangle=
\left[\sqrt{(-m^2)}{\cal M}^K \right]^\frac{1}{K+1}
= \mu_K\sim (m_{soft}{\cal M}^K)^\frac{1}{K+1},
\end{equation}
where $m_{soft}={\cal O} (|m|)={\cal O} (M_Z)$ is a typical soft supersymmetry
breaking scale.
While $\vev{s}$ is an intermediate scale VEV, the mass $M_S$ of
the physical field $s$ is still on the order of the soft SUSY breaking scale:
For running mass domination,
\begin{equation}
M_S^2\equiv\left.\frac{d^2V}{ds^2}\right|_{s=\langle s \rangle}=
\left.\left(\beta_{m^2}+\frac{1}{2}\mu\frac{d}{d\mu}\beta_{m^2}\right)
\right|_{\mu=\langle s\rangle}\simeq \beta_{m^2}\sim\frac{m^2_{soft}}{16\pi^2},
\end{equation}
while in the NRO--controlled case,
\begin{equation}
M_S^2=2(K+1)(-m^2)\sim m^2_{soft}.
\end{equation}
What powers $P'_i$ in (\ref{pval})
for first and second generation suppression factors
in an NRO--dominated model could produce
an up--type quark mass ratio of order $10^{-5}:10^{-3}:1$?
From eq.\ (\ref{veveq}), we see the suppression factors become
\begin{equation}
\left(\frac{m_{soft}}{M }\right)^{\frac{P'_i}{K+1}},
\label{prat}
\end{equation}
where the coefficient $\alpha_K$ has been absorbed into
the definition of the mass scale $M\equiv {\cal M}/{\cal C}_K$.
The mass suppression factors for specific $P'$ (in the range $0$ to $5$)
and $K$ (in the range of $1$ to $7)$ are given in Table \ref{masstable2}.
From this table we find that the choices $P'_1= 2$ and $P'_2= 1$ in tandem with
$K=5$ or $K=6$ (for the self--interaction terms of $S$)
can, indeed, reproduce the required mass ratio. These values of $K$
invoke an intermediate scale $\vev{S}$ around
$8\times 10^{14}$ GeV to $2\times 10^{15}$ GeV,
The first and second generation down--quark suppression factors
can be similarly realized.
However, unless $tan\ \beta\equiv \frac{\vev{H_2}}{\vev{H_1}}\gg 1$,
the intra--generational mass ratio of
$10^{-2}: 10^{-2}: 1$ for $m_{\tau}$, $m_{b}$, and
$m_{t}$ is not realizable from a $K= 5$ or $6$ NRO singlet term.
For $tan\ \beta \sim 1$, $m_{\tau}$, $m_{b}$ are
too small to be associated with a renormalizable coupling ($P=0$)
like that assumed for $m_{t}$,
but are somewhat larger
than predicted by $P= 1$ for $K=5$ or $K=6$.
Instead, $m_b$ and $m_{\tau}$ might be
associated with a different NRO involving the VEV of an entirely different
singlet.
In that event, Table \ref{masstable2} suggests another
flat direction $S'$ (formed from a second singlet pair $S_1'$ and $S_2'$),
with a $K=7$ self--interaction NRO and $P'_3=1$ suppression factor for
$m_b$ and
$m_{\tau}$.\footnote{An intermediate scale VEV $\vev{S}$
can also solve the $\mu$ problem through a superpotential term
$W_{\mu}\sim \hat{H_1} \hat{H_2}\hat{S}\left({\hat{S}\over M }\right)^{P_{\mu}}$.
With NRO--dominated $\langle S\rangle \sim (m_{soft} {M }^K)^{\frac{1}{K+1}}$,
the effective Higgs $\mu$--term takes the form,
$ \mu_{eff} \sim m_{soft} \left( {m_{soft}\over M } \right)^{\frac{P-K}{K+1}}$.
The phenomenologically preferred choice among this class of
terms is clearly $P=K$:
this yields a $K$--independent ${\mu}_{eff} \sim m_{soft}$.}
\section{III. Realization of Quark Mass Hierarchy in String Models}
In string models, one problem generally appears at the string
scale that must be resolved ``before'' possible intermediate
scale flat direction VEVs can be investigated.
That is, most four--dimensional quasi--realistic
$SU(3)_C\times SU(2)_L\times U(1)_Y$ string models
contain an anomalous $U(1)_A$ (meaning ${\rm Tr} Q_A\ne 0$) \cite{KNCF}.
In fact, in a generic charge basis, a string model with an
Abelian anomaly may actually contain not just one, but
several anomalous $U(1)$ symmetries.
However, all anomalies can all be transferred into a single
$U(1)_A$ through the {\it unique} rotation
\begin{equation}
U(1)_{\rm A} \equiv c_A\sum_n \{{\rm Tr} Q_{n}\}U(1)_n,
\label{rotau1}
\end{equation}
with $c_A$ a normalization factor.
The remaining non--anomalous components of the original set of $\{U(1)_n\}$
may be rotated into a complete orthogonal basis $\{U(1)_a\}$.
The standard anomaly cancellation mechanism
\cite{DSW,ADS} breaks $U(1)_{\rm A}$ at the string scale, while simultaneously
generating a FI $D$--term,
\begin{equation}
\xi\equiv \frac{g^2_s M_P^2}{192\pi^2}{\rm Tr} Q_A\, ,
\label{fid}
\end{equation}
where $g_{s}$ is the string coupling
and $M_P$ is the reduced Planck mass,
$M_P\equiv M_{Planck}/\sqrt{8 \pi}\approx 2.4\times 10^{18}$.
The FI $D$--term breaks spacetime supersymmetry unless it is cancelled by
appropriate VEVs $\vev{X_p}$
of scalars $X_p$ that carry non--zero anomalous charge,
\begin{equation}
\vev{D}_{\rm A} \equiv \sum_j Q^{(A)}_j |\vev{X_p}|^2 +
\xi
= 0\,\, .
\label{anomd}
\end{equation}
Generalizations of $D$-- and $F$--constraints for $S_{1,2}$
(i.e., of eqs.\ (\ref{dif}) and (\ref{ff}))
are imposed on possible VEV directions $\{\vev{X_p}\}$:
$\vev{D}_{a} \equiv \sum_{p} Q^{(a)}_{p}|\vev{X_p}|^2 = 0$ and
$\vev{F_{p}} \equiv \vev{\frac{\partial W}{\partial \hat{X}_p}} = 0;
\,\, \vev{W} =0$.
My colleagues and I at Penn have developed
methods for systematically determining \cite{dfset,GCM}
and classifying \cite{CCEEL2} $D$-- and $F$--flat
directions in string models.
We have applied this process \cite{CCEELW,GC1} to the
free fermionic three generation $SU(3)_C\times SU(2)_L\times U(1)_Y$ models
(all of which contain an anomalous $U(1)_A$)
introduced in refs.\ \cite{FNY}, \cite{AF} and \cite{CHLM}.
For each model, we have determined the anomaly cancelling
flat directions that preserve hypercharge,
only involve VEVs of non--Abelian singlet fields, and
are $F$--flat to all orders in the non--renormalizable superpotential.
Flat directions in Model 5 of \cite{CHLM} have particularly received
our attention \cite{CCEELW}.
In Model 5 we investigated the physics implications of
various non--Abelian singlet flat directions.
After breaking the anomalous $U(1)_A$, all of these flat directions
left one or more additional $U(1)_a$ unbroken at the string scale.
For each flat direction, the complete set of
effective mass terms and effective trilinear superpotential
terms in the observable sector were computed {\it to all orders} in the
VEV's of the fields in the flat direction.
The ``string selection--rules'' disallowed a large number of couplings
otherwise allowed by gauge invariance,
resulting in a massless spectrum with a large number
of exotics,\footnote{Recently it was shown \cite{CFN}
that free fermionic construction can actually provide string models
wherein {\it all} MSSM exotic states gain near--string--scale masses
via flat direction VEVs, leaving only a string--generated MSSM in the
observable sector below the string scale.
The model of ref.\ \cite{FNY} was presented as the first known with these
properties.}
which in most cases are excluded by experiment.
This signified a generic flaw of these models.
Nevertheless, we found the resulting trilinear couplings of the
massless spectrum to possess a number of interesting features which
we analyzed for two representative flat directions. We investigated
the fermion texture; baryon-- and lepton--number violating
couplings; $R$--parity breaking; non--canonical $\mu$ terms;
and the possibility of electroweak and intermediate scale symmetry breaking
scenarios for a $U(1)'\in \{ U(1)_a\}$.
The gauge coupling predictions were obtained in the electroweak scale case.
We found $t-b$ and $\tau-\mu$ fermion mass
universality, with the string scale Yukawa couplings $g$ and $g/\sqrt{2}$,
respectively.
Fermion textures existed for certain flat directions,
{\it but only in the down--quark sector.}
Lastly, we found
baryon-- and lepton-- number violating couplings that could
trigger proton--decay, $N-{\bar N}$ oscillations, leptoquark interactions
and $R$--parity violation, leading to the absence of a stable LSP.
\section{IV. Comments}
I have discussed how intermediate VEVs hold
the potential to yield quasi--realistic quark mass ratios.
Four--dimensional string models usually require
cancellation of the FI $D$--term contribution from an anomalous $U(1)_A$
by near string scale VEVs.
Since some non--anomalous $U(1)_a$ are simultaneously broken at
the string scale by these VEVs,
which $U(1)_a$ might be associated with intermediate scale VEVs
strongly depends on the particular set of (near) string--scale VEVs chosen.
As our investigations into flat directions have demonstrated,
various choices for flat VEV directions can drastically alter
low energy phenomenology.
The textures of the quark mass matrices can be strongly effected by choice
of flat direction since
textures are wrought by effective mass terms in the non--renormalizable
superpotential.
\section{Acknowledgements}
G.C. thanks the organizers of QCD '98 for a very enjoyable and educational
conference.
\hfill\vfill\eject
\begin{table}
\vskip 2.0truecm
\begin{tabular}{ccccccccccc}
$m_u$
&$:$
&$m_c$
&$:$
&$m_t$
&$=$
&$3\times 10^{-5}$
&$:$
&$7\times 10^{-3}$
&$:$
&$1$
\\
\hline
$m_d$
&$:$
&$m_s$
&$:$
&$m_b$
&$=$
&$6\times 10^{-5}$
&$:$
&$1\times 10^{-3}$
&$:$
&$3\times 10^{-2}$
\\
\hline
$m_e$
&$:$
&$m_{\mu}$
&$:$
&$m_{\tau}$
&$=$
&$0.3\times 10^{-5}$
&$:$
&$0.6\times 10^{-3}$
&$:$
&$1\times 10^{-2}$
\\
\end{tabular}
\caption{Fermion mass ratios with the top quark mass normalized to $1$.
The values of $u-$, $d-$, and $s$-quark masses used in the ratios
(with the $t$-quark mass normalized to $1$ from an assumed mass of $170$ GeV)
are estimates
of the $\overline{\rm MS}$ scheme current-quark masses at a scale
$\mu\approx 1$ GeV.
The $c$- and $b$-quark masses are pole masses.}
\label{masstable1}
\end{table}
\begin{table}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
~& $P^{(')}$ & $K=1$ & $K=2$ & $K=3$ & $K=4$ & $K=5$
& $K=6$ & $K=7$\\
\hline
\hline
$\left(\frac{m_{soft}}{M }\right)^{\frac{1}{K+1}}$
&
& $2\times 10^{-8}$
& $7\times 10^{-6}$
& $1\times 10^{-4}$
& $8\times 10^{-4}$
& $3\times 10^{-3}$
& $6\times 10^{-3}$
& $1\times 10^{-2}$
\\
\hline
$\langle S\rangle$ (GeV)
&
& $5\times 10^{9}$
& $2\times 10^{12}$
& $4\times 10^{13}$
& $2\times 10^{14}$
& $8\times 10^{14}$
& $2\times 10^{15}$
& $3\times 10^{15}$
\\
\hline
& $K-1$
& $5\times 10^{7}$
& $1\times 10^{5}$
& $7\times 10^{3}$
& $1\times 10^{3}$
& $400$
& $200$
& $90$
\\
$\frac{\mu_{eff}}{m_{soft}}$
& $K$
& 1
& 1
& 1
& 1
& 1
& 1
& 1
\\
& $K+1$
& $2\times 10^{-8}$
& $7\times 10^{-6}$
& $1\times 10^{-4}$
& $8\times 10^{-4}$
& $3\times 10^{-3}$
& $6\times 10^{-3}$
& $1\times 10^{-2}$
\\
\hline
& $0$
& 1
& 1
& 1
& 1
& 1
& 1
& 1
\\
& $1$
& $2\times 10^{-8}$
& $7\times 10^{-6}$
& $1\times 10^{-4}$
& $8\times 10^{-4}$
& $3\times 10^{-3}$
& $6\times 10^{-3}$
& $1\times 10^{-2}$
\\
& $2$
& $3\times 10^{-16}$
& $5\times 10^{-11}$
& $2\times 10^{-8}$
& $6\times 10^{-7}$
& $7\times 10^{-6}$
& $4\times 10^{-5}$
& $1\times 10^{-4}$
\\
$\frac{m_{Q,L}}{\langle H_i\rangle} $
& $3$
& $6\times 10^{-24}$
& $3\times 10^{-16}$
& $2\times 10^{-12}$
& $5\times 10^{-10}$
& $2\times 10^{-8}$
& $2\times 10^{-7}$
& $2\times 10^{-6}$
\\
& $4$
& $1\times 10^{-31}$
& $2\times 10^{-21}$
& $3\times 10^{-16}$
& $4\times 10^{-13}$
& $5\times 10^{-11}$
& $1\times 10^{-9}$
& $2\times 10^{-8}$
\\
& $5$
& $2\times 10^{-39}$
& $2\times 10^{-26}$
& $5\times 10^{-20}$
& $3\times 10^{-16}$
& $1\times 10^{-13}$
& $9\times 10^{-12}$
& $2\times 10^{-10}$
\\
\end{tabular}
\caption{Non-Renormalizable
MSSM mass terms via $\langle S \rangle$. For
$m_{soft}\sim 100$ GeV, $M \sim 3\times 10^{17}$ GeV.}
\label{masstable2}
\end{table}
\hfill\vfill\eject
\def\NPB#1#2#3{{\it Nucl.\ Phys.}\/ {\bf B#1} (#2) #3}
\def\NPBPS#1#2#3{{\it Nucl.\ Phys.}\/ {{\bf B} (Proc.\ Suppl.) {\bf #1}}
(19#2) #3}
\def\PLB#1#2#3{{\it Phys.\ Lett.}\/ {\bf B#1} (#2) #3}
\def\PRD#1#2#3{{\it Phys.\ Rev.}\/ {\bf D#1} (#2) #3}
\def\PRL#1#2#3{{\it Phys.\ Rev.\ Lett.}\/ {\bf #1} (#2) #3}
\def\PRT#1#2#3{{\it Phys.\ Rep.}\/ {\bf#1} (#2) #3}
\def\MODA#1#2#3{{\it Mod.\ Phys.\ Lett.}\/ {\bf A#1} (#2) #3}
\def\IJMP#1#2#3{{\it Int.\ J.\ Mod.\ Phys.}\/ {\bf A#1} (#2) #3}
\def\nuvc#1#2#3{{\it Nuovo Cimento}\/ {\bf #1A} (#2) #3}
\def\RPP#1#2#3{{\it Rept.\ Prog.\ Phys.}\/ {\bf #1} (#2) #3}
\def{\it et.\ al\/}{{\it et.\ al\/}}
\def\upenngrpa{G.\ Cleaver, M.\ Cveti\v c, L.\ Everett, J.R.\ Espinosa,
and P.\ Langacker}
\def\upenngrpb{G.\ Cleaver, M.\ Cveti\v c, L.\ Everett, J.R.\ Espinosa,
P.\ Langacker, and J.\ Wang}
\def\bibitem{\bibitem}
|
2202.08514
|
\section{Keywords}
\section{Introduction}
\subsection{Background}
Deep neural networks, especially convolutional neural networks (CNNs) has led to big accomplishments in the computer vision field \cite{rotation}. For real-world problems, there is a scarcity of data for training purposes, so expensive efforts concerning time and resources are required to provide these labelled training data. This problem led to a big increase in the interest of researchers in using unsupervised feature learning for solving visual understanding tasks with lack of availability of labelled data \cite{aet}.
The basic idea in SSL is to produce some supervisory signals to solve assigned tasks. This task may include representation of the data or auto labelling of the data. SSL provides us with the advantage of training networks without requiring extensive labelling of the images. In order to utilize the strengths of SSL techniques over the period of time, major area of work has been done in the domain of developing various pretext tasks.
\subsection{Previous studies}
Multiple survey and review papers on self-supervised techniques have been published. One such work \cite{surveydeep} describes deep learning-based self-supervised general visual feature learning methods from images or videos.A survey on augmentation techniques was published in late 2019 but it majorly focused on application of augmentation techniques including geometric transformation, color space augmentations, filters and feature space augmentations. Another paper \cite{surveygencon} explained self-supervised learning technique in various fields including computer vision,graph learning and natural language processing based on generative, contrastive and mix of generative and contrastive objectives . Another paper \cite{surveycontrast} discusses the use of contrastive learning approach with self supervised methods.Basic idea in this technique is to assemble similar samples near each other in relevant to other samples with dissimilarities.
\subsection{Problem statement and motivation}
To the best of our ability, we were unable to find a review or survey paper that solely focuses on self-supervised learning (SSL) techniques that use geometric transformations (GT). Although many such works have been reviewed in review papers, these have not been covered in depth. For instance, one of the review papers \cite{surveydeep} talks about the popular rotation prediction SSL technique, but doesn't go into the details of it and doesn't mention works which have improvised on the method. Also, it doesn't talk about the much successful work of autoencoding geometric transformations \cite{aet}. This lack of extensive review of geometric transformations based SSL techniques show a gap in the literature which we aim to fill.
We believe an extensive review that purely focuses on geometric transformation based self-supervised techniques is necessary due to 2 reasons. (1) Geometric transformations have proved to be simple yet powerful supervisory signals in unsupervised representation learning. (2) Many successful works have used geometric transformations from different paradigms such as autoencoding and classification. To enable a detailed and in-depth understanding of SSL with geometric transformations we present a concise survey encompassing multiple aspects including algorithmic details as well as futuristic unresolved problems.
\subsection{Proposed approach and contributions}
This work contains thorough review and comparison of 6 SSL models that use GT. Multiple avenues including preprint servers, conference proceedings and journals were searched for shortlisting relevant papers. Our contributions can be summarized in 4 points. (1) Provide the reader with an in-depth review of geometric transformation based SSL techniques. (2) Highlight the importance and success of using such methods. (3) Discuss the shortcomings in such techniques and the relevant problems. (4) Explain trend setting research as well as future direction.
We organize the rest of the paper as follows. In section 3 we briefly discuss overview of existing methods in detail, where we describe methods that are based on image rotation prediction as well as using auto encoding transformations. In section 4 we present results along with qualitative and quantitative comparison of the performance of the techniques on CIFAR-10 \cite{cifar} and ImageNet \cite{imagenet} datasets. In section 5 we derive insights and explain the results of comparing the different approaches. Section 6 contains conclusion and future directions.
\section{Overview of existing methods}
\subsection{Methods based on predicting geometric transformations}
In order to extract useful features from images in unsupervised fashion, RotNet \cite{rotation} is trained to predict the rotation by multiples of 90 degrees applied to the input images.
Features learned through this method are capable of generalizing well in various tasks. But the features learned are biased with regards to rotation transformation and does not help in various tasks which are in favour of invariance in rotation. Also, rotation is not determinable for all images in practice and therefore this method does not perform well with orientation agnostic images. To address these shortcomings, an SSL technique \cite{decoupling} that decouples the learned features using the task of predicting rotation along with the task of task discriminating the instances was presented. The learned representations comprises of 2 parts in which one is discriminative to rotation and other is unrelated to rotation as shown in Figure \ref{fig:decoupling}. In another work called ExemplarCNN \cite{exemplarcnn}, authors trained the CNN to differentiate among a group of surrogate classes. To create such classes, various transformations are applied picture patch sampled randomly. This algorithm strikingly performed well.
\begin{figure}[h]
\caption{Depiction of the rotation feature decoupling method. The ouput from neural network is a decoupled meaningful feature representation comprising sections that are related and unrelated to rotation. The rotation related part is trained by predicting rotations applied to image. A PU learning problem is modelled here with using the noisy rotation labels, that is trained to learn instance weights in order to decrease the impact of pictures that are rotation ambiguous. The second part is trained using loss that penalises distance in order to impose rotation irrelevance along with a task of discriminating instances by the use of classification without parameters}
\includegraphics[width=13cm]{images/decoupling.png}
\centering
\label{fig:decoupling}
\end{figure}
\subsection{Methods based on autoencoding geometric transformations}
For a randomly sampled transformation, the self-supervised learning technique of Auto-Encoding Transformation (AET) \cite{aet} attempts to quantify it just using the learned features precisely as the output. The main theory here is that the transformation can be quantified if the learned features are able store the important characteristics of original images and images after transformation. A variant of this approach is Autoencoding Variational Transformations (AVT) \cite{avt}. Provided images after transformation, AVT trains the autoencoder by increasing the shared features between learned representations and transformations.
In order to quantify the divergence of predicted transformations from their labelled equivalents, deterministic and probabilistic AETs depend upon Euclidean distance. But, this is a contentious assumption as a set of transformations usually stay on a curved manifold instead of residing in a flat Euclidean space. To solve this issue, the authors developed AETv2 \cite{aetv2}, which uses the geodesic distance to distinguish the way in which an image is transformed towards the manifold of a set of transformations, and use its length to quantify the difference between transformations.
\begin{figure}[h]
\caption{Depiction of the way in which AETv2 \cite{aetv2} is trained from start to finish. In order to make the output matrix $\mathbf{T}^{-1} \hat{\mathbf{T}}$ from the decoder of transformation possess a unit determinant, it is normalized, and it should be noted that the projection of $\mathbf{T}^{-1} \hat{\mathbf{T}}$ onto $\mathbf{S O}(3)$ is followed in order to calculate the geodesic distance and also the loss of projection in order for the model to be trained}
\includegraphics[height=7cm]{images/aetv2.png}
\centering
\label{fig:aetv2}
\end{figure}
\section{Results}
\subsection{CIFAR-10}
Comparison of the performance of the models in the downstream task of object recognition on CIFAR-10 \cite{cifar} measured by error rates produced interesting results, as shown in Table \ref{tab:cifar}. AETv2 \cite{aetv2} model produced best results with an error rate of 7.44. It can also be seen that RotNet \cite{rotation} was a breakthrough work, that made much progress compared to previous traditional methods, bringing down the error rate to 8.84. While rotation provides a simple, yet powerful supervisory signal, autoencoding multiple transformations has proved to be a slightly better method in feature representation learning.
\begin{table}[h]
\caption{ Comparative study between various selected methods using self supervised technique on CIFAR-10 \cite{cifar} data set.Two supervised methods including NIN and random Init along with conv contains same architecture with only difference that one is supervised where as the other one is trained in such a way that first two blocks are initialized and kept frozen in training process.}
\centering
\begin{tabular}{cc}
\hline
Model & Error rate \\ \hline
NIN-Supervised (Lower Bound) & 7.20 \\
Random Init. + conv (Upper Bound) & 27.50 \\ \hline
Roto-Scat + SVM \cite{rotoscat} & 17.7 \\
DCGAN \cite{dcgan} & 17.2 \\
ExamplarCNN \cite{exemplarcnn} & 15.7 \\
Scattering \cite{scattering} & 15.3 \\
RotNet + conv \cite{rotation} & 8.84 \\
AETv1 + conv \cite{aet} & 7.82 \\
AVT + conv \cite{avt} & 7.75 \\
AETv2 + conv \cite{aetv2} & \textbf{7.44} \\ \hline
\end{tabular}
\label{tab:cifar}
\end{table}
\subsection{ImageNet}
Some interesting insights could be derived from the results of the methods' performance in the image classification task on ImageNet \cite{imagenet} compared using top-1 accuracy, as presented in Table \ref{tab:imagenet}. While AETv2 performed better when the network was trained with up to 3 convolutional blocks, RotNet performed better with more than 3 convolutional blocks. This could be due to the simplicity of the RotNet model. Generally, using more than 3 convolutional blocks produced a gradual decrease in object recognition accuracy, which we believe is because the feature learnt in these layers start to become more specific on the pretext task. Furthermore, we see that increased depth of models led to better performance in object recognition with regards to the feature maps produced by earlier layers. We believe this is because a deeper model enables the features of layers early on to be less peculiar to the pretext task.
\begin{table}[h]
\caption{Comparison of Top-1 accuracy using linear layers on ImageNet \cite{imagenet} data set. For comparison of self supervised models Alex Net is used. This classifier is trained with various depth of convolution layers containing features maps with 9000 elements finally.Upper bound and lower bounds of self supervised model's performances for supervised and random models are also shown for comparison.}
\centering
\begin{tabular}{cccccc}
\hline
Model & Conv1 & Conv2 & Conv3 & Conv4 & Conv5 \\ \hline
ImageNet Labels (Upper Bound) \cite{rotation} & 19.3 & 36.3 & 44.2 & 48.3 & 50.5 \\
Random (Lower Bound) \cite{rotation} & 11.6 & 17.1 & 16.9 & 16.3 & 14.1 \\
Random rescaled \cite{randomrescaled} (Lower Bound) & 17.5 & 23.0 & 24.5 & 23.2 & 20.6 \\ \hline
BiGAN \cite{bigan} & 17.7 & 24.5 & 31.0 & 29.9 & 28.0 \\
RotNet \cite{rotation} & 18.8 & 31.7 & 38.7 & 38.2 & 36.5 \\
Rotation feature decoupling \cite{decoupling} & 19.3 & 33.3 & 40.8 & \textbf{41.8} & \textbf{44.3} \\
AETv1 \cite{aet} & 19.2 & 32.8 & 40.6 & 39.7 & 37.7 \\
AVT \cite{avt} & 19.5 & 33.6 & 41.3 & 40.3 & 39.1 \\
AETv2 \cite{aetv2} & \textbf{19.6} & \textbf{34.1} & \textbf{41.9} & 40.4 & 37.9 \\ \hline
\end{tabular}
\label{tab:imagenet}
\end{table}
\section{Discussion}
A thorough comparison of the methods have shown that AETv2 \cite{aetv2} performs best in terms of learning relevant features from images, and being able to contribute to lower error rates in object recognition on ImageNet \cite{imagenet} and CIFAR-10 \cite{cifar} datasets. As shown in Figure \ref{fig:aetloss}, it can be seen that the path of loss of prediction of transformation follows a similar path of that of error of classification and top-1 accuracy on CIFAR-10 \cite{cifar} and ImageNet \cite{imagenet}. This indicates that predicting transformations better implies better result of classification making use of learned representations. This validates the choice of AET and its variants for the supervision of learning feature representations.
One downside of using transformations is that they might leave behind any low-level visual artifacts that are easily detectable which will lead the CNN to learn easy features without practical value in the tasks of vision perception, e.g., to implement scale and aspect ratio image transformations, image resizing routines that leave easily detectable image artifacts would have to be used. The models' performance on the CIFAR-10 \cite{cifar} and ImageNet \cite{imagenet} datasets can still be improved. More combinations of transformations could be experimented with to improve performance. Moreover, attention based models could be coupled with the transformations to facilitate better feature learning.
\begin{figure}
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/aetcifar.png}
\caption{CIFAR-10 \cite{imagenet}}
\label{fig:aetcifar}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{images/aetimagenet.png}
\caption{ImageNet \cite{imagenet}}
\label{fig:aetimagenet}
\end{subfigure}
\hfill
\caption{The two plots show the change in error rate and top-1 accuracy with the change in AET \cite{aet} loss as training progressed on the CIFAR-10 \cite{imagenet} and ImageNet \cite{imagenet} datasets.}
\label{fig:aetloss}
\end{figure}
\section{Conclusion}
Although multiple papers have reviewed contrastive and generative SSL techniques there is none that solely focuses on SSL techniques that use GT. Also, such methods have not been covered in depth in papers where they have been reviewed. This is a gap in the literature which we aim to fill through this work. Our motivation to present this work is that GT have shown to be powerful supervisory signals in unsupervised representation learning. Moreover, many works which used GT in SSL techniques have found tremendous success, but have not gained much attention. The work has extensively reviewed 6 approaches on self-supervised representation learning using image transformations that are based on predicting rotation as well as auto encoding transformations. We compared the models based on their performance in the downstream task of object recognition on CIFAR-10 \cite{cifar} and ImageNet \cite{imagenet} datasets, measured by their error rates and top-1 accuracy. The review provided insights as to which models worked better and why. It has also showed that image transformations are powerful supervisory signals for feature representation learning. We also believe this paper will help researchers in inventing new GT based SSL techniques.
\clearpage
\bibliographystyle{abbrv}
|
math/9712219
|
\section{Introduction} \label{intro}
Let ${F_n}$ denote a
free group of rank $n$. The group $\o({F_n})$ contains
mapping class groups of
compact
surfaces and maps onto $GL(n,{\mathbb Z})$. It is perhaps
not surprising
that $\o({F_n})$ behaves at times like a mapping class
group and at times
like a linear group. J. Birman, A. Lubotzky, and J.
McCarthy
\cite{blm:s2a} showed that solvable subgroups of
mapping class groups
are finitely generated and virtually abelian. Of course,
$GL(3,{\mathbb Z})$
contains the Heisenberg group which is solvable but
not virtually abelian. In
this paper, we show
that, with respect to the nature of solvable subgroups,
$\o({F_n})$ behaves more
like mapping class groups.
\begin{thm}\label{main} Every solvable subgroup of
$\o({F_n})$
has a subgroup of index at most $3^{5n^2}$ that is
finitely generated and free
abelian.\end{thm}\noindent The rank of an abelian
subgroup
of $\o({F_n})$ is bounded by $vcd(\o({F_n}))=2n-3$ for
$n>1$ \cite{cv:moduli}.
Since $Aut({F_n})$ embeds in $Out(F_{n+1})$, solvable
subgroups of
$Aut({F_n})$ are also virtually abelian.
Theorem~\ref{main} complements
\cite{bfh:tits2} where we show that $\o({F_n})$ satisfies
the Tits
Alternative, i.e. that subgroups of $\o({F_n})$ are either
virtually solvable
or contain a free group of rank 2.
H. Bass and A. Lubotzky \cite{bl:niltech} have shown
that
solvable subgroups of $Out({{F_n}})$ are virtually
polycyclic. In particular, they
are finitely
generated. We include an independent proof of this
fact for completeness and because
the ingredients of our proof are needed for the proof
of Theorem~\ref{main}.
The starting point for this paper is a short exact
sequence from \cite{bfh:tits1} and \cite{bfh:tits2}.
Begin with a solvable subgroup
${\cal H}$ of $\o({F_n})$. After passing to a finite index
subgroup we may assume that ${\cal H}$
acts trivially on
$H_1({F_n};{\mathbb Z}/3{\mathbb Z})$. By Theorem 8.1
of
\cite{bfh:tits1} and Proposition 3.5 of
\cite{bfh:tits2} there is an exact sequence
$$1\to{\cal H}_0\to{\cal H}\overset
\Omega \to{\mathbb Z}^b\to 1$$ where ${\cal H}_0$ is $UPG$
(defined below).
There are two parts to the proof of
Theorem~\ref{main}. First we show that ${\cal H}_0$ is
abelian by constructing an embedding $\Phi : {\cal H}_0 \to
{\mathbb Z}^r$. Then we show that $\Phi$ extends
to a
homomorphism $\Phi' : {\cal H} \to {\mathbb Z}^r$.
The
direct sum of $\Omega$ and $\Phi'$ is an
embedding of ${\cal H}$ into ${\mathbb
Z}^{b+r}$ showing that ${\cal H}$ is finitely generated and
free abelian.
Our approach is motivated by the special case that
${\cal H}$ is realized as a subgroup of the mapping class
group of a compact surface $S$. The
surface $S$ decomposes into a union of
annuli $A_1,\dots,A_r$ and subsurfaces $S_i$ of
negative Euler characteristic;
virtually every $\eta \in
{\cal H}$ is represented by a homeomorphism $f : S \to S$
that restricts to a Dehn twist on
each $A_j$ and that preserves each $S_i$. If $\eta
\in {\cal H}_0$ then each $f|S_i$ is the
identity. The homomorphisms $\Phi$ and $\Phi'$
are defined
by taking their $j^{th}$ coordinates to be the
number of twists that occurs across $A_j$. For a
further discussion of the
geometric case, see Example~\ref{Dehn}.
We study an outer automorphism ${\cal \eta}$
through its lifts $\hat
\eta : C_{\infty} \to C_{\infty}$ to the Cantor set at infinity $C_{\infty}$ or
equivalently (see
Subsection~\ref{lifts}) through the automorphisms
$\Phi : F_n \to F_n$ that
represent $\eta$. The $C_{\infty}$ point of view simplifies
certain
proofs because it allows us to consider
fixed
\lq directions\rq\ in $F_n$ that are not periodic
and therefore do not come from
fixed elements of $F_n$.
In the course of proving Theorem~\ref{main} we
prove the following result which is
of independent interest. We state it here in terms of
automorphisms although we prove
it in terms of $C_{\infty}$.
\begin{proposition}\label{lifting to Aut} Every
abelian subgroup
${\cal H} \subset Out(F_n)$ has a virtual lift $\tilde {\cal H}
\subset Aut(F_n)$. If $\gamma$ is
a non-trivial primitive element of $F_n$ that is
fixed, up to conjugacy, by each
element of ${\cal H}$ then $\tilde
{\cal H}$ can be chosen so that each element of $\tilde {\cal H}$
fixes $\gamma$.
\end{proposition}
The paper is organized as follows. In
section~\ref{prelim} we establish notation and
record known results for future reference. In
sections~\ref{A} and \ref{fin gen} we
prove that
${\cal H}_0$ is finitely generated and free abelian and that
the
above exact sequence is
virtually central. In
section~\ref{lifting} we prove
Proposition~\ref{lifting to Aut} and in
section~\ref{proof of main} we prove
Theorem~\ref{main}.
\section{Notation and Preliminaries} \label{prelim}
\subsection{Lifts to $C_{\infty}$} \label{lifts}
We assume that $F_n$ is identified with
$\pi_1(R_n,*)$ where $R_n$ is the rose with $n$
petals and with vertex
$*$. Let $\tilde R_n$ be the universal cover of
$R_n$ and let $\tilde
*$ be a preferred lift of $*$. The space of ends of
$\tilde R_n$ is a Cantor set that we denote $C_{\infty}$.
A {\it marked graph} is a graph $G$ with a preferred
vertex $v$
along with a homotopy equivalence $\tau : (R_n,*)
\to
(G,v)$ that identifies $\pi_1(G,v)$ with
$\pi_1(R_n,*)$ and so also
with $F_n$.
Denote the space of ends of
$\Gamma$ by ${\cal E}(\Gamma)$. The
marking homotopy equivalence lifts to an
equivariant map $\tilde \tau :
(\tilde R_n,\tilde *) \to (\Gamma,\tilde v)$ that induces
a
homeomorphism from $C_{\infty}$ to ${\cal E}(\Gamma)$
(See for example section 3.2 of
\cite{bfh:tits1}). We use this homeomorphism to
identify ${\cal E}(\Gamma)$ with $C_{\infty}$
and so for the rest of this paper refer to the space of
ends of $\Gamma$ as $C_{\infty}$.
An outer automorphism $\eta$
of $F_n$ can be represented, in many ways, by a
homotopy equivalence $f : G \to G$ of a marked
graph.
More precisely, $f : G \to G$ can be chosen so that
when
$\pi_1(G,v)$ is identified with $F_n$, the
outer automorphism of
$\pi_1(G,v)$ determined by $f$ agrees with $\eta$.
A preferred vertex $\tilde v $ in the
universal cover
$\Gamma$ of $G$ provides an identification of the
group ${\cal T}$ of covering
translations of $\Gamma$ with $\pi_1(G,v)$ and so
with $F_n$. The
action of
${\cal T}$ on $\Gamma$ extends to an action (of
$F_n$) on $C_{\infty}$ by homeomorphisms. If $T$
is non-trivial, then the endpoints of its axis are the
only fixed points for the action
of
$T$ on $C_{\infty}$.
Suppose that $f : G \to G$ represents $\eta$ and that
$\tilde f :
\Gamma \to \Gamma$ is a lift of $f$. For each $T
\in {\cal T}$ there exists a unique $T' \in
{\cal T}$ such that $\tilde f T = T'
\tilde f$. This defines an automophism $T \mapsto
T'$ of
${\cal T}$ and so an automorphism $\Phi : F_n
\to F_n$. It is easy to check that the outer
automorphism
class of $\Phi$ is $\eta$ and that as
$\tilde f$ varies over all lifts of $f$, $\Phi$ varies
over all automorphisms representing
$\eta$. The subgroup
${\cal T}(\tilde f)
\subset {\cal T}$ of covering translations that
commute with $\tilde f$ corresponds to the
fixed subgroup
$Fix(\Phi) \subset F_n$. Since $Fix(\Phi)$ is
quasiconvex and finitely
generated, the closure in
$C_{\infty}$ of the endpoints of axes of elements of
${\cal T}(\tilde f)$ is identified with the space of ends of
$Fix(\Phi)$ \cite{co:bcc}.
Each $\tilde f : \Gamma \to \Gamma$ extends to a
homeomorphism $\hat f: C_{\infty} \to C_{\infty}$ . Denote the
group of $F_n$-equivariant homeomorphisms of
$C_{\infty}$ by $EH(C_{\infty})$. The composite
$\Phi \mapsto \tilde f \mapsto \hat f$ defines an
injective homomorphism
from $Aut(F_n)$ to $EH(C_{\infty})$ that is independent of
the choice of homotopy
equivalence $f : G \to G$ representing $\eta$. (See for
example section 3.2 of
\cite{bfh:tits1}). We will
sometimes write
$\hat
\eta$ instead of
$\hat f$ where $\eta
\in Out(F_n)$ is the outer automorphism
determined by $f : G \to G$.
If
$\tilde L \subset \Gamma$ is a line with endpoints $P$
and $Q$, we denote the line
with endpoints $\hat f(P)$ and $\hat f(Q)$ by $\tilde
f_\#(\tilde L)$. If $\tilde L$ is
the axis of $T \in {\cal T}$ then $\tilde f_\#(\tilde L)$ is
the axis of the
covering translation $T'
\in {\cal T}$ satisfying $\tilde f T = T'
\tilde f$.
We conclude this subsection by recording some facts
for future reference. If $\tilde {\cal H}
\subset Aut(F_n)$ is a lift of ${\cal H} \subset Out(F_n)$
then we denote the corresponding
lift to $EH(C_{\infty})$ by $\hat {\cal H}$. For any
subgroup
${\mathbb F}
\subset F_n$, the closure in
$C_{\infty}$ of the endpoints of the axes of elements in
${\mathbb F}$ is denoted by
$C({\mathbb F})$ and is naturally identified with the
space of ends of ${\mathbb F}$. Although the $UPG$
property referred to in the
following lemma is not defined until the next section,
it is convenient to
place this result here. This part of the lemma is
quoted from
\cite{bfh:tits2} so there is no danger of circular
reasoning.
\begin{lemma} \label{rank>1} Suppose that ${\cal H}$ is a
subgroup of Out$(F_n)$ and that
${\mathbb F}$ is an ${\cal H}$-invariant (up to conjugacy)
subgroup of $F_n$ that is its own
normalizer. Then
\begin{enumerate}
\item There is a well-defined restriction
${\cal H}|{\mathbb F}\subset$ Out(${\mathbb
F})$.
\item If ${\cal H}$ is $UPG$, then ${\cal H}|{\mathbb F}$ is
$UPG$.
\item If ${\mathbb F}$ has rank at least two, then
any lift $\widetilde{{\cal H}|{\mathbb F}}
\subset Aut({\mathbb F})$ of ${\cal H}|{\mathbb F}$
extends uniquely to a lift
$\tilde {\cal H} \subset Aut(F_n)$.
\item If every element of a lift
$\widehat{{\cal H}|{\mathbb F}} \subset EH(C({\mathbb
F}))$
of ${\cal H}|{\mathbb F}$ fixes at least three
points then $\widehat{{\cal H}|{\mathbb F}}$ extends
uniquely to a lift $\hat {\cal H} \subset
EH(C_{\infty})$
\end{enumerate}
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{rank>1}} Since
${\mathbb F}$ is its own normalizer,
two automorphisms of $F_n$ that preserve
${\mathbb F}$ are conjugate by an element of
$F_n$ if and only if they are conjugate by an element
of ${\mathbb F}$. Part (1) follows
immediately. Part (2) is Lemma 4.13 of
\cite{bfh:tits2}. If ${\mathbb F}$ has rank at
least two, then there are no non-trivial inner
automorphisms that pointwise fix ${\mathbb
F}$. It follows that the restriction homomorphism
from the subgroup of
$Aut(F_n)$ representing $\eta \in Out(F_n)$ to the
subgroup of
$Aut({\mathbb F})$ representing $\eta|{\mathbb
F}$ is injective. Part (3) follows easily.
For part (4), note that a non-trivial
covering translation does not fix more than two
points in $C_{\infty}$ and so two lifts $\hat
\eta_1$ and $\hat \eta_2$ of $\eta$ that agree on
three points must be equal. \qed
\subsection{Relative train track maps}
We study an element ${\cal \eta} \in Out(F_n)$
through its lifts $\hat \eta
: C_{\infty} \to C_{\infty}$. We analyze the $\hat \eta$'s by
representing $\eta$ as a homotopy
equivalence $f : G \to G$ of a marked graph with
particularly nice properties
and studying the corresponding $\hat f$'s. In this
subsection we recall some
properties of $f : G \to G$.
A {\it filtered graph} is a marked graph along
with a filtration $\emptyset
= G_0 \subset G_1 \subset \dots \subset G_K = G$
where each $G_i$ is obtained
from $G_{i-1}$ by adding a single edge $E_i$. We
reserve the words path and
loop for immersions of the interval and the circle
respectively. If $\rho$ is a
map of the interval or the circle into $G$ or
$\Gamma$, then $[\rho]$ is the unique
path or loop that is homotopic to $\rho$ rel
endpoints if any. We say that a
homotopy equivalence
$f : G
\to G$ {\it respects the filtration} if each
$f(E_i) = E_iu_{i,f}$ for some loop $u_{i,f} \subset
G_{i-1}$. Let $FHE(G,{\cal V})$\ be the group (Lemma 6.2 of
\cite{bfh:tits2}) of homotopy classes,
relative to vertices, of filtration respecting homotopy
equivalences of $G$. There is
a natural map from $FHE(G,{\cal V})$\ to $Out(F_n)$. We say
that $\eta \in Out(F_n)$ is $UPG$ (for
unipotent with polynomial growth) if it is in the
image of $FHE(G,{\cal V})$\ for some
$G$. We say that a subgroup of $Out(F_n)$ is $UPG$
if each of its elements
is. The main theorem of
\cite{bfh:tits2} states that every $UPG$ subgroup
${\cal H}_0$ lifts to a subgroup
${\cal K}$ of $FHE(G)$ for some
$G$. We say that ${\cal K}$ is a {\it Kolchin
representative} of ${\cal H}_0$.
In general we will use ${\cal K}$ to denote a subgroup
of $FHE(G,{\cal V})$.
If
$\tilde L
\subset
\Gamma$ is a line and $l$ is
the highest parameter value for which $\tilde L$ crosses
a lift of $E_l$,
then define the {\it highest edge splitting}
of $\tilde L = \dots \tilde \sigma_{-1}\cdot \tilde
\sigma_0 \cdot \tilde
\sigma_1\dots$ by subdividing at the initial vertex
of each lift of $E_l$ that
is crossed, in either direction, by $\tilde L$. We refer to
the
vertices that
determine this decomposition as the {\it splitting
vertices} of the highest
edge splitting. If $\tilde f : \Gamma \to \Gamma$ is a
lift of $f \in FHE(G,{\cal V})$ and if
$\tilde L$ is
$\tilde f_\#$-invariant, then Lemma 5.
\ of
\cite{bfh:tits1}
implies that
$[\tilde f(\tilde
\sigma_j)]=
\tilde \sigma_{j+r}$ for some $r$ and all $j$. Roughly
speaking,
$\tilde f$ acts on $\tilde L$ by translating the highest edge
splitting by $r$ units.
We will need the following fixed point results.
\begin{lemma} \label{only one line}
\begin{itemize}
\item For any filtered graph $G$ and distinct
$P_1,P_2,P_3 \in
C_{\infty}$, there is a line $\tilde L$ in $\Gamma$
connecting $P_i$ to $P_j$ for some $1 \le i < j \le 3$
with the following property: If
$f
\in
FHE(G,{\cal V})$ and $P_1,P_2,P_3 \in Fix(\hat f)$ then $\tilde f$
fixes each highest
edge splitting vertex in $\tilde L$.
\item If $f \in FHE(G,{\cal V})$ and $\tilde f : \Gamma \to
\Gamma$ is fixed point free,
then $\hat f$ fixes exactly two points.
\end{itemize}
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{only one line}}
For the first item, let $\tilde L_{i,j} \subset \Gamma$
be the line connecting
$P_i$ to $P_j$ and let $l_{i,j}$ be the highest
parameter value for which $\tilde L_{i,j}$
crosses a lift of
$E_{l_{i,j}}$. Assuming without loss that $l_{1,2} \ge
l_{1,3},l_{2,3}$, let $\tilde L =
\tilde L_{1,2}$ and let
$\tilde v$ be any highest edge splitting vertex of $\tilde L$.
Suppose that $f \in FHE(G,{\cal V})$
and that $\hat f$ fixes each
$P_i$. If $\tilde f$ does not fix $\tilde v$ then $\tilde f$
translates the highest edge
splitting vertices of
$\tilde L$ away from one endpoint of $\tilde L$, say $P_1$,
and toward the other,
$P_2$. It follows that the highest edge splitting of
$\tilde L$ is bi-infinite and hence
that
$l_1 = l_2 =l_3$. This implies that $\tilde f$ translates
the highest edge splitting
vertices of $\tilde L_{2,3}$ away from $P_3$ and
toward $P_2$. But now $\tilde f$ translates the highest
edge splitting vertices of $\tilde
L_{1,3}$ away from $P_1$ and away from $P_3$.
This
contradiction verifies the first item.
We assume now that $f \in FHE(G,{\cal V})$ and that $\tilde f$ is
fixed point free. By the first item, it suffices to show
that $\hat f$ fixes
at least two points.
Implicit in Proposition 6.21 of \cite{bfh:tits1} is the
existence of a half-infinite ray
$\tilde R_+ \subset \Gamma$ with highest edge
splitting $\tilde R_+ = \tilde \sigma_0
\cdot \tilde \sigma_1\dots$ such that $[\tilde f(\tilde
\sigma_j)]=
\tilde \sigma_{j+r}$ for some $r>0$ and all $j \ge 0$.
Let $\tilde v_j$ be the initial
vertex of $\tilde \sigma_j$. Since
$\tilde f$ restricts to a bijection of vertices, there are
unique vertices
$\tilde v_j$ so that $\tilde f(\tilde v_j) = \tilde v_{j+r}$ for all
$j \in {\mathbb Z}$. Let
$\tilde \sigma_j$ be the path connecting $\tilde v_{j}$ to
$\tilde v_{j+1}$ for $j < 0$ and note
that
$[\tilde f(\tilde
\sigma_j)]= \tilde \sigma_{j+r}$ for all $j$. If $l$ is
the highest parameter value for which $\tilde R_+$
crosses a lift of $E_l$, then each $\tilde
\sigma_j$ contains exactly one lift of $E_l$ and these
lifts are distinct for distinct
$j$. (The $j < 0$ case follows from the $j \ge 0$ case
which holds by
construction.) It follows that $\dots \tilde \sigma_{-
1}\cdot \tilde
\sigma_0 \cdot \tilde \sigma_1\dots$ is an
embedded line whose
endpoints are both fixed by $\hat f$. \qed
\begin{lemma}\label{finite lifts} Up to conjugation
by covering translations, each
$\eta \in Out(F_n)$ has only finitely
many lifts $\hat \eta : C_{\infty} \to C_{\infty}$ whose fixed point
set contains at least three points.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{finite lifts}}
Represent
$\eta$ by a homotopy equivalence $f : G \to G$ of
some marked graph and let $\tilde f :
\Gamma \to
\Gamma$ be a lift whose extension $\hat f$ fixes at
least three points. Given
$P_1,P_2,P_3
\in Fix(\hat f)$, let $\tilde v \in \Gamma$ be the
unique vertex contained in
each of the lines connecting two of the $P_i$'s. The
bounded cancellation lemma
\cite{co:bcc} implies that there is a bound, depending
on $f$ but not on the choice of
the lift $\tilde f$ or the choice of points $P_i$, to the
length of the path $\tilde \sigma$
connecting
$\tilde v$ to
$\tilde f(\tilde v)$. In particular, the projected image
$\sigma$ takes on only finitely many
values as we vary the lift $\tilde f$ and the choice of
points $P_i$.
Suppose that $\tilde f_i$, $i=1,2$,
are lifts of $f$ and that there
exist lifts $\tilde v_i$ of a vertex $v$ and $\tilde
\sigma_i$ of a path $\sigma$ such that
$\tilde
\sigma_i$ is the path connecting $\tilde v_i$ to $\tilde
f_i(\tilde v_i)$. The covering
translation $T :
\Gamma \to \Gamma$ that carries $\tilde v_1$ to
$\tilde v_2$ satisfies $\tilde f_2 T = T \tilde
f_1$ and so conjugates $\hat f_1$ to $\hat f_2$.
\qed
\section{Property A} \label{A}
In this section we prove that a finitely generated,
solvable
$UPG$ subgroup ${\cal H}_0$ embeds in some ${\mathbb Z}^r$ and
hence that every solvable $UPG$
subgroup is free abelian. Some of the arguments
proceed by induction on the
skeleta $G_i$ of a filtered graph $G$. For this reason,
we do not assume in this
section that ${\cal K}$ is a Kolchin representative of some
${\cal H}_0$ but only that {\it ${\cal K}$ is
a finitely generated solvable subgroup of $FHE(G,{\cal V})$ with
the following feature: if $E_i$
is not a loop then some $u_{i,f}$ is non-trivial.} In
particular, we do
not assume that ${\cal K}$ injects into $Out(\pi_1(G))$
and we allow $G$ to have valence one
vertices. Note that if some $u_{i,f}$ is non-trivial,
then the terminal vertex of
$E_i$ is contained in a loop in $G_{i-1}$ and so must
have valence at least two in
$G_{i-1}$. Thus if $E_i$ is the first edge to contain a
vertex $v$, then either
$E_i$ is a loop or $E_i$ has $v$ as initial vertex. In
either case, $v$
is the initial vertex of $E_i$.
The group of lifts $\tilde f :
\Gamma \to \Gamma$ [respectively $\hat f : C_{\infty} \to
C_{\infty}$] of elements of ${\cal K}$ has a
natural projection to ${\cal K}$. By an {\it action of ${\cal K}$
on $\Gamma$ [respectively $C_{\infty}$]
by lifts} we mean a section of this projection. In
other words, an action by lifts is an assignment $f
\mapsto \tilde f$ [respectively $f
\mapsto \hat f]$ that respects composition. Every
action $s$ of ${\cal K}$ on $\Gamma$ by
lifts determines an action $\hat s$ of ${\cal K}$ on $C_{\infty}$ by
lifts and vice-versa.
For each edge $E_i \subset G$ choose, once and for
all, a lift $\tilde E_i^*
\subset \Gamma$. Define $s_i(f) : \Gamma \to
\Gamma$ to be the unique lift of $f
\in {\cal K}$ that fixes the initial endpoint of $\tilde E_i^*$
and note that {\it $s_i$ is an
action of ${\cal K}$ on $\Gamma$ by lifts}. Denote the
terminal endpoints of $E_i$ and
$\tilde E_i^*$ by $v_i$ and $\tilde v_i$ respectively.
\begin{defn} \label{propA} If $E_i$ is not a
component of $G_i$, denote the
component of
$G_{i-1}$ that contains $v_i$ by $B_i$ and the
(necessarily
$s_i(f)$-invariant) copy of the universal cover of
$B_i$ that contains $\tilde
v_i$ by $\Gamma_{i-1}$.
We say that ${\cal K}$ (and the choice of the $\tilde
E_i^*$'s) satisfies {\it Property
A} (for abelian) if:
\begin{itemize}
\item For each $i$, either $E_i$ is a component of
$G_i$
(in which case $E_i$ is a loop and $u_{i,f}$ is trivial
for all $f \in {\cal K}$) or
$\tilde v_i$ is a highest edge splitting vertex in a line
$\tilde L_i \subset
\Gamma_{i-1}$ that is
$s_i(f)_\#$-invariant for all $f \in {\cal K}$.
\item If $\tilde L_i$ and $\tilde L_j$ have the same
projection in $G$,
then $\tilde L_i =
\tilde L_j$ and
$\tilde v_i =
\tilde v_j$.
\end{itemize}
\vspace{2in}
If $\tilde L_i$ projects to an indivisible loop
$\alpha_i$ and if some $u_{i,f}$ is non-trivial, then
we say (abusing notation
slightly) that $\alpha_i$ is an {\it essential axis} and
that $E_i$ is an {\it
essential edge}. The set of essential axes is denoted
${\it A({\cal K})}$ and the set of essential edges is denoted
${\it E({\cal K})}$. If the
essential axis $\alpha$ is associated to $m_{\alpha}$
essential edges, then we
say that
$m_{\alpha}$ is the {\it multiplicity} of $\alpha$. In
the analogy with the mapping
class group of a compact surface, $A({\cal K})$ is the set of
reducing curves and ${\cal K}$ can be
chosen so that each $m_{\alpha} = 1$.
For each essential axis $\alpha$, let $\tilde
v_{\alpha}$ be the
preferred vertex of the preferred lift $\tilde \alpha$
($=\tilde v_i \in \tilde L_i$
for any
$\tilde L_i$ that projects to $\alpha$), let $T_{\alpha}$
be an indivisible
covering translation whose axis is $\tilde \alpha$ and
let $s_{\alpha}(f)$ be
the lift of $f \in {\cal K}$ that fixes $\tilde v_{\alpha}$.
Note that {\it $s_{\alpha}$ defines an action of ${\cal K}$
on $\Gamma$ by lifts}. Note also that if we think of
$\alpha$ as a closed path
with both endpoints at the projected image $v$ of
$\tilde v$, then $[f(\alpha)] =
\alpha$. It follows that $s_{\alpha}(f)_\#(\tilde
\alpha) = \tilde \alpha$ and hence
that $s_{\alpha}(f)$ commutes with $T_{\alpha}$.
\end{defn}
\begin{example} \label{Dehn} We set notation for
this geometric example as shown
below: $M$ is the orientable genus two surface with
one boundary component; $A \subset
M$ is an embedded non-peripheral annulus with
boundary components $X$ and $E_1$;
$S = M \setminus A$; $G \subset M$ is
the embedded graph (spine of $S$) shown below with
vertices $v_1$
and $v_2$ and edges $E_1, \dots, E_5$; ${\cal H}_0 \cong
\mathbb Z$ is the subgroup
of the mapping class group of $M$ generated by the
Dehn twist across the
annulus
$A$; and ${\cal K} \cong \mathbb Z \subset FHE(G,{\cal
V})$ is generated by $E_i
\mapsto E_i$ for $i \ne 2$ and $E_2 \mapsto
E_2E_1$.
\vspace*{2.5in}
The universal cover of $M$ is identified with a
convex subset of the hyperbolic
plane
$H$ in the Poincare disk model. Choose a lift $\tilde L$
of $E_1$ and let $\tilde A$ be the
lift of
$A$ that contains $\tilde L$. On
the left [respectively right] of $\tilde A$ is a copy $\tilde
S_1$ [respectively $\tilde S_2$] of
the universal cover of $S$. Given $D \in {\cal H}_0$, let
$\tilde D_i : \tilde M \to \tilde M$ be the
lift that is the identity on $\tilde S_i$.. Then $\hat
D_i$ fixes the endpoints of
$\tilde S_i$ and
$\hat D_1$ and $\hat D_2$ are the only lifts of $D$
that fix the endpoints of $\tilde L$
and at least one other point. Note also that if $D$ is a
Dehn twist of order $k$ around
$A$ then $\tilde D_1 = T^k \tilde D_2$ where $T$ is the
indivisible covering translation that
preserves $\tilde L$.
\vspace*{2.5in}
On the graph level, there is a single essential edge
$E_2$ with
essential axis $\alpha =
E_1$. The lifts $\tilde D_1$ and $\tilde D_2$ correspond to
$s_{\alpha}(f)$ and
$s_2(f)$ respectively.
The free group $\pi_1(S)$ is
generated by $E_2E_1 \bar E_2,
E_3E_1\bar E_3, E_4$ and $E_5$. The group ${\cal H}_0$
can be extended to a non-$UPG$
abelian group ${\cal H}$ by adding a generator that
restricts to a pseudo-Anosov homeomorphism
on $S$. This can be represented by a relative train
map $f : G \to G$ that is the identity on $E_1$ and
$E_2$ and that has a single
exponentially growing stratum with edges
$E_3,E_4$ and $E_5$. \end{example}
\vspace{.1in}
The following lemma justifies our notation.
\begin{lemma}\label{akr implies abelian} If ${\cal K}$
satisfies property A,
then there is an injective homomorphism
$\Phi_{{\cal K}} : {\cal K} \to {\mathbb Z}^r$
where
$r$ is the cardinality of $E({\cal K})$. In particular, ${\cal K}$ is
free abelian.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{akr implies
abelian}} If $E_i$ is an
essential edge with essential axis $\alpha$ then
$s_i(f)$ and $s_{\alpha}(f)$ are
lifts of $f$ that commute with $T_{\alpha}$ and so
differ by an iterate of
$T_{\alpha}$. Define a homomorpism $\phi_{{\cal K}}^i
: {\cal K} \to {\mathbb Z}$ by
$s_i(f) = T_{\alpha}^{\phi_{{\cal K}}^i(f)} s_{\alpha}(f)$
and define $\Phi_{{\cal K}}$ to be the
product of the
$\phi_{{\cal K}}^i$'s. Then $f$ is in the kernel of
$\Phi_{{\cal K}}$ if and only
if $f(E_i) = E_i$ for each essential edge $E_i$ if and
only if $u_{i,f}$ is
trivial for each essential edge $E_i$.
If $f \ne$ identity, then there is a smallest
parameter value $i>1$ for
which
$u_i(f)$ is non-trivial. Since $f|G_{i-1}$ is
the identity, $s_i(f)$
restricts to a non-trivial covering translation of
$\Gamma_{i-1}$. The line
$\tilde L_i
\subset
\Gamma_{i-1}$ is $s_i(f)_\#$-invariant so must be
the
axis of that covering
translation. Thus $E_i \in E({\cal K})$ and
$f$ is not in the kernel of $\Phi_K$. This proves
that $\Phi_K$ is
injective.
\qed
\vspace{.1in}
The following lemma produces a pair of fixed
points in $C_{\infty}$ or equivalently a fixed line in
$\Gamma$.
\begin{lemma} \label{two fixed points} Suppose
that $\hat \psi_1,
\dots, \hat \psi_m : C_\infty \to C_\infty$ are lifts
of elements of a
finitely generated UPG subgroup. If the
$\hat
\psi_j$'s commute, then
$\cap_{j=1}^m$Fix($\hat \psi_j)$ contains at least
two points.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{two fixed
points}} Choose a Kolchin
representative ${\cal K}$ of the UPG subgroup. There are
elements
$f_j \in {\cal K}$ and commuting lifts $\tilde f_j : \Gamma
\to \Gamma$ so that $\hat f_j = \hat
\psi_j$ for $j=1,\dots,m$.
If $\tilde f_1$ is fixed point free then
(Lemma~\ref{only one line}) Fix($\hat f_1)$
is a pair of points
$\{P,Q\}$. Since
$\hat f_j$ commutes with $\hat f_1$, $\hat f_j$
setwise preserves
$\{P,Q\}$ and we need only show that $\tilde f_j$ does
not reverse the
orientation on the line $\tilde L$ connecting $P$ and
$Q$. Let $\tilde L = \dots \tilde
\sigma_{-1}\cdot \tilde \sigma_0 \cdot \tilde
\sigma_1\dots$ be highest edge splitting of $\tilde L$. If
$\tilde f_j$ reverses the
orientation on $\tilde L$, then for some $j$, $[\tilde f(\tilde
\sigma_j)]$ equals
$\tilde \sigma_j$ with its orientation reversed. The
projected image $\sigma_j$ determines
a conjugacy class in $F_n$ that is periodic but not
fixed under the action of the outer
automorphism determined by $f$. This contradicts
Proposition 4.5 of
\cite{bfh:tits2}.
Suppose next that $Fix(\tilde f_1) \ne \emptyset$
and that the group ${\cal T}(\tilde
f_1)$ of covering translations that commute with
$\tilde f_1$ is trivial. The
fixed point set of
$f_j$, and hence of $\tilde f_j$, is a union of vertices
and edges. Since
${\cal T}(\tilde f_1)$ is trivial,
each vertex
$v \in G$ has at most one lift $\tilde v \in Fix(\tilde
f_1)$. Since $\tilde f_j$ commutes
with $\tilde f_1$, it preserves
$Fix(\tilde f_1)$. We conclude that each vertex in
$Fix(\tilde f_1)$ is fixed by each
$\tilde f_j$. (Recall that $f_j$ fixes each vertex in $G$.)
The edges with initial vertex in
$Fix(\tilde f_1)$ project to distinct edges in $G$. Let $\tilde
E_k$ be the unique
such edge with minimal
$k$, let $B'$ be the component of
$G_{k-1}$ that contains the terminal endpoint of
$E_k$ and let
$\Gamma_{k-1}' \subset \Gamma$ be the copy of
the universal cover of $B'$ that
contains the terminal endpoint of $\tilde E_k$. Since
each $f_j$ maps an initial segment
of $E_k$ to an initial segment of $E_k$,
$\Gamma_{k-1}'$ is $\tilde
f_j$-invariant for each $j$. By our choice of $k$, the
restriction $\tilde
f_1|\Gamma_{k-1}'$ is fixed point free. We can
now repeat the argument of the
first case on the restriction of the $\tilde f_j$'s to
$\Gamma_{k-1}'$.
\vspace{2in}
Finally, suppose that ${\cal T}(\tilde f_1)$ is non-
trivial. Identify
${\cal T}(\tilde f_1)$ with a subgroup ${\mathbb F}$ of
$F_n$. The
space of ends of
${\mathbb F}$ is the closure
$C({\mathbb F}) \subset $ Fix($\hat f_1) \subset
C_{\infty}$ of the endpoints
of axes for elements of ${\cal T}(\tilde f_1)$. Since $\tilde
f_j$ commutes with
$\tilde f_1$, the automorphism of ${\cal T}$
determined by $\tilde f_j$ preserves
${\cal T}(\tilde f_1)$ and
$\hat f_j = \hat \psi_j$ preserves $C({\mathbb F})$.
By
Lemma~\ref{rank>1}, the
$\psi_j|{\mathbb F}$'s are contained in a $UPG$
subgroup of Out(${\mathbb F}$).
We argue by induction on
$m$, the $m=1$ case following from the fact that
$C({\mathbb F})$ contains at
least two points. Suppose that $m > 1$. By the
inductive hypothesis, there exist $P,Q \in
C({\mathbb F})$ that are fixed by
the $m-1$ maps
$\hat f_2|C({\mathbb F}),\dots,\hat
f_m|C({\mathbb F})$. Since $\hat
f_1|C({\mathbb F})$ is the identity, $P$ and
$Q$ are fixed by each
$\hat f_j$.\qed
\vspace{.1in}
\begin{cor}\label{fg solv} Every finitely generated
solvable UPG subgroup ${\cal H}_0$
has a Kolchin representative that
satisfies property A; in particular,
${\cal H}_0$ is free abelian.
\end{cor}
\noindent{\bf Proof of Corollary~\ref{fg solv}}
Choose a Kolchin representative
${\cal K} \subset FHE(G,{\cal V})$ for ${\cal H}_0$. We work our way up
the strata
$E_i$, modifying ${\cal K}$ so that it satisfies
property A. Denote the restriction of
${\cal K}$ to $G_i$ by
${\cal K}_i$. Since $G_1$ is a single edge, ${\cal K}_1$ has
property A; we may assume by
induction that ${\cal K}_{i-1}$ satisfies property A and is
therefore abelian.
If $E_i$ is a
component of $G_i$ then ${\cal K}_i$ satisfies
property A. We may therefore assume that
$\Gamma_{i-1}$ and the action
$\hat s_i$ of
${\cal K}$ on
$C_{\infty}$ are defined. The ends of $\Gamma_{i-1}$
define a subset $C_{\infty}^*
\subset C_{\infty}$.
Let $R :
{\cal K}_i \to {\cal K}_{i-1}$ be the restriction
homomorphism. If $R$ is an
isomorphism, then ${\cal K}_i$ is abelian. Choose
generators
$g_1,\dots,g_m$ for ${\cal K}_i$ representing
$\psi_1,\dots,\psi_m \in {\cal H}_0$, and let
$\hat \psi_j^* =
\hat s_i(g_j)|C_{\infty}^*$. Note that the
$\hat \psi_j^*$'s are lifts of elements
of the $UPG$ subgroup ${\cal H}_0|\pi_1(G_{i-1})$ and
that, since $s_i$ is an action
and the $g_j$'s commute, the
$\hat \psi_j^*$'s commute. Lemma~\ref{two fixed
points} therefore produces
$P_i,Q_i \in C_{\infty}^*$ that are fixed by each
$\hat s_i(g_j)$ and hence by
$\hat s_i(f)$ for each $f \in {\cal K}$. The line $\tilde L_i$
connecting $P_i$ to $Q_i$
is $s_i(f)_\#$-invariant for each $f \in {\cal K}$.
If $\tilde L_i$ has the same image in $G$ as $\tilde L_j$
for some $j < i$, then
after replacing $\tilde E_i^*$ by a translate if necessary,
we may assume that $\tilde
L_i = \tilde L_j$. Choose a highest edge splitting vertex
$\tilde w_i$ of $\tilde L_i$
and apply the sliding operation of \cite{bfh:tits1}
simultaneously to
each
$f \in
{\cal K}$ reattaching $\tilde E_i^*$ so that its terminal vertex
is $\tilde w_i$. This does
not change
$s_i(f)|\Gamma_{i-1}$ and
respects the group structure of ${\cal K}$. A new Kolchin
representative (still
called ${\cal K}$) is produced that agrees with the old one
on $G_{i-1}$ and has the
additional feature that $\tilde w_i$ is the terminal
vertex of $\tilde E_i^*$; in
particular, the first condition of property A is
satisfied. Since
$\tilde w_i$ can be any highest edge splitting vertex, we
may assume without loss
that the second condition of property A is also
satisfied.
Suppose now that the
kernel $K$ of $R$ is non-trivial. Each
$f^* \in K$ satisfies $E_j \mapsto E_j$ for $1 \le j
\le i-1$ and $E_i \mapsto
E_iu_{i,f^*}$. Thus the restriction of $s_i(f^*)$ to
$\Gamma_{i-1}$
is the covering translation determined by the lift of
$u_{i,f^*}$ beginning at
$\tilde v_i$. The assignment
$f^*
\mapsto u_{i,f^*}$ defines a \lq suffix\rq\
homomorphism from $K$ into the free
group
$\pi_1(G,v_i)$. Since
${\cal K}$, and hence $K$, is solvable, the image of this
homomorphism is
isomorphic to $Z$. Thus the non-trivial
$s_i(f^*)|\Gamma_{i-1}$'s
have a common axis $\tilde L_i$. Choose $f^*$ so that
$s_i(f^*)|\Gamma_{i-1}$ is
non-trivial. For each $f \in
{{\cal K}}$, $s_i(ff^*f^{-1})|\Gamma_{i-1}$ is
a non-trivial covering translation with axis
$s_i(f)_\#(\tilde L_i)$.
Since $K$ is normal in ${\cal K}$,
$s_i(f)_\#(\tilde L_i) = \tilde L_i$. We have verified that
$\tilde L_i$ is
$s_i(f)_\#$-invariant for each $f \in {\cal K}$. The proof
now concludes as in
the previous case.
\qed
\section{Abelian subgroups are finitely generated}
\label{fin gen}
In this section we prove that an abelian UPG
subgroup ${\cal H}_0$ is finitely
generated. Section~\ref{A} and the exact sequence
$1\to{\cal H}_0\to{\cal H}\overset
\Omega \to{\mathbb Z}^b\to 1$ then implies that every
solvable subgroup of Out($F_n$) is
finitely generated. This fact was originally proved by
H. Bass and A. Lubotzky
\cite{bl:niltech}. We also show that the above
sequence is a virtually
central extension.
If ${\cal K}$ is a Kolchin representative of ${\cal H}_0$ that
satisfies property A, $ f
\in
{\cal K}$ and $\Phi_{{\cal K}} : {\cal K} \to
{\mathbb Z}^r$ is the embedding of Lemma~\ref{akr
implies abelian}, then the
coordinates of $\Phi_{{\cal K}}(f)$ are defined in terms of
the $s_i(f)$'s and the
$s_{\alpha}(f)$'s. Our goal is to recognize these lifts
of $f$ by their induced
actions on $C_{\infty}$, thus removing their
dependence on the choice of ${\cal K}$.
We begin placing an
additional restriction (the second item below) on our
Kolchin graphs and justifying our
assumption from section~\ref{A} (the first item
below).
\noindent \begin{lemma} \label{conditioned} For
every finitely generated UPG subgroup
${\cal H}_0$ there is a Kolchin representative ${\cal K}$ with the
following properties.
\begin{itemize}
\item If $E_i$ is not a loop then $u_{i,f}$ is non-
trivial for some $f \in {\cal K}$.
\item Every vertex of $G$ is the initial vertex of at
least two edges.
\end{itemize}
\end{lemma}
\noindent {\bf Proof of Lemma~\ref{conditioned}}
Start with any Kolchin representative
${\cal K}$. After restricting to a subgraph if necessary, we
may assume that $G$ has
no valence one vertices. We will show that if either
of the two properties fail, then we
can replace
$G$ by a graph with fewer edges. This process
terminates after finitely many steps to
produce the desired Kolchin representative.
If the first property fails, then $E_i$ is
pointwise fixed for each $f \in
{\cal K}$ and we may collapse it to a point.
Suppose then
that the first property holds, that $v$ is a vertex and
that
$E_i$ is the first edge that contains $v$. If $E_i$ is
not a loop, then
$u_{i,f}$ is a non-trivial loop in
$G_{i-1}$ containing the terminal endpoint of $E_i$
for some $f \in {\cal K}$. Since loops
are assumed to be immersed, they cannot pass
through valence one vertices and $v$ must
be the initial vertex of
$E_i$. For the same reason, $v$ must
also be the initial vertex of the second edge that is
attached to it.
Suppose then that $E_i$ is a loop and that
$E_{j_0}, \dots ,E_{j_m}$
are the other edges that contain $v$. If the second
property fails then
$E_{j_0}, \dots, E_{j_m}$ are non-loops with $v$ as
terminal endpoint and
$u_j(f) = E_i^{k_j(f)}$ for $j=j_0,\dots,j_m$.
Redefine ${\cal K}$ by
replacing each $k_j(f)$ with $k_j(f)-k_{j_0}(f)$. This
can be achieved
on the graph level by sliding $v$ around $E_i$
$j_0(f)$ times. This
has no effect on the outer automorphism determined
by $f$ and respects the
group structure on ${\cal K}$. The edge
$E_{j_0}$ is now fixed by each $f \in {\cal K}$ and so can
be collapsed to
point. \qed
\begin{defn} We say that a Kolchin representative
${\cal K}$ for ${\cal H}_0$ is an
{\it abelian Kolchin representative} if it satisfies property A and the
conclusions of
Lemma~\ref{conditioned}.
\end{defn}
We now turn to the task of determining, from the
action of ${\cal K}$ on
$C_{\infty}$ by lifts, if the axis of a given covering
translation projects to an
element of $A({\cal K})$.
Recall that in the analogy with the mapping class
group of a
compact surface $M$, $A({\cal K})$ corresponds to the set
of reducing curves in the minimal
reduction. Such reducing curves are completely
characterized as follows (See
Example~\ref{Dehn}, the proof of
Proposition~\ref{lifting to Aut} or
\cite{ht:surfaces}): The free homotopy class
determined by a closed curve $\alpha
\subset M$ is an element of the minimal reducing
set for the mapping class represented
by a homeomorphism $h : M
\to M$ if and only if for some (and hence each)
covering translation $T : \tilde M \to \tilde
M$ corresponding to $\alpha$, there are two lifts $\tilde
h_1,
\tilde h_2 : \tilde M \to \tilde M$ that commute with $T$
and whose extensions $\hat h_i$ over
the \lq circle at infinity\rq\ fix at least three points.
(If the free homotopy class
of $\alpha$ is fixed by $h$ but $\alpha$ is not one of
the reducing curves then there is
one such lift
$\tilde h$.) The analogous result in the non-geometric
case is given in
Corollary~\ref{independence of K}.
\begin{defn} For any $\psi \in {\cal H}_0$ and covering
translation $T$, define
$IL(\psi,T)$ (for Interesting Lifts) to be the set of lifts
$\hat \psi : C_{\infty} \to C_{\infty}$ that commute
with $T$ and fix
at least three points.
\end{defn}
Recall that $\alpha \in A({\cal K})$ has a preferred lift
$\tilde \alpha$ and an
indivisible covering translation $T_{\alpha}$ with
axis equal to $\tilde \alpha$.
The next lemma states that the actions $s_{\alpha}$
and $s_i$ produce
interesting lifts.
\begin{lemma}\label{interesting} Suppose that ${\cal K}$
is an abelian Kolchin representative\ and that $\alpha$ is
the essential axis for $E_i \in E({\cal K})$. Then
$\bigcap_{f \in {\cal K}}Fix(\hat
s_{\alpha}(f))$ and
$\bigcap_{f \in {\cal K}}Fix(\hat s_i(f))$ each contain at
least three points. In
particular, if
$f$ represents $\psi \in {\cal H}_0$, then $\hat
s_{\alpha}(f),\hat s_i(f) \in
IL(\psi,T_{\alpha})$.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{interesting}}
There is a preferred
topmost splitting vertex $\tilde v_{\alpha} \in\tilde
\alpha$.
Choose an edge $\tilde E_j$ with initial vertex $\tilde
v_{\alpha} $; by Lemma~\ref{conditioned}, we may
assume that $E_j \ne \alpha $.
If $u_{j,f}$ is trivial for all $f \in {\cal K}$, then $E_j$ is a
loop that is fixed by
each
$f
\in {\cal K}$. In this case, let $R_{\alpha}$ be an
endpoint of the axis for
$E_j$ that contains $\tilde E_j$. If some
$u_{j,f}$ is non-trivial, then $\tilde L_j$ is defined and
we
choose $R_{\alpha}$ to be an endpoint of the
translate $\tilde L_j'$ of $\tilde L_j$
associated to
$\tilde E_j$. In either case, each $\hat s_{\alpha}(f)$
fixes $R_{\alpha}$.
\vspace*{3in}
The proof for $s_i(f)$ is similar. By
Lemma~\ref{conditioned}
there exists an edge $\tilde E_l \ne \tilde E_i^*$, with the
same initial endpoint as
$\tilde E_i^*$. Define $R_i \in$ Fix($\hat s_i(f))$ as in
the previous case using $ E_l$
in place of $ E_j$.\qed
\vspace{.1in}
If $f \in {\cal K}$ represents $\psi \in {\cal H}_0$, then we use
$\hat s(f)$ and $\hat
s(\psi)$ interchangably. We refer to the
$\hat s_i(\psi)$'s and the
$\hat s_{\alpha}(\psi)$'s as the {\it canonical lifts}
of $\psi$. The next lemma and
corollary show that $A({\cal K})$ depends only on ${\cal H}_0$
and not on the choice of ${\cal K}$ and
that one can decide if $\hat \psi$ is canonical from
its action on $C_{\infty}$.
\begin{lemma} \label{axes are canonical} Suppose
that ${\cal K}$ is an abelian Kolchin representative\ and
that $\tilde L$ is a line with endpoints $P,Q \in C_{\infty}$.
Suppose further that:
\begin{itemize}
\item $\tilde f : \Gamma \to \Gamma$ is a lift of
some $f \in {\cal K}$
\item Fix($\hat f$) contains $P,Q$ and at least one
other point.
\item $\tilde f$ does not fix the highest edge splitting
vertices of $\tilde
L$ .
\end{itemize}
Then there exists $i$ and a covering translation $T$
such that $T(\tilde
L) =
\tilde L_i$ and $\tilde f = T^{-1} s_i(f) T $.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{axes are
canonical}} By
Lemma~\ref{only one line}, Fix($\tilde f) \ne
\emptyset$. Choose an arc $\tilde \sigma$
that intersects Fix($\tilde f$) only in its initial vertex,
say $\tilde
p_1$, and that intersects the splitting vertices of $\tilde
L$ only in its terminal
vertex, say $\tilde v$. Let $\tilde E_i'$ be the first edge of
$\tilde
\sigma$ and let $\tilde p_2$ be its terminal endpoint.
Since
$\tilde f$ fixes
$\tilde p_1$ but not
$\tilde p_2$, $u_{i,f}$ is non-trivial and $\tilde L_i$ is
defined. Let $T$ be
the covering translation that carries $\tilde E_i'$ to
$\tilde E_i^*$ and let $\tilde
L_i' = T^{-1}(\tilde L_i)$ be the translate of $\tilde L_i$
associated to $\tilde E_i'$.
By construction, the ends of $\tilde L_i'$ are $\hat f$-
invariant.
\vspace{2in}
If
$\tilde L_i'
\ne
\tilde L$ then there is an endpoint, say $S$, of $\tilde L_i'$
that is
neither $P$ nor $Q$. Lemma~\ref{only one line}
implies that the highest
edge splitting vertices of either $\tilde L_{PS}$ (= the
line connecting $P$
to $S$) or
$\tilde L_{SQ}$ are fixed. But $\tilde L_{PS}$ consists of a
segment of $\tilde L$,
a segment of $\tilde L_i'$ and perhaps a segment of
$(\tilde \sigma \setminus \tilde E_i)$. The last segment
contains no fixed
vertices. By construction, the highest edge splitting
vertices of the
other segments are not fixed. Thus the highest edge
splitting vertices of $\tilde L_{PS}$ are not fixed. The
symmetric argument
applies to $\tilde L_{SQ}$ and yields the desired
contradiction. We conclude
that $\tilde L_i' = \tilde L$ and that $T(\tilde v_i) = \tilde p_2 =
\tilde v$. Since
$T^{-1} s_i(f) T$ and $\tilde f$ both fix $\tilde p_1$, they
must be equal.
\qed
\vspace{.1in}
\begin{cor} \label{independence of K}If ${\cal K}$ is an
abelian Kolchin representative\ of ${\cal H}_0$, then:
\begin{itemize}
\item A covering translation $T$ corresponds to an
essential axis of ${\cal K}$ if and
only if
$IL(\psi,T)$ contains at least two elements for some
$\psi \in {\cal H}_0$.
\item For each $\alpha \in A({\cal K})$ and each $\psi
\in {\cal H}_0$,
$IL(\psi,T_{\alpha}) = \{\hat s_i(\psi): E_i$ is an
essential edge with axis
$\alpha\} \cup \{ \hat s_{\alpha}(\psi)\}$
\item $A({\cal K}) = A({\cal H}_0)$ depends only on ${\cal H}_0$ and
not on
${\cal K}$.
\end{itemize}
\end{cor}
\noindent{\bf Proof of Corollary~\ref{independence
of K}} The
first and second items are a direct consequence of
Lemma~\ref{interesting} and
Lemma~\ref{axes are canonical}. The third item
follows from the first.
\qed
\begin{lemma} \label{fg} Every abelian UPG
subgroup ${\cal H}_0$ is finitely
generated.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{fg}} Choose an
increasing sequence
${\cal H}_1 \subset {\cal H}_2 \subset
\dots$ of finitely generated subgroups whose union
is ${\cal H}_0$. The
cardinality of $A({\cal H}_j)$ and the multiplicities of the
elements of $A({\cal H}_j)$ are
uniformly bounded (by the maximum number of
edges in a marked graph with the property
that the terminal vertex of each edge has valence at
least three.) Since $A({\cal H}_j)
\subset A({\cal H}_{j+1})$, we may assume after passing
to a subsequence, that $A({\cal H}_j)$
and the multiplicities are independent of $j$. In
particular, there is a fixed $r$ so
that
$\Phi_j :{\cal H}_j \to {\mathbb Z}^r$ where
$\Phi_j$ is the embedding of
Lemma~\ref{akr implies abelian} and where ${\cal
H}_j$ has been identified with a
Kolchin representative ${\cal K}_j$.
Fix $\psi \in {\cal H}_1, \alpha \in A({\cal H}_1)$ and $j
\ge 1$; Let $IL(\psi,T_{\alpha}) =
\{b_1, b_2,\dots \}$. For each $b_k$ and $b_l$, there
is an integer $p(k,l)$ such that
$b_lb_k^{-1} = T_{\alpha}^{p(k,l)}$. For any fixed $f
\in {\cal K}$,
Corollary~\ref{independence of K} implies that each
coordinate
$\phi_j^i(f)$ of $\Phi_j(f)$ is one of the $p(k,l)$'s.
In
particular,
$\Phi_j(f)$ takes on only finitely many values as $j$
varies.
Let $g_1,\dots,g_q$ be generators of ${\cal H}_1$.
After passing to a
subsequence we may assume that $\Phi_j(g_i)$ is
independent of $j$ for $i=1\dots
q$. The lattice $\Phi_j({\cal H}_1)$ is therefore
independent of $j$. It is
contained with finite index in a maximal lattice $L$
of rank $q$. The lattice
$\Phi_j({\cal H}_j)$ has rank $q$ and contains
$\Phi_j({\cal H}_1)$ so is contained in $L$. In
particular, the index of
${\cal H}_1$ in
${\cal H}_j$ is uniformly bounded. It follows that
${\cal H}_j = {\cal
H}_{j+1}$ for all sufficiently large $j$.
\qed
\vspace{.1in}
We close this section by showing that
$1\to{\cal H}_0\to{\cal H}\overset
\Omega\to{\mathbb Z}^b\to 1$ is a virtually central
extension; the proof is a variation on that
of Lemma~\ref{fg}.
\begin{lemma}\label{virtually central} $A({\cal H}_0)$ is
${\cal H}$ invariant (up to conjugacy).
There is a finite index subgroup ${\cal H}' \subset {\cal H}$
whose actions on ${\cal H}_0$ and on $A({\cal H}_0)$
are trivial.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{virtually
central}} For each $\psi \in
{\cal H}_0$, each $\alpha \in A({\cal H}_0)$ and each lift $\hat
\eta$ of each $\eta \in {\cal H}$,
conjugation by
$\hat \eta$ sends
$IL(\psi,T_{\alpha})$ to
$IL(\psi',T')$ where $\psi' = \eta \psi \eta^{-1}$ and
$T' = \hat \eta T_{\alpha}
\hat \eta ^{-1}$. In particular, $IL(\psi,T_{\alpha})$
and $IL(\psi',T')$
have same cardinality. Corollary~\ref{independence
of K} therefore
implies that the axis of $T'$ projects to an element of
$A({\cal H}_0)$ and so
$A({\cal H}_0)$ is invariant under the action of $\eta$.
Since
$A({\cal H}_0)$ is finite, there is a finite index subgroup
for which this action is trivial.
We assume now that the action on $A({\cal H}_0)$ is
trivial. Choose $\hat
\eta$ so that
$T' = T_{\alpha}$ or equivalently so that $\hat \eta$
commutes with
$T_{\alpha}$. Choose an abelian Kolchin representative\
${\cal K}$ for ${\cal H}_0$ and let $\Phi :{\cal H}_0 \to {\mathbb
Z}^{r}$
be the embedding of Lemma~\ref{akr implies
abelian}.
For any
pair of elements
$b_k,b_l
\in
IL(\psi,T_{\alpha})$, there is an integer
$p(k,l)$ such that $b_lb_k^{-1} = T_{\alpha}^{p(k,l)}$.
Let $P(\psi)$ be
the finite collection of integers that occur as $p(k,l)$'s.
Since
conjugation by $\hat \eta$ carries
$IL(\psi,T_{\alpha})$ to
$IL(\psi',T_{\alpha})$ and since $\hat \eta $
commutes with $T_{\alpha}$,
$P(\psi)=P(\psi')$. By construction,
$\Phi(\psi')$ therefore takes on only finitely many
values as $\eta$ varies over
${\cal H}$ and $\psi \in {\cal H}_0$ is fixed. Since $\Phi$ is an
embedding, $\psi'$ takes on
only finitely many values. After passing to a finite
index subgroup we may
assume that the action of
${\cal H}$ by conjugation on $\psi$ is trivial. After
applying this argument to a
finite generating set for ${\cal H}_0$, we see that the action
of ${\cal H}$ on ${\cal H}_0$
is virtually trivial.\qed
\section{Proof of Proposition~\ref{lifting to Aut}}
\label{lifting}
The following lemma produces interesting lifts for
(iterates of) individual elements
of $Out(F_n)$.
\begin{lemma} \label{lift one} Suppose that $n \ge
2$ and that $\eta \in Out(F_n)$.
After replacing
$\eta$ by an iterate if necesssary, there is a lift
$\hat\eta \in EH(C_{\infty})$
that fixes at least three points. If $\gamma$ is a
non-trivial
primitive element of $F_n$ that is fixed (up to
conjugacy) by $\eta$ and if $T$ is a
covering translation corresponding to $\gamma$,
then we may choose $\hat\eta$ to commute
with $T$.
\end{lemma}
G. Levitt and M. Lustig inform us that they are
developing techniques
to understand the dynamics of
automorphisms of hyperbolic groups on the boundary
of the group, and that
they are able to provide an alternate
proof of this lemma.
\noindent{\bf Proof of Lemma~\ref{lift one}} We
assume at first that $\gamma$ and $T$
are given.
The case that $\eta$ is
realized as an isotopy class of a surface
homeomorphism $h : S \to S$ is well
known (see for example Lemma 3.1 of
\cite{ht:surfaces}): The Thurston classification
theorem implies, after replacing
$h$ by an iterate if necessary, that $S$ divides along
annuli into
subsurfaces with negative Euler characteristic on
which $h$ is either the
identity or is pseudo-Anosov. Assuming that the
reduction is done along the
minimal number of annuli, we may choose the
curve representing $\gamma$ to lie
in one of the subsurfaces $S_i$. If
$h|S_i$ is the identity, then it has a lift that fixes the
ends determined by
$S_i$ (See Example~\ref{Dehn}). If $h|S_i$ is
pseudo-Anosov then $\gamma$
determines a boundary component of $S_i$ and
there is a lift fixing $\tilde
\gamma$ and the endpoints of the singular leaves of
the pseudo-Anosov foliations
associated to that boundary component.
We now turn to the general case and argue by
induction. Since
every outer automorphism of $F_2$ is realized as a
surface isotopy class, the
preceding argument handles the $n=2$ case. We may
therefore assume that the
lemma holds for free groups of rank less than $n$.
The smallest free factor ${\cal F}(\gamma)$ that
contains
$\gamma$ \cite{bfh:tits1} is $\eta$-invariant (up to
conjugacy). If $1 < $ rank$({\cal
F}(\gamma)) < n$, then the inductive hypothesis
provides a lift of $\eta|{\cal
F}(\gamma)$ with the desired properties. Extending
this lift (Lemma~\ref{rank>1}) to
all of $F_n$ completes the proof. We may therefore
assume that rank$({\cal F}(\gamma))$
is either 1 or
$n$.
Choose an improved relative train track map $f :
G \to G$
representing an iterate of $\eta$. Theorem 6.4
of \cite{bfh:tits1} contains a list of all the properties of
$f : G \to G$
that are used in this proof.
The conjugacy class of $\gamma$ determines a loop
in $G$ that we also
call
$\gamma$.
If $\cal F(\gamma)$ has rank $n$, then $\gamma$
must cross an edge in
the highest stratum of $G$. If this stratum is
exponentially growing, then
Theorem 6.4 of \cite{bfh:tits1} implies that there is
an
$\eta$-invariant (up to conjugacy) subgroup
${\mathbb F}$ that is its own normalizer and
with the following additional property: There is a
conjugacy between the the outer
automorphism $\eta|{\mathbb F}$ and a
pseudo-Anosov mapping class $h : S \to S$.
Moreover, the conjugacy carries
$\gamma$ to a boundary component of $S$. Since
${\mathbb F}$ has rank at
least two, lifts of $\eta|{\mathbb F}$ extend uniquely
to lifts of $\eta$. We
may therefore assume that
${\mathbb F} = F_n$ and hence that
$\eta$ is represented by $h$. We are now reduced to
a previous case.
If the top
stratum $G_m$ is not exponentially growing, then
$G_m$ is a single edge $E_m$.
The loop $\gamma$ splits (Lemma 5.2 of
\cite{bfh:tits1}) at the initial vertex $v$ of
$E_m$ each time that it crosses
$E_m$ in either direction. We may therefore think
of $\gamma =\gamma_1\cdot
\dots \cdot \gamma_r$ as a concatenation of
Nielsen paths
based at
$v$ (i.e. each $[f(\gamma_i)] = \gamma_i$). We
claim that there are two distinct
$\gamma_l$'s. If not, then, since
$\gamma$ is indivisible, $\gamma = \gamma_1$ is
of the form $E_m \delta$, $\delta \bar
E_m$ or
$ E_m \delta \bar E_m$ for some path $\delta
\subset G_{m-1}$, where $\bar E_m$ is
$E_m$ with its orientation reversed. In the first
two cases $\gamma$ is a free factor
and in the last case
$\gamma$ is freely homotopic to $\delta$. Each of
these contradicts our assumption that
$\cal F(\gamma)$ has rank $n$ and so verify our
claim..
The axis $Ax(T) =
\dots
\tilde
\gamma_1 \tilde \gamma_2 \dots $ of $T$
decomposes as a concatenation of lifts of the
$\gamma_l$'s. Let $\tilde v$ be the initial vertex of
$\tilde \gamma_1$ and let $\tilde f$ be
the lift of $f$ that fixes $\tilde v$. Since the
$\gamma_l$'s are Nielsen paths, $\tilde f$
fixes each concatenation point in the decomposition
$Ax(T) = \dots \tilde
\gamma_1 \tilde \gamma_2 \dots $. In particular,
$\hat \eta = \hat f$ fixes the
endpoints of
$Ax(T)$ and so commutes with $T$. Now think of
$\gamma_1$ as a loop and extend $\tilde
\gamma_1$ to the axis $Ax(T_1)$ of a covering
translation by concatenating
translates of $\tilde \gamma_1$. Then $\hat \eta =
\hat f$ also fixes the endpoints
of $Ax(T_1)$.
It remains to consider the case that $\cal
F(\gamma)$ has rank one. We may
assume that the first stratum $G_1 = \gamma$ is a
single edge; let $v$ be the
vertex of $G_1$.
Suppose that an edge of an exponentially growing stratum\ $H_r$ is
attached to $v$. After replacing $f$ by
an iterate if necessary there is an edge $E$ of $H_r$
with initial vertex $E$ such
that $f(E) = E \cdot \beta$ splits into the
concatenation of $E$ with some non-trivial
path $\beta$. Let
$\tilde v$ be a lift of $v$ in the axis of $T$, let $\tilde f :
\Gamma \to \Gamma$ be the
lift of $f$ that fixes $\tilde v$ and let $\tilde E$ be the lift
of $E$ with initial vertex
$\tilde v$. Then
$\tilde f(\tilde E) =\tilde E \cdot \tilde \beta$ splits into the
concatenation of $\tilde E$
and some other non-trivial path and so each $[\tilde
f^k (\tilde E)]$ is a
proper initial segment of $[\tilde f^{k+1} (\tilde E)]$. It
follows that the
$[\tilde f^k (\tilde E)]$'s converge to an invariant ray
whose endpoint $R$ is fixed by
$\tilde f$. Let $\hat \eta = \hat f$.
If there are no exponentially growing strata
attached to $v$, then there are no
zero strata attached to $v$ and each edge $E_i$ that
is attached to $v$ is its own stratum $H_i$ and
satisfies $f(E_i) =E_iu_i$ for
some path $u_i \subset G_{i-1}$. If $v$ is the initial
vertex of $E_i$ and
either $E_i$ is a loop or $u_i$ is non-trivial, then we
define
$\hat \eta$ as in the proof of
Lemma~\ref{interesting}: Let $\tilde v$ be
a lift of $v$ in the axis of $T$, let $\tilde f : \Gamma
\to \Gamma$ be the
lift of $f$ that fixes $\tilde v$ and let $\tilde E$ be the lift of
$E$ with initial
vertex $\tilde v$. If $u_i$ is trivial then $R$ is the
endpoint of the axis for
$E_i$ that contains $\tilde E_i$. If $u_i$ is non-trivial
then $R$ is the
endpoint of the invariant ray $\tilde u_i\cdot[\tilde f(\tilde
u_i)]\cdot[\tilde f^2(\tilde u_i)]\cdot
\dots$.
In the remaining case, every $E_i$ attached to $v$
is either a fixed
non-loop or has $v$ as its terminal endpoint.
Proceeding as in the proof of
Lemma~\ref{conditioned} we can reduce the
number of edges in $G$ and arrive at
one of our previous cases.
This completes the proof when $\gamma$ and
$T$ are given. It remains to consider
the case that $\eta$ does not act periodically on any
conjugacy class in $F_n$. By
induction on $n$, we may assume that $\eta$ does
not act periodically on the
conjugacy class of any proper free factor in $F_n$.
We may therefore assume our improved
relative train track map $f : G \to G$ has only one
stratum and that this stratum is
exponentially growing.
After passing to a further iterate if necessary, we may
assume
that there is a vertex $v$ and two edges $E_1$ and
$E_2$ initiating at $v$ such that
$f(E_i) = E_i
\cdot \beta_i$ splits into the concatenation of $E_i$
with some non-trivial path
$\beta_i$. Choose a lift $\tilde v$ of $v$, let $\tilde f$ be
the lift of $f$ that fixes
$\tilde v$ and let $\tilde E_i$ be the lift of $E_i$ initiating
at $\tilde v$. As above, $[\tilde
f^k (\tilde E_i)]$ converges to an invariant ray whose
endpoint $R_i$ is fixed by $\hat f$.
Moreover, the bounded cancellation lemma
\cite{co:bcc} and the fact that the lengths
of $[\tilde f^k (\tilde E_i)]$ grow exponentially in $k$
imply that $R_i$ is an attracting
fixed point for the action of $\hat f$ on $C_{\infty}$.
We have shown that some iterate of $\eta$ has a
lift with at least two attracting
fixed points. Applying this to $\eta^{-1}$, we
conclude (suppressing the iterate
for notational simplicity) that some $\hat{\eta}$ has
at least two repelling fixed points. If $\tilde\eta$ has
a fixed point, then
the preceding argument shows,after passing to an
iterate if necessary, that there are
also at least two attracting fixed points. Suppose then
that $\tilde\eta$ is
fixed point free. During the proof of Proposition 6.21
of \cite{bfh:tits1} we show
that there exists $\tilde x \in \Gamma$ such that $\tilde x,
\tilde\eta(\tilde x),
\tilde\eta^2(\tilde x) \dots$ is an infinite sequence in
an embedded ray $\tilde B$;
the endpoint of $\tilde B$ is fixed by $\hat\eta$ and is
not one of the
repelling fixed points. \qed
\vspace{.1in}
We prove Proposition~\ref{lifting to Aut} in the
following equivalent form.
\vspace{.1in}
\noindent {\bf Proposition~\ref{lifting to Aut}} {\it
Every abelian subgroup
${\cal H} \subset Out(F_n)$ has a virtual lift $\hat {\cal H}
\subset EH(F_n)$. If $\gamma$ is
a non-trivial primitive element of $F_n$ that is
fixed, up to conjugacy, by each
element of ${\cal H}$ and if $T$ is a covering translation
corresponding to $\gamma$, then
$\hat {\cal H}$ can be chosen so that each element
commutes with $T$}.
\vspace{.1in}
\noindent{\bf Proof of Proposition ~\ref{lifting to
Aut}} We argue by induction on the
rank of the free abelian group ${\cal H}$. If ${\cal H}$ has rank
one with generator $\eta$, then
$\hat {\cal H}$ is determined by choosing a lift $\hat
\eta$ that commutes with $T$ if $T$ is
given. We may now assume that the rank of ${\cal H}$ is
at least two and that the lemma holds
for all ranks less than that of ${\cal H}$.
By Lemma~\ref{lift one}, there is an element $\eta
\in {\cal H}$ and a lift $\hat \eta$
that fixes at least three points and that commutes
with $T$ if $T$ is given. Let
$C = Fix(\hat \eta$) and let ${\cal T}(C)$ be the group
of covering
translations that preserve $C$. As described in
subsection~\ref{lifts}, $\hat \eta$
determines an automorphism $\Phi : F_n \to F_n$
whose fixed subgroup
${\mathbb F}$ corresponds to ${\cal T}(C)$ under
the identification of $F_n$ with
${\cal T}$. (See the proof of Lemma~\ref{T(C)} for
further details.) If
$\gamma$ and
$T$ are given, then $T \in {\cal T}(C)$ and
$\gamma \in {\mathbb F}$.
We first show, after passing to a subgroup of ${\cal H}$
with finite index, that every
$\mu \in {\cal H}$ has a lift $\hat \mu$ that commutes
with $\hat \eta$ and with $T$ if $T$
is given. In particular, $C$ is $\hat\mu$-invariant,
${\mathbb F}$ is ${\cal H}$-invariant
(up to conjugacy) and, if $\gamma$ is given then
each element of ${\cal H}|{\mathbb F}$ fixes
$\gamma$ up to conjugacy.
Suppose at first that $\gamma$ and $T$ are not
given. We say that two lifts
$\hat \eta_1$ and $\hat \eta_2$ of
$\eta$ are equivalent if $\hat \eta_1 = T_1 \hat
\eta_2 T_1^{-1}$ for some covering
translation $T_1$. Let $IL(\eta)$ be the set of
equivalence classes of lifts $\hat \eta$
that fix at least three points. By Lemma~\ref{finite
lifts}, $IL(\eta)$ is finite.
Since ${\cal H}$ is abelian, ${\cal H}$ acts by conjugation on
$IL(\eta)$. After passing to a
subgroup of finite index, we may assume that this
action is trivial. Thus for any lift
$\hat \mu_1$ of $\mu
\in {\cal H}$, there is a covering translation $T_1$ such
that $\hat \mu_1 \hat \eta \hat
\mu^{-1}_1 = T_1 \hat \eta T_1^{-1}$. Thus $\hat
\mu = T_1^{-1}\hat
\mu_1$ commutes with $\hat \eta$.
Suppose now that $\gamma$ and $T$ are given.
Each $\mu \in {\cal H}$ has a lift $\hat
\mu$ that commutes with $T$. Since ${\cal H}$ is
abelian, $\hat
\mu \hat \eta \hat \mu^{-1}$ is a lift of $\eta$ that
commute with $T$ and so $\hat
\mu \hat \eta \hat \mu^{-1} = T^a\hat \eta$ for
some $a$. Let $\hat \eta_k =
\hat \mu^k \hat \eta \hat \mu^{-k} = \hat
\mu \hat \eta_{k-1} \hat \mu^{-1} = T^{ak}\hat
\eta$. Then each $\hat \eta_k$ is
conjugate to $\hat \eta$ and so fixes at least three
points. On the other hand, there
are only finitely many values of $l$ for which
$Fix(T^l\hat \eta) \ne Fix(T)$. (This
follows from: (i) $\hat \eta$ fixes $Fix(T) = \{P,Q\}$
and so cannot move points very
near $P$ to points very near $Q$; and (ii) $T$ acts co-
compactly on $C_{\infty}
\setminus \{P,Q\}$.) We conclude that $a = 0$ and
hence that $\hat \mu$ commutes
with $\hat \eta$.
The proof now divides into cases, depending on the
rank of ${\cal T}(C)$. If ${\cal
T}(C)$ is the trivial group, then each $\mu$ has a
unique lift $\hat \mu$ that
preserves $C$ and so the assignment $\mu \mapsto
\hat \mu$ defines $\hat {\cal H}$.
Suppose next that ${\cal T}(C)$ has rank one. Since
${\mathbb
F}$ is ${\cal H}$-invariant (up to conjugacy), we may
assume that $T$
is given and generates ${\cal T}(C)$.
We claim that $C$ contains only finitely many
$T$-orbits. This is a special case of the main theorem
of \cite{co:bcc}; the argument
in this case is short so we include it for completeness.
Let $P$ and
$Q$ be the endpoints of the axis of
$T$. Since $T$ acts co-compactly on $C_{\infty} \setminus
\{P,Q\}$, it suffices to show that $P$ and $Q$ are the
only accumulation
points of $C$. Suppose to the contrary that $S$ is an
accumulation
point other than $P$ and $Q$. Arguing as in the
proof of Corollary~\ref{finite
lifts}, using triples of points in $C$ limiting on $S$,
we find lifts $\tilde v_i \in
\Gamma$ of a vertex $v \in G$ so that the covering
translation $T_i$ that carries
$\tilde v_1$ to $\tilde v_i$ commutes with $\hat \eta$.
But then $T_i$ is a
multiple of $T$ in contradictiction to the assumption
that $T_i(\tilde
v_1) \to S$. This verifies our claim.
The action of ${\cal H}$ on the finitely many $T$-orbits
of $C$ is well
defined. After passing to a finite index subgroup, this
action is trivial. Choose a
point $R \in C \setminus\{P,Q\}$. Composing
$\hat \mu$ with an iterate of $T$, we may assume
that $\hat \mu$ fixes $R$. The assignment $\mu
\mapsto \hat \mu$ defines
$\hat {\cal H}$.
Finally suppose that ${\cal T}(C)$ has rank at least
two. As noted above,
if $\gamma$ is given then it is fixed, up to
conjugacy, by each element of
${\cal H}^* = {\cal H}|{\mathbb F}$. Since
${\mathbb F}$ is the fixed subgroup of an
automorphism
$\Phi$ representing $\eta$, the image of
$\eta$ in
${\cal H}^*$ is trivial. Thus the rank of ${\cal H}^*$ is less
than that of ${\cal H}$ and by
induction, there is a lift $\widehat{{\cal H}^*} \subset
EH(C)$ to elements that commute with
$T$ if $T$ is given. By Lemma~\ref{rank>1},
$\widehat{{\cal H}^*}$ extends to the
desired lift $\hat {\cal H}$. \qed
\vspace{.1in}
\section{Proof of Theorem~\ref{main}} \label{proof
of main}
We may assume without loss that ${\cal H}_0$ is non-
trivial and, by
Lemma~\ref{virtually central}, that ${\cal H}$ acts
trivially on
${\cal H}_0$ and on $A({\cal H}_0)$. Let ${\cal K}$ be an
abelian Kolchin representative\ for ${\cal H}_0$. Throughout this section $\hat s =
\hat s_i$ or $\hat
s_{\alpha}$ and $T_{\alpha}$ is its
associated covering translation.
Before proving the next lemma we show that it
implies Theorem~\ref{main}.
\begin{lemma} \label{extend} The lift $\hat s({\cal H}_0)
\subset EH(C_{\infty})$ virtually extends to a
lift $\hat S({\cal H})\subset EH(C_{\infty})$ all of whose
elements commute with
$T_{\alpha}$.
\end{lemma}
\vspace{.1in}
\noindent{\bf Proof of Theorem~\ref{main}} We
may assume that the extensions
$\hat S_i$ and
$\hat S_{\alpha}$ of $\hat s_i$ and $\hat
s_{\alpha}$ produced by
Lemma~\ref{extend} are defined on all of ${\cal H}$.
Define a homomorphism
$\phi_i' : {\cal H} \to {\mathbb Z}$ by $S_i(\psi) =
T_{\alpha}^{\phi_i'(\psi)}
S_{\alpha}(\psi)$ and note that the product $\Phi' :
{\cal H}
\to
{\mathbb Z}^r$ of the $\phi_i'$'s extends the embedding
$\Phi : {\cal H}_0 \to {\mathbb Z}^r$ of
Lemma~\ref{akr implies abelian}. Define $\Psi : {\cal H}
\to
{\mathbb Z}^{b+r}$ to be the product of $\Omega$ and
$\Phi'$. Since ${\cal H}_0$ is the
kernel of $\Omega$ and $\Phi'|{\cal H}_0 = \Phi$ is an
embedding, $\Psi$ is an
embedding.
We now know that solvable subgroups ${\cal H}$ of
$\o({F_n})$ are finitely generated and
virtually abelian.
By passing to a subgroup of index at
most
$D(n)$ where
$D(n):=|GL(n,{\mathbb Z}/3{\mathbb Z})|< 3^{n^2}$, we may assume
that the image of ${\cal H}$ in
$GL(n,\Z/3\Z)$ is trivial.
Thus,
${\cal H}$ has a subgroup of index at most this number
that is a torsion free
Bieberbach group of $vcd$
at most
$vcd(\o({F_n}))=2n-3$ (see
\cite{cv:moduli}). Also, a Bieberbach group of $vcd$
at most
$n$ has a subgroup of index at most $D(n)$ that is
free abelian (see, for
example, \cite{lc:bg}).
Thus, ${\cal H}$ has a free abelian subgroup of index at
most $D(n)D(2n-3)< 3^{5n^2}$.
This completes
the proof of the Theorem~\ref{main}.
\qed
\vspace{.1in}
We are now reduced to Lemma~\ref{extend}. The
proof uses Proposition~\ref{lifting to
Aut} and follows the general line of the proof of
Proposition~\ref{lifting to
Aut}.
Let $C = \bigcap_{f \in {\cal K}}Fix(\hat s(f)$) and let
${\cal T}(C)$ be the group of
covering translations that preserves $C$. Note that
${\cal T}(C)$ contains $T_{\alpha}$
and so has rank at least one. Let ${\mathbb F}$ be the
subgroup of $F_n$ corresponding
to ${\cal T}(C)$ and let $\tilde s({\cal H}_0) \subset Aut(F_n)$
be the lift of ${\cal H}_0$ corresponding
to $\hat s({\cal H}_0) \subset EH(C_{\infty})$.
\begin{lemma}\label{T(C)} ${\mathbb F}$ is the
fixed subgroup $\{ \gamma \in F_n : \tilde
\eta(\gamma) =
\gamma$ for each $\tilde \eta \in \tilde
s({\cal H}_0)\}$ of $\tilde
s({\cal H}_0)$.
\end{lemma}
\noindent{\bf Proof of Lemma~\ref{T(C)}} If the
endpoints of the axis
of $T \in {\cal T}$ are contained in $C$, then they
are fixed by each $\hat
s(f)$ and so
$T$ commutes with each $s(f) : \Gamma \to
\Gamma$. Thus $T$ commutes with each $\hat
s(f)$ and each
$Fix(\hat s(f))$ is $T$-invariant. It follows that $T
\in {\cal T}(C)$.
Conversely, if $P$ and $Q$ are, respectively, the
backward and forward endpoints
of the axis of $T$ then
$\lim_{n \to \infty}T^n(R) =Q$ and $\lim_{n \to
\infty}T^{-n}(R) =P$ for all $R \in C_{\infty}
\setminus \{P,Q\}$. If $T(C) = C$ then $C$ must
contain $P$ and $Q$.
We have shown that $T \in {\cal T}(C)$ if and
only if the endpoints of the axis
of $T$ are contained in $C$. By construction, the
latter condition is equivalent
to the endpoints of the axis of $T$ being fixed by each
$\hat s(f)$. The lemma
now follows from the definition of $\tilde
s({\cal H}_0)$.\qed
\vspace{.1in}
\noindent{\bf Proof of Lemma~\ref{extend}} Given
$\mu \in {\cal H}$, choose a lift $\hat \mu$
that commutes with $T_{\alpha}$. Suppose also that
$\hat \eta \in \hat s
({\cal H}_0)$ is given. Since ${\cal H}$ acts trivially on ${\cal H}_0$,
$\hat
\mu
\hat
\eta
\hat \mu^{-1}$ is a lift of $\eta$ that commute with
$T_{\alpha}$ and so $\hat
\mu \hat \eta \hat \mu^{-1} = T_{\alpha}^a\hat
\eta$ for some $a$. Arguing exactly as
in the proof of Proposition~\ref{lifting to Aut}, we
conclude that $\hat \mu$ commutes
with $\hat \eta$. It follows that
$C$ is
$\hat
\mu$-invariant, that ${\mathbb F}$ is ${\cal H}$-
invariant up to conjugacy and that each
element of
${\cal H}|{\mathbb F}$ fixes $\alpha$ up to conjugacy.
The proof now divides into cases, depending on the
rank of ${\cal T}(C)$. Suppose
that ${\cal T}(C)$ has rank one. We claim that $C$
contains only finitely many
$T_{\alpha}$-orbits. Let $P$ and $Q$ be the
endpoints of the axis of
$T_{\alpha}$. Since $T_{\alpha}$ acts co-compactly
on $C_{\infty} \setminus
\{P,Q\}$, it suffices to show that $P$ and $Q$ are the
only accumulation
points of $C$. Suppose to the contrary that $S$ is an
accumulation
point other than $P$ and $Q$. Lemma~\ref{only one
line}, applied to triples of
points in $C$ limiting on $S$, implies that there are
vertices $\tilde v_i \in \Gamma$
that limit on $S$ and that are fixed by $\tilde s(f)$ for
each
$f \in {\cal K}$. There is no loss in
assuming that the $v_i$'s are all lifts of the same
vertex in $G$. The covering
translation
$T_i$ that carries
$\tilde v_1$ to $\tilde v_i$ commutes with each $s(f)$
and so must be a
multiple of $T_{\alpha}$. But this contradicts the
assumption that $T_i(\tilde
v_1) \to S$. This verifies our claim.
There is a well defined action of ${\cal H}$ on the
finitely many $T$-orbits of $C$.
After passing to a finite index subgroup, this action is
trivial. By
Lemma~\ref{interesting} there exists $R \in C_{\infty}$ that
is not an endpoint of the axis of
$T_{\alpha}$. Composing
$\hat \mu$ with an iterate of $T$, we may assume
that $\hat \mu$ fixes $R$. The assignment $\eta
\mapsto \hat \eta$ defines
$\hat S$.
We may now assume that ${\cal T}(C)$ has rank
at least two. By
Lemma~\ref{rank>1}, the restriction ${\cal H}^*$ of ${\cal H}$
to Out( ${\mathbb F}$) is well
defined. By Lemma~\ref{T(C)}, the restriction
${\cal H}_0^*$ of ${\cal H}_0$ to Out(${\mathbb F}^*)$ is
trivial. Restricting the exact sequence
$1\to{\cal H}_0\to{\cal H}\overset
\Omega \to{\mathbb Z}^b\to 1$ to ${\mathbb F}^*$ we see
that ${\cal H}^*$ is abelian. By
Propositon~\ref{lifting to Aut}, there is a virtual lift
$\widehat{S^*} \subset EH(C)$
of ${\cal H}^*$ such that each element of $\widehat{S^*}$
commutes with $T_{\alpha}$. Let $\hat
S$ be the unique extension of $\widehat{S^*}$ to a
virtual lift of ${\cal H}$. Since
${\cal H}_0^*$ is trivial,
$\hat S(\psi)$ restricts to the identity on $C$ for each
$\psi \in
{\cal H}_0$. Thus $\hat S(\psi)$ and
$\hat s(\psi)$ agree on $C$ and so must agree
everywhere. \qed
\providecommand{\bysame}{\leavevmode\hbox
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|
physics/9712054
|
\section*{Abstract}
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\begin{document}
\title{On degree zero semistable bundles over an elliptic curve}
\author{C.~I.~Lazaroiu$^a$}
\maketitle
\vbox{
\centerline{Department of Physics}
\centerline{Columbia University}
\centerline{New York, NY 10027}
\medskip
\bigskip
}
\abstract{Motivated by the study of heterotic string compactifications
on elliptically fibered Calabi-Yau manifolds,
we present a procedure for testing semistability and identifying
the decomposition type of degree zero holomorphic vector bundles over a
nonsingular elliptic curve. The algorithm requires explicit knowledge
of a basis of sections of an associated `twisted bundle'.}
\vskip 3.5 in
$^a$ lazaroiu@phys.columbia.edu
\pagebreak
\section{Introduction}
In the study of a certain class of heterotic string compactifications one
encounters the following:
{\bf Problem:}
{\em Given a smooth elliptic curve $E$ and a degree zero holomorphic
vector bundle $V$ over $E$, find a practical algorithm to
determine whether $V$ is semistable.
In this case, find a maximal decomposition of $V$ in indecomposable
subbundles. }
This question appeared in \cite{CGL} in the course of an investigation of the
relation between $(0,2)$ heterotic string compactifications and $F$-theory.
Since the most accessible bundle data are usually
its global holomorphic sections, we will be interested in solving this problem
by finding a characterization of semistability and of the decomposition type
in terms of properties of a basis of sections of a certain bundle
associated to $V$. Our main result is Theorem \ref{final_theorem} in section
2.3.
{\bf Physical motivation:}
A $(0,2)$ heterotic compactification is characterized by a Calabi-Yau manifold
$Z$ and a stable holomorphic vector bundle $V$ over $Z$ \cite{Witten_old}.
If one is interested in models having a potential $F$-theory dual
\cite{V,VM,FMW}, one takes
$Z$ to be elliptically fibered and with a section. In this case, for
a certain component of the moduli space, there exists an alternate
description of stable vector bundles over $Z$ in terms of pairs
$(\Sigma,L)$ where $\Sigma$ is the spectral cover of $V$ and $L$ is a line
bundle over $\Sigma$ \cite{FMW,FMW_math,Donagi_Markman}.
Such data are easier to manipulate then the
abstract bundle data. On the other hand, there exists an accessible class of
$(0,2)$
compactifications, namely those realized via $(0,2)$ linear sigma models
\cite{Witten_phases,Distler_notes,MDG,CDG}.
In this case, $V$ is presented as the sheaf cohomology of a monad defined
over $Z$, while $Z$ itself is realized as a complete intersection in a
toric variety \cite{BB}. This leads to the problem, studied in \cite{CGL},
of translating between these alternative
presentations of $V$ in the $(0,2)$ linear case.
The main condition for $V$ to admit a spectral cover
description is that its restriction $V|_E$ to the generic elliptic fibre $E$
of $Z$ be semistable. In order to carry out the task of \cite{CGL}, one needed
a method to test this condition for a given bundle $V$. This proves
essential in organizing the wealth of models that can be built. An important
point, which was tangentially mentioned in \cite{CGL}, is that the
above condition often fails to hold, even for $(0,2)$ models which seem to be
physically well-defined. One also finds a significant number of models
for which the condition is satisfied but $V|_E$ does not fully decompose as a
direct sum of line bundles. Discriminating between such cases can be achieved
by the methods of the present paper. On the other hand, the method of
\cite{CGL} was justified only for the case when $V|_E$ is semistable and
fully decomposable. Here we remedy this by providing a systematic discussion
of the general situation.
{\bf Mathematical context:}
The main results we need date back to a classical paper of Atiyah
\cite{Atiyah}
\footnote{Background material can be found for example in \cite{Potier}.}
.
Fix a nonsingular elliptic curve $E$ with a distinguished point $p$.
Let ${\cal E}(r,0)$ be the set of (holomorphic equivalence classes of)
{\em indecomposable} holomorphic vector bundles of rank $r$ and degree zero
over $E$. Any element $V \in {\cal E}(r,0)$ is of
the form $V=L\otimes F_r$ with $L\in \operatorname{Pic}^0(E)$ a degree zero line bundle
uniquely determined by $V$ and satisfying $L^r={\rm det} V$.
Here $F_r$ is the unique
element of ${\cal E}(r,0)$ with $h^0 \neq 0$. One has $h^0(F_r)=1$.
The bundles $F_r$ can be defined inductively by $F_1:=O_E$ and by the
fact that $F_r$ is the unique nontrivial extension :
\begin{equation}
0\longrightarrow F_{r-1}\longrightarrow F_r\longrightarrow
O_E\longrightarrow 0
\end{equation}
of $F_{r-1}$ by $O_E$. The Riemann-Roch theorem gives
$h^1(F_r)=h^0(F_r)=1$. It is known that $F_r$ is semistable for all
$r$.
For any holomorphic vector bundle of degree zero and rank $r$ over
$E$, consider a maximal decomposition as a direct sum of holomorphic
subbundles:
\begin{equation}
V=\oplus_{j=1..k}{V_j}
\end{equation}
If $V$ is semistable, then we necessarily have ${\rm deg} V_j\le 0$ for all
$j=1..k$ and since $0={\rm deg} V=\sum_{j=1..k}{{\rm deg} V_j}$, it follows that
${\rm deg} V_j=0$ for all $j$. If $r_j:={\rm rank} V_j$, we thus have
$V_j \in {\cal E}(r_j,0)$ and $V_j = L_j \otimes F_{r_j}$, with
$L_j \in \operatorname{Pic}^0(E)$. Thus :
\begin{equation}
\label{splitting}
V=\oplus_{j=1..k}{L_j \otimes F_{r_j}}
\end{equation}
Note that $ \sum_{j=1..k}{r_j}=r$.
Conversely, if such a decomposition of $V$ exists, then, since all terms are
semistable and of slope $0$, a standard result
(see \cite[p17,Cor.~7]{Seshadri}) assures us that $V$ is
semistable and of degree zero. The idea of our approach will be to use
(\ref{splitting}) in order to simultaneously check semistability and
determine the maximal decomposition, thus avoiding the difficult problem of
testing semistability independently.
The sequence of pairs $(r_j,L_j) (j=1..k)$ will be called the
{\em decomposition type} (or {\em splitting type}) of $V$.
By using the distinguished point $p \in E$ to write $L_j \approx O(q_j-p)$
for some
$q_j \in E$, we can identify this data with the sequence of pairs
$(r_j,q_j)$, modulo the choice of $p$. Obviously the splitting type
determines $V$ up to isomorphism.
Part of this information is encoded by what we will call the
{\em spectral divisor} $\Sigma_V$ of $V$, defined by :
\begin{equation}
\Sigma_V:=r_1q_1+...+r_kq_k \in {\rm Div}(E)
\end{equation}
Note that some of the points $q_j$ may coincide. If $q_1=...=q_{j_1}:=Q_1~$,
... ,$~q_{j_1+...+j_{l-1}+1}=..=q_{j_1+...+j_l}:=Q_l$
(with $j_1+...+j_l=k$), then
$\Sigma_V=\rho_1Q_1+...+\rho_l Q_l$ where
\begin{equation}
\nonumber
\rho_i=\sum_{j_1+...+j_{i-1}+1 \le i \le j_1+...+j_i}{r_i}.
\end{equation}
In particular, $\Sigma_V$ cannot discriminate between direct
factors of the type
$O(Q_1)\otimes(F_{r_1}\oplus ... \oplus F_{r_{j_1}})$ and factors
of the type $O(Q_1)\otimes F_{r_1+...+r_{j_1}}$. In fact, it is easy to see
\footnote{Since the only stable bundles of slope zero over an elliptic
curve are the degree zero line bundles, any Jordan-Holder (JH) filtration of
$V$ is by subbundles of consecutive dimension. The (isomorphism class of) the
associated graded bundle
$gr(V)$ is independent of the choice of the JH filtration.
If $V$ decomposes as above, the natural JH filtrations of $F_{r_i}$
induce a JH filtration of $V$ in the obvious way. The associated graded
bundle is $gr(V)=O(Q_1-p)^{\oplus \rho_1}\oplus...\oplus
O(Q_l-p)^{\oplus \rho_l}$. Therefore, $\Sigma$ depends only on $gr(V)$, i.e.
only on the $S$-equivalence class of $V$.}
that $\Sigma_V$ depends only on the $S$-equivalence class of
$V$. Two degree zero semistable vector bundles having the same spectral
divisor need not have the same splitting type.
The explicit computation of $\Sigma_V$ was the main task of \cite{CGL}.
In that paper, a solution of this problem was presented only for the
`fully split' case (this is rigorously formulated in Section 3). A by-product
of the study we undertake here is a simple generalization of the method of
\cite{CGL} for determining the spectral divisor
(see Corrolary \ref{spectral_div} in section 2.3).
We will often consider the `twisted' bundle $V':=V \otimes O(p)$,
which has degree $r$ and slope $1$.
If $V$ is semistable, one has the following
\begin{Lemma}
\label{cohom_dim}
Let $V$ be a degree zero semistable vector bundle over $E$.
Then $h^0(V')={\rm rank} V$ and $h^1(V')=0$.
\end{Lemma}
{\em Proof:}
By (\ref{splitting}), we have $h^0(V')=
\sum_{j=1..k}{h^0(O(q_j) \otimes F_{r_j})}$.
Since $O(q_j) \otimes F_{r_j}$ is indecomposable and of positive degree,
a result
of \cite{Atiyah} shows that $h^0(O(q_j) \otimes F_{r_j})=
{\rm deg} O(q_j)\otimes F_{r_j}=r_j$ and the Riemann-Roch theorem gives
$h^1(O(q_j) \otimes F_{r_j})=0$. This implies the conclusion.
$\Box$
As input data for the resolution of our problem we will assume explicit
knowledge of a basis of sections of $V'$. This is typically easily computed,
at least if $V$ is presented as the sheaf cohomology of a monad. \footnote{
Indeed, in that case one can consider the $O(p)$-twisted monad.
The long exact
cohomology sequence of the twisted monad will collapse due to the fact that
$h^1(V')=0$. This is one of the nice properties of $V'$.}
The plan of this paper is as follows.
In section 2 we study semistability and the decomposition type for a degree
zero
holomorphic vector bundle $V$ over $E$. We formulate necessary and sufficient
conditions on a basis of sections of $V'$ in order for $V$ to be
semistable; this will also indicate its decomposition type.
In particular, we obtain a simple receipt for the spectral
divisor. We also consider the spectral divisor in the monad case and
propose a `moduli problem'.
In section 3 we consider the fully decomposable (`fully split') case.
We present a criterion for
identifying fully decomposable and semistable vector bundles of degree zero
over $E$, together with an algorithmic implementation. This is the main case
considered in \cite{CGL}. The novelty here is that the algorithm we
give tests semistability of $V$ (and at the same time determines its
spectral divisor and its decomposition type, thus describing $V$ completely
in the language of \cite{Atiyah});
in \cite{CGL}, the focus was on computing $\Sigma_V$ and
$V$ was {\em assumed} to be semistable and fully decomposable
in order to simplify the presentation.
We also explain how one can analyze $V$ by starting from more
general twists. This is necessary in practice in cases when one
cannot easily compute the sections of bundles over $E$ twisted by $O(p)$.
\footnote{In the set-up of \cite{CGL}, one is interested in smooth
elliptic curves
realized as complete intersections in a toric variety ${\Bbb P}$.
In this case, one can easily compute the sections of $V\otimes L_E$, for
restrictions $L_E$ of
reflexive sheaves $L$ over ${\Bbb P}$.
If $O(p)$ is not such a restriction then the
sections of $V \otimes O(p)$ are not easily accessible.
For example, if $E$ is realized as a cubic in ${\Bbb P}^2$, the line bundle
$O_{{\Bbb P}^2}(1)$ over ${\Bbb P}^2$ restricts to a degree three line bundle
$O(p_1+p_2+p_3)$ over $E$,
and one can apply the methods of section 3 to the twisted bundle
$V':=V\otimes O(p_1+p_2+p_3)$.}.
With the physics oriented reader in mind, the discussion of section 3 is
carried out by a direct approach and can be
read independently of the rest of the paper; it is intended as a technical
companion of \cite{CGL}.
{\bf Notation and terminology:} If $s$ is a regular section of a holomorphic
vector bundle,
then $(s)$ denotes the zero divisor (divisor of zeroes) of $s$.
${\rm Div}(E)$ is the free abelian
group of divisors on $E$. If $D \in {\rm Div}(E), D=\sum_{j=1..k}{n_j p_j}$, with
$n_j \in {\Bbb Z}, p_j \in E$, then ${\rm supp} D$ denotes the set $\{p_j|j=1..k\}$.
All vector bundles and their morphisms are holomorphic. For any vector space
$A$, ${\rm Gr} ^k(A)$ denotes the grassmanian of $k$-dimensional subspaces of $A$.
If $S \in A$ is a subset, then $<S>$ denotes the linear span of $S$.
For any holomorphic bundle $R$ over $E$, ${\rm Gr} ^k(R)$ denotes the set of
rank $k$ holomorphic subbundles of $R$. $\sim$ denotes linear equivalence of
divisors and $\operatorname{Pic}(E)$ the Picard group of $E$. If $r$ is an integer, then
$\operatorname{Pic}^r(E)$ is the set of isomorphism classes of degree $r$ line bundles over
$E$; it is only a sub{\em set} of $\operatorname{Pic}(E)$, except for $r=0$, when it is a
subgroup. We say that a filtration $0=K_0\subset K_1\subset ... \subset K_r=U$
of an $r$-dimenisonal vector space $U$ is {\em nondegenerate} if
$K_{i-1} \neq K_i$ for all $i=1..r$. Then $K_i$ have consecutive dimensions.
For a holomorphic bundle $V$,
we denote by $\mu(V):={\rm deg} V/{\rm rank} V$ its slope (nomalized degree).
{\bf Intuitive idea}
The starting point for our analysis is the fact that the
twisted bundles $F'_r$ are given recursively as nontrivial extensions
of $O(p)$ by itself. By Lemma \ref{cohom_dim}, the associated cohomology
sequences collapse and this provides a very good handle on the behaviour
of $F'_r$. In terms of the local behaviour of sections, the difference between
$F'_r$ and the completely trivial extension $O(p)^{\oplus r}$ is manifest only
at the point $p$. In both cases, the bundles admit a basis of $r$ sections
whose values are linearly independent at each point of $E$ except $p$. At
this point, the behaviour in the two cases is dramatically different. While
in the completely decomposable case the values of all sections vanish
simultaneously at $p$ along linearly independent `directions', in the case
of $F'_r$ only one of them vanishes, while the others have linearly
independent values. In the latter case, however, the `direction' of the first
section approaches the space spanned by the values of the others as we
approach $p$ on $E$, and at the point $p$ it lies in that space.
The behaviour of the
sections of $V'$ can be obtained essentially by a `linear superposition' from
the behaviour of its indecomposable factors.
Most of what follows
consists in developing enough technology in order to make these ideas precise.
This being understood, the physics-oriented reader may at first consider
only the
first part of subsection 2.1, the statements of Theorems \ref{F_criterion} and
\ref{final_theorem} in section 2 and of Theorem \ref{theorem_fully_split}
in section 3 and the associated algorithm.
\section{General analysis}
Let $V$ be a degree zero holomorphic vector bundle over a smooth
elliptic curve $E$. Fix a point $p \in E$ and define $V':=V\otimes O(p)$.
We present a criterion for deciding whether $V$ is
semistable and, in this case, for determining its splitting type.
This criterion requires explicit knowledge of a basis of holomorphic
sections of $V'$.
The plan of this section is as follows. In subsection 1 we discuss
a notion of order of incidence of a holomorphic section on a subbundle.
Since this discussion does not require assuming ${\rm deg} V=0$, we will
present it for a general holomorphic vector bundle over $E$. In subsection 2
we use these concepts to describe the sections of the bundles
$F'_r$. In subsection 3 we give our characterization of
degree zero semistable bundles.
\subsection{Incidence order of holomorphic sections on subbundles}
In this subsection let $W$ be a rank $r$ holomorphic vector bundle
over $E$ and let $T$ be a rank $r_0$ holomorphic subbundle of $W$.
Any {\em nonzero} regular section of $W$ defines a unique line
subbundle
$L_s$ of $W$ in the following way (see \cite{Atiyah}).
For each $t \in {\rm supp} (s)$, choose
a local holomorphic coordinate $z$ on $E$ centered at $t$.
Let $\nu_t$ be the degree of vanishing of $s$ at $t$. Then
$\exists \lim_{e\rightarrow p}{z^{-\nu_t}s(e)} :={\hat s}(t)$,
where ${\hat s}(t) \in W_t - \{ 0 \}$.
We define $(L_s)_e:=<s(e)>$, for all $e \in E - {\rm supp} (s)$ and
$(L_s)_t:=<{\hat s}(t)>$ for all $t \in {\rm supp} (s)$. Note that changing the local
holomorphic coordinate $z$ to another local holomorphic coordinate $z'$
centered at $t$ will change
${\hat s}(t)$ to ${\hat s}'(t)=\lim_{e \rightarrow p}
(z(e)/z'(e))^{\nu_t}{\hat s}(t)$.
Thus, the vector ${\hat s}(t)$ is defined up to multiplication by a
nonzero complex number. In particular, $(L_s)_t$ is
well-defined. By using the local triviality of $W$ or by the argument given in
\cite{Atiyah}, one can convince oneself that $L_s$ is a
holomorphic subbundle of $W$.
Note that $L_{\lambda s}=L_s$, $\forall \lambda \in C^*$, so that
we have a well-defined map ${\Bbb P} H^0(W) \longrightarrow {\rm Gr} ^1(W)$ from
the projectivisation of $H^0(W)$ to the set of holomorphic line
subbundles of $W$.
Since $s$ is a holomorphic section of $L_s$, it follows that $L_s$ is
holomorphically equivalent to $O(s)$,
where $O(s)$ is the line bundle on $E$ associated to the divisor
$(s) = \sum_{t \in {\rm supp} (s)}{\nu_t t}$. In particular, we have
${\rm deg} L_s ={\rm deg} (s) =\sum_{ t \in {\rm supp} (s)}{ \nu_t }=
\sum_{e \in E}{{\rm deg} s(e)}$, where we define ${\rm deg} s(e)$ to be $\nu_e$, if
$e \in {\rm supp} (s)$ and $0$ otherwise.
For each $e \in E$, we have a natural linear map
$\phi_e :H^0(W) \longrightarrow W_e$ given by
$\phi_e(s):=s(e),\forall s \in H^0(W)$ (the evaluation map at $e$).
We denote its image and kernel by
$R_e:=\phi_e(H^0(W)) \subset W_e$,
$K_e:={\rm ker} \phi_e \subset H^0(W)$
and we define
$r_e(W):={\rm dim}_{{\Bbb C}} R_e$,
$d_e(W):={\rm dim}_{{\Bbb C}} K_e$.
We have $r_e(W)+d_e(W)=h^0(W)$ at any point $e \in E$.
Define a subspace $N_e$ of $W_e$ by
$ N_e:=<\{{\hat s}(e)|s \in K_e\}> \subset W_e$
(if $s=0$, we define ${\hat s}(e)$ to be zero).
It is easy to see that changing the
linear coordinate $z$ does not affect $N_e$. Note that
$N_e=\sum_{s\in K_e}{(L_s)_e}$.
In general, the subspaces $N_e,R_e$
of $W_e$ may intersect and their sum need not generate $W_e$.
Define ${\cal Z}(W):=\{t \in E | K_t \neq 0\}$.
If $W$ is semistable then we must have ${\rm deg} s={\rm deg} L_s \le \mu(W)$.
Since $s$ is regular, we also have ${\rm deg} s \ge 0$ .
Then ${\rm deg} s \in \{ 0,..,[\mu(W)] \}$, where $[~]$ denotes the integer part.
In particular, we have ${\rm deg} s(e) \le \mu(W)$ for all $e \in E$.
\begin{Proposition}
\label{isom}
Suppose that $W$ is semistable and of slope $1$ .
Let $e \in E$ and fix a local coordinate $z$ around $e$ on $E$.
Then the map $s \in K_e \rightarrow {\hat s} \in N_e$ is a ${\Bbb C}$-linear
isomorphism. In particular, we have ${\rm dim}_{{\Bbb C}} N_e=d_e$.
\end{Proposition}
{\em Proof:} By the above, we see that any $s \in K_e -\{0\}$ must have
a {\em simple} zero at $e$. If $s_1,s_2 \in K_e$ and
$\alpha_1,\alpha_2 \in {\Bbb C}$, let $s:=\alpha_1 s_1 + \alpha_2 s_2$. Then
$\exists \lim_{e' \rightarrow e}{z^{-1}s(e')}=\alpha_1 {\hat s}_1(e) +
\alpha_2 {\hat s}_2(e)$. If
$\alpha_1 {\hat s}_1(e) + \alpha_2 {\hat s}_2(e)=0$,
then $s$ must be zero (otherwise $s$ would have degree $>1$ at $e$).
In this case ${\hat s}(e)=0$ by definition. If
$\alpha_1 {\hat s}_1(e) + \alpha_2 {\hat s}_2(e)\neq 0$, then
${\hat s}(e)=\alpha_1 {\hat s}_1(e) + \alpha_2 {\hat s}_2(e)$.
Thus in both cases we have
${\hat s}(e)=\alpha_1 {\hat s}_1(e) + \alpha_2 {\hat s}_2(e)$,
which shows linearity. If $s \in K_e$, then by definition
${\hat s}(e)$ is zero only if $s=0$. This shows injectivity.
Surjectivity is obvious. $\Box$
What follows is a generalization of the previous classical discussion.
\begin{Definition}
Let $s \in H^0(W)$ be a holomorphic section of $W$.
Consider the holomorphic section ${\overline s}$ of the
quotient bundle $W/T$, naturally induced by $s$. We say that $s$ is
{ \em incident of order (degree) $d$ on $T$ at a point $e \in E$} if
${\overline s}$ has a
zero of order (exactly) $d$ at $e$. In this case, we write ${\rm deg} _Ts(e):=d$
and we call it {\em the incidence order(degree)} of $s$ on $T$ at e.
\end{Definition}
Note that we have $s(e) \in T_e$ iff ${\rm deg} _Ts(e) > 0 $. Intuitively,
${\rm deg} _Ts(e)$ characterizes `how fast' $s(e') \in W_{e'}$ approaches the
subspace $T_{e'}$ of $W_{e'}$ as $e'$ approaches $e$ on $E$.
If $s \in H^0(T) \subset H^0(W)$, then ${\overline s}$ is identically
zero, so the degree of incidence of $s$ on $T$ is not defined for such $s$
at any point of $E$. If $s \in H^0(W) - H^0(T)$, then ${\overline s}$ is a
nonzero section of $W/T$ and the associated divisor $({\overline s})$
is a finite set of points of $E$.
Therefore, the set $Z_T(s):=\{ e \in E | {\rm deg} _Ts(e) >0 \}=
\{ e \in E | s(e) \in T \}={\rm supp} ({\overline s})$ is finite for all sections
$s\in H^0(W) - H^0(T)$. In particular,
${\rm deg} _Ts(e)$ is well defined in this case at all points $e \in E$. Thus, for
all $s \in H^0(W) - H^0(T)$, we can define the {\em total degree of $s$
along T} by ${\rm deg} _Ts:=\sum_{e \in Z_s(T)}{{\rm deg} _Ts(e)}={\rm deg} {\overline s}$.
For $T={\bf 0}$ (the null subbundle of $W$) we have
${\overline s}=s$ so
${\rm deg} _{\bf 0}s(e)={\rm deg} s(e) $ and the above definition reduces
to the usual one.
\begin{Proposition}
\label{induced_section}
Let $M$ be a holomorphic subbundle of $T$ and $s \in H^0(W) -H^0(T)$.
Let $q \in E$ an arbitrary point. Let $\sigma$ be the section of $W/M$
induced by $s$ via the canonical projection $W\stackrel{p}{\rightarrow}W/M$.
Then ${\rm deg} _{T}s(q)={\rm deg} _{T/M}\sigma(q)$
\end{Proposition}
{\em Proof:} Oviously $T/M$ is a subbundle of $W/M$ and $\sigma \in
H^0(W/M)-H^0(T/M)$. $s$ and $\sigma$ induce the same section ${\overline s}$
of $W/T$ via the canonical projections $W\rightarrow W/T$ and
$W/M \rightarrow (W/M)/(T/M)\approx W/T$.
Therefore:
${\rm deg} _{T/M}\sigma(q)={\rm deg} {\overline s}(q)={\rm deg} _Ts(q)$.
$\Box$
We have the following :
\begin{Proposition}
\label{degree_bound}
Suppose that $W$ is semistable of normalized degree $\mu(W)$ and that
$T$ has normalized degree $\mu(T)=\mu(W)$.
Then we have ${\rm deg} _Ts \leq \mu(W)$ for all $s \in H^0(W) - H^0(T)$.
\end{Proposition}
{\em Proof:} Indeed, $T$ is in this case obviously semistable
(since $W$ is semistable) and thus $W/T$ is semistable of normalized degree
$\mu(W/T)=\mu(W)$ (see, for example Proposition 8 on page 18 of
\cite{Seshadri}).
Then ${\overline s}$ must have have total degree at most equal to
$\mu(W/T)=\mu(W)$ in order for
$L_{\overline s}\approx O(s)$ not to destabilize
$W/T$. Then use ${\rm deg} _Ts={\rm deg} {\overline s}$. $\Box$
For $W$ semistable and $\mu(T)=\mu(W)=1$, this
shows that a section $s \in H^0(W) - H^0(T)$ either does not
intersect $T$ or intersects it at exactly one point, the incidence
degree of $s$ at that point being exactly one.
We now give an alternative description of the incidence degree,
which is more practical from a computational point of view.
\begin{Proposition}
Let $s \in H^0(W) - H^0(T)$ and $q \in E$.
Consider a local holomorphic frame $(s_1 ... s_{r_0})$ of $T$ around $q$.
Then
\begin{equation}
\label{frame_criterion}
{\rm deg} s(q)={\rm deg} s\wedge s_1\wedge ... \wedge s_{r_0}(q)
\end{equation}
\end{Proposition}
{\em Proof:}
Let $U$ be an open neighborhood of $q$ such that the exact sequence
\begin{equation}
\label{split_sequence}
0 \longrightarrow T|_U \stackrel {j}{\longrightarrow} W|_U
\stackrel{p}{\longrightarrow}(W/T)|_U
\longrightarrow 0
\end{equation}
splits in the holomorphic category. Let $u :(W/T)|_u \longrightarrow W|_U$ be
a holomorphic injection such that
$W|_U=j(T|_U) \oplus u((W/T)|_U)$.
We identify $T$ with $j(T)$ via $j$ and $(W/T)|_U$ with
$u((W/T)|_U)$
via $u$. We can assume that $U$ is small enough so that all 3 bundles
involved are trivial above $U$.
Let $s_1...s_{r_0}$ be a local holomorphic frame of $T$ above
$U$ and $s_{r_0+1}...s_r$ a frame of
$(W/T)|_U\equiv u((W/T)|_U)$. Then $s_1...s_r$ is a local frame
of $W$ above $U$.
Write $s(e)=\sum_{i=1..r}{f_i(e)s_i(e)}$ with $f_i \in {\cal O}_U$.
Then
\begin{equation}
\nonumber
{\overline s}(e)=\sum_{i=r_0+1...r}{f_i(e)s_i(e)}.
\end{equation}
and
\begin{equation}
\nonumber
s(e)\wedge s_1(e) \wedge ... \wedge s_{r_0}(e)=
\sum_{i=r_0+1...r}{f_i(e)s_i(e) \wedge s_1(e) \wedge ... \wedge s_{r_0}(e)}
\end{equation}
The statement
${\rm deg} _Ts(q)=d$ is equivalent to
$\exists \lim_{e \rightarrow q}{z^{-d}{\overline s}(e)} \neq 0$,
which is equivalent to
$\exists \lim_{e \rightarrow q}{z^{-d}{\overline f}(e)} \neq 0$,
where
${\overline f}:= (f_{r_0+1}...f_r) \in \oplus_{i=r_0+1 ...r}{{\cal O}_U}$.
This in turn is equivalent to $\exists \lim_{e \rightarrow q}{z^{-d}s(e)
\wedge s_1(e) \wedge ... \wedge s_{r_0}(e)}\neq 0$.
$\Box$
Now let $s \in H^0(W) -H^0(T)$ and $q \in E$.
The associated section ${\overline s} \in H^0(W/T)$ defines a line
subbundle
$L_{\overline s} \subset W/T$ as above.
In particular, at
the point $q$ we have a $1$-dimensional subspace $(L_{\overline s})_q$ of
the fibre $(W/T)_q=W_q/T_q$.
We define $W_s(q)$ to be the $(r_0+1)$-dimensional subspace of $W_q$
which
induces $(L_{\overline s})_q$, i.e. the
preimage of $(L_{\overline s})_q$ via the natural surjection
$W_q\stackrel{p_q}{\longrightarrow} W_q/T_q $. For $e \in E -Z_T(s)$
we obviously have $W_s(q)=<s(e)>\oplus T_e$. The following gives
an analogue of this decomposition for points $ q \in Z_T(s)$:
\begin{Proposition}
\label{osc_vector}
Let $s \in H^0(W) - H^0(T) $ and $q \in E$.
Let $z$ be any local holomorphic coordinate on $E$, centered at $q$.
The following are equivalent :
(a) ${\rm deg} _Ts(q)=d$
(b) There exist {\em local} holomorphic sections ${\tilde s}$ of $W$
around $q$
and ${s_0}$ of $T$ around $q$ such that :
(b1) $s(e)=z^d{\tilde s}(e) + s_0(e) $ for all $e$ sufficiently close
to $q$
(b2) ${\tilde s}(q) \in W_q - T_q $
In this case, we have $W_s(q)=<{\tilde s}(q)> \oplus T_q$.
\end{Proposition}
{\em Proof:}
Assume $(a)$ holds and
consider a neighborhood $U$ of $q$ such that the
sequence \ref{split_sequence} splits. Since ${\rm deg} {\overline s}(q)=d$,
we can choose $U$ small enough so that there exists
a holomorphic section $\sigma$ of $W/T$ above $U$ such that
${\overline s}(e)=z^{d}\sigma(e), \forall e \in U$ and
$\sigma(q) \neq 0$.
Then there exists a holomorphic section ${\tilde s}:=u \circ
\sigma$ of $W|_U$,
such that ${\overline {\tilde s}}=p({\tilde s})=\sigma$. Thus
$p(s-z^d{\tilde s})=0$, so that $s(e) -z^d {\tilde s}(e) \in T_e$,
$\forall e \in U$. Since $s(e) -z^d {\tilde s}(e)$ is holomorphic,
this gives a holomorphic section $s_0$ of $T|_U$ such that
$s=z^d{\tilde s}+s_0$ and $(b1)$ holds.
Moreover, $\sigma(q) \neq 0$ implies
${\tilde s}(q) \in W_q -T_q$ and thus
$(b2)$ holds.
Since ${\overline s}(e)=z^d \sigma(e)$, we have
${\hat {\overline s}}(q)=\sigma(q)$ so that
$\sigma(q) \in (L_{\overline s})_q$. Thus
${\tilde s}(q) \in p_q^{-1}((L_{\overline s})_q)=W_s(q)$ and
$W_s(q)=<{\tilde s}(q)> \oplus T_q$.
The converse implication is trivial in view of the previous proposition.
$\Box$
Note that ${\tilde s}$, $s_0$ cannot,
in general, be extended beyond a neighborhood of $q$.
Also note that ${\tilde s}(q)$ is only determined modulo $T_q$ and
modulo a constant multiplicative factor
(from the choice of the local holomorphic coordinate $z$ around $q$).
\begin{Definition}
Let $s \in H^0(W)-H^0(T)$. Define
$W_{s,T}=\sqcup_{e \in E}{W_s(e)}$.
Then $W_{s,T}$ has a natural structure of holomorphic vector bundle over $E$
and $s \in H^0(W_{s,T})$ while $T$ is a holomorphic subbundle of $W_{s,T}$.
\end{Definition}
{\em Proof:}
A holomorphic trivialization of $W_{s,T}$ is obtained as follows. For $U$
an open set such that $U \cap Z_T(s)=\Phi$, choose a local frame
$s_1..s_{r_0}$ of $T$ over $U$ and trivialize $W_{s,T}$ over $U$ by using the
local frame $s_1...s_{r_0},s$. For $U$ such that $U \cap Z_T(s)={q}$
( a single point), by choosing $U$ small enough and picking a local
holomorphic coordinate $z$ on $E$, one one can write $s(e)=z^d{\tilde s}(e) +
s_0(e)$ as before, where $d={\rm deg} _T(s)$ and ${\tilde s}$ does not meet $T$
over $U$.
Then ${\tilde s}$ is a local
holomorphic
section of $W$ above $U$ and one can trivialize $W_{s,T}$ over
$U$ by using
$s_1...s_{r_0},{\tilde s}$, where $s_1...s_{r_0}$ is a local holomorphic frame
of $T$ above $U$. The holomorphic
compatibility of the various local trivializations is immediate.
$\Box$
Intuitively, the fibre $W_{s,T}(q)=<{\tilde s}(q)> \oplus T_q$ for
$q \in Z_T(s)$ is the correct `limit' of the fibres
$W_{s,T}(e)=<s(e)> \oplus T_e$ as $e \rightarrow q$. The section $s$
determines a line subbundle $L_s$ of $W$ and we have $W_{s,T}=L_s\oplus T$.
For $T={\bf 0}$
(the null subbundle of $W$), we obviously have $W_{s,{\bf 0}}=L_s$.
This is a generalization of the construction of $L_s$.
Now suppose that the set ${\cal Z}(W)$ is finite
\footnote{We will see that this is the case if $W=V\otimes O(p)$ with
$V$ semistable and of degree zero}.
In this case, if $s_1...s_k \in H^0(W)$ are ${\Bbb C}$-linearly independent
sections of
$W$, then
they are also ${\Bbb C}$-linearly independent at the generic point of
$E$ (i.e. $s_1(e)...s_k(e)$ are linearly independent in $W_e$ for
a generic $e \in E$); then we can define
inductively $W_{s_1...s_k}:=W_{s_k,W_{s_1...s_{k-1}}}$, with
$W_{s_1}:=L_{s_1}$. Indeed, one can easily see that
$s_2 \in H^0(W) -H^0(W_{s_1})$ and (by induction)
$s_j \in H^0(W)-H^0(W_{s_1...s_{j-1}})$,
$\forall j=2..k$, due to the generic
linear independence of $s_1..s_k$. $W_{s_1...s_k}$ is a rank $k$
vector bundle and $s_1..s_k$ are sections of $W_{s_1..s_k}$ which are
linearly independent at the generic point. Intuitivley, $W_{s_1...s_k}$ is
the subbundle of $W$ `spanned' by $s_1..s_k$.
If $(\sigma_1...\sigma_k)^t=A(s_1...s_k)^t$, with $A \in GL(k,{\Bbb C})$ a constant
nondegenerate matrix, then it is easy to see that $W_{\sigma_1..\sigma_k}=
W_{s_1...s_k}$. Indeed, $s_1 ... s_k$ are sections of
$W_{\sigma_1..\sigma_k}$ (since $\sigma_1...\sigma_k$ are) so that
$W_{\sigma_1...\sigma_k,e}=W_{s_1..s_k,e}$ for all $e$ with $s_1(e)...s_k(e)$
linearly independent. For $q\in E$ such that $s_1(q)...s_k(q)$ are linearly
dependent, one can consider vectors
${\tilde \sigma}_1(q),..,{\tilde \sigma_k}(q)$,
determined by local sections $\sigma_j$ of $W_{\sigma_1...\sigma_j}$ and
$\sigma_{0j}$ of $W_{\sigma_1...\sigma_{j-1}}$ via the conditions:
$\sigma_j(e)=z^{deg_{W_{\sigma_1...\sigma_{j-1}}}\sigma_j(q)}
{\tilde \sigma}_j(e)+
\sigma_{0j}(e)$ for $e$ close to $q$ and
$\sigma_j(q) \in W_{\sigma_1..\sigma_{j},q}-W_{\sigma1...\sigma_{j-1},q}$.
Note that ${\tilde \sigma}_1(q),..,{\tilde \sigma_k}(q)$ are linearly
independent.
These vectors obviously belong to $W_{s_1...s_k}(q)$ since for $e \neq q$
they are related to
$\sigma_1...\sigma_k$ (and thus to $s_1...s_k$) by linear
combinations of these vectors and since subbundles of $W$ are closed
in the total space of $W$.
Thus $W_{\sigma_1...\sigma_k}(q)=
<{\tilde \sigma}_1(q),..,{\tilde \sigma_k}(q)> \subset W_{s_1..s_k}(q)$
and they must coincide since they have the same dimension. Therefore,
$W_{s_1...s_k}$ depends only on the subspace $<s_1...s_k>$ of $H^0(W)$.
Thus, {\rm if} ${\cal Z}(W)$ {\rm is finite} then we have a natural map :
\begin{equation}
\psi_k:{\rm Gr} ^k(H^0(W))\rightarrow {\rm Gr} ^k(W).
\end{equation}
An alternative way to understand this is as follows (cf. \cite{Atiyah}).
If ${\cal Z}(W)$ is
finite, then given a $k$-dimensional
subspace $K$ of
$H^0(W)$, $\phi_e(K)$ defines a rational section $f$
of ${\bf G}{\rm r}^k(W)$, where ${\bf G}{\rm r}^k(W)$ is
the bundle obtained by taking the grassmannian ${\rm Gr}^k(W_e)$ of $W_e$ as
the fibre above each $e \in E$. Singularities of this section may appear
only at a point $e$ where $\phi_e(K)$ fails to be $k$-dimensonal, i.e.
at the points $e_1..e_s$ of $E$ where the values of a system $s_1..s_k$
of sections of $W$ giving a
basis of $K$ fail to be linearly independent. Loosely speaking,
one may worry that at such points
there is no `completion' of the set $\{\phi_e(K)| e \in E-\{e_1..e_s\}\}$
which makes it into the total space of a holomorphic vector bundle.
This does not happen for the following reason.
With the natural structure,
${\bf G}{\rm r}^k(W)$ is a complete variety and a classical result
implies that $f$ must be regular. Thus $f$ determines a
subbundle of $W$, which clearly coincides with $W_{s_1...s_k}$.
Again assuming ${\cal Z}(W)$ to be finite,
suppose that we are given a filtration
${\cal K}:
0:=K_0 \subset K_1 \subset ...\subset K_{k-1} \subset K_k$
of a subspace $K_k$ of $H^0(W)$, such that
${\rm dim} _{{\Bbb C}}K_j=j$, $\forall j=0..r$.
Associated to ${\cal K}$ via $\psi$ there is a filtration
${\cal W}({\cal K}):
0:=W_0 \subset W_1 \subset ...\subset W_{k-1} \subset
W_k$
by holomorphic subbundles with ${\rm rank} W_j=j$, $\forall j=0..r$.
If $s_j \in K_j-K_{j-1}$ for all $j=1..k$, then it
is obvious that the integers
$\delta^{\cal K}_j(t):=
{\rm deg} s_1\wedge...\wedge s_j(t)$ $(t \in {\cal Z}(W), j=1..r)$
depend only on ${\cal K}$.
It is also easy to see -- by using Proposition
\ref{osc_vector} -- that
${\rm deg} _{W_{j-1}}s_j(t) =
\delta^{\cal K}_j(t)-\delta^{\cal K}_{j-1}(t)$, where we let
$\delta^{\cal K}_0(t)$ be equal to $0$.
\subsection{The space of sections of the bundles $F'_r$}
Let $F'_r:=F_r \otimes O(p)$. $F'_r$ is a rank $r$ indecomposable
and semistable bundle of slope $1$. Since $F_r$ is semistable and of degree
zero, we have $h^0(F'_r)=r$ and
$h^1(F'_r)=0$. Recall from \cite{Atiyah} that we have exact sequences:
\begin{equation}
0\longrightarrow F_{k} \stackrel{i}{\longrightarrow}F_r
\stackrel{p}{\longrightarrow} F_l \longrightarrow 0
\end{equation}
for all $k,l \geq 0$ with $k+l=r$.
Below we will use their twisted version :
\begin{equation}
\label{twisted}
0\longrightarrow F'_{k} \stackrel{i}{\longrightarrow}F'_r
\stackrel{p}{\longrightarrow} F'_l \longrightarrow 0
\end{equation}
For $l=1$, we obtain the twisted version of the defining sequences of $F'_r$:
\begin{equation}
\label{twisted_1}
0\longrightarrow F'_{r-1} \stackrel{i}{\longrightarrow}F'_r
\stackrel{p}{\longrightarrow} O(p) \longrightarrow 0
\end{equation}
while for $k=1$ this gives :
\begin{equation}
\label{twisted_2}
0\longrightarrow O(p) \stackrel{j}{\longrightarrow} F'_r \stackrel{p}
{\longrightarrow} F'_{r-1}\longrightarrow 0
\end{equation}
Since $H^1(F'_k)=0$,
the exact cohomology sequence associated to (\ref{twisted})
collapses to:
\begin{equation}
\label{short}
0\longrightarrow H^0(F'_{k}) \stackrel{i_*}{\longrightarrow}H^0(F'_r)
\stackrel{p_*}{\longrightarrow} H^0(F'_l) \longrightarrow 0
\end{equation}
Being an exact sequence of vector spaces, this must split. Therefore, there
must exist ${\Bbb C}$-bases $<\sigma_1...\sigma_k>$ of $H^0(F'_{k})$,
$<\sigma_{k+1}...\sigma_r>$ of $H^0(F'_l)$ and $<s_1...s_r>$ of $H^0(F'_r)$
such that $j_*(\sigma_i)=s_i, \forall i=1..k$ and
$p_*(s_j)=\sigma_j, \forall j=k+1..r$.
\begin{Proposition}
\label{ker_prop}
For any $r\geq 1$, we have $d_p(F'_r)=1$ and
$d_e(F'_r)=0$ for all $e \in E-\{p\}$.
\end{Proposition}
{\em Proof:}
The sequence (\ref{twisted_2}) shows that $d_p(F'_r) > 0$.
Now suppose that $d_p(F'_r)>1$. Then there exist two linearly independent
sections $s_1,s_2$ of $F'_r$ such that $s_1(p)=s_2(p)=0$. Let $L_i=L_{s_i}$
be the associated line subbundles of $F'_r$. Since $F'_r$ is semistable and
of degree 1, we must have ${\rm deg} L_i=1$ and $(s_i)=p_i$.
Hence $\exists \lim_{e \rightarrow q}{s_i(e)/z}=
{\hat s}_i(q) \neq 0$.
Suppose that ${\hat s}_1(p),{\hat s}_2(p)$ are linearly dependent.
Then we can write ${\hat s}_1(p)=\alpha {\hat s}_2(p)$ with $\alpha \in
{\Bbb C}^*$. The section $s:=s_1 -\alpha s_2$ is then nonzero (since $s_1,s_2$ are
${\Bbb C}$-linearly independent) and we obviously have ${\rm deg} s(p)\ge 2$,
which contradicts semistability
of $F'_r$. Thus, it must be the case that ${\hat s}_1(q),{\hat s}_2(q)$ are
linearly independent.
Now suppose there exists $e_0 \in E-\{p\}$ such that $s_1(e_0)$ and
$s_2(e_0)$ are
linearly dependent. Write $s_1(e_0)=\beta s_2(e_0)$, with $\beta \in {\Bbb C}^*$.
Then the section $s'=s_1 - \beta s_2$ vanishes both at $e_0$ and at $p$
and so ${\rm deg} s'(p)\ge 2$, again contradicting semistability of $F'_r$.
It follows
that $s_1(e), s_2(e)$ are linearly independent for all $ e \in E -\{p\}$.
From these two facts we immediately see that the subbundle sum $L_1 + L_2$
is {\em direct}. Since $(s_i)=p$, we also have $L_i\approx O(p)$; thus
we have a holomorphic subbundle $L_1 \oplus L_2 =O(p) \oplus O(p)$ of
$F'_r$.
Twisting by $O(-p)$, this gives a trivial subbundle of rank two
$I_2 \subset F_r$. Since $h^0(I_2)=2$, this would imply $h^0(F_r)\geq 2$,
a contradiction. This finishes the proof of the first statement.
Now let $e \in E-\{p\}$. To show that $K_e=0$, we
proceed by induction on $r$, using the sequence (\ref{twisted_2}).
For $r=1$ the statement is obvious. Suppose the statement holds for $r-1$, but
fails for $r$. Then there exists a nonzero section $s$ of
$F'_r$ such that $s(e)=0$. We cannot have
$s \in H^0({\rm im} j)$ since that would imply
${\rm deg} s \geq 2$ (as $e \neq p$),
which contradicts semistability of
$F'_r$. Thus ${\overline s}: = p_*(s)$ is a nonzero section of $F'_{r-1}$.
Since $s(e)=0$, we have ${\overline s}(e)=0$, so that $K_e(F'_{r-1}) \neq 0$.
This is impossible by the induction hypothesis.
$\Box$
Consider the commutative
group structure $(E,\oplus)$ on $E$ with zero element $p$.
If $q_1,q_2 \in E$, then $q_1\oplus q_2$ is defined to be the unique
point $q$ of $E$ such that $(q_1)+(q_2)\sim (q)+(p)$, i.e.
$O(q_1+q_2)\approx O(q+p)$. Thus $(q_1\oplus q_2)\sim (q_1)+(q_2)-(p)$.
Then $(q_1\oplus ...\oplus q_r) \sim (q_1)+...+(q_r)-(r-1)p$.
Let $T_r^{(p)}(E)$ be the $r$-torsion
subgroup of $(E,p)$, i.e. the set of points $t \in E$ such that
$rt=0$ in $(E,\oplus)$, which is equivalent to $r(t)\sim r(p)$, i.e
$O(rt)\approx O(rp)$. The map $q \in E \rightarrow O(q-p) \in \operatorname{Pic}^0(E)$
is a group isomorphism from $(E,\oplus)$ to $\operatorname{Pic}^0(E)$, which maps
$T_r^{(p)}(E)$ to the subgroup
$U_r:=\{L \in \operatorname{Pic}^0(E)| L^r\approx O_E\}\subset \operatorname{Pic}^0(E)$ of
roots of order $r$ of $O_E$. We have $U_r \approx ({\Bbb Z}_r)^2$.
\begin{Proposition}
Let $r >0$. The isomorphism classes of {\em indecomposable}
bundles $A'$ which can be presented
as extensions :
\begin{equation}
0\longrightarrow I_{r-1} \stackrel{j}{\longrightarrow} A'
\stackrel{p}{\longrightarrow} O(rp)\longrightarrow 0
\end{equation}
of
$O(rp)$ by the trivial rank $r-1$ bundle $I_{r-1}$ are in bijective
correspondence with $U_r$. More precisely, each such bundle $A'$ is of the
form:
\begin{equation}
A'=O(q)\otimes F_r=L\otimes F'_r
\end{equation}
where $q \in T_r^{(p)}(E)$ and $L:=O(q-p)\in U_r$. Here
$F'_r:=F\otimes O(p)$.
\end{Proposition}
Note that we are {\em not} considering extension classes, but isomorphism
classes of bundles which can be presented as extensions.
{\em Proof:}
{\em Show that $F'_r$ fit into such sequences}
Use induction on $r$. For $r=1$ the statement is obvious (with $I_0={\bf 0}$).
Suppose the statement holds for $r-1$, so that there is an exact sequence:
\begin{equation}
\label{foo_seq1}
0 \longrightarrow I_{r-2} \stackrel{j}{\longrightarrow} F'_{r-1} \stackrel{p}
{\longrightarrow} O((r-1)p) \longrightarrow 0
\end{equation}
Let $s_1...s_{r-1}$ be a basis of $H^0(I_{r-2})$ and $s_{r-1}$ a section of
$ F'_{r-1}$ such that $s_1...s_{r-1}$ is a basis of $H^0(F'_{r-1})$ and
(using \ref{short}) such that $p_*(s_{r-1}) \in H^0(O(p))-\{0\}$.
Then $s_1(e)...s_{r-2}(e)$ are linearly independent for all $e \in E$
By Proposition \ref{ker_prop}, we have that $s_{r-1}(e) \in F'_{r-1,e} -
<s_1(e)...s_{r-2}(e)>=F'_{r-1,e}-I_{r-1,e}$ for all $e \neq p$.
Now use the recursive definition (\ref{twisted_1}) of $F'_r$.
This shows that we can choose $s_r \in H^0(F'_r)$ such that
$s_1...s_r$ is a basis of $H^0(F'_r)$ and such that the induced section
${\overline s}_r \in H^0(O(p))$ has zero divisor $({\overline s}_r)=(p)$.
Since $s_{r-1}(p)=0$, Proposition \ref{ker_prop} applied to $F'_r$ shows that
$s_1(e)...s_{r-2}(e),s_r(e)$ are linearly independent for {\em all} $e \in E$,
while $s_1(e)...s_{r-2}(e),s_{r-1}(e)$,\\ $s_r(e)$ are linearly independent
for
$e \neq p$. Thus $s_1(e)...s_{r-2}(e),s_r(e)$ determine a trivial subbundle
$I_{r-1}$ of $F'_r$ and $s_{r-1}(e)$ belongs to this subbundle iff $e=p$
(where $s_{r-1}(p)=0$).
It follows that the induced section ${\overline s}_{r-1}$ of the line bundle
$L:=F'_r/I_{r-1}$ vanishes only at $p$. Since ${\rm deg} F'_r=r$, we have
${\rm deg} L=r$
so that ${\rm deg} ({\overline s}_{r-1})=r$. Therefore
$({\overline s}_{r-1})=rp$ and $L\approx O(rp)$. This gives an exact sequence :
\begin{equation}
\label{foo_seq2}
0 \longrightarrow I_{r-1} \stackrel{j}{\longrightarrow} F'_r \stackrel{p}
{\longrightarrow} O(rp) \longrightarrow 0
\end{equation}
{\em Show that $O(q)\otimes F_r$ for $q \in T^{(p)}_r(E)$ are also
extensions of $O(rp)$ by $I_{r-1}$}
Since $q \in T^{(p)}_r(E)$, we have $O(rq)\approx O(rp)$. Combined with
(\ref{foo_seq2})(applied for $p$ substituted with $q$),
this gives the desired statement.
{\em Show that any indecomposable $A'$ which can be presented as such an
extension is of this form}
If $A'$ is an extension of $O(rp)$ by $I_{r-1}$, then
${\rm det} A'\approx O(rp)$. If $A'$ is indecomposable then $A:=A'\otimes O(-p)$
belongs to ${\cal E}(r,0)$, so that
$A\approx O(q-p)\otimes F_r$ for some $q \in E$.
($q$ is uniquely determined by $A$). Then $A'\approx O(q)\otimes F_r$,
so that ${\rm det} A'\approx O(rq)$. Thus we must have $O(rq)\approx O(rp)$
i.e. $q \in T_r^{(p)}(E)$. This finishes the proof. $\Box$
\begin{Theorem}
\label{F_criterion}
Let $V$ be a degree zero holomorphic vector bundle of rank $r$ over $E$
and let $V':=V\otimes O(p)$. The following statements are equivalent :
(a) $V$ is holomorphically equivalent to $O(q)\otimes F_r$, where $q$ is a
point of $E$
(b) There exists a ${\Bbb C}$-basis $(s_1 ... s_r)$ of $H^0(V')$ with the following
properties :
(b1) $s_1(e) ... s_r(e)$ is a basis of $V'_e$ for all
$e \in E - \{ p \}$
(b2) $s_1(p)=0$ and
$s_2(p),...,s_r(p)$ are linearly independent in $V'_p$
(b3) ${\rm deg} s_1\wedge s_2\wedge ... \wedge s_j (p)= j$ for all $j=1..r$.
(c) The following conditions are satisfied :
(c1) $h^0(V')=r$
(c2) ${\cal Z}(V')=\{p\}$
(c3) There exists a {\em nondegenerate} filtration
\begin{equation}
{\cal K}: 0=K_0\subset K_1 \subset ... \subset K_r:=H^0(V')
\end{equation}
of $H^0(V')$, with associated filtration
\begin{equation}
0 =W_0\subset W_1 \subset ... \subset W_r:=V'
\end{equation}
of $V'$, having the properties :
(c31) $K_j=\{s \in H^0(V') | s(p) \in (W_{j-1})_p\}$
(i.e. $K_j=\phi_p^{-1}(W_{j-1})$), $\forall j=1..r$
(c32) $\delta^{\cal K}_j(p)=j, \forall j=1..r$
Moreover, in this case we have $W_j\approx F'_j$ and
$K_j=H^0(W_j)\approx H^0(F'_j)$ for all $j=1..r$.
\end{Theorem}
Note that $s_2...s_r$ generate a trivial subbundle $I_{r-1}$ of $F'_r$ (since
they are everywhere linearly independent), while the section $s_1$ is incident
on $I_{r-1}$ at $p$ in order $r$. This is in agreement with the previous
proposition. The precise manner of incidence of $s_1$ on $I_{r-1}$ is
controlled by condition $(b3)$.
Note that $(c31)$ acts as an inductive definition of the filtration ${\cal K}$.
For $j=1$, $(c31)$ gives $K_1={\rm ker} \phi_p=K_p(V')$. The map
$\psi_1:{\rm Gr}^1(H^0(V'))\rightarrow {\rm Gr}^1(V')$ gives the subbundle
$W_1=\psi_1(K_1)$. Then $(c32)$ for $j=2$ defines $K_2$, the map $\psi_2$
gives $W_2=\psi_2(K_2)$ and so on.
In particular, ${\cal K}$ is naturally associated to $F'_r$
\footnote{Of course, $F'_r$ are only determined up to isomorphism. Naturality
heer means that such an isomorphism is compatible with the filtrations
${\cal K}$}.
It is easy to see from the proof of the theorem below that ${\cal K}$ is
nothing
other then the cohomology filtration induced by the standard Jordan-Holder
filtration of $F'_r$ :
\begin{equation}
\label{bundle_filtration}
0\longrightarrow F'_1\longrightarrow F'_2 \longrightarrow ...\longrightarrow
F'_{r-1}\longrightarrow F'_r
\end{equation}
Indeed, (\ref{bundle_filtration}) has the partial sequences:
\begin{equation}
0\longrightarrow F'_{j-1}\longrightarrow F'_j \longrightarrow O(p)
\longrightarrow 0
\end{equation}
(for $j=2...r$). Since $H^1(F'_{j-1})=0, \forall j=2..r$, these give the
exact sequences :
\begin{equation}
0\longrightarrow H^0(F'_{j-1})\longrightarrow H^0(F'_j) \longrightarrow
H^0(O(p))\longrightarrow 0
\end{equation}
which combine to give the filtration :
\begin{equation}
0\longrightarrow H^0(F'_1)\longrightarrow H^0(F'_2) \longrightarrow ...
\longrightarrow H^0(F'_{r-1})\longrightarrow H^0(F'_r)
\end{equation}
of $H^0(F'_r)$. This can be identified with the filtration ${\cal K}$ in the
theorem.
{\em Proof:}
{\em Show that (a) implies (b) }
We proceed by induction on $r$. For $r=1$, the statement is
trivial.
Let $r\geq 2$ and suppose the statement holds for $r-1$.
By the above discussion, we can choose bases
$\sigma_1...\sigma_{r-1}$ of $H^0(F'_{r-1})$,
$\sigma_r$ of $H^0(O(p))$ and $s_1...s_r$ of $H^0(F'_r)$ such that
$j_*(\sigma_1)=s_1...j_*(\sigma_{r-1})=s_{r-1}$ and $p_*(s_r)=\sigma_r$.
Since the result holds for $r-1$, we can further assume that
$\sigma_1...\sigma_{r-1}$ satisfy the properties $(b)$ for $r$ replaced with
$r-1$. Since $p_*(s_r)=\sigma_r$ and $(\sigma_r)=p$, it folows that
$s_r(e) \in (F'_r)_e-j_e((F'_{r-1})_e), \forall e \in E -\{p\}$, while
$s_r(p) \in j_p((F'_{r-1})_p)$. Since $j_e$ is injective for all $e \in E$,
and since $\sigma_1 ... \sigma_{r-1}$ satisfy $(b1)$, we see that
$s_1(e)... s_r(e)$ are linearly independent for all $ e \in E-\{p\}$, so
that $s_1 ... s_r$ satisfy $(b1)$.
By $(b2)$ for $\sigma_1...\sigma_{r-1}$ we obtain that $s_1(p)=0$
and $s_2(p)...s_{r-1}(p)$ are linearly independent.
Now suppose that $s_r(p) \in <s_2(p) ... s_{r-1}(p)>$. Then
$s_r(p)=\alpha_2 s_2(p) +.. + \alpha_{r-1}s_{r-1}(p)$. Then
$s:=s_r -\alpha_2 s_2 -.. - \alpha_{r-1}s_{r-1}$ is a regular section
of $F'_r$ which vanishes at $p$. Since $s_r$ is linearly independent of
$s_1...s_{r-1}$, it is clear that $s$ is linearly independent of
$s_1...s_{r-1}$. In particular, $s$ is linearly independent of $s_1$.
This implies that we have two linearly independent sections $s_1$, $s$ of
$F'_r$, both vanishing at $p$. Since this is impossible by virtue of
Proposition \ref{ker_prop}, it follows that $s_2(p) ... s_r(p)$ are
linearly independent and $(b2)$ holds.
Since $p_*(s_r)=\sigma_r$ has a simple zero at $p$, it follows
that $s_r$ vanishes in order $1$ along the subbundle
$j_*(F'_{r-1})$ of $F'_r$. Since $(b3)$ holds for $F'_{r-1}$ by the induction
hypothesis, we also know that $s_j$ vanishes in order $1$ along the
subbundle $W_j$ of $F'_{r-1}$, where $W_j=W_{s_1...s_j}$, for all $j=1..r-1$.
In particular, we have $s_j(e)=z{\tilde s}_j(p) + s_{0j}(e)$, with
$s_{0j} \in H^0(W_{j-1})$
for all $j=1..r-1$, and all $e$ sufficiently close to $p$. This implies that
$s_1(e) \wedge ... \wedge s_{r-1}(e) = z^{r-1}{\tilde s}_1(e) \wedge ...
\wedge {\tilde s}_{r-1}(e)$ so that
${\tilde s}_1(e) \wedge ...\wedge {\tilde s}_{r-1}(e) \neq
0$ for $e$ near $p$. This shows that ${\tilde s}_1, ..., {\tilde s}_{r-1}$
give a local holomorphic frame of $F'_{r-1}$ in a vicinity of $p$.
Then by Proposition \ref{frame_criterion}, we must have
${\rm deg} {\tilde s}_1\wedge ...\wedge {\tilde s}_{r-1}\wedge s_r (p)=1$,
so that ${\rm deg} s_1\wedge ... \wedge s_r (p)=r$. Thus $(b3)$
holds for $F'_r$.
Thus $(a)$ implies $(b)$.
{\em Show that (b) implies (c)}
Assume $(b)$ holds. Then $(c1)$ and $(c2)$ are obvious.
We can construct a filtration:
\begin{equation}
{\cal K}: 0:=K_0\subset K_1:=<s_1> \subset K_2:=<s_1,s_2> \subset ...\subset
K_r:=H^0(V')
\end{equation}
of $H^0(V')$, and an associated filtration :
\begin{equation}
{\cal W}: 0:=W_0\subset W_1 \subset W_2 \subset ...\subset W_r:=V'
\end{equation}
of $V'$, as explained in the previous subsection.
Let us analyze the situation at the point $p$.
{\em Claim:} For each $j=1..r$, we have ${\rm deg} _{W_{j-1}}s_j(p)=1$
and $s_2(p)...s_j(p)$ is a ${\Bbb C}$-basis of $(W_{j-1})_p$.
We prove the claim by induction on $j$. For $j=1$ we have
$W_{j-1}=W_0={\bf 0}$ and, by $(b3)$, we have ${\rm deg} _{W_0}s_1(p)={\rm deg} s_1 (p)=1$.
The second part of the claim is trivial in this case.
Now let $j \in \{2...r\}$ and assume that the claim is true for all $j'<j$.
Fix a local coordinate $z$ on $E$, centered at $p$.
By Proposition \ref{osc_vector}, we can write :
\begin{equation}
s_k(e)=z{\tilde s}_k(e) + s_{0k}(e), ~\forall k=1..j-1
\end{equation}
for all $e$ sufficiently close to $p$, where
${\tilde s}_k(p) \in V'_p-(W_{k-1})_p$ and $s_{0k}$ is a local section of
$W_{k-1}$. Then $s_1(e)\wedge ...\wedge s_{j-1}(e)=z^{j-1}{\tilde s}_1(e)
\wedge ... \wedge {\tilde s}_{j-1}(e)$ for $e$ close to $p$. By $(b3)$, we
have ${\tilde s}_1(p)\wedge ... \wedge {\tilde s}_{j-1}(p) \neq 0$ and by
continuity ${\tilde s}_1(e)\wedge ... \wedge {\tilde s}_{j-1}(e) \neq 0$
for $e$ close to $p$. Thus ${\tilde s}_1 ... {\tilde s}_{j-1}$ is a
local holomorphic frame of $W_{j-1}$ around $p$.
We obtain :
\begin{equation}
\nonumber
s_1(e)\wedge ... \wedge s_j(e)=z^{j-1}{\tilde s}_1(e)\wedge ... \wedge
{\tilde s}_{j-1}(e)\wedge s_j(e)
\end{equation}
(for $e$ close to $p$), which together with $(b3)$ gives :
\begin{equation}
\nonumber
\label{foo}
{\rm deg} {\tilde s}_1\wedge ... \wedge {\tilde s}_{j-1}\wedge s_j (p) = 1
\end{equation}
Since ${\tilde s}_1 ... {\tilde s}_{j-1}$ is a
local holomorphic frame of $W_{j-1}$ around $p$, this shows, by Proposition
\ref{frame_criterion}, that ${\rm deg} _{W_{j-1}}s_j(p)=1$.
Since
${\tilde s}_1(p)\wedge ... \wedge {\tilde s}_{j-1}(p)\wedge s_j(p)=0$
by (\ref{foo}), it follows that
$s_j(p)\in <{\tilde s}_1(p) ... {\tilde s}_{j-1}(p)>$ \\ $=(W_{j-1})_p$.
By the induction hypothesis, $s_2(p)...s_{j-1}(p)$ is a basis of
$(W_{j-2})_p \subset (W_{j-1})_p$,
so that $s_2(p) .... s_{j-1}(p) \in (W_{j-1})_p$.
Thus, the vectors $s_2(p)....s_j(p)$ all belong to the $j$-dimensional
vector space $(W_{j-1})_p$. Since they are linearly independent by $(b2)$,
they must form a basis of this subspace. This finishes the proof of the claim.
Since
$\delta^{\cal K}_j(p)-\delta^{\cal K}_{j-1}(p)={\rm deg} _{W_{j-1}}s_j(p)$,
the first part of the claim implies $(c32)$.
The second part of the claim is easily seen to imply $(c31)$.
Thus $(b)$ implies $(c)$.
{\em Show that (c) implies (a)}
Again proceed by induction on $r$. For $r=1$ the statement is immediate.
Now let $r >1 $ and suppose that $(c)\Rightarrow (a)$ holds for $r-1$.
Also assume that $V'$ satisfies $(c)$.
Since ${\cal K}$ is nondegenerate, we have ${\rm dim}_{{\Bbb C}}K_j=j$ for all $j=1..r$.
In particular, $K_1$ is a line bundle. By $(c31)$ and $(c31)$ we have
$K_1\approx O(p)$. Define $W':=V'/K_1$. We have an exact sequence:
\begin{equation}
\label{s_1}
0 \longrightarrow W_1\stackrel{j}{\longrightarrow}V'\stackrel{p}{\longrightarrow} W'
\longrightarrow 0
\end{equation}
To show $(a)$ it suffices to show that $W'\approx F'_{r-1}$ and that
(\ref{s_1}) is nonsplit. By the induction hypothesis,
to show $W'\approx F'_{r-1}$ it suffices to show that $W'$ satisfies $(c)$
for $r-1$. We proceed to do this.
{\em Show that $W'$ satisfies $(c1)$.}
Since $H^1(O(p))=0$, (\ref{s_1}) gives :
\begin{equation}
\label{s_2}
0 \longrightarrow H^0(O(p))\stackrel{j_*}{\longrightarrow}H^0(V')
\stackrel{p_*}{\longrightarrow} H^0(W') \longrightarrow 0
\end{equation}
Thus $h^0(W')=r-1$.
{\em Show that $W'$ satisfies $(c2)$.}
For each $e \in E -\{p\}$ we have a commutative diagram with exact rows:
\begin{equation}
\begin{array}{ccccccccc}
0 & \longrightarrow & H^0(K_1) & \stackrel{j_*}\longrightarrow &H^0(V')
&\stackrel{p_*}{\longrightarrow} & H^0(W') & \ \longrightarrow 0 \\
\ & \ & \phi^{O(p)}_e \downarrow & \ & \phi_e \downarrow
& \ & \phi'_e \downarrow & \ \\
0 & \longrightarrow & K_{1,e} & \stackrel{j_e}{\longrightarrow} & V'_e &
\stackrel{p_e}{\longrightarrow} &
W'_e & \longrightarrow & 0 \\
\end{array}
\end{equation}
where the vertical arrows represent the evaluation maps.
$\phi^{O(p)}_e$ is trivially an isomorphism, while $\phi_e$ is
an isomorphism since $V'$ satisfies $(c1)$ and
$(c2)$. Thus $\phi'_e$ is an isomorphism. We will see below that $\phi_p$
is not injective. Thus $W'$ satisfies $(c2)$.
{\em Show that $W'$ satisfies $(c31)$ and $(c32)$.}
First we show that $K_j=H^0(W_j)$ for all $j=1..r$. To see this, note that
$(c31)$ implies $H^0(W_{j-1}) \subset K_j$ for all $j$. This inclusion is
{\em strict} (otherwise $\phi_e|_{K_j}$ for $e \neq p$ would coincide with the
evaluation map of $W_{j-1}$; since $\phi_e$ is injective and ${\rm dim}_{{\Bbb C}}K_j=j$,
this would contradict the rank theorem). We also trivially have
$K_j \subset H^0(W_j)$ for all $j$. This gives
$H^0(W_{j-1}) \subset K_j\stackrel{\neq}{\subset}H^0(W_j)$ for all $j$
and since ${\rm dim}_{{\Bbb C}}K_j=j$ we obtain $K_j=H^0(W_j)$.
${\cal K}$ induces a filtration ${\cal K}'$:
\begin{equation}
0=K'_0\subset K'_1\subset ...\subset K'_{r-1}
\end{equation}
by $K'_j:=p_*(K_{j+1})$ for all $j=1..r-1$. By (\ref{s_2}) we have
$K'_{r-1}=H^0(W')$ and ${\rm dim}_{C}K'_j=j$ for all $j=1..r-1$. On the other hand,
the filtration ${\cal W}$ of $V'$ induces a nondegenerate
filtration ${\cal W}'$ of $W'$:
\begin{equation}
0=W'_0\subset W'_1\subset ...\subset W'_{r-1}=W'
\end{equation}
by $W'_j:=p(W'_{j+1})$.
For each $j=1..r-1$ we have a commutative diagram:
\begin{equation}
\begin{array}{ccccccccc}
0 & \longrightarrow & K_1 & \stackrel{j_*}{\longrightarrow} & K_{j+1}
&\stackrel{p_*}{\longrightarrow}& K'_j & \longrightarrow 0 \\
\ & \ & \phi_p \downarrow & \ & \phi_p \downarrow
& \ & \phi'_p \downarrow & \ \\
0 & \longrightarrow & K_{1,p} &
\stackrel{j_p}{\longrightarrow} & W_{j,p} &
\stackrel{p_p}{\longrightarrow} &
W'_{j-1,p} & \longrightarrow & 0 \\
\end{array}
\end{equation}
(we have $\phi'_p(K'_j)=\phi'_p(p_*(K_{j+1}))=p_p(\phi_p(K_{j+1}))\subset
p_p(W_{j,p})=W'_{j-1,p}$ where we used $(c31)$ for $V'$ ).
Commutativity of the second square gives $p_*^{-1}(\phi_p^{' -1}(W'_{j-1,p}))=
\phi_p^{-1}(p_p^{-1}(W'_{j-1,p}))=\phi_p^{-1}(W_{j,p})=K_{j+1}$ where we used
$(c31)$ for $V'$. Thus $\phi_p^{-1}(W'_{j-1,p})$\\$=
p_*(K_{j+1})=K'_j$ and ${\cal K}',{\cal W}'$ satisfy $(c31)$.
Now pick $s_j \in K_j-K_{j-1}$ for all $j=1..r$ and let
$\sigma_j:=p_*(s_{j+1})$ for all $j=1..r-1$. Then
$\sigma_j \in K'_j -K'_{j-1}$ and ${\rm deg} _{W'_{j-1}}\sigma_j (p)={\rm deg} _{W_j}
s_{j+1}(p)$ by Proposition \ref{induced_section}. Using $(c32)$ for $V'$
and $\delta^{\cal K}_j(p)-\delta^{\cal K}_{j-1}(p)={\rm deg}_{W_{j-1}}\sigma_j(p)$,
this immediately implies $(c32)$ for $W'$. (In particular, we have
${\rm ker} \phi'_p=K'_1 \neq 0$, as announced above).
Now suppose that (\ref{s_1}) is split. Then
$V' \approx O(p) \oplus F'_{r-1}$. Since $O(p)$ and
$F'_{r-1}$ both posess nonzero
sections which vanish at $p$,
this immediately gives two linearly independent sections of $V'$ which vanish
at $p$. But $(c3)$ implies $d_p(V')=1$, which gives a
contradiction. Thus, (\ref{s_1}) cannot split and we must have
$V'\approx F'_r$
and $V\approx F_r$. Thus $(c)$ implies $(a)$.
To prove the last statement of the theorem it suffices to note that
each of the bundles $W_j$ in $(c)$ also satisfies $(c)$ for the
appropriate rank. $\Box$
It is now possible to analyze the freedom in the choice of ${\tilde s}_j$
and define a notion of canonical bases of $H^0(F'_r)$ by imposing further
conditions on $s_1..s_r$. This leads to a concrete description of the
endomorphisms of $F'_r$ via their induced action on $H^0(F'_r)$, which
can then be used to analyze the endomorphisms of a general
degree zero semistable
bundle by using the results of the next subsection.
Since this is not directly related to the main focus of the present paper, we
will not proceed down that path.
\subsection{The main theorem}
The results of the previous subsection immediately lead to:
\begin{Theorem}
\label{final_theorem}
Let $V$ be a degree zero holomorphic vector bundle of rank $r$ over $E$ and
$V'=V \otimes O(p)$. Let $\phi_e$ be the evaluation map of $V'$ and
$S:={\cal Z}(V'):=\{t \in E | K_t(V') \neq 0\}$.
The following are equivalent :
(a) $V$ is semistable
(b0) $h^0(V')=r$
(b1) The set $S$ is finite. Let
$d_t:={\rm dim}_{{\Bbb C}}K_t(V')$ for all $t \in S$.
(b3) There exists a direct sum decomposition :
\begin{equation}
H^0(V')=\oplus_{t \in S}{\oplus_{i=1..d_t}{K^{(i)}_{r_{t,i}}(t)}}
\end{equation}
with ${\rm dim}_{{\Bbb C}}K^{(i)}_{r_{t,i}}(t)=r_{t,i}$ and {\em nondegenerate}
filtrations :
\begin{equation}
{\cal K}^{(i)}(t) \ : \
0=K^{(i)}_0(t)\subset K^{(i)}_1(t)\subset ...\subset K^{(i)}_{r_{t,i}}(t)
\end{equation}
with $\psi$-associated bundle filtrations:
\begin{equation}
{\cal W}^{(i)}(t) \ : \
0=W^{(i)}_0(t)\subset W^{(i)}_1(t)\subset ...\subset W^{(i)}_{r_{t,i}}(t)
\end{equation}
with the properties :
(b31) We have
$V'_t=R_t(V')\oplus \oplus_{i=1..d_t}{(W^{(i)}_{r_{t,i}}(t))_t}$, for all
$t \in S$.
(b32) $\delta^{{\cal K}^{(i)}(t)}_s(t)=s$ for all $t \in S$, all $i=1..d_t$
and all $s=1..r_{t,i}$
(b33) The induced filtrations
$0=\phi_t(K^{(i)}_1(t))\subset ...\subset \phi_t(K^{(i)}_{r_{t,i}}(t))$
in $V'_t$ are nondegenerate for all $t\in S$ and $i=1..d_t$
(c) The following conditions are satisfied:
(c1) $h^0(V')=r$
(c2) The set $S$ is finite. Let $d_t={\rm dim}_{{\Bbb C}}
K_t(V')$, $\forall t \in S$
(c3) There exists a basis $(s^{(i)}_{t,j})_{t \in S, i=1..d_t, j=1..r_{t,i}}$
of $H^0(V')$ ($\sum_{t \in S, i=1..d_t}{r_{t,i}}=r$) with the properties :
(c31) ${\rm deg} (\Lambda_{i=1..d_t, t' \in S}
{s^{(i)}_{t',1}\wedge...\wedge s^{(i)}_{t',r_{t',i}}})(t)=
\sum_{i=1..d_t}{r_{t,i}}$,
$\forall t \in S$
(c32) $(s^{(i)}_{t,j})_{j=2...r_{t,i}}$ are
linearly independent for all $t\in S$ and all $i=1..d_t$.
(c33) ${\rm deg} (s^{(i)}_{t,1}\wedge...\wedge s^{(i)}_{t,j})(t)=j$,
$\forall t \in S, \ \forall i=1..d_t, \ \forall j=1..r_{t,i}$
In this case, we have:
$V' \approx \oplus_{t \in S}{\oplus_{i=1 ... d_t}{O(t)\otimes F_{r_{t,i}}}}$
\end{Theorem}
The proof should be rather obvious by now. Instead of writing down all of its
details, let us try to make the statement of the theorem
look less formidable. Clearly the
bundles $W^{(i)}_{r_{t,i}}(t)$ are isomorphic to $O(t)\otimes F_{r_{t,i}}$,
while $W^{(i)}_j(t)\approx O(t)\otimes F_j$ give their canonical filtrations.
$d_t$ is the number of different indecomposable bundles which
multiply $O(t)$ in the decomposition of $V'$. These bundles are just
$W^{(i)}_{r_{t,i}}(t)$, and have ranks $r_{t,i}$ (of which some may coincide).
Conditions $(b32)$ and $(b33)$ or, equivalently, conditions $(c32)$
and $(c33)$ are needed to assure that
$W^{(i)}_{r_t,i}(t)\approx F_{r_{t,i}}$. Conditions $(b31)$,
respectively $(c31)$ are needed in order to have a {\em direct} factor of
the form
$O(t)\otimes \oplus_{i=1..d_t}{F_{r_{t,i}}}$ in the decomposition of $V'$.
Note that the spectral divisor is:
\begin{equation}
\Sigma_V=\sum_{t \in S}{\sum_{i=1..d_t}{r_{t,i}~t}}
\end{equation}
We immediately obtain
\footnote{This result can also be obtained without making use of Theorem
\ref{final_theorem}}:
\begin{Corrolary}
\label{spectral_div}
Let $V$ be a degree zero semistable holomorphic vector bundle over $E$ and
$V'=V \otimes O(p)$. Let $s_1...s_r$ be a ${\Bbb C}$-basis of $H^0(V')$.
Then the spectral divisor of $V$ is given by :
\begin{equation}
\label{sp_div}
\Sigma_V=(s_1\wedge ... \wedge s_r)
\end{equation}
\end{Corrolary}
{\em Proof:} Since $(s_1\wedge ... \wedge s_r)$ is independent of the choice
of the basis of sections $s_1...s_r$, we can choose $s_1..s_r$ to have
the properties listed in $(c)$ of Theorem \ref{final_theorem}. Then the
conclusion is obvious.
$\Box$
This shows that the spectral divisor can be computed by an obvious
adaptation of the methods of \cite{CGL} even in the general case.
However, the divisor $(s_1\wedge...\wedge s_r)$ alone cannot give us enough
information to test
semistability and/or determine the splitting type.
Starting from the above theorem, it is relatively straightforward to develop
an algorithm for testing semistability of $V$ and determining its splitting
type by doing a series of simple manipulations on an arbitrary basis of
$H^0(V')$. Instead of presenting the algorithm in its full generality (which
requires introducing a slightly tedious amount of notation), we will
show explicitly how this can be implemented in the simpler case when one is
interested in identifying degree zero {\em fully decomposable} semistable
bundles. This is explained in section 3 below.
\subsection{The spectral divisor in the monad case and a `moduli problem'}
In this subsection we consider the case when $V$ is given by the cohomology
of a monad:
\begin{equation}
\label{monad}
0 \longrightarrow \oplus_{j=1..s}{O_E}\stackrel{f}{\longrightarrow}
\oplus_{a=1..m}{O(D_a)}\stackrel{g}{\longrightarrow}O(D_0)\longrightarrow 0
\end{equation}
Here $D_a,D_0$ are some divisors on $E$.
We define the twisted bundles and exact sequences as before.
We denote all twisted objects by a prime. As usual, we twist by $O(p)$ with $p$
an arbitrary point on $E$.$p$ is fixed throughout the following discussion.
We have $m=r+s+1$ where $r:={\rm rank} V$.
Write (\ref{monad}) as the pair of exact sequences :
\begin{equation}
\label{sequence1}
0 \longrightarrow {\rm ker} g \hookrightarrow \oplus_{a=1..m}{O(D_a)}\stackrel{g}
{\longrightarrow}O(D_0)\longrightarrow 0
\end{equation}
\begin{equation}
\label{sequence2}
0 \longrightarrow\oplus_{j=1..s}{O_E}\stackrel{f}{\longrightarrow}{\rm ker} g
\stackrel{p}{\longrightarrow}V{\longrightarrow}0
\end{equation}
By taking degrees we obtain :
\begin{equation}
\label{degrees}
{\rm deg} V=\sum_{a=1..m}{{\rm deg} D_a}-{\rm deg} D_0={\rm deg} ({\rm ker} g)
\end{equation}
We have:
\begin{Proposition}
\label{ker_ss}
The following are equivalent :
(a) $V$ is semistable and of degree zero
(b) ${\rm ker} g$ is semistable and of degree zero
\end{Proposition}
Proof:
Assume that $(a)$ holds. Then the sequence (\ref{sequence2}) shows that
${\rm ker} g$ is an extension of $\oplus_{j=1..s}{O_E}$ by $V$.
As both these bundles are semistable and of slope zero, a standard result of
Seshadri (see. for example, \cite{Seshadri}) immediately entails $(b)$.
Assume $(b)$ holds. Then (\ref{sequence2}) shows that $V={\rm coker} f$
and since $\oplus_{j=1..s}{O_E}$ and ${\rm ker} g$ are both semistable and of slope
zero we can use another result of Seshadri to obtain $(a)$.
$\Box$
This proposition reduces the study of semistabilty of $V$ to that
of ${\rm ker} g$. In particular, we see that
semistability of $V$ depends only on the properties of the map $g$ and on
the bundles $\oplus_{a=1..m}{O(D_a)}$ and $O(D_0)$.
For the following we assume that $\oplus_{a=1..m}{{\rm deg} D_a}={\rm deg} D_0:=d$.
with $d \ge 0$. We let $d_a:={\rm deg} D_a$.
Then (\ref{degrees}) assures us that ${\rm deg} V={\rm deg} {\rm ker} g=0$.
Now {\em suppose that $V$ is semistable} .
Then by Proposition \ref{ker_ss} ${\rm ker} g$ is also semistable .
Then Lemma \ref{cohom_dim} assures us that $H^1({\rm ker} g')=0$.
Noting that $H^1(O(p))$ also vanishes by the Riemann-Roch theorem, it follows
that by twisting the two exact sequences above and taking cohomology we obtain
two {\em short} exact sequences :
\begin{equation}
\label{cohomology1}
0 \longrightarrow H^0({\rm ker} g') \hookrightarrow \oplus_{a=1..m}{H^0(O(D'_a))}
\stackrel{g_*}{\longrightarrow}H^0(O(D'_0))\longrightarrow 0
\end{equation}
\begin{equation}
\label{cohomology2}
0\longrightarrow\oplus_{j=1..s}{H^0(O(p))}\stackrel{f_*}{\longrightarrow}
H^0({\rm ker} g')\stackrel{p_*}{\longrightarrow}H^0(V'){\longrightarrow}0
\end{equation}
where $D'_a:=D_a+p,D'_0:=D_0+p$ and we denoted $f\otimes id, g\otimes id$ by
the same letters for simplicity.
The collapse of the cohomology sequence associated to
(\ref{sequence1}) is a direct consequence of the semistability of ${\rm ker} g$.
Since $d+1$ is positive, the Riemann-Roch theorem tells us that
$h^0(O(D'_0))={\rm deg} (D'_0)={\rm deg} D_0+1=d+1$. Since ${\rm ker} g$ is semistable
and of degree zero, Lemma \ref{cohom_dim} gives $h^0({\rm ker} g')=
{\rm rank} ({\rm ker} g)=r+s=m-1$; then (\ref{cohomology1}) gives
$h^0(\oplus_{a=1..m}O(D'_a))=m+d$. This last fact is not a consequence of
Riemann-Roch unless $d_a$ are all nonnegative.
\begin{Proposition}
\label{cover_rel}
Let $\Sigma_{{\rm ker} g}$ and $\Sigma_V$ be the spectral divisors of ${\rm ker} g$,
respectively $V$.
Then $\Sigma_{{\rm ker} g}=\Sigma_V+sp$.
\end{Proposition}
{\em Proof:}
Since (\ref{cohomology2}) is an exact sequence of
vector spaces, it must split. We can thus choose a
basis $v_1 ... v_{r+s}$ of
${\rm ker} g'$ with the properties :
(1)$v_1=f_*(w_1)...v_s=f_*(w_s)$, where $w_1...w_s$ is a
basis of $A:=\oplus_{j=1..s}{H^0(O(p))}$
(2)$p_*(v_{s+1}):=u_1...p_*(v_{s+r}):=u_r$ is a basis of $H^0(V')$
The canonical isomorphism ${\rm det} ({\rm ker} g') \approx {\rm det} (A)\otimes {\rm det} (V')$
maps the section $v_1\wedge ..\wedge v_{r+s}\in H^0({\rm det} ({\rm ker} g'))$ into
the the section $(w_1\wedge...\wedge w_s)\otimes (u_1\wedge ...\wedge u_r)
\in H^0({\rm det} A)\otimes H^0({\rm det} V')\subset H^0({\rm det} A \otimes {\rm det} V')$.
Thus:
\begin{equation}
\nonumber
\Sigma_{{\rm ker} g}=(v_1\wedge...\wedge v_{r+s})=
(w_1\wedge...\wedge w_s\otimes u_1\wedge ...\wedge u_r)
=(w_1\wedge...\wedge w_s) + (u_1\wedge ...\wedge u_r)=
sp+\Sigma_V
\end{equation}
where in the first and last line we used the corrolary to
Theorem \ref{final_theorem}.
$\Box$
The relation between $\Sigma_{{\rm ker} g}$ and the bundle
$B:=\oplus_{a=1..m}{O(D'_a)}$ is more complicated. The reason is that
there is no simple connection between the local behaviour of the sections
of ${\rm ker} g$ and the sections of $B$\footnote{This happens because typically
we have $d_a > 0$ for some $a$. Then $O(D'_a)$ is quasi-ample (the evaluation
map is surjective everywhere), which to complications.}.
To extract more information about ${\rm ker} g$, one has to undertake
a more
detailed study based on the properties of the map $g$. In particular,
one would like to find necessary and sufficient conditions on $g$ such that
${\rm ker} g$ is semistable and describe the associated moduli space of $g$.
Although we will not
attempt this here, let us formulate the geometric set-up of the problem.
Theorem \ref{final_theorem} reduces the semistability condition for
${\rm ker} g$ to conditions on the subspace $W:=H^0({\rm ker} g')$ of $U:=H^0(B)$.
We have ${\rm dim}_{{\Bbb C}}U=m+d$ and semistability requires that ${\rm dim}_{{\Bbb C}}W=m-1$.
Let $H_e:={\rm ker} \phi^B_e$, for all $ e\in E$. Note that
$\phi_e^{{\rm ker} g'}=\phi_e^{B}|_W$, so that
${\rm ker} \phi_e^{{\rm ker} g'}=H_e \cap W$.
Suppose for simplicity that all $D_a$ are effective and that
${\rm Card}\{a \in \{1..m\} | d_a=0\}=\nu$. Since $\phi_e^B=\oplus_{a=1..m}
{\phi_e^{O(D'_a)}}$ for all $e \in E$, we have ${\rm ker} (\phi_e^B)=
\oplus_{a=1..m}{{\rm ker} \phi_e^{O(D'_a)}}$.
With our assumptions, we have ${\rm codim}_{{\Bbb C}}({\rm ker} \phi^{O(D'_a)}_e)=1$
for all $a$ with $d_a>0$ and all $e \in E$, while for all $a$ with $d_a=0$
(i.e. $D_a=0,~ O(D'_a)=O(p)$) we have
${\rm codim}_{{\Bbb C}}({\rm ker} \phi^{O(D'_a)}_e)=1$ for $e \neq p$ and
${\rm codim}_{{\Bbb C}}({\rm ker} \phi^{O(D'_a)}_p)=0$.
Thus ${\rm codim}_{{\Bbb C}}H_e=m$ for all $e \neq p$ while ${\rm codim}_{{\Bbb C}}H_p=m-\nu$.
Note that ${\rm dim}_{C}W+{\rm dim}_{{\Bbb C}}H_e = {\rm dim}_{{\Bbb C}}U-1$ for $e \neq p$ while
${\rm dim}_{{\Bbb C}}W+{\rm dim}{{\Bbb C}}H_p ={\rm dim}_{{\Bbb C}}U +\nu-1$. For given divisors $D_a$ and a given
map $g$, $W \cap H_e$ will have fixed dimension $D$ for almost all points
$e \in E$. The points where the dimension of this intersection increases
correspond to the points of the set ${\cal Z}({\rm ker} g')$.
If $\nu > 1$, it follows that ${\rm dim}_{{\Bbb C}} W \cap H_p \ge \nu-1$.
On the other hand, we cannot deduce any simple
lower bound on
${\rm dim}_{{\Bbb C}}W \cap H_e$ for $e \neq p$.
Geometrically, we are given a map $H:E \rightarrow Sbsp(U)$,
$H(e):=H_e, \forall e \in E$
from $E$ to the set of subspaces of the $m+d$-dimensional ${\Bbb C}$-vector space
$U$. The precise form of this map is completely fixed by the bundle $B$.
As $e$ varies in $E$, $H_e$ describes a complicated trajectory in $Sbsp(U)$.
Generically on $E$, $H_e$ has codimension $m$, except at the point $e=p$ where
it has codimension $m-\nu$.
Giving a semistable subbundle of $B$ of the form ${\rm ker} g$ requires
giving the $m-1$ dimensional subspace $W$ of $U$, with the property that
it is complementary to $H_e$ for a generic $e\in E$ and satisfying the
other conditions in Theorem \ref{final_theorem}. The precise position of
$W$ inside $U$ is controlled by the map $g$.
It is not hard to see that the remaining
conditions in the theorem can be expressed in terms of
`incidence relations' constraining the `speeed of incidence' of $W$ on $H_e$
as $e \rightarrow t_i$; this is similar to the discussion of Section 2.
The set-up above allows us to reduce the problem of determining the maps
$g$ giving
a semistable ${\rm ker} g$ to a problem
in linear algebra and analysis. In particular, it is ideal for
extracting information about `moduli'.
The `trajectory' of $H_e$ is, however,
rather complicated in general and the problem may be quite difficult in
practice. It
would be interesting to investigate this further.
\section{The fully split case}
\subsection{Twist by $O(p)$}
Let $(E,p)$ be an elliptic curve with a marked point
and $V$ a holomorphic bundle of degree zero and rank $r$ on $E$.
Let $V':=V\otimes O(p)$.
We say that $V$ is {\em fully split} if there exists a decomposition
$V=\oplus_{j=1..r}{L_j}$ of $V$ into a direct sum of line bundles $L_j$.
We present an algorithm for determining whether a given
degree zero holomorphic vector bundle $V$ is semistable {\em and} fully split.
The algorithm requires explicit knowledge of $H^0(V')$ and
allows for the {\rm det} ermination of the line bundles $L_j$ up to
holomorphic equivalence.
\begin{Theorem}
\label{theorem_fully_split}
Let $V$ be a degree zero holomorphic vector bundle of rank $r$
over $E$. Let $R_e:=R_e(V'),~r_e:={\rm dim}_{{\Bbb C}}R_e,~K_e:=K_e(V')$ and
$d_e:={\rm dim}_{{\Bbb C}}K_e$ for any $e \in E$.
The following statements are equivalent :
(a) V is semistable and fully split
(b) $V'$ satisfies all of the following conditions :
(b0) $h^0(V')=r$
(b1) The set $S:={\cal Z}(V')= \{ t \in E | r_t < r \}$ is finite
(b2) For all $t \in S$, all holomorphic sections of $V'$ belonging to
$K_t - {0}$ have degree $1$ at $t$
(b3) For each $t \in S $ we have $V'_t = R_t \oplus N_t$
(b4) We have $H^0(V')=\oplus_{t \in S}{K_t}$
(c) $V'$ satisfies (b0),(b1), (b4) and the condition that there exits a
basis $(s_1..s_r)$ of $H^0(V')$ such that :
(b23) ${\rm deg} s_1\wedge s_2 \wedge ... \wedge s_r (t)= d_t,\forall t \in S$
Moreover, in this case we have
$V \approx \oplus_{t \in S}{O(t-p)^{\oplus d_t}}$.
\end{Theorem}
Note that if $(b23)$ holds for a basis of $H^0(V')$ then it will hold
for any other basis.
{\em Proof:}
{\em Show that (a) implies (b):}
Assume $(a)$ holds and write $V=\oplus_{i=1..r}{L_i}$ with
$L_i \in \operatorname{Pic}^0(E)$.
Then $L_i\approx O(q_i -p)$ $(q_i \in E)$ and $V'=\oplus_{i=1..r}{L'_i}$,
with $L'_i=L_i \otimes O(p)\approx O(q_i)$.
We know that $(b0)$ holds by Lemma \ref{cohom_dim}.
Let $s_i \in H^0(L'_i)-\{0\}$. Then $s_1 ... s_r$ is a ${\Bbb C}$-basis of $H^0(V')$.
We obviously have $S=\cup_{i=1..r}{\{q_i\}}$, so $(b1)$ holds.
Since $V'$ is semistable of slope $1$ we see that $(b2)$ also holds (cf.
the remark before Proposition \ref{isom}).
Now suppose there are two distinct points $t_1,t_2 \in S$ such that
$K_{t_1}\cap K_{t_2} \neq \{0\}$. Let $s \in K_{t_1}\cap K_{t_2}$.
Then $s(t_1)=s(t_2)=0$ and, since $s$ is regular, we must have ${\rm deg} L_s\geq2$,
which contradicts semistability of $V'$. Thus the sum $\sum_{t \in S}{K_t}$
is direct. On the other hand, any $s \in H^0(V')$ is a linear combination
$s =\sum_{i=1..r}{\alpha_i s_i}$ $(\alpha_i \in {\Bbb C})$. Since $s_i \in K_{q_i}$,
we have $ s \in \sum_{t \in S}{K_t}$. Thus $(b4)$ holds.
To show $(b3)$, note that $L'_i=L_{s_i}$ (in the notation of subsection 2.1).
Fixing $t\in S$, we clearly have
$R_t=\oplus_{i; q_i\neq t}{(L'_i)_t}$, $K_t=\oplus_{i; q_i = t}{H^0(L'_i)}$
and $N_t=\oplus_{i; q_i=t}{(L'_i)_t}$.
Since $V'_t=\oplus_{i=1..r}{(L'_i)_t}$, we have $V'_t=R_t \oplus N_t$.
{\em Show that (b) implies (a):}
Let $d_t:={\rm dim}_{{\Bbb C}}K_t$ $(t \in S)$.
By $(b4)$, we can choose a ${\Bbb C}$-basis $(s^{(t)}_j)_{t \in S, j=1..d_t}$
of $H^0(V')$ such that $(s^{(t)}_{j})_{j=1..d_t}$ is a ${\Bbb C}$ -basis of
$K_t$ for each $t \in S$. By $(b4)$ and $(b2)$, each section $s^{(t)}_i$
has exactly one zero on $E$, namely at $t$, and this zero is simple
(the unicity of this zero easily follows from $(b4)$). Therefore the line
bundles $L^{(t)}_j:=L_{s^{(t)}_j}$ have degree $1$ and we have
$L^{(t)}_j \approx O(t)$. In particular, for all $j=1..d_t$ we have
$s^{(t)}_j(t)=0$ and ${\hat s}^{(t)}_j (t) \neq 0, \forall j=1..d_t $
and $s^{(t)}_j(t')\neq 0, \forall t' \in S - \{t \}$.
Moreover, $(b3)$ implies that $s^{(t')}_j(t) (t' \in S -\{t\},j=1..d_{t'})$
and ${\hat s}^{(t)}_j(t) (j=1..d_t)$ form a basis of $V'_t$. Therefore, we
have $V'_t = \oplus_{t' \in S, j=1..d_{t'}}{(L^{(t')}_j)_t}$,
$\forall t \in S$.
On the other hand, for all $e \in E - S$ we have ${\rm dim}_{{\Bbb C}}R_e=r$.
Since $(s^{(t)}_j(e))_{t \in S, j=1..d_t}$ obviously generate $R_e$
and since $(b4)$ implies that
${\rm Card}\{s^{(t)}_j | t \in S, j=1..d_t\}=r$, it must be the case that
$(s^{t}_j(e))_{t \in S, j=1..d_t}$ is a ${\Bbb C}$-basis of $V'_e$,
for all $e \in S -t$. Therefore, we also have
$V'_e = \oplus_{t \in S, i=1..d_t}{(L^{(t)}_i)_e}$, for $e \in E-S$.
Therefore,
$V'=\oplus_{t \in S, j=1..d_t}{L^{(t)}_j}$. Since each component of this sum
has slope $1$, it follows that $V'$ is semistable and of slope $1$,
while $V=V' \otimes O(-p)$ is semistable and of degree zero.
We also have $V'\approx \oplus_{t \in S}{O(t)^{d_t}}$ and
$V\approx \oplus_{t \in S}{O(t-p)^{d_t}}$.
{\em Show that (b) and (c) are equivalent}
For this, assume that $(b0)$,
$(b1)$ and $(b4)$ hold. Then we show that $(b2)$ and $(b3)$ together are
equivalent to $(b23)$.
Remember that ${\rm deg} s_1\wedge...\wedge s_r(e)$ does not depend on the
choice of the ${\Bbb C}$-basis of $H^0(V')$. Enumerating
$S=\{t_1..t_k\}$ we can assume
that $(s_i)_{d_1+...+d_{j-1}+1\le i \le d_1+...+d_j}$ is a ${\Bbb C}$-basis
of $K_{t_j}$ for all $j=1..k$. Since the argument is similar for each
$j$, let us focus on $t_1:=t$. Then $s_1...s_{d_t}$ is a
basis of $K_t$ and for $i=1..d_t$ we have $s_i(e)=z\sigma_i(e)$ for all
$e$ close to $t$, where $\sigma_i$ are local holomorphic sections
of $V'$ around $t$.
{\em Claim 1}: If $(b0)$, $(b1)$ and $(b4)$ hold then
$s_{d_t+1}(t)...s_r(t)$ is a basis of $R_t$.
Indeed, since $s_1(t)=...s_{d_t}(t)=0$, we clearly have that
$s_{d_t+1}(t)...s_r(t)$ generate $R_t$. If $\alpha_{d_t+1}s_{d_t+1}(t)+
... + \alpha_rs_r(t)=0$ is zero a linear combination, then
the section $s:= \alpha_{d_t+1}s_{d_t+1}+... + \alpha_r s_r$ of $V'$
vanishes at $t$ so that it belongs to $K_t$. Since our basis $s_1...s_r$
is `adapted' to the decomposition $(b4)$, $s$ then gives an element
of $K_t \cap (\sum_{t' \in S -\{t\}}{K_{t'}})$, which must be zero
since the sum in $(b4)$ is direct. Since $s_{d_t+1}...s_r$ are ${\Bbb C}$-linearly
independent, this implies that $\alpha_{d_t+1}=..=\alpha_r=0$. Thus
$s_{d_t+1}(t)...s_r(t)$ are linearly independent and the claim is proven.
{\em Claim 2}: If $(b0)$, $(b1)$ and $(b4)$ hold then the following are
equivalent:
($\alpha$) $(b2)$ holds at $t$
($\beta$) $\sigma_1(t)...\sigma_{d_t}(t)$ are linearly independent
In this case, $\sigma_1(t)...\sigma_{d_t}(t)$ form a basis of $N_t$.
To prove this, first assume that $(b2)$ holds at $t$. Consider
a zero linear combination
$\alpha_1\sigma_1(t) +...+\alpha_{d_t}\sigma_{d_t}(t)=0$.
If the section $s:=\alpha_1s_1(t) +...+\alpha_{d_t}s_{d_t}$ would be nonzero,
then it would have vanishing degree at least $2$ at $t$. This would contradict
$(b2)$. Therefore, we must have $s=0$ and $\alpha_1=..\alpha_{d_t}=0$.
This proves that ($\alpha$) implies ($\beta$).
Now assume that $(\beta)$ holds and consider a section $s \in K_t -\{ 0\}$.
Then $s=\alpha_1s_1(t) +...+\alpha_{d_t}s_{d_t}$ for some $\alpha_i \in {\Bbb C}$
so that $s(e)=z(\alpha_1\sigma_1(e) +...+\alpha_{d_t}\sigma_{d_t}(e))$
for $e$ close to $t$. Since $s$ is not the zero section, at least one
$\alpha_i$ is nonzero and $(\beta)$ implies that
$\alpha_1\sigma_1(t) +...+\alpha_{d_t}\sigma_{d_t}(t)$ is nonzero. Thus
$s$ has degree $1$ at $t$ and $(\alpha)$ holds.
Assume that the equivalent conditions $(\alpha)$, $(\beta)$ hold and
show that $\sigma_1(t) ... \sigma_{d_t}(t)$ generate $N_t$.
We have $N_t:=<A>$, where $A:=\{{\hat s}(t)| s \in K_t\}$.
If $s \in K_t-\{0\}$, the above arguments show that ${\hat s}(t)$
belongs to $<\sigma_1(t) ... \sigma_{d_t}(t)>$, and this is also
trivially true for $s=0$ (since ${\hat s}(t)=0$ by definition in this case).
Therefore we have $A \subset <\sigma_1(t) ... \sigma_{d_t}(t)>$
and $\sigma_1(t) ... \sigma_{d_t}(t)$ generate $N_t$. This finishes the proof
of Claim 2.
Now return to the proof of the theorem.
Since $s_1(e) \wedge ... \wedge s_r(e)=
z(\sigma_1(e) \wedge ... \wedge \sigma_{d_t}(e) \wedge s_{d_t+1}(e)\wedge ...
\wedge s_r(e))$ for $e$ close to $t$, $(b23)$ is equivelent to the statement
that $\sigma_1(t) ...\sigma_{d_t}(t), s_{d_t+1}(t)... s_r(t)$ is a basis
of $V'$. By Claim 1, linear independence of $s_{d_t+1}(t)... s_r(t)$ is
automatic and $<s_{d_t+1}(t)... s_r(t)>=R_t$. By Claim 2, linear
independence of $\sigma_1(t) ...\sigma_{d_t}(t)$ is equivalent to
$(b2)$ and in this case $<\sigma_1(t) ...\sigma_{d_t}(t)>=N_t$.
Then $<\sigma_1(t) ...\sigma_{d_t}(t), s_{d_t+1}(t)... s_r(t)>=V'_t$
is equivalent to $(b3)$. $\Box$
Let us explain how one can test $(b4)$. Suppose that $(b0),(b1)$ hold and let
$s_1...s_r$ be an arbitrary ${\Bbb C}$-basis of $H^0(V')$.
For each $t \in S$, consider the $d_t$ -dimensional
subspace $P_t$ of ${\Bbb C}^r$ of linear relations among $s_1(t) ... s_r(t)$:
$P_t:=\{a:=(a_1 ... a_r) \in {\Bbb C}^r | a_1 s_1(t) + .. + a_r s_r(t)=0 \}$
Choose vectors $a^{(t,j)}\in {\Bbb C}^r$
$ (t \in S, j=1..d_t)$ such that, for each $t \in S$
$(a^{(t,j)})_{j=1..d_t}$ is a basis of $P_t$.
Let $\zeta^{(t,j)}:=\sum_{i=1..r}{a^{(t,j)}_i~s_i} \in H^0(V')$.
Then $(\zeta^{(t,j)})_{j=1..d_t}$ is a basis of $K_t$ for all $t \in S$.
In particular, we have $d_t={\rm dim}_{{\Bbb C}}P_t$.
Clearly $(b4)$ is equivalent to the condition:
\begin{equation}
{\Bbb C}^r=\oplus_{t \in S}{P_t}
\end{equation}
Chosing an enumeration $S=\{t_i|i=1..k\}$ of $S$,
we can form a matrix $A\in Mat(d,r,C)$, whose lines are given by the
vectors $(a^{(t_i,j)})_{i=1..k, j=1..d_t}$.
Then $(b4)$ is equivalent to the conditions $d=r$ and ${\rm det} A \neq 0$.
Therefore, we obtain the following
\
Algorithm :
\
Suppose $V$ is a rank $r$ and degree zero holomorphic vector bundle over $E$.
Let $p \in E$ arbitrary and define $V':=V\otimes O(p)$.
\
Step 1:
Obtain a basis $(s_1 ... s_n)$ of $H^0(V')$.
\
Step 2:
If $n \neq r $ then $V$ is not semistable. Otherwise, continue with
Step 3.
\
Step 3:
Let $\delta := s_1 \wedge ... \wedge s_r \in H^0(\Lambda^r V')$.
If $\delta = 0 $ then $V$ is not semistable (this follows from
the main theorem in section 2). Otherwise, the set
$S:={\rm supp} (\delta)$
is finite. In this case, enumerate $S=\{t_1 .. t_k\}$ and continue with
Step 4.
\
Step 4 :
For each $t \in S$, determine $d_t={\rm dim}_{{\Bbb C}}K_t$\footnote{
In general we can determine $d_t$ as $d_t={\rm dim}_{{\Bbb C}}P_t$. In the monad case,
$V'$ has a natural embedding into a direct sum of line bundles and
$d_t$ can be determined directly by considering the rank of a matrix of
sections as in \cite{CGL}}.
Then $V'$ is semistable and fully split iff each of
the following conditions is satisfied:
(a) $\sum_{t \in S}{d_t}=r$
(b) ${\rm deg} s_1\wedge ... \wedge s_r(t)=d_t$ for all $t \in S$
(c) The matrix $A$ is nonsingular
In this case, we have $V' \approx \oplus_{t \in S}{O(t)^{\oplus d_t}}$.
In particular, the spectral divisor of $V$ is given by :
\begin{equation}
\Sigma_{V}=\sum_{t \in S}{d_t~t}= (s_1 \wedge ... \wedge s_r)
\end{equation}
Note that ${\rm supp} \Sigma_V =S$.
\subsection{More general twists}
Let $V$ be a fully split semistable vector bundle of degree zero over $E$.
Then $V=\oplus_{j=1..r}{L_j}$ with $L_j \in \operatorname{Pic}^0(E)$.
Let $D=p_1+...+p_h$ be an effective divisor on $E$, where $p_1...p_h$ are
{\em mutually distinct} points on $E$.
We use $p_1$ as a base point of $E$. Then we can write
$L_j\approx O(q_j-p_1)$ with $q_j \in E$.
Define:
\begin{equation}
V':=V \otimes O(D)=\oplus_{j=1..r}{L'_j}
\end{equation}
where $L'_j:=L_j \otimes O(D)\approx O(q_j+p_2+...+p_h)$.
Since ${\rm deg} L'_j=h$, we have $h^0(L'_j)=h$ and $h^0(V')=rh$.
The Riemann-Roch theorem gives $h^1(V')=0$. The spectral divisor
of $V$ is $\Sigma_V=\sum_{j=1..r}{q_j}$. Let $S:={\rm supp} \Sigma_V$.
For each $q \in S$, let $S_q=\{j \in \{1..r\} | q_j=q\}$
and $d_q:={\rm Card}S_q$.
Then $L'_j \approx O(q+p_2+..+p_h)$ for all $j \in S_q$.
\begin{Lemma}
\label{lemma_S}
Let $q \in E$ be arbitrary. The set :
\begin{equation}
G_q(D):=\{s \in H^0(O(q+p_2+...+p_h)) | s(p_j)=0, \forall j=2..h\}
\end{equation}
is a one-dimensional subspace of the ${\Bbb C}$-vector space
$H^0(O(q+p_2+...+p_h))$.
Moreover, for any $s \in S_q(D)-\{0\}$ we have :
\begin{equation}
(s)=q+p_2+p_3+...+p_h
\end{equation}
\end{Lemma}
{\em Proof:}
Obviously the zero section belongs to $G_q(D)$.
Now let $s \in G_q(D) -\{0\}$. Since $s(p_2)=...=s(p_h)=0$, we have :
\begin{equation}
\label{foo2}
(s)=D_s +p_2+...+p_h
\end{equation}
with $D_s$ an effective divisor. Since $s \in H^0(O(q+p_2+...+p_h))$,
we have ${\rm deg} (s)={\rm deg} (q+p_2+...+p_h)=h$. But ${\rm deg} (s)={\rm deg} D_s +(h-1)$ by
(\ref{foo2}). Thus ${\rm deg} D_s$=1. Since $D_s$ is effective this implies $D_s=q'$
for some $q' \in E$.
On the other hand, $s \in H^0(O(q+p_2+...+p_h))$ implies
$(s)\sim q+p_2+...+p_h$, where $\sim$ denotes linear equivalence. Together
with (\ref{foo2}), this gives $q' \sim q$. If $q' \neq q$, this would imply
$E \approx {\Bbb P}^1$ by a classical theorem. Thus we must have $q'=q$ and
$(s)=q+p_2+...+p_h$ for all $(s) \in G_q(D)-\{0\}$. By a standard argument
this implies that any $s' \in G_q(D)-\{0\}$ is of the form $s'=\lambda s$
with $\lambda \in {\Bbb C}^*$ a constant. Thus $G_q(D)$ is a one dimensional
${\Bbb C}$-vector space.
$\Box$
Let :
\begin{equation}
G_j:=\{s \in H^0(L'_j) | s(p_2)=...=s(p_h)=0 \}\approx G_{q_j}(D)
\end{equation}
($j=1..r$).
By Lemma \ref{lemma_S}, $G_j$ are one-dimensional subspaces of $H^0(V')$.
Define :
\begin{equation}
G:=\{s \in H^0(V') | s(p_2)=...=s(p_h)=0\} \subset H^0(V')
\end{equation}
and:
\begin{equation}
G(q)=\oplus_{j \in S_q}{G_j}\subset G
\end{equation}
(for all $q \in S$).
\begin{Proposition}
We have :
\begin{equation}
\label{grad}
G=\oplus_{j=1..r}{G_j}=\oplus_{q \in S}{G(q)}
\end{equation}
In particular, $G$ is an $r$-dimensional subspace of the $rh$ dimensional
${\Bbb C}$-vector space $H^0(V')$.
Moreover, for any $s\in G-\{0\}$ we have the alternative :
Either
(a) $(s)=q+p_2+...+p_h$ for some $q\in S$
or
(b) $(s)=p_2+...+p_h$
\noindent If $(a)$ holds then $s \in G(q)$ for some $q \in S$,
while if $(b)$ holds then $s \in G-\cup_{q \in S}{G(q)}$.
\end{Proposition}
Here the equalities are between divisors and {\em not} between divisor
classes. That is, the equality in $(a)$ and $(b)$ is to be taken at face value
and {\em not} in the sense of linear equivalence.
{\em Proof:}
Since $V'=\oplus_{j=1..r}{L'_j}$, the statement $G=\oplus_{j=1..r}{G_j}$ is
obvious.
Now let $s \in G-\{0\}$. By Proposition \ref{degree_bound}, we have:
\begin{equation}
\label{deg}
{\rm deg} (s)\le \mu(V')=h
\end{equation}
Since $(s)$ is effective and $p_2...p_r \in {\rm supp} (s)$,
there are only two possibilities :
(a) $(s)=q+p_2+...+p_h$ for some $q \in E$ (note that $q$ may belong to the
set $\{ p_2...p_h\}$), and in this case ${\rm deg} (s)=h$
(b) $(s)=p_2+...+p_h$, and in this case ${\rm deg} (s)=h-1$.
\noindent Using $G=\oplus_{j=1..r}G_j$, we can write:
\begin{equation}
\label{dir_sum}
s=\oplus_{j=1..r}{s_j}
\end{equation}
where $s_j \in G_j$.
From $s_j \in G_j \subset H^0(L'_j)$, we obtain
$(s_j)=q_j+p_2+...+p_h ~{\rm \ unless \ } s_j=0$.
Since (\ref{dir_sum}) is a direct sum, we have :
\begin{equation}
\label{ineq}
(s_j) \geq (s) {\rm \ unless \ } s_j=0
\end{equation}
for all $j=1..r$ \footnote{This means that ${\rm deg} (s_j)(e) \ge {\rm deg} (s)(e)$ for
all $e \in E$.}. Indeed, $s$ can have a zero of order $m$ at $e\in E$
iff each $s_j$ has a zero of order at least $m$ at $e$.
In case $(a)$, (\ref{ineq}) shows that $(s_j)=(s)$ for all $j=1..r$ with
$s_j$ different from zero. This set is nonvoid iff $q \in S$ and in this
case we obtain $s=\sum_{j \in S_q}{s_j} \in G(q)$.
In case $(b)$, we cannot have $s \in G(q)$ for any $q$, since obviously this
would imply $(s) \ge q+p_2+...+p_h$, a contradiction. $\Box$
\begin{Definition}
A ${\Bbb C}$-basis $\sigma_1...\sigma_r$ of $G$ is called {\em canonical} if
${\rm deg} \sigma_j=h$, for each $j=1..r$.
\end{Definition}
By the previous proposition,
a basis of $G$ is canonical iff it is adapted to the
graduation (\ref{grad}) of $G$, i.e. iff it is of the form
$(\sigma^{q}_j)_{q \in S, j=1..d_q}$ with $(\sigma^{q}_j)_{j=1..d_q}$
bases of $G(q)$.
\begin{Corrolary}
Let $s_1...s_r$ be an arbitrary basis of $G$. Then
the spectral divisor of $V$ is given by :
\begin{equation}
\Sigma_V=(s_1\wedge ....\wedge s_r)-r(p_2+...+p_d)
\end{equation}
\end{Corrolary}
{\em Proof: } Indeed, if $\sigma_1...\sigma_r$ is a canonical basis of $G$ then we have
$(s_1\wedge ... \wedge s_r)=(\sigma_1\wedge ... \wedge \sigma_r)=
\sum_{j=1..r}{q_j}+r(p_2+...+p_r)=\Sigma_V+r(p_2+...+p_r)$. $\Box$
This reduces the problem of determining the spectral divisor of
$V$ to finding a basis of $G$. In the monad case,
that can be easily accomplished
by an obvious modification of the methods of \cite{CGL}.
It is now straightforward to formulate an analogue of Theorem
\ref{theorem_fully_split}, in which $H^0(V')$ is replaced by $G$,
whose proof is almost identical. Since this brings no new concepts to bear,
we will not insist.
There is a also a relatively straightforward generalization of the above
to the non-fully-split case. A detailed statement would be rather
lengthy and will not be given here.
\bigbreak\bigskip\bigskip\centerline{{\bf Aknowledgements}}\nobreak
The author would like to thank T.~M.~Chiang for valuable comments on the
manuscript. This work was supported by the DOE grant DE-FG02-92ER40699B
and by a C.U. Fister Fellowship.
|
astro-ph/9712162
|
\section*{Introduction}
The prodigious luminosities and compact sizes of quasars led theorists to
a model in which gas falls into a supermassive black hole through a
geometrically thin, optically thick accretion disk. If quasars contain
a hole of about $10^{8-9} M_\odot$ which is accreting at near the
Eddington limit, then their spectra should peak in the ultraviolet, as
observed \cite{shi78}. However, so far this paradigm has failed to explain
many other details of quasar spectra: the calculated spectra are too narrow
in range of frequency (an extra power law is needed to fit the spectrum
\cite{sun89}); have too strong a feature at the Lyman edge \cite{ant89};
and have too large polarization at the wrong polarization angle \cite{sto84}.
Part of the reason for the failure might be due to oversimplified disk
spectra calculations, which until now have neglected the effects of metal
line opacity. We present here a preliminary calculation including
metal line opacities; the calculation, however, is not fully
self-consistent, but simply illustrates that the opacity of metal lines
can play an important role in shaping accretion disk spectra.
We have computed the spectrum and polarization of a
disk with the following parameters: mass of black
hole ${\rm M_{BH}=2\times 10^{9} M_\odot}$, accretion rate
${\rm \dot{M}=1 M_\odot/}$year, and
spin of black hole $a=0.998 {\rm M_{BH}}$. This corresponds to a
luminosity of 0.072 ${\rm L_{Edd}}$.
We computed the vertical
structure at each radius using TLUSDISK \cite{hub97}. Using
this structure, we then computed the spectrum and polarization using
the code SYNSPEC \cite{hub94} combined with the polarization code of
Blaes and Agol \cite{bla96}. Finally, we convolved the spectra
from each radius with a relativistic transfer function \cite{ago97}.
The vertical structure (i.e. temperature, density, flux) is computed taking
into account the continuum opacity of hydrogen and helium in non-LTE, and
then this structure is used to compute the opacity due to metal lines
assumed to be in LTE at the calculated temperature and density.
We assume a 30 km/s Doppler width for all lines. We take the outer edge of the
disk to be $50 {\rm GM_{BH}/c^2}$. We take into account
departures from LTE for the 9 levels of HI, 14 levels of HeI, and
14 levels of HeII. So far, we have only included metal lines between
$200-3000{\rm\AA}$, assuming solar abundance. We have not yet
taken into account the effects of bound-bound transitions on the hydrogen
and helium number densities. Because the atmosphere structure calculation
did not include metal lines in the radiative equilibrium condition, the spectrum
is not fully self-consistent. Hence there is an artificial reduction in flux.
The spectra for different inclination angles are presented in figure~\ref{fig1},
comparing spectra models with and without metal lines.
\begin{figure}
\centerline{\epsfig{file=figure1.ps,width=5in}}
\vspace{10pt}
\caption{Comparison of the flux ($F_\nu$) at three different viewing angles.
}\label{fig1}
\end{figure}
The flux falls more steeply in the UV when metals are included; above
$10^{14.8}$~Hz for the face-on disk, $f_\nu\propto \nu^{-0.3}$ approximately.
This may change when the calculation is done self-consistently (i.e. when
flux is conserved). The models are identical below $10^{15}$~Hz since metal
lines are not included in that wavelength range. The spectral slope at smaller
frequencies changes due to the small outer cutoff radius we used.
The ultraviolet
polarization is decreased significantly by the line opacity (figure
\ref{fig2}).
For this nearly edge-on disk, the polarization
is between 1.2-1.8\% in the observable region. This is reduced from the
maximum value of 3-4\% assuming pure electron scattering with relativistic
effects included. If quasars are preferentially seen closer to face-on,
then their ultraviolet polarization will be
even lower, and the small observed polarization could be due to scattering off
of material at larger radii.
\begin{figure}
\centerline{\epsfig{file=figure2.ps,width=5in,height=3.52in}}
\vspace{10pt}
\caption{Comparison of the flux ($\lambda F_\lambda$) and percent polarization
for the model viewed
nearly edge-on. The polarization angle is not shown since it stays
roughly constant within 5$^\circ$ for all three models in this wavelength
range. Note that for this viewing angle, the Lyman edge is so highly smeared
that it is not visible, and the continuum is quite hard.}\label{fig2}
\end{figure}
Inclusion of metal opacity reduces the bump near the Lyman edge which is
present in the H/He continnum-only face-on disk model (figure \ref{fig3}).
\begin{figure}
\centerline{\epsfig{file=figure3.ps,width=5in}}
\vspace{10pt}
\caption{Flux versus wavelength near the Lyman edge
for a face-on ($\cos i=0.98$) disk. The Lyman edge feature is in emission in
the inner parts of the disk (which are strongly redshifted), and absorption
in the outer parts (which are less redshifted), which causes the bump
redward of the Lyman edge. When metal lines are included, the photosphere
redward of the Lyman edge is brought closer to the surface where the
temperature is higher/lower for an edge in emission/absorption, reducing
the contrast across the Lyman edge.}\label{fig3}
\end{figure}
This may be why it is difficult to find Lyman edge features in quasars.
The main contributors to the metal line opacity in this region are
Fe, Ni, Mn, and S. There is a
broader dip blueward of the Lyman edge due to metal lines, mostly Fe. Note
that the flux is different in the two cases because we haven't included the
lines in the atmosphere structure calculation, so the total flux is not
constant.
In summary, metal line opacities can reduce the polarization, change the
spectral shape, and reduce the Lyman edge jump in accretion disk model
spectra for quasars. We need to see whether these effects still
occur in models which include the lines in calculating the disk structure.
To make a better comparison of the models with and without metals, we will
construct line-blanketed disk structure models including lines from a wider
range of wavelengths and bound-bound transitions for H and He, and we will
include the
contribution of the disk at larger radii. The metal line opacity will change
the temperature equilibrium and the radiative acceleration, and thus will
be important for calculating the disk structure, which will in turn affect
the continuum shape.
|
1107.2639
|
\section{Introduction} \label{sec:introduction}
The following is a classical result of Ryser, which provides a necessary and sufficient condition for completability of partial latin squares where the filled cells form a rectangle. In this and subsequent results, we shall assume that the rectangle is in the upper left corner of the square. This is merely for convenience, and in fact Theorem \ref{ryser} holds for any partial latin square that can be put into this form by permuting its rows and columns.
\begin{theorem} \emph{(Ryser, 1951 \cite{bipartite, ryser})} Let $P$ be a partial latin square of order $n$ whose filled cells are those in the upper left $r\times s$ rectangle $R$, for some $r,s \in \{1, \dots, n\}$. Then $P$ is completable if and only if \[\nu(\sigma) \ge r + s - n\] for each symbol $\sigma \in \{1, \dots, n\}$, where $\nu(\sigma)$ is the number of times that $\sigma$ appears in $R$. \label{ryser} \end{theorem}
This set of $n$ inequalities as known as \emph{Ryser's Condition}. In this paper, we consider another condition, introduced by Hilton and Johnson, that is known as \textit{\hc}\ \cite{hiltonjohnson2}. Bobga and Johnson \cite{bobgaphd, bobgajohnson} observed that for partial latin squares where the filled cells form a rectangle, \hc\ is equivalent to Ryser's Condition.\footnote{In fact, this follows from a result of Hilton and Johnson \cite{hiltonjohnson}, who show that the $n$ inequalities of Ryser's Condition can be replaced by a single inequality. They did not state their result in terms of \hc, but they clearly realized this very quickly.}
To state \hc\, we first need some definitions. Given a partial latin square, a symbol is said to be \textit{missing} from a row (or column) if it does not appear in a filled cell of that row (or column). Given a symbol $\sigma \in \{1, \dots, n\}$, a cell is said to \textit{support} $\sigma$ if the cell either contains $\sigma$, or the cell is empty and $\sigma$ is missing from the cell's row and column. A set of cells is said to be \textit{independent} if no two of the cells belong to the same row or column; and if each cell in the set supports $\sigma$, the set is said to be an \textit{independent set for $\sigma$}.
Let $P$ be a partial latin square of order $n$. Given a set $T$ of cells of $P$, and a symbol $\sigma \in \{1, \dots, n\}$, let $\alpha(\sigma, T)$ denote the size of the largest subset of $T$ that is an independent set for $\sigma$. Then the \textit{\hil}\ for $T$, denoted \HI{T}, is the inequality
\begin{equation*}
\sum_{\sigma = 1}^n \alpha(\sigma, T) \ge \size{T}. \label{plshalls}
\end{equation*}
The partial latin square $P$ is said to satisfy \textit{\hc}\ if for each set $T$ of cells of $P$, the \hil\ \HI{T} is satisfied.\footnote{\hc\ can actually be defined in a more general setting; namely graphs whose vertices are equipped with colour lists. See \cite{hoffmanjohnson} for a survey.}
It is not hard to show that \hc\ is a \textit{necessary} condition for completability of partial latin squares (see Lemma~\ref{hconlyif}). Some time ago Cropper asked whether in fact \hc\ is a \textit{sufficient} condition \cite{cropper}. John Goldwasser provided a negative answer, giving a partial latin square (see Figure~\ref{goldwasser}) that satisfies \hc\ but is not completable \cite{ghp}. However, Cropper's question served to stimulate interest in the area, and a number of papers have appeared recently (e.g. \cite{bobgaphd, bobgajohnson, ghp, hiltonvaughan}).
\begin{figure}
\begin{center} \begin{tikzpicture}
[matrix of nodes/.style={minimum size=7mm, execute at begin cell=\node\bgroup, execute at end cell=\egroup; }]
\draw[color=gray,step=7mm] (0,0) grid (4.2,4.2);
\node [matrix, matrix of nodes] at (2.1,2.1) {
1 & 2 & 3 & 4 & 5 & 6 \\
3 & 6 & 1 & 2 & 4 & 5 \\
5 & 4 & 2 & 6 & 3 & 1 \\
2 & 5 & & & & \\
4 & 1 & & & & \\
6 & 3 & & & & \\ };
\end{tikzpicture} \end{center}
\caption{Goldwasser's square} \label{goldwasser} \end{figure}
As noted above, in the case of partial latin squares where the filled cells form a rectangle, \hc\ is both a necessary and sufficient condition for completability. So we have the following theorem.
\begin{theorem} Let $P$ be a partial latin square of order $n$ whose filled cells are those in the upper left $r\times s$ rectangle, for some $r,s \in \{1, \dots, n\}$. Then $P$ is completable if and only if $P$ satisfies \hc. \label{pls1} \end{theorem}
So while \hc\ is not, in general, a sufficient condition for a partial latin square to be completable, if we restrict our attention to partial latin squares where the filled cells form a rectangle, it is both a necessary and sufficient condition for completability. In light of Theorem~\ref{pls1} it seems sensible to ask if there are other classes of partial latin square for which \hc\ is also a sufficient condition for completability.
Bobga and Johnson considered this question, and found that \hc\ is also a sufficient condition for completability in the case of partial latin squares where the filled cells form a rectangle with one empty cell inside \cite{bobgaphd, bobgajohnson}.
In Section \ref{sec:ryser}, we shall prove the following generalization of their result.
\begin{theorem} Let $P$ be a partial latin square of order $n$ whose filled cells are those in the upper left $r\times s$ rectangle, for some $r,s \in \{1, \dots, n\}$, except for $t$ cells in this rectangle that are empty, with the condition that there is no more than one of these empty cells in each column. Then $P$ is completable if and only if $P$ satisfies \hc. \label{pls5} \end{theorem}
To the extent that Theorem~\ref{pls1} is a restatement of Ryser's Theorem (Theorem~\ref{ryser}), Theorem~\ref{pls5} can be considered as a generalization of Ryser's Theorem. It seems likely that Theorem~\ref{pls5} is not the end of the matter, and that more general results along these lines are possible. For example, one could consider partial latin squares where the filled cells form a rectangle, except for a $2 \times t$ rectangle of empty cells inside. We do not know if \hc\ is a sufficient condition for completability in this case.
In Section \ref{sec:complexity}, we shall consider \hc\ and computational complexity. In this section we use some standard notions from theoretical computer science, for which we refer the reader to the book by Garey and Johnson \cite{gareyjohnson}. An obvious computational problem to study is the following.
\begin{problem} Let $P$ be a partial latin square. Decide if $P$ satisfies \hc. \label{hcprob} \end{problem}
Unfortunately we are not able to say much about the complexity of this problem. It may be the case that Problem~\ref{hcprob} is in P, but at present we cannot even show that it is in NP. In Section \ref{sec:preliminaries}, we show that it is possible to check each \hil\ in polynomial time (see Lemma~\ref{hilpoly}), and so if we have a partial latin square that does \textit{not} satisfy \hc, this fact can be verified in polynomial time. Thus Problem~\ref{hcprob} is in co-NP.
The following problem was shown to be NP-complete by Colbourn \cite{colbourn} (see also \cite{eastonparker}).
\begin{problem} Let $P$ be a partial latin square. Decide if $P$ is completable. \label{plsprob} \end{problem}
We can consider the following variant of Problem~\ref{plsprob}.
\begin{problem} Let $P$ be a partial latin square that satisfies \hc. Decide if $P$ is completable. \label{plshcprob} \end{problem}
The set of partial latin squares that are ``yes'' instances of Problem~\ref{plsprob} is the same as the set of partial latin squares that are ``yes'' instances of Problem~\ref{plshcprob}. The difference is that in Problem~\ref{plshcprob}, the input is restricted to partial latin squares that satisfy \hc. Thus Problem~\ref{plshcprob} is an example of a \textit{promise problem}, as the input partial latin square is ``promised'' to satisfy \hc. (See \cite{goldreich} for a survey of promise problems.) We shall prove the following.
\begin{theorem} Problem~\ref{plshcprob} is NP-hard. \label{plsnph} \end{theorem}
Theorem~\ref{plsnph} suggests that knowing that a partial latin square satisfies \hc\ may not be very helpful in determining its completability. In Section \ref{sec:complexity} we shall give a reduction from an NP-complete hypergraph colouring problem to Problem~\ref{plsprob}, with the property that its image is contained in the set of partial latin squares that satisfy \hc. From the existence of this reduction, we can deduce that Problem~\ref{plshcprob} is NP-hard, and obtain a new proof that Problem~\ref{plsprob} is NP-complete.
The structure of this paper is as follows. In Section~\ref{sec:preliminaries} we prove some basic results and quote some classical theorems. In Section~\ref{sec:ryser} we give the proof of Theorem~\ref{pls5} and in Section~\ref{sec:complexity} we give the proof of Theorem~\ref{plsnph}.
\section{Preliminaries} \label{sec:preliminaries}
In this section we give some basic results that will be needed later. First we give the standard result that \hc\ is a necessary condition for a partial latin square to be completable.
\begin{lemma} Let $P$ be a partial latin square of order $n$. Then $P$ is completable only if $P$ satisfies \hc. \label{hconlyif} \end{lemma}
\begin{proof} Let $P^*$ be a completion of $P$, and let $T$ be a set of cells of $P^*$. For each symbol $\sigma \in \{1, \dots, n\}$, let $T_\sigma$ be the subset of the cells of $T$ that contain $\sigma$. Summing over all symbols, we have $T_1 + \dots + T_n = \size{T}$. As each set of cells $T_\sigma$ is an independent set, we have $\alpha(\sigma, T) \ge \size{T_\sigma}$. Therefore
\[ \sum_{\sigma=1}^n \alpha(\sigma, T) \ge \sum_{\sigma=1}^n \size{T_\sigma} = \size{T}, \] and so \HI{T} holds. Since this is true for all sets $T$, it follows that $P$ satisfies \hc. \end{proof}
We also need the following standard lemma.
\begin{lemma} Let $P$ be a partial latin square of order $n$, $T$ a set of cells, and $f \in T$ a filled cell. Then \HI{T} holds if and only if \HI{T-f} holds. \label{t-f} \end{lemma}
\begin{proof} Assume that \HI{T} holds. As the cell $f$ is filled, it only supports one symbol, and so it can only contribute 1 to the quantity
\[\sum_{\sigma=1}^n \alpha(\sigma, T), \]
and so
\[ \sum_{\sigma=1}^n \alpha(\sigma, T-f) \ge \left( \sum_{\sigma=1}^n \alpha(\sigma, T) \right) - 1 \ge \size{T} - 1 = \size{T-f}, \]
and so \HI{T-f} holds. Conversely, suppose that \HI{T-f} holds, and that $f$ contains a symbol $\sigma \in \{1, \dots, n\}$. As no other cells of $T$ in the same row or column as $f$ support $\sigma$, the size of a maximum independent set of cells of $T$ that support $\sigma$ is exactly one greater than the size of such a set in $T-f$. So
\[ \sum_{\sigma=1}^n \alpha(\sigma, T) = \left( \sum_{\sigma=1}^n \alpha(\sigma, T-f) \right) + 1 \ge \size{T-f} + 1 = \size{T}, \]
and so \HI{T} holds. \end{proof}
Because of Lemma~\ref{t-f}, if one wishes to determine if a partial latin square satisfies \hc, it is sufficient to verify that the \hil\ for each set of \textit{empty} cells is satisfied. So we have the following theorem.
\begin{theorem} Let $P$ be a partial latin square. $P$ satisfies \hc\ if and only if the \hil\ is satisfied by each set of empty cells.
\label{empty} \end{theorem}
A \textit{vertex cover} of a graph $G$ is a set of vertices $C$, such that $C$ contains at least one vertex of each edge of $G$. We shall need the following classical theorem.
\begin{theorem} \emph{(K\"onig--Egerv\'ary, 1931 \cite{bipartite, bondymurty, diestel, konig})} Let $G$ be a bipartite graph. The size of a maximum matching in $G$ is equal to the size of a minimum vertex cover. \label{konig-egervary} \end{theorem}
It can be easily verified that Goldwasser's square (see Figure~\ref{goldwasser}) is incompletable. To see that it satisfies \hc, we can use the following lemma, which will also prove useful in Section \ref{sec:complexity}.
\begin{lemma} Let $P$ be a partial latin square. Suppose that a symbol $\sigma$ is missing from $k$ columns and $k$ rows of $P$ but is present in all the other rows and columns of $P$, and that we have a set of empty cells $T$ which contains $t$ cells that support $\sigma$. Then $\alpha(\sigma, T) \ge \ceiling{t/k}$. \label{toverk} \end{lemma}
\begin{proof} Let $G$ be the bipartite graph on $2k$ vertices, defined as follows. There are $k$ vertices $r_1, \dots, r_k$, representing the rows from which $\sigma$ is missing, and $k$ vertices $c_1, \dots, c_k$, representing the columns from which $\sigma$ is missing. For each cell of $T$ that supports $\sigma$, we place an edge between the vertex that represents the cell's row, and the vertex that represents the cell's column. Thus matchings in $G$ correspond to independent sets of cells in $P$. By Theorem~\ref{konig-egervary}, the size of a maximum matching in $G$ is equal to the size of the smallest vertex cover. The maximum degree of $G$ is at most $k$, so a set of $s$ vertices is incident with at most $sk$ edges. Since $G$ has $t$ edges, a vertex cover of $G$ must contain at least $\ceiling{t/k}$ vertices. Hence there is a matching in $G$ of size $\ceiling{t/k}$, and therefore $\alpha(\sigma, T) \ge \ceiling{t/k}$. \end{proof}
Note that in the preceding proof we observed that $\alpha(\sigma, T)$ is equal to the size of a maximum matching in a certain bipartite graph. So we can determine if \HI{T} holds by computing the size of a maximum matching in $n$ bipartite graphs, one for each choice of symbol $\sigma$. Since the size of a maximum matching in a bipartite graph on $N$ vertices can be determined in $O(N^3)$ time (see e.g. \cite[Chapter 20]{schrijver}), we have the following result.
\begin{lemma} Let $P$ be a partial latin square, and $T$ a set of cells. Then \HI{T} can be determined in $O(n^4)$ time. \label{hilpoly}
\end{lemma}
In Goldwasser's square, each symbol is missing from 2 rows and 2 columns, and each empty cell supports 2 symbols. Let $T$ be a set of empty cells of Goldwasser's square. For each $i \in \{1, \dots, 6\}$, let $a_i$ be the number of cells of $T$ that support $i$. As each cell of $T$ supports 2 symbols we have \begin{equation} \sum_{i=1}^6 a_i = 2\size{T} \label{2T} \end{equation} But then \begin{align*}
\sum_{i=1}^6 \alpha(i, T) &\ge \sum_{i=1}^6 \ceiling{a_i / 2} &&\text{(by Lemma~\ref{toverk})} \\
&\ge \tfrac{1}{2} \sum_{i=1}^6 a_i \\ &= \size{T}, &&\text{(by (\ref{2T}))}
\end{align*} and so \HI{T} holds. By Theorem~\ref{empty}, \hc\ holds if the \hil\ holds for each set of empty cells. Since this holds for all sets $T$, it follows that Goldwasser's square satisfies \hc. The following theorem formalizes this argument.
\begin{theorem} Let $P$ be a partial latin square of order $n$. For all $\sigma \in \{1, \dots, n\}$, let $\nu(\sigma)$ denote the number of times that $\sigma$ appears in $P$. For each empty cell $b$ of $P$ we let $S(b)$ denote the set of symbols supported by $b$. Suppose we have
\begin{equation} \sum_{\sigma \in S(b)} \frac{1}{n - \nu(\sigma)} \ge 1 \; \; \text{for each empty cell $b$ of $P$.} \label{atleast1} \end{equation}
Then $P$ satisfies \hc. \label{atleast1t} \end{theorem}
\begin{proof} By Theorem~\ref{empty}, \hc\ holds if the \hil\ holds for each set of empty cells. Let $T$ be a set of empty cells of $P$. For each symbol $\sigma \in \{1, \dots, n\}$, let $T_\sigma$ denote the subset of $T$ consisting of the cells that support $\sigma$. Then
\begin{align*}
\sum_{\sigma = 1}^n \alpha(\sigma, T) &\ge
\sum_{\sigma = 1}^n \frac{\size{T_\sigma}}{n - \nu(\sigma)} &&\text{(by Lemma~\ref{toverk})} \\
&= \sum_{b \in T} \sum_{\sigma \in S(b)} \frac{1}{n - \nu(\sigma)} \\
&\ge \sum_{b \in T} 1 &&\text{(by (\ref{atleast1}))} \\
&= \size{T},
\end{align*} and so \HI{T} holds.
\end{proof}
Finally, we state four classical results that will be needed in due course.
\begin{theorem} \emph{(Hall, 1936 \cite{bipartite, bondymurty, diestel, hall})} Let $G$ be a bipartite graph with bipartition $(A, B)$. There is a matching in $G$ which covers $A$ if and only if each subset $A' \subseteq A$ has at least $\size{A'}$ neighbours. \label{hall2} \end{theorem}
\begin{theorem} \emph{(Dulmage and Mendelsohn, 1958 \cite{bipartite, dulmagemendelsohn})} Let $G$ be a bipartite graph with bipartition $(A, B)$ and suppose $M_1$ and $M_2$ are two matchings in $G$. Then there is a matching $M \subseteq M_1 \cup M_2$ such that $M$ covers all the vertices of $A$ covered by $M_1$ and all the vertices of $B$ covered by $M_2$. \label{dulmage} \end{theorem}
For a graph $G$, the maximum degree will be denoted $\Delta(G)$ and the chromatic index will be denoted $\chi'(G)$.
\begin{theorem} \emph{(K\"onig, 1916 \cite{bipartite, bondymurty, diestel, konig})} Let $G$ be a bipartite graph. Then $\chi'(G) = \Delta(G)$. \label{konig} \end{theorem}
The final theorem, of Ford and Fulkerson, is commonly known as the \textit{Max-flow Min-cut Theorem}. An \textit{integral flow} is a flow in which the flow along each edge is an integer.
\begin{theorem} \emph{(Ford and Fulkerson, 1956 \cite{bondymurty, diestel, fordfulkerson})} Let $G$ be a directed graph with integral edge capacities and two distinguished vertices $\alpha$ (the ``source'') and $\omega$ (the ``sink''). Then the size of a maximum flow between $\alpha$ and $\omega$ is equal to the minimum size of a cut that separates these two vertices. Moreover, there is a maximum flow that is integral. \label{maxflow} \end{theorem}
It is a common student exercise to deduce Theorem~\ref{hall2} (Hall's Theorem) from Theorem~\ref{maxflow}. We shall use a method in the proof of Theorem~\ref{pls5} that was inspired by this exercise.
\section{A generalization of Ryser's Theorem} \label{sec:ryser}
Theorem~\ref{pls1} states that Ryser's Condition is equivalent to \hc. In this section, we give a proof of Theorem~\ref{pls1}, in the same spirit as that given by Bobga and Johnson \cite{bobgaphd, bobgajohnson}. The proof will serve as a template for the more difficult proof of Theorem~\ref{pls5}. In the proof we show that Ryser's Condition is in fact equivalent to a single \hil: \HI{H}, where $H$ is the set of cells in the top $r$ rows.
To prove Theorem~\ref{pls1} we need the following lemma.
\begin{lemma} Let $P$ be a partial latin square of order $n$ whose filled cells are those in the upper left $r\times s$ rectangle $R$, for some $r,s \in \{1, \dots, n\}$. Let $H$ be the set of cells in the top $r$ rows, and for each symbol $\sigma \in \{1, \dots, n\}$ let $\nu(\sigma)$ denote the number of times that $\sigma$ appears in $R$. Then
\[ \alpha(\sigma, H) = \min\{r, \ \nu(\sigma)+n-s\}. \]
\label{alphasigma} \end{lemma}
\begin{proof} Fix a symbol $\sigma \in \{1, \dots, n\}$. Let $S_1$ be the set of cells in $R$ that contain $\sigma$. $S_1$ is an independent set of size $\nu(\sigma)$. In $H$ there are $n-s$ columns with empty cells, and $\sigma$ is missing from $r-\nu(\sigma)$ rows. From the cells in these $n-s$ columns and $r-\nu(\sigma)$ rows we can select an independent set $S_2$ of size $\min\{r-\nu(\sigma), \ n-s\}$. Then $S = S_1 \cup S_2$ is an independent set of size $\min\{r, \ \nu(\sigma)+n-s\}$. So $\alpha(\sigma, H) \ge \min\{r, \ \nu(\sigma)+n-s\}$. Moreover, $\alpha(\sigma, H) \le r$ as there are $r$ rows in $H$; also $\alpha(\sigma, H) \le \nu(\sigma)+n-s$ as no independent set for $\sigma$ can use more than $\nu(\sigma)+n-s$ columns. So in fact we have $\alpha(\sigma, H) = \min\{r, \ \nu(\sigma)+n-s\}$. \end{proof}
We can now prove Theorem~\ref{pls1}.
\begin{proof}[Proof of Theorem~\ref{pls1}] If $P$ is completable then it satisfies \hc\ by Lemma~\ref{hconlyif}. Conversely, suppose $P$ satisfies \hc. Then \HI{H} holds, which means that
\[ \sum_{\sigma = 1}^n \alpha(\sigma, H) \ge rn,\]
which can only be the case if $\alpha(\sigma, H) = r$ for each $\sigma \in \{1, \dots, n\}$. By Lemma~\ref{alphasigma} we have $\nu(\sigma) + n - s \ge r$ for each $\sigma \in \{1, \dots, n \}$, and so $P$ satisfies Ryser's Condition. Thus $P$ is completable by Theorem~\ref{ryser}. \end{proof}
\begin{figure}
\begin{center} \begin{tikzpicture}
\begin{scope}[scale=1.2]
\draw[black] (0,-1) rectangle (4,3);
\fill[gray] (0,1) rectangle (3,3);
\draw[black] (0,-1) rectangle (4,3);
\begin{scope}[xshift=6cm]
\draw[black] (0,-1) rectangle (4,3);
\fill[gray] (0,1) rectangle (3,3);
\fill[white] (0.4,2.4) rectangle +(0.6,0.2)
(1,2.2) rectangle +(0.6,0.2)
(1.6,2) rectangle +(0.6,0.2);
\draw[black] (0,-1) rectangle (4,3);
\end{scope}
\end{scope}
\end{tikzpicture} \end{center}
\caption{The shapes of partial latin square considered in Theorems \ref{pls1} and \ref{pls5}.} \label{shapes12}
\end{figure}
The following lemma is a generalization of Lemma~\ref{alphasigma}, and is essential to the proof of Theorem~\ref{pls5}.
\begin{lemma} Let $P$ be a partial latin square of order $n$ whose filled cells are all in the upper left $r\times s$ rectangle $R$, for some $r,s \in \{1, \dots, n\}$, although at most one cell in each column inside the rectangle may be empty. Let $J$ be a subset of these empty cells. Let $H$ be the set of cells in the top $r$ rows, and for each $\sigma \in \{1, \dots, n\}$ let $\nu(\sigma)$ denote the number of times that $\sigma$ appears in $R$, and let $\rho(\sigma)$ be the number of rows in which there is an empty cell in $R-J$ that supports $\sigma$. Then
\[ \alpha(\sigma, H-J) = \min\{r, \ \nu(\sigma)+\rho(\sigma)+n-s\}. \]
\label{alphasigma2} \end{lemma}
\begin{proof} Fix a symbol $\sigma \in \{1, \dots, n\}$. Let $S_1$ be the set of cells in $H$ that contain $\sigma$, plus one empty cell from $R-J$ that supports $\sigma$ from each row that contains such a cell. $S_1$ is an independent set of size $\nu(\sigma) + \rho(\sigma)$. We can find an independent set $S_2$ of size $\min\{r-\nu(\sigma)-\rho(\sigma), \ n-s\}$ from the cells in the rightmost $n-s$ columns that are in the $r - \nu(\sigma) - \rho(\sigma)$ rows that have no cells in $S_1$. Then $S = S_1 \cup S_2$ is an independent set for $\sigma$ of size $\min\{r, \ \nu(\sigma) + \rho(\sigma) + n - s\}$. So $\alpha(\sigma, H-J) \ge \min\{r, \ \nu(\sigma)+ \rho(\sigma) + n - s\}$. Moreover, $\alpha(\sigma, H-J) \le r$ as there are $r$ rows in $H$; also $\alpha(\sigma, H-J) \le \nu(\sigma) + \rho(\sigma) + n - s$ as no independent set for $\sigma$ can use more than $\nu(\sigma) + \rho(\sigma) + n - s$ columns. So in fact we have $\alpha(\sigma, H-J) = \min\{r, \ \nu(\sigma) + \rho(\sigma) + n - s\}$. \end{proof}
We can now prove Theorem~\ref{pls5}.
\begin{proof}[Proof of Theorem~\ref{pls5}] If $P$ is completable then it satisfies \hc\ by Lemma~\ref{hconlyif}. Conversely, suppose $P$ satisfies \hc.
Let $H$ be the set of $rn$ cells in the first $r$ rows of $P$, and let $B$ be the set of empty cells contained within $R$. We shall give a procedure for finding a completion of $P$. The procedure consists of three steps, which we now outline.
\begin{enumerate}[{Step} 1.]
\item A partial latin square $Q_1$ is constructed from $P$, by filling \textit{some} of the cells of $B$, in such a way that each symbol appears at least $r + s - n$ times within $R$. This step will be shown to be possible because the \hil\ \HI{H-B'} holds for each subset $B' \subseteq B$. The main tool used to show this will be Theorem~\ref{maxflow} (the Max-flow Min-cut Theorem).
\item A partial latin square $Q_2$ is constructed from $P$ by filling \textit{all} the cells of $B$. This step will be performed entirely independently to Step 1; $Q_2$ will not depend on $Q_1$ in any way. For $Q_2$ there will be no conditions put on the number of times each symbol must appear in $R$. This step will be shown to be possible because the \hil\ \HI{B'} holds for each subset $B' \subseteq B$. The main tool used to show this will be Theorem~\ref{hall2} (Hall's Theorem).
\item A partial latin square $Q$ is constructed from $Q_1$ and $Q_2$, in which all the cells of $B$ are filled, and for which Ryser's Condition holds. The main tool used will be Theorem~\ref{dulmage} (the Dulmage-Mendelsohn Theorem). It follows from Theorem~\ref{ryser} (Ryser's Theorem) that $Q$ is completable.
\end{enumerate}
Note that we shall not need to make use of the fact that \textit{all} the Hall Inequalities are satisfied; it will suffice to make use of those inequalities that are of the form \HI{B'} or \HI{H-B'} for some $B' \subseteq B$.
\begin{center} Step 1. \end{center}
In this step, we shall describe a procedure for constructing a partial latin square $Q_1$ from $P$ by filling in \textit{some} of the cells of $B$ in such a way that each symbol $\sigma \in \{1, \dots, n\}$ appears at least $r + s - n$ times in $R$.
By Lemma~\ref{alphasigma2}, first with $J = B$, and second with $J = \emptyset$, we have
\[ \alpha(\sigma, H-B) = \min \{r, \ \nu(\sigma) + n - s\} \]
and
\[ \alpha(\sigma, H) = \min \{r, \ \nu(\sigma) + \rho(\sigma) + n - s\}, \] where $\rho(\sigma)$ is the number of rows in which $\sigma$ is supported by a cell of $B$. \HI{H} implies that for each $\sigma \in \{1, \dots, n\}$ we have $\alpha(\sigma, H) = r$, and therefore
\[ \nu(\sigma) + \rho(\sigma) + n - s \ge r. \]
For each symbol $\sigma \in \{1, \dots, n\}$ let $\mu(\sigma)$ be the least non-negative integer such that
\begin{equation} \nu(\sigma) + \mu(\sigma) \ge r + s - n. \label{nplusm} \end{equation}
Thus if $\sigma$ already appears at least $r + s - n$ times in $R$ we have $\mu(\sigma) = 0$; otherwise we have $\mu(\sigma) > 0$ and
\begin{equation} \nu(\sigma) + \mu(\sigma) = r + s - n. \label{nplusmeq} \end{equation}
Note that for each $\sigma \in \{1, \dots, n\}$ we have
\begin{equation} \mu(\sigma) \le \rho(\sigma). \label{mger} \end{equation}
Let
\[ u = \sum_{\sigma = 1}^n \mu(\sigma). \]
If we can fill in $u$ cells of $B$ using each symbol $\sigma \in \{1, \dots, n\}$ $\mu(\sigma)$ times, then in the resulting partial latin square, by (\ref{nplusm}), each symbol will appear at least $r + s - n$ times in $R$, and we will have the required partial latin square $Q_1$. We shall now show that this can always be done.
If a symbol $\sigma$ has $\mu(\sigma) = 0$ then no copies of this symbol need be placed in $B$. In fact, we only need be concerned with those symbols $\sigma$ for which $\mu(\sigma) > 0$; and for these symbols equation (\ref{nplusmeq}) holds.
Consider a directed graph $G$ with edge-capacities, that has vertex set $U \cup X \cup B \cup \{\alpha, \omega\}$ where
\[ U = \{ \sigma : \sigma \in \{1, \dots, n\} \text{\ and\ } \mu(\sigma) > 0\}, \]
\[ X = \{ (\sigma, w) : w \in \{1, \dots, r\} \text{\ and $\sigma$ is supported by at least one empty cell in row $w$} \} ,\] and $\alpha$ and $\omega$ are two additional vertices, which can be thought of as a ``source'' and a ``sink''. The source will have zero in-degree and the sink will have zero out-degree; we shall consider cuts in $G$ that separate $\alpha$ from $\omega$.
In $G$, $\alpha$ is joined to each vertex $\sigma \in U$ by an edge of capacity $\mu(\sigma)$, the direction being from $\alpha$ to $\sigma$. Each vertex $\sigma \in U$ is joined to all the vertices in $X$ of the form $(\sigma, w)$ for some $w \in \{1, \dots, r\}$ by edges of capacity 1, the direction being from $U$ to $X$. Each vertex $(\sigma, w) \in X$ is joined to all $b \in B$ where $b$ is a cell in row $w$ that supports $\sigma$, with edges of capacity $u$, the direction being from $X$ to $B$. Each vertex in $B$ is joined to $\omega$ by an edge of capacity 1, the direction being from $B$ to $\omega$. (See Figure~\ref{pls5fig}.)
\begin{figure}
\begin{center} \begin{tikzpicture}[scale=0.8,decoration={markings, mark=at position 0.66 with {\arrow{latex}}}]
\draw[black, postaction={decorate}] (0,2)--(2.7,3);
\draw[black, postaction={decorate}] (0,2)--(2.7,1);
\draw[black, postaction={decorate}] (0,2)--(2.7,2);
\draw[black, postaction={decorate}] (3.3,1)--(5.7,0.5);
\draw[black, postaction={decorate}] (3.3,2)--(5.7,2);
\draw[black, postaction={decorate}] (3.3,3)--(5.7,3.5);
\draw[black, postaction={decorate}] (6.3,0.5)--(8.7,0.8);
\draw[black, postaction={decorate}] (6.3,2)--(8.7,2);
\draw[black, postaction={decorate}] (6.3,3.5)--(8.7,3.2);
\draw[black, postaction={decorate}] (9.3,0.8)--(12,2);
\draw[black, postaction={decorate}] (9.3,2)--(12,2);
\draw[black, postaction={decorate}] (9.3,3.2)--(12,2);
\draw[draw=black, rounded corners=7] (2.7,0.5) rectangle (3.3,3.5);
\draw[draw=black, rounded corners=7] (5.7,0) rectangle (6.3,4);
\draw[draw=black, rounded corners=7] (8.7,0.3) rectangle (9.3,3.7);
\draw (0,2) node[left] {$\alpha$} (12,2) node[right] {$\omega$};
\fill (0,2) circle (2pt) (12,2) circle (2pt);
\draw (3,2) node {$U$};
\draw (6,2) node {$X$};
\draw (9,2) node {$B$};
\end{tikzpicture} \end{center}
\caption{The directed graph $G$ from Theorem~\ref{pls5}.} \label{pls5fig}
\end{figure}
\vspace{5pt}
\begin{claim} It is possible to fill some of the cells of $B$ using each symbol $\sigma \in U$ $\mu(\sigma)$ times if and only if there is a $u$-flow in $G$ from $\alpha$ to $\omega$. \end{claim}
\begin{proof} Suppose there is such a partial filling of $B$. For each instance that a symbol $\sigma \in U$ is placed in cell $b \in B$ of row $w$, we can create a 1-flow in $G$ from $\alpha$ to $\omega$ by sending a flow of 1 from $\alpha$ to $\sigma \in U$ to $(\sigma, w) \in X$ to $b \in B$ to $\omega$. The sum of all these 1-flows gives a $u$-flow from $\alpha$ to $\omega$.
Conversely, suppose that there is a $u$-flow from $\alpha$ to $\omega$. By Theorem~\ref{maxflow} (the Max-flow Min-cut Theorem) there is such a flow that is integral. All the edges from $\alpha$ to $U$ must carry a flow equal to their capacity, and there must be $u$ paths from $U$ to $\omega$ each carrying 1-flows. These flows indicate how the cells of $B$ can be filled. The fact that the edges between $U$ and $X$ have capacity $1$ ensures that each symbol is placed at most once in each row. This proves Claim~1. \renewcommand{\qedsymbol}{} \end{proof}
By Claim~1, to show that $Q_1$ can be constructed, it will suffice to show that there is a $u$-flow in $G$ from $\alpha$ to $\omega$. So suppose, for a contradiction, that there does not exist such a $u$-flow in $G$. Then by Theorem~\ref{maxflow} (the Max-flow Min-cut Theorem) there must be a cut $T$ in $G$ of size less than $u$ that separates $\alpha$ from $\omega$. $T$ cannot contain any of the edges between $X$ and $B$ as each of these edges has capacity $u$. Also $T$ cannot contain \textit{all} the edges between $\alpha$ and $U$ (or it would have size $u$).
Let $U' \subseteq U$ be the set of vertices that $\alpha$ is joined to with an edge that is not in $T$. (See Figure~\ref{pls5fig2}.) For each $\sigma \in U'$ let $c(\sigma)$ be the number of edges joining $\sigma$ to $X$ that are in $T$. We may suppose that for all $\sigma \in U'$ we have $c(\sigma) < \mu(\sigma)$ as otherwise we can create a new cut $T'$ from $T$, where $\size{T'} \le \size{T}$, by removing the edges joining $\sigma$ to $X$ and adding the edge joining $\alpha$ to $\sigma$ instead.
Let $B' \subseteq B$ be the set of vertices in $B$ that are joined to vertices in $U'$ by paths containing no edges of $T$. Because the cut $T$ separates $\alpha$ from $\omega$ all the edges that join $B'$ to $\omega$ must be in $T$. Since we have assumed that $T$ has size less than $u$ we must have
\[ \sum_{\sigma \in U - U'} \mu(\sigma) + \sum_{\sigma \in U'} c(\sigma) + \size{B'} < u = \sum_{\sigma \in U} \mu(\sigma), \]
so that
\begin{equation} \sum_{\sigma \in U'} \left( \mu(\sigma) - c(\sigma) \right) > \size{B'}. \label{keyineq} \end{equation}
By Lemma~\ref{alphasigma2} we have for each $\sigma \in U$,
\begin{align*} \alpha(\sigma, H-B') &= \min\{r, \ \nu(\sigma) + \rho'(\sigma) + n - s\} \\
&\le \nu(\sigma) + \rho'(\sigma) + n - s,
\end{align*}
where $\rho'(\sigma)$ is the number of rows in which $\sigma$ is supported by a cell in $B - B'$. Then by (\ref{nplusmeq}) we have
\begin{equation}
\alpha(\sigma, H-B') \le r - \mu(\sigma) + \rho'(\sigma). \label{nurhodash}
\end{equation}
\begin{figure}
\begin{center} \begin{tikzpicture}[scale=0.8,decoration={markings, mark=at position 0.66 with {\arrow{latex}}}]
\draw[black, postaction={decorate}] (0,2)--(2.7,2.8);
\draw[black, postaction={decorate}] (0,2)--(2.7,1.2);
\draw[black, postaction={decorate}] (0,2)--(2.7,2);
\draw[black, postaction={decorate}] (3.3,1.2)--(5.7,1);
\draw[black, postaction={decorate}] (3.3,2)--(5.7,2);
\draw[black, postaction={decorate}] (3.3,2.8)--(5.7,3);
\draw[black, postaction={decorate}] (6.3,1)--(8.7,1.5);
\draw[black, postaction={decorate}] (6.3,2)--(8.7,2);
\draw[black, postaction={decorate}] (6.3,3)--(8.7,2.5);
\draw[black, postaction={decorate}] (9.3,1.5)--(12,2);
\draw[black, postaction={decorate}] (9.3,2)--(12,2);
\draw[black, postaction={decorate}] (9.3,2.5)--(12,2);
\draw[draw=black, rounded corners=7] (2.7,0.5) rectangle (3.3,3.5);
\draw[draw=black] (2.7,1.2) rectangle (3.3,2.8);
\draw[draw=black, rounded corners=7] (5.7,0) rectangle (6.3,4);
\draw[draw=black] (5.7,1) rectangle (6.3,3);
\draw[draw=black, rounded corners=7] (8.7,0.3) rectangle (9.3,3.7);
\draw[draw=black] (8.7,1.5) rectangle (9.3,2.5);
\draw (0,2) node[left] {$\alpha$} (12,2) node[right] {$\omega$};
\fill (0,2) circle (2pt) (12,2) circle (2pt);
\draw (3,2) node {$U'$};
\draw (9,2) node {$B'$};
\end{tikzpicture} \end{center}
\caption{The sets $U'$ and $B'$ from Theorem~\ref{pls5}.} \label{pls5fig2}
\end{figure}
Consider a row $w$ that contains at least one empty cell $b \in B - B'$ that supports $\sigma$. Each such row contributes $1$ to $\rho'(\sigma)$. The occurrence of cell $b \in B-B'$ in row $w$ supporting $\sigma$ corresponds to a unique path joining $\sigma \in U'$ to $b \in B-B'$ and passing through $(\sigma, w)$ in $X$. Such a path must contain one of the $c(\sigma)$ edges of $T$ joining $\sigma \in U'$ to $X$. Therefore for $\sigma \in U'$ we have
\[\rho'(\sigma) \le c(\sigma), \]
and therefore by (\ref{nurhodash}),
\[ \alpha(\sigma, H-B') \le r - \mu(\sigma) + c(\sigma). \]
So \begin{align*}
\sum_{\sigma = 1}^n \alpha(\sigma, H-B') &\le \sum_{\sigma \in U-U'} r + \sum_{\sigma \in U'} \left( r - \mu(\sigma) + c(\sigma) \right) \\
&\le rn - \sum_{\sigma \in U'} \left( \mu(\sigma) - c(\sigma) \right) \\
&< rn - \size{B'}, & \text{(by (\ref{keyineq}))}
\end{align*} which contradicts \HI{H-B'}.
Hence there must be a $u$-flow from $\alpha$ to $\omega$, and so by Claim~1 $Q_1$ can be constructed.
\begin{center} Step 2. \end{center}
In this step, we shall describe a procedure for constructing a partial latin square $Q_2$ from $P$ by filling \textit{all} the cells of $B$.
This can be done by filling the cells of $B$ row by row. We define the bipartite graphs $G_1, \dots, G_r$ as follows. For $w \in \{1, \dots, r\}$, $G_w$ is the graph with vertex set $S \cup B_w$ where $S = \{1, \dots, n\}$ is the set of symbols and $B_w$ is the subset of $B$ consisting of the members of $B$ that are in row $w$; where a cell $b \in B_w$ is joined to each symbol in $S$ that it supports. Note that some of the graphs $G_1, \dots, G_r$ may have no edges; in this case they can be disregarded. Note that the graphs are constructed in such a way that a matching in $G_w$ that covers $B_w$ corresponds to a filling of the cells $B_w$ where each cell is filled with a symbol that it supports and no two cells are filled with the same symbol.
\begin{claim} There is a matching in $G_w$ that covers $B_w$. \end{claim}
\begin{proof} Suppose not. Then by Theorem~\ref{hall2} (Hall's Theorem) there is a subset $B' \subseteq B_w$ with neighbour set $Y \in S$ such that $\size{Y} < \size{B'}$. But for any symbol $\sigma \in \{1, \dots, n\}$ we have $\alpha(\sigma, B') \le 1$, so \HI{B'} implies that the cells of $B'$ support at least $\size{B'}$ symbols, which contradicts $B'$ having fewer than $\size{B'}$ neighbours in $G_w$. This proves Claim 2. \renewcommand{\qedsymbol}{} \end{proof}
It follows that it is possible to fill all the cells of $B_w$ for each $w \in \{1, \dots, r\}$, and so in this way $Q_2$ can be constructed.
\begin{center} Step 3. \end{center}
In this step, we shall describe a procedure for constructing a partial latin square $Q_3$ from $Q_1$ and $Q_2$, in which all the cells of $B$ are filled, and each symbol appears at least $r + s - n$ times within $R$. We shall make further use of the bipartite graphs $G_1, \dots, G_r$ defined in Step 2.
For each row $w \in \{1, \dots, r\}$ that contains cells in $B$, the way that the cells of $B_w$ are filled in $Q_1$ and $Q_2$ give two matchings in $G_w$, $M_1$ and $M_2$. By Theorem~\ref{dulmage} (the Dulmage-Mendelsohn Theorem), there is a matching $M_w \subseteq M_1 \cup M_2$ that covers all the vertices in $B_w$ that are covered by $M_2$ (i.e{.} in fact, \textit{all} the vertices of $B_w$), and all the vertices in $S$ that are covered by $M_1$.
Then $Q$ can be created by taking $P$ and filling the cells of $B$ according to the matching $M_w$ for each row $w$. Since for each $w \in \{1, \dots, r\}$, $B_w$ contains all the symbols that it does in $Q_1$, each symbol $\sigma \in U$ appears at least $\mu(\sigma)$ times in $B$. It follows that in $Q$ each symbol appears at least $r + s - n$ times.
In $Q$, all the cells of $B$ are filled, and each symbol appears at least $r + s - n$ times. Thus $Q$ satisfies Ryser's Condition, and so by Theorem~\ref{ryser} it is possible to fill in the empty cells of $Q$ to create a latin square. This proves Theorem~\ref{pls5}. \end{proof}
\section{Complexity questions} \label{sec:complexity}
In this section we give the proof of Theorem~\ref{plsnph}. We need the following theorem of Kratochv\'il \cite{kratochvil}. A \textit{$(k$-in-$m)$-colouring} of an $m$-uniform hypergraph is a colouring of the vertices with red and blue such that each edge contains exactly $k$ red vertices and $m - k$ blue vertices.
\begin{theorem} For every $q\ge 3$, $m\ge 3$ and $1\le k\le m-1$, the problem of deciding $(k$-in-$m)$-colourability of $q$-regular $m$-uniform hypergraphs is NP-complete. \label{kratochvil} \end{theorem}
So in particular, the following problem is NP-complete.
\begin{problem} Let $H$ be a 4-uniform 4-regular hypergraph. Decide if $H$ is 2-in-4 colourable. \label{2-in-4} \end{problem}
A partial latin square of order $n$ is said to be \textit{L-shaped} if all the cells are filled except for those in the upper left $r\times s$ rectangle, for some $r,s \in \{1, \dots, n\}$. (This condition on the shape is opposite to that required for Theorems \ref{ryser} and \ref{pls1}.)
\begin{problem} Let $P$ be an L-shaped partial latin square. Decide if $P$ is completable. \label{lshapeprob} \end{problem}
\begin{figure}
\begin{center} \begin{tikzpicture}
\begin{scope}[scale=0.8]
\draw[gray,step=2cm] (0,0) grid (8,8);
\draw (1,7.5) node {$a_{j,0}$} (1,7) node {$a_{j,1}$} (1,6.5) node {$b_{i,k,0}$}
(3,7.5) node {$a_{j,0}$} (3,7) node {$a_{j,1}$} (3,6.5) node {$b_{i,k,0}$}
(5,7.5) node {$a_{j,0}$} (5,7) node {$a_{j,1}$} (5,6.5) node {$b_{i,k,1}$}
(7,7.5) node {$a_{j,0}$} (7,7) node {$a_{j,1}$} (7,6.5) node {$b_{i,k,1}$}
(1,4.66) node {$b_{i,k-1,5}$} (1,5.33) node {$b_{i,k,0}$}
(3,4.66) node {$b_{i,k,4}$} (3,5.33) node {$b_{i,k,0}$}
(5,4.66) node {$b_{i,k,4}$} (5,5.33) node {$b_{i,k,1}$}
(7,4.66) node {$b_{i,k,5}$} (7,5.33) node {$b_{i,k,1}$}
(1,3.33) node {$b_{i,k-1,5}$} (1,2.66) node {$b_{i,k,2}$}
(3,3.33) node {$b_{i,k,4}$} (3,2.66) node {$b_{i,k,2}$}
(5,3.33) node {$b_{i,k,4}$} (5,2.66) node {$b_{i,k,3}$}
(7,3.33) node {$b_{i,k,5}$} (7,2.66) node {$b_{i,k,3}$}
(1,0.5) node {$a_{j,2}$} (1,1) node {$a_{j,3}$} (1,1.5) node {$b_{i,k,2}$}
(3,0.5) node {$a_{j,2}$} (3,1) node {$a_{j,3}$} (3,1.5) node {$b_{i,k,2}$}
(5,0.5) node {$a_{j,2}$} (5,1) node {$a_{j,3}$} (5,1.5) node {$b_{i,k,3}$}
(7,0.5) node {$a_{j,2}$} (7,1) node {$a_{j,3}$} (7,1.5) node {$b_{i,k,3}$};
\end{scope}
\begin{scope}[xshift=9cm, scale=0.8]
\draw[gray,step=1cm] (0,0) grid (4,4);
\draw[black] (0.6,0.6) rectangle +(0.8,0.8)
(2.6,0.6) rectangle +(0.8,0.8)
(0.6,2.6) rectangle +(0.8,0.8)
(2.6,2.6) rectangle +(0.8,0.8)
(1.6,1.6) rectangle +(0.8,0.8)
(0.1,1.6)--(0.4,1.6)--(0.4,2.4)--(0.1,2.4)
(3.9,1.6)--(3.6,1.6)--(3.6,2.4)--(3.9,2.4);
\end{scope}
\end{tikzpicture} \end{center}
\caption{The symbols in the $4 \times 4$ square of $M$ whose top-left cell is $(4i,4j)$ (left), and the positions of the symbols in $B$ indicated by rectangles (right).} \label{4x4} \end{figure}
We shall show that Problem~\ref{lshapeprob} is NP-complete, by giving a reduction from Problem~\ref{2-in-4} to Problem~\ref{lshapeprob}.\footnote{The reduction is a \textit{Karp reduction}, sometimes called a ``polynomial transformation''. All this terminology is discussed in \cite{gareyjohnson}.} The reduction has the property that its image is contained in the set of L-shaped partial latin squares that satisfy \hc.
\begin{lemma} Problem~\ref{lshapeprob} is NP-complete. Moreover, there is a reduction from Problem~\ref{2-in-4} to Problem~\ref{lshapeprob} that maps 4-uniform 4-regular hypergraphs to L-shaped partial latin squares that satisfy \hc. \label{plsred} \end{lemma}
An $r \times s$ \textit{latin rectangle} is an $r \times s$ array in which each cell is filled, and no symbol appears more than once in any row or column. A \textit{framework} is a tuple \[R = (r, s, t, R_1, \dots, R_r, C_1, \dots, C_s)\] where $r$, $s$ and $t$ are positive integers, and $R_1, \dots, R_r, C_1, \dots, C_s$ are subsets of $\{1, \dots, t\}$. The sets $R_1, \dots, R_r$ are called the \textit{row lists}, and the sets $C_1, \dots, C_s$ are called the \textit{column lists}.\footnote{Our frameworks are a special case of the ``latin frameworks'' from \cite{colbourn, eastonparker} and the ``patterned holes'' from \cite{lindnerrodger}.}
A \textit{latinization} of a framework $R = (r, s, t, R_1, \dots, R_r, C_1, \dots, C_s)$ is an $r \times s$ latin rectangle where each symbol that appears in row $i$ belongs to $R_i$, for each $1 \le i \le r$, and each symbol that appears in column $j$ belongs to $C_j$, for each $1 \le j \le s$.
An L-shaped partial latin square $P$ is said to \textit{realize} $R$ (or be a \textit{realization} of $R$), if $R_i$ is the set of symbols missing from row $i$, for each $1 \le i \le r$, and $C_j$ is the set of symbols missing from column $j$, for each $1 \le j \le s$. Thus if $P$ realizes $R$, the upper left $r \times s$ rectangle of any completion of $P$ is a latinization of $R$, and \textit{vice-versa}.
If $P$ is an L-shaped partial latin square, where the upper left $r \times s$ rectangle is empty, then we can create a framework $R = (r, s, t, R_1, \dots, R_r, C_1, \dots, C_s)$, for which $P$ is a realization, in the following natural way. For each $1 \le i \le r$, let $R_i$ be the set of symbols missing from row $i$, and for each $1 \le j \le s$, let $C_j$ be the set of symbols missing from column $j$. Such a framework will have the following properties.
\begin{enumerate}[(i)]
\item $\size{R_i} = s$ for each $1 \le i \le r$.
\item $\size{C_j} = r$ for each $1 \le j \le s$.
\item Each symbol appears the same number of times in the row lists as it does in the column lists.
\end{enumerate}
If a framework satisfies conditions (i), (ii) and (iii), it is said to be \textit{balanced}. It turns out that given any balanced framework $R$, there is an L-shaped partial latin square $P$ that realizes $R$.
\begin{figure} \begin{center}
\begin{tikzpicture}
\draw (0,1.5) node {$\begin{pmatrix}
1&0&0&1&1&1\\
1&1&0&0&1&1\\
1&1&1&0&0&1\\
1&1&1&1&0&0\\
0&1&1&1&1&0\\
0&0&1&1&1&1
\end{pmatrix}$};
\begin{scope}[xshift=-5.5cm,yshift=2.1cm]
\draw[rotate=0] (0,0.4) ellipse (2.1 and 1);
\draw[rotate=60] (0,0.4) ellipse (2.1 and 1);
\draw[rotate=120] (0,0.4) ellipse (2.1 and 1);
\draw[rotate=180] (0,0.4) ellipse (2.1 and 1);
\draw[rotate=240] (0,0.4) ellipse (2.1 and 1);
\draw[rotate=300] (0,0.4) ellipse (2.1 and 1);
\fill[rotate=30] (0,1.2) circle(3pt);
\fill[rotate=90] (0,1.2) circle(3pt);
\fill[rotate=150] (0,1.2) circle(3pt);
\fill[rotate=210] (0,1.2) circle(3pt);
\fill[rotate=270] (0,1.2) circle(3pt);
\fill[rotate=330] (0,1.2) circle(3pt);
\end{scope}
\end{tikzpicture} \caption{An example of a 4-uniform 4-regular hypergraph $H$, where the edges are drawn as ellipses (left), and its incidence matrix $D$, under a suitable labelling of the vertices and edges (right).} \label{rh} \end{center} \end{figure}
\begin{figure} \begin{center}
\begin{tikzpicture} \input{halls_diagram.tex}
\end{tikzpicture} \caption{The position of the symbols from $B$ in $M$.} \label{rh2}
\end{center} \end{figure}
\begin{theorem} Let $R = (r, s, t, R_1, \dots, R_r, C_1, \dots, C_s)$ be a balanced framework. For any $n \ge \max \{t, r+s\}$, there is a partial latin square $P$ of order $n$ that realizes $R$, and such a partial latin square $P$ can be found in polynomial time. \label{fr1} \end{theorem}
\begin{proof} Fix an $n$ such that $n \ge \max \{t, r+s\}$. We shall give a procedure, consisting of two steps, for constructing an L-shaped partial latin square $P$ of order $n$ that realizes $R$. Initially, suppose $P$ is a partial latin square of order $n$ with all cells empty. In the first step, we fill the cells in the rightmost $n-s$ columns of the top $r$ rows of $P$, and in the second step, we fill the cells in the bottom $n-r$ rows.
Step 1 is as follows. Consider the bipartite graph $G_1$ with bipartition $(A, B)$, where the vertices in $A$ are labelled $a_1, \dots, a_n$ and correspond to the symbols $1, \dots, n$; the vertices in $B$ are labelled $b_1, \dots, b_r$ and correspond to rows $1, \dots, r$ of $P$; and where there is an edge between $a_i$ and $b_j$ if and only if symbol $i$ is not in $R_j$, for each $1 \le i \le n$ and $1 \le j \le r$.
Thus $a_i$ has degree $r - \nu(i)$, where $\nu(i)$ is the number of times that the symbol $i$ occurs in the row lists, for each $1 \le i \le n$; and $b_j$ has degree $n - s$, as there are $s$ symbols in the row list $R_j$, for each $1 \le j \le r$. The maximum degree $\Delta(G_1)$ is $n - s$ for otherwise there would be some $i$ for which $r - \nu(i) > n - s$, which contradicts the assumption that $n \ge r + s$.
By Theorem~\ref{konig} (K\"onig's Theorem), $G_1$ has a proper edge-colouring with the colours $1, \dots, n-s$. So the cells in the rightmost $n-s$ columns of the top $r$ rows of $P$ can be filled according to this edge-colouring: symbol $i$ is placed in cell $(j, s + k)$ if and only if $a_i$ is joined to $b_j$ with an edge of colour $k$, for each $1 \le i \le n$, $1 \le j \le r$, $1 \le k \le n - s$. At this point the number of times the symbol $i$ either appears in the top $r$ rows of $P$ or in one of the row lists $R_1, \dots, R_r$ is $r$, for each $1 \le i \le n$. And because the framework $R$ is balanced, the number of times $i$ appears in the column lists $C_1, \dots, C_s$ or in the top $r$ rows of $P$ is also $r$, for each $1 \le i \le n$.
Step 2 is as follows. Consider the bipartite graph $G_2$ with bipartition $(C, D)$, where the vertices in $C$ are labelled $c_1, \dots, c_n$ and correspond to the $n$ columns of $P$, the vertices in $D$ are labelled $d_1, \dots, d_n$ and correspond to the symbols $1, \dots, n$, and where there is an edge between $d_i$ and $c_j$ if and only if $i$ is not in $C_j$ (when $1 \le j \le s$) or $i$ does not appear in the top $r$ cells of column $j$ of $P$ (when $s < j \le n$) for each $1 \le i \le n$ and $1 \le j \le n$. Each vertex in $G_2$ has degree $n-r$, so by Theorem~\ref{konig} (K\"onig's Theorem) we can give $G_2$ a proper edge-colouring using the colours $1, \dots, n - r$. We can now fill the bottom $n - r$ rows of $P$ according to this edge-colouring: symbol $i$ in placed in cell $(r + k, j)$ if and only if $d_i$ is joined to $c_j$ with an edge of colour $k$, for each $1 \le i \le n$, $1 \le j \le n$, $1 \le k \le n - r$.
This procedure creates an L-shaped partial latin square $P$, where the symbols missing from rows $1, \dots, r$ are those in the row lists $R_1, \dots, R_r$ and the symbols missing from columns $1, \dots, s$ are those in the column lists $C_1, \dots, C_s$. In other words, $P$ is a realization of $R$. The procedure involves edge-colouring two graphs, each with at most $2n$ vertices, which can be done in $O(n^3)$ time (see e.g. \cite[Chapter 20]{schrijver}). Since $n$ is polynomial in the size of the framework, the procedure can be performed in polynomial time. \end{proof}
Given a framework $R$, the \textit{admissible symbol array} $M = M(R)$ is an $r \times s$ array in which $M_{ij} = R_i \cap C_j$ for each $1 \le i \le r$ and $1 \le j \le s$. Thus a latinization of $R$ can be described as an $r \times s$ latin rectangle $L$ for which $L_{ij} \in M_{ij}$ for each $1 \le i \le r$ and $1 \le j \le s$. We are now ready to prove Lemma~\ref{plsred}.
\begin{proof}[Proof of Lemma~\ref{plsred}]
Let $H$ be a 4-uniform 4-regular hypergraph of order $u$. We shall give a procedure for constructing an L-shaped partial latin square $Q$ such that $Q$ is completable if and only if $H$ is 2-in-4 colourable. $Q$ will be an L-shaped partial latin square of order $n = 4u^2 + 12u$, in which the upper left $4u \times 4u$ square is empty, and the rest of the cells are filled. In addition, $Q$ will satisfy \hc.
Note that since $H$ is 4-uniform and 4-regular, $H$ has the same number of vertices as edges. So $H$ has $u$ vertices and $u$ edges, which we assume are labelled $v_0, \dots, v_{u-1}$ and $e_0, \dots, e_{u-1}$ respectively. We shall describe the construction from $H$ of a balanced framework
\[R = R(H) = (4u, 4u, n, R_1, \dots, R_{4u}, C_1, \dots, C_{4u}), \]
with the property that $R$ is latinizable if and only if $H$ is 2-in-4 colourable. Instead of describing $R$ directly, we describe its admissible symbol array $M$.
$M$ is a $4u \times 4u$ array whose entries are subsets of $\{1, \dots, n\}$. For convenience we consider the set $\{1, \dots, n\}$ to be the union of three sets $A$, $B$ and $C$, whose members are labelled as follows. $A$ consists of $a_{j,k}$ for each $0 \le j \le u - 1$ and $0 \le k \le 3$; $B$ consists of $b_{i,j,k}$ for each $0 \le i \le u - 1$, $0 \le j \le 3$ and $0 \le k \le 5$; and $C$ consists of $c_{i,j,k}$ for each $0 \le i \le u - 1$, $0 \le j \le u - 1$, $0 \le k \le 3$ for which $v_i$ is not incident with $e_j$.
Thus $\size{A} = 4u$, $\size{B} = 24u$, $\size{C} = 4u(u-4)$ and $n = \size{A} + \size{B} + \size{C}$.
We can consider the cells of $M$ as consisting of a $u \times u$ grid of $4 \times 4$ squares. In the construction of $M$, the entries in the $4 \times 4$ square whose top-left cell is $(4i, 4j)$ are determined by whether vertex $v_i$ is incident with the edge $e_j$.
If $v_i$ is incident with $e_j$, and $e_j$ is the $(k+1)$st edge in the ordering $e_0, \dots, e_{u-1}$ that is incident with $v_i$, then the cells of the $4 \times 4$ square whose top-left cell is $(4i, 4j)$ contain some symbols from $A$ and some from $B$. Each cell in the top row contains the symbols $a_{j,0}$ and $a_{j,1}$, and each cell in the bottom row contains the symbols $a_{j,2}$ and $a_{j,3}$. The cells contain the symbols $b_{i,k,l}$, for each $l \in \{0, \dots, 5\}$, and $b_{i,k-1,5}$ (where subtraction is taken modulo 4), in the manner shown in Figure~\ref{4x4}. The right-hand diagram gives a simplified picture, where the positions of the symbols in $B$ are indicated by rectangles; for each symbol in $B$, there is a rectangle whose corners are located in the cells that contain it.
If $v_i$ is not incident with $e_j$, then the cells of the $4 \times 4$ square whose top-left cell is $(4i, 4j)$ contain some symbols from $C$; namely the symbols $c_{i,j,0}$, $c_{i,j,1}$, $c_{i,j,2}$ and $c_{i,j,3}$.
Figure~\ref{rh} gives an example of the a 4-uniform 4-regular hypergraph $H$ and its incidence matrix. Figure~\ref{rh2} gives a simplified illustration of the array $M$, constructed from $H$, where the positions of the symbols from $B$ are indicated by rectangles.
It is easy to verify that the cells in each row of $M$ contain $4u$ symbols, and the cells in each column of $M$ contain $4u$ symbols. Each symbol in $A$ and $C$ belongs to 16 cells, in 4 rows and 4 columns. Each symbol in $B$ belongs to 4 cells, in 2 rows and 2 columns. It is not hard to construct a balanced framework $R$ that has $M$ as its admissible symbol array.
\begin{figure} \begin{center}
\begin{tikzpicture}
\input{halls_slopes.tex}
\begin{scope}[yshift=3cm, xshift=7.2cm, xscale=-1]
\input{halls_slopes.tex}
\end{scope}
\end{tikzpicture} \caption{Symbols in ``red'' position (top) and ``blue'' position (bottom).} \label{rhslope}
\end{center} \end{figure}
\begin{claimx} $R$ can be latinized if and only if $H$ is 2-in-4 colourable. \end{claimx}
\begin{proof} First suppose that $R$ is latinizable, and that $L$ is a latinization of $R$. Consider the symbols $b_{i,k,l}$, where $0 \le i \le u - 1$ is fixed, and $k$ and $l$ range over $\{0, \dots, 3\}$ and $\{0, \dots, 5\}$ respectively. These symbols are naturally associated with the vertex $v_i$. The first point to note is that the position of just one of these symbols in $L$ determines the position of all of them. Moreover, there are just two different ways in which they can be placed, called ``red position'' and ``blue position''. These are illustrated in Figure~\ref{rhslope}, where the dots indicate the positions of symbols from $B$.
We can colour the vertices of $H$ red or blue according to whether their associated symbols in $B$ are in red position or blue position. We claim that this in fact gives a 2-in-4 colouring of $H$. Consider the symbols $a_{j,k}$, where $0 \le j \le u - 1$ is fixed, and $k$ ranges over $\{0, \dots, 3\}$. Each of these symbols must appear four times in the columns $4j, \dots, 4j+3$ of $L$. But this can only happen if amongst the symbols in $B$ corresponding to the vertices incident with $e_j$, two are in red position, and two are in blue position. An example of how they might be placed is illustrated in Figure~\ref{rhcolumn}. Hence from $L$ we can create a 2-in-4 colouring of $H$, and so $H$ is 2-in-4 colorable.
\begin{figure} \begin{center} \begin{tikzpicture}
\tikzstyle{every node}=[font=\small]
\input{halls_column.tex}
\fill[gray!20] (0,0) rectangle +(0.6,0.6)
(0,4.2) rectangle +(0.6,0.6)
(0,4.8) rectangle +(0.6,0.6)
(0,9) rectangle +(0.6,0.6)
(1.2,0) rectangle +(0.6,0.6)
(1.2,4.2) rectangle +(0.6,0.6)
(1.2,4.8) rectangle +(0.6,0.6)
(1.2,9) rectangle +(0.6,0.6)
(0.6,1.8) rectangle +(0.6,0.6)
(0.6,2.4) rectangle +(0.6,0.6)
(0.6,6.6) rectangle +(0.6,0.6)
(0.6,7.2) rectangle +(0.6,0.6)
(1.8,1.8) rectangle +(0.6,0.6)
(1.8,2.4) rectangle +(0.6,0.6)
(1.8,6.6) rectangle +(0.6,0.6)
(1.8,7.2) rectangle +(0.6,0.6);
\draw[gray,step=0.600cm] (0,0) grid (2.400,9.600);
\path (0.3,9.3) node {$a_{j,0}$}
(0.9,6.9) node {$a_{j,0}$}
(1.5,4.5) node {$a_{j,0}$}
(2.1,2.1) node {$a_{j,0}$}
(1.5,9.3) node {$a_{j,1}$}
(2.1,6.9) node {$a_{j,1}$}
(0.3,4.5) node {$a_{j,1}$}
(0.9,2.1) node {$a_{j,1}$}
(0.9,7.5) node {$a_{j,2}$}
(0.3,5.1) node {$a_{j,2}$}
(2.1,2.7) node {$a_{j,2}$}
(1.5,0.3) node {$a_{j,2}$}
(2.1,7.5) node {$a_{j,3}$}
(1.5,5.1) node {$a_{j,3}$}
(0.9,2.7) node {$a_{j,3}$}
(0.3,0.3) node {$a_{j,3}$};
\end{tikzpicture} \caption{A possible arrangement of the symbols $a_{j,k}$ for $0 \le k \le 3$.} \label{rhcolumn}
\end{center} \end{figure}
Conversely, suppose $H$ is 2-in-4 colourable. Then we can choose a 2-in-4 colouring of $H$, and place the symbols of $B$ in red or blue position according to whether their associated vertices in $H$ are red or blue. This leaves empty cells in which the symbols $a_{j,k}$ for all $0 \le j \le u - 1$ and $0 \le k \le 3$ can be placed. The symbols of $C$ can be placed without any difficulty, as the symbols $c_{i,j,0}$, $c_{i,j,1}$, $c_{i,j,2}$ and $c_{i,j,3}$ for a fixed $0 \le i \le u - 1$ and $0 \le j \le u - 1$ can only be placed in the $4 \times 4$ square whose top-left cell is $(4i, 4j)$. We can fill this square with any latin square on these four symbols. Hence $R$ is latinizable. This proves claim 1. \renewcommand{\qedsymbol}{} \end{proof}
We have described the construction of a balanced framework $R$ that is latinizable if and only if $H$ is 2-in-4 colourable. We can now use the procedure given in Theorem~\ref{fr1} to find an L-shaped partial latin square $Q$ of order $n = 4u^2 + 12u$ that realizes $R$. Thus $Q$ is completable if and only if $H$ is 2-in-4 colourable.
It remains to show that $Q$ satisfies \hc. Due to the way in which we constructed $R$, and subsequently $Q$, each empty cell supports either: (i) two symbols from $B$, (ii) one symbol from $B$ and two symbols from $A$, or (iii) four symbols from $C$. Each symbol from $B$ is missing from two rows and columns of $Q$, and each symbol from $A$ and $C$ is missing from four rows and columns of $Q$. Hence the conditions of Theorem~\ref{atleast1t} are satisfied, and $Q$ satisfies \hc.
The computation of the framework $R$ from $H$ can clearly be performed in polynomial time, and as shown in the proof of Theorem~\ref{fr1}, $Q$ can be computed from $R$ in polynomial time.
\end{proof}
A Turing machine equipped with an oracle for deciding Problem~\ref{plshcprob} could decide Problem~\ref{2-in-4} in polynomial time by transforming instances of Problem~\ref{2-in-4} into instances of Problem \ref{plshcprob}, using the reduction given in Lemma~\ref{plsred}, and then calling the oracle. Since Problem \ref{2-in-4} is NP-complete, it follows that Problem \ref{plshcprob} is NP-hard.
There are a number of variants of Problem~\ref{plshcprob} that can be shown to be NP-hard using the reduction given in Lemma~\ref{plsred}. For example, for any $\epsilon > 0$ we have the following two problems.
\begin{problem} Let $P$ be a partial latin square that satisfies \hc\ where the proportion of empty cells is less than $\epsilon$. Decide if $P$ is completable. \label{plsepsilone} \end{problem}
\begin{problem} Let $P$ be a partial latin square that satisfies \hc\ where the proportion of empty cells is greater than $1 - \epsilon$. Decide if $P$ is completable. \label{plsepsilonf} \end{problem}
In fact both these problems are NP-hard.
\begin{theorem} Problems \ref{plsepsilone} and \ref{plsepsilonf} are NP-hard. \label{epsilontheorem} \end{theorem}
\begin{proof} In Lemma~\ref{plsred}, we gave a reduction from Problem~\ref{2-in-4} to Problem~\ref{plshcprob}, that maps a 4-regular 4-uniform hypergraph $H$ on $u$ vertices to an L-shaped partial latin square of order $n = 4u^2 + 12u$, where the upper left $4u \times 4u$ square is empty. So all but a finite number of hypergraphs are mapped to partial latin squares for which the proportion of empty cells is less than $\epsilon$. It follows that Problem~\ref{plsepsilone} is NP-hard.
We can also consider a variant of the reduction given in Lemma~\ref{plsred}, where after constructing the partial latin square we delete the symbols in the bottom right $(n - 4u) \times (n - 4u)$ square. It is not difficult to see that if some symbols are deleted from a partial latin square that satisfies \hc, the resulting partial latin square must also satisfy \hc. So this reduction maps 4-regular 4-uniform hypergraphs to partial latin squares that satisfy \hc, and all but a finite number of hypergraphs are mapped to partial latin squares for which the proportion of empty cells is greater than $1 - \epsilon$. Hence Problem~\ref{plsepsilonf} is also NP-hard. \end{proof}
\section{Concluding remarks}
The most obvious open question is whether Problem~\ref{plshcprob} is in NP, and indeed, whether it is in P. \hc\ can also be defined for graphs whose vertices are equipped with colour lists (see e.g. \cite{hoffmanjohnson}). Problem~\ref{plshcprob} is a special case of the following problem.
\begin{problem} Let $G$ be a perfect graph (see e.g. \cite{bondymurty}) whose vertices have colour lists. Decide if $G$ satisfies \hc.
\end{problem}
It would also be interesting to find more classes of partial latin square for which \hc\ is a necessary and sufficient condition for completability. This has been done to some extent in \cite{bghj}, but there are no doubt more results that could be obtained along these lines. For example, \hc\ may be a necessary and sufficient condition for completability in the case of a partial latin square for which the filled cells form a rectangle with a $2 \times t$ rectangle of empty cells inside.
|
1107.2861
|
\section{Introduction}
The problem of acceleration in physical systems driven by time-periodic
external forces, both in classical and quantum mechanics, has a rather
long history though the results in the latter case are much less complete.
One of the most prominent examples which initiated a lot of efforts
in this field is the so called Fermi accelerator. On the basis of
a theory due to Fermi to explain the acceleration of cosmic rays \cite{Fermi}
Ulam formulated a mathematical model describing a massive particle
bouncing between two infinitely heavy walls while one of the walls
is oscillating \cite{Ulam}. A thorough analysis finally did not fully
confirm the expectations, however \cite{ZaslavskiiChirikov,LiebermanLichtenberg,Pustylnikov}.
The model has also been reformulated in the framework of quantum mechanics
\cite{Karner}.
More models of this sort have been studied in detail so far but we
just mention one of them, the so called electron cyclotron resonance.
One readily finds that electrons in a uniform magnetic field can gain
energy from a microwave electric field whose frequency is equal to
the electron cyclotron frequency. Because of an unlimited energy increase
the relativistic effects cannot be neglected in a complete analysis.
But even the relativistic model admits a quite explicit characterization
of the resonant solution for a transverse circularly polarized electromagnetic
wave propagating along the uniform magnetic field \cite{RobertsBuchsbaum}.
In experimental arrangements the heated electrons are confined in
a magnetic mirror field. Consequently, as they move along a flux tube
of the mirror field they are exposed to the resonance heating only
in a restricted region \cite{Seidl,Grawe,JaegerLichtenbergLieberman}.
This acceleration mechanism is widely used in plasma physics.
Here we wish to discuss, on the quantum level, a model sharing some
features with the preceding one. We again consider a charged particle
placed in a uniform magnetic field. In our model the situation is
simplified, however, in the sense that the particle is confined to
a plane perpendicular to the magnetic field. Instead of a transverse
electromagnetic wave propagating along the uniform field we apply,
as an external force, an oscillating Aharonov-Bohm flux. The frequency
of oscillations $\Omega$ again coincides with the cyclotron frequency
$\omega_{c}$ or, more generally, it may be an integer multiple of
$\omega_{c}$.
The Aharonov-Bohm effect itself received a tremendous attention as
a genuinely quantum phenomenon \cite{AharonovBohm}, and almost all
its possible aspects have been studied in the time-independent case.
For example, a careful analysis can be found in \cite{Ruijsenaars}.
On the other hand, the time-dependent case represents an essentially
more difficult mathematical problem and it has been treated so far
only marginally in a few papers \cite{Lee_etal,AgeevDavydovChirkov,AschHradeckyStovicek,AschStovicek}.
The model we propose has already been studied in the framework of
classical mechanics \cite{AschKalvodaStovicek}. It turns out that
a resonance acceleration again exists but it has some remarkable new
features if compared to the standard electron cyclotron resonance.
If $\Omega$ is an integer multiple of $\omega_{c}$, then the classical
trajectory eventually reaches an asymptotic domain where it resembles
a spiral whose circles pass very closely to the singular flux line
and, at the same time, their radii expand with the rate $t^{1/2}$
as $t$ approaches infinity. The particle moves along the circles
approximately with frequency $\omega_{c}$ while its energy increases
linearly with time. Denoting by $\mathcal{E}(t)$ the energy depending
on time, an important characteristic of the dynamics is the acceleration
rate which is computed in \cite{AschKalvodaStovicek} and is given
by the formula\begin{equation}
\gamma_{\text{acc}}:=\lim_{t\to\infty}\frac{\mathcal{E}(t)}{t}=\frac{e\omega_{c}}{4\pi}\,|\Phi'(\tau)|.\label{eq:acc_rate_class}\end{equation}
Here $\tau$ is a real number which is expressible in terms of some
asymptotic parameters of the trajectory.
The purpose of the current paper is to demonstrate that one can derive
a formula analogous to (\ref{eq:acc_rate_class}) also in the framework
of quantum mechanics. To this end and because of complexity of the
problem, we restrict ourselves to the case when the AB flux depends
on time as a sinusoidal function. To this system we apply the quantum
averaging method getting this way an approximate time evolution for
which we observe a resonance effect whose principal characterization
is again a linear increase of energy.
Let us now be more specific. We consider a quantum point particle
of mass $M$ and charge $e$ moving on the plane in the presence of
a homogeneous magnetic field of magnitude $B$. For definiteness,
all constants $M$, $e$, $B$ are supposed to be positive. Assume
further that the particle is driven by an Aharonov-Bohm magnetic flux
concentrated along a line intersecting the plane in the origin and
whose strength $\Phi(t)$ is oscillating with frequency $\Omega$.
In the time-independent case, the Hamiltonian corresponding to a homogeneous
magnetic field and a constant Aharonov-Bohm flux of magnitude $\Phi_{0}$
reads\[
\frac{\hbar^{2}}{2M}\!\left(-\frac{1}{r}\,\partial_{r}r\partial_{r}+\frac{1}{r^{2}}\!\left(-i\partial_{\theta}-\frac{e\Phi_{0}}{2\pi\hbar}+\frac{eBr^{2}}{2\hbar}\right)^{\!2}\right)\]
where $(r,\theta)$ are polar coordinates on the plane, and the Hilbert
space in question is $L^{2}(\mathbb{R}_{+}\times S^{1},r\mbox{d}r\mbox{d}\theta)$.
Making use of the rotational symmetry of the model we restrict ourselves
to a fixed eigenspace of the angular momentum $J_{3}=-i\hbar\partial_{\theta}$
with an eigenvalue $j_{3}\hbar$, $j_{3}\in\mathbb{Z}$. Put \[
p:=j_{3}-e\Phi_{0}/(2\pi\hbar).\]
Then this restriction leads to the radial Hamiltonian
\begin{equation}
H(p)=\frac{\hbar^{2}}{2M}\!\left(-\frac{1}{r}\,\partial_{r}r\partial_{r}+\frac{1}{r^{2}}\!\left(p+\frac{eBr^{2}}{2\hbar}\right)^{\!2}\right)\label{eq:def_H_p}\end{equation}
in $\mathscr{H}=L^{2}(\mathbb{R}_{+},r\mbox{d}r)$. Without loss of generality,
we can assume that $p>0$ (note that $H(-p)-H(p)$ is a constant).
The boundary conditions at the origin are chosen to be the regular
ones (then $H(p)$ is the so called Friedrichs self-adjoint extension
of the symmetric operator defined on compactly supported smooth functions).
Let us note that if $0<p<1$, then more general boundary conditions
are admissible \cite{ExnerStovicekVytras} but here we confine ourselves
to the above standard choice.
Let \[
\omega_{c}=eB/M\]
be the cyclotron frequency. The operator $H(p)$ has a simple discrete
spectrum, the eigenvalues are\begin{equation}
E_{n}(p)=\hbar\omega_{c}(n+p+1/2),\quad n=0,1,2,\ldots,\label{eq:E_n_p}\end{equation}
with the corresponding normalized eigenfunctions \begin{equation}
\varphi_{n}(p;r)=c_{n}(p)\, r^{p}\, L_{n}^{(p)}\!\left(\frac{eBr^{2}}{2\hbar}\right)\exp\!\left(-\frac{eBr^{2}}{4\hbar}\right)\label{eq:basis_varphi_n}\end{equation}
where \[
c_{n}(p)=\left(\frac{eB}{2\hbar}\right)^{\!(p+1)/2}\left(\frac{2\, n!}{\Gamma(n+p+1)}\right)^{\!1/2}\]
are the normalization constants and $L_{n}^{(p)}$ are the generalized
Laguerre polynomials.
Thus our main goal is to study the time evolution governed by the
periodically time-dependent Hamiltonian $H(a(t))$ where \[
a(t)=p+\epsilon f(\Omega t)\]
and $f(t)$ is a $2\pi$-periodic continuously differentiable function,
$\Omega>0$ is a frequency and $\epsilon$ is a small parameter. This
means that the Aharonov-Bohm flux is supposed to depend on time as\begin{equation}
\Phi(t)=\Phi_{0}-(2\pi\hbar\epsilon/e)\, f(\Omega t).\label{eq:Phi_t}\end{equation}
Without loss of generality one can assume that \begin{equation}
\int_{0}^{2\pi}f(t)\mbox{d}t=0.\label{eq:aver_f_0}\end{equation}
As discussed in \cite{AschHradeckyStovicek}, for the values $0<p<1$
the domain of $H(a(t))$ in fact depends on $t$, and this feature
makes the discussion from the mathematical point of view a bit more
complicated. Nevertheless, the time evolution is still guaranteed
to exist.
\section{The Floquet operator and the quasienergy\label{sec:Floquet_quasienergy}}
Let $U(t,t_{0})$ be the propagator (evolution operator) associated
with $H(a(t))$; it is known to exist \cite{AschHradeckyStovicek}.
An important characteristic of the dynamical properties of the system
is the time evolution over a period which is described by the Floquet
(monodromy) operator $U(T,0)$, with $T=2\pi/\Omega$. We are primarily
interested in the asymptotic behavior of the mean value of energy
\[
\langle U(T,0)^{N}\psi,H(p)U(T,0)^{N}\psi\rangle\]
for an initial condition $\psi$ as $N$ tends to infinity while focusing
on the resonant case when \begin{equation}
\Omega=\mu\omega_{c}\ \,\text{for some}\ \mu\in\mathbb{N}.\label{eq:resonance}\end{equation}
A basic tool in the study of time-dependent quantum systems is the
quasienergy operator \[
K=-i\hbar\partial_{t}+H(a(t))\]
acting in the so called extended Hilbert space which is, in our case,
\[
\mathscr{K}=L^{2}((0,T)\times\mathbb{R}_{+},r\mbox{d}t\mbox{d}r).\]
The time derivative is taken with the periodic boundary conditions.
This approach, very similar to that usually applied in classical mechanics,
makes it possible to pass from a time-dependent system to an autonomous
one. The price to be paid for it is that one has to work with more
complex operators on the extended Hilbert space.
An important property of the quasienergy consists in its close relationship
to the Floquet operator \cite{Howland,Yajima}. In more detail, if
$\psi(t,r)\in\mathscr{K}$ is an eigenfunction or a generalized eigenfunction
of $K$, $K\psi=\eta\psi$, which also implies that $\psi(t+T,r)=\psi(t,r)$,
then the wavefunction $e^{-i\eta t/\hbar}\psi(t,r)$ solves the Schr\"odinger
equation with the initial condition $\psi_{0}(r)=\psi(0,r)$. It follows
that $U(T,0)\psi_{0}=e^{-i\eta T/\hbar}\psi_{0}$. Thus from the spectral
decomposition of the quasienergy one can deduce the spectral decomposition
of the Floquet operator.
Let \[
K_{0}=-i\hbar\partial_{t}+H(p)\]
be the unperturbed quasienergy operator. Its complete set of normalized
eigenfunctions is \[
\{T^{-1/2}e^{im\Omega t}\varphi_{n}(p;r);\, m\in\mathbb{Z},n\in\mathbb{Z}_{+}\}\]
(here $\mathbb{Z}_{+}=\{0,1,2,\ldots\}$ stands for nonnegative integers,
the wave functions $\varphi_{n}(p;r)$ are defined in (\ref{eq:basis_varphi_n})),
with the corresponding eigenvalues $m\hbar\Omega+E_{n}(p)$. Thus
$K_{0}$ has a pure point spectrum which is in the resonant case (\ref{eq:resonance})
infinitely degenerated.
To take into account these degeneracies we perform the following transformation
of indices. Denote by $[x]$ and $\{x\}$ the integer and the fractional
part of a real number $x$, respectively, i.e. $x=[x]+\{x\}$, $[x]\in\mathbb{Z}$
and $0\leq\{x\}<1$. Furthermore, let \[
\rho(\mu,k)=\mu\,\{k/\mu\}\]
be the remainder in division of an integer $k$ by $\mu$. The transformation
of indices is a one-to-one map of $\mathbb{Z}\times\mathbb{Z}_{+}$
onto itself sending $(m,n)$ to $(k,\ell)$, with \begin{equation}
k=k(m,n):=\mu m+n,\ \ell=\ell(m,n):=[n/\mu],\label{eq:indices_kl_mn}\end{equation}
and, conversely,\begin{equation}
m=m(k,\ell):=[k/\mu]-\ell,\ n=n(k,\ell):=\mu\ell+\rho(\mu,k).\label{eq:indices_mn_kl}\end{equation}
Using the new indices $(k,\ell)$ we put\begin{equation}
\Psi_{k,\ell}(p;t,r)=T^{-1/2}\, e^{im(k,\ell)\Omega t}\,\varphi_{n(k,\ell)}(p;r).\label{eq:basis_Psi_kl}\end{equation}
Then the vectors $\Psi_{k,\ell}$, $(k,\ell)\in\mathbb{Z}\times\mathbb{Z}_{+}$,
form an orthonormal basis in the extended Hilbert space $\mathscr{K}$. For
a fixed integer $k\in\mathbb{Z}$ let $P_{k}$ be the orthogonal projection
onto the subspace in $\mathscr{K}$ spanned by the vectors $\Psi_{k,\ell}$,
$\ell\in\mathbb{Z}_{+}$. Then \begin{equation}
K_{0}=\sum_{k\in\mathbb{Z}}\lambda_{k}P_{k}\text{ where }\lambda_{k}=\hbar\omega_{c}(k+p+1/2).\label{eq:def_K0}\end{equation}
Furthermore, using the basis $\{\Psi_{k,\ell}\}$ one can identify
$\mathscr{K}$ with the Hilbert space $\ell^{2}(\mathbb{Z}\times\mathbb{Z}_{+})$.
In particular, partial differential operators in the variables $t$
and $r$ like the quasienergy are identified in this way with matrix
operators. In the sequel we denote matrix operators by bold uppercase
letters.
\section{The quantum averaging method}
The full quasienergy operator $K=K(\epsilon)$ depends on the small
parameter $\epsilon$. Let us write $K(\epsilon)$ as a formal power
series, $K(\epsilon)=K_{0}+\epsilon K_{1}+\epsilon^{2}K_{2}+\ldots$.
In our case,\begin{equation}
K_{1}=f(\Omega t)\hbar\omega_{c}\!\left(\frac{\hbar p}{M\omega_{c}r^{2}}+\frac{1}{2}\right)\!,\ K_{2}=\frac{f(\Omega t)^{2}\hbar^{2}}{2Mr^{2}}\,,\label{eq:def_K1_K2}\end{equation}
and $K_{3}=K_{4}=\ldots=0$. The ultimate goal of the quantum averaging
method in the case of resonances is a unitary transformation resulting
in a partial (block-wise) diagonalization of $K(\epsilon)$. Thus
one seeks a skew-Hermitian operator $W(\epsilon)$ so that $e^{W(\epsilon)}K(\epsilon)e^{-W(\epsilon)}$
commutes with $K_{0}$ which is the same as saying that it commutes
with all projections $P_{k}$. This goal is achievable in principle
through an infinite recurrence which in practice should be interrupted
at some step. Here we shall be content with the first order approximation.
Let us introduce the block-wise diagonal part of an operator $A$
in $\mathscr{K}$ as \[
\mathop\mathrm{diag}\nolimits A:=\sum_{k\in\mathbb{Z}}P_{k}AP_{k}.\]
Thus $\mathop\mathrm{diag}\nolimits A$ surely commutes with $K_{0}$. The off-diagonal part
is then defined as $\mathop\mathrm{offdiag}\nolimits A:=A-\mathop\mathrm{diag}\nolimits A$. Developing formally in
$\epsilon$ one has $W(\epsilon)=\epsilon W_{1}+O(\epsilon^{2})$
and\[
e^{W(\epsilon)}K(\epsilon)e^{-W(\epsilon)}=K_{0}+\epsilon K_{1}+\epsilon\,[W_{1},K_{0}]+O(\epsilon^{2}).\]
Choosing $W_{1}$ as \[
W_{1}=\sum_{k_{1},k_{2},k_{1}\neq k_{2}}(\lambda_{k_{1}}-\lambda_{k_{2}})^{-1}P_{k_{1}}K_{1}P_{k_{2}}\]
one has \begin{equation}
[W_{1},K_{0}]=-\mathop\mathrm{offdiag}\nolimits K_{1}\label{eq:comm_W1_K0}\end{equation}
and \[
e^{W(\epsilon)}K(\epsilon)e^{-W(\epsilon)}=K_{0}+\epsilon\mathop\mathrm{diag}\nolimits K_{1}+O(\epsilon^{2}).\]
Let us note that the solution is also expressible in terms of averaging
integrals, and this explains the name of the method \cite{Scherer_I,JauslinGuerinThomas}.
In more detail, one has\begin{equation}
\mathop\mathrm{diag}\nolimits A=\lim_{\tau\to\infty}\,\frac{1}{\tau}\int_{0}^{\tau}e^{-iuK_{0}/\hbar}A\, e^{iuK_{0}/\hbar}\,\mbox{d}u\label{eq:diagA_int}\end{equation}
and\begin{equation}
W_{1}=\lim_{\tau\to\infty}\,\frac{i}{\hbar\tau}\int_{0}^{\tau}(\tau-u)e^{-iuK_{0}/\hbar}\mathop\mathrm{offdiag}\nolimits(K_{1})e^{iuK_{0}/\hbar}\,\mbox{d}u.\label{eq:W1_int}\end{equation}
After switching on the perturbation, any unperturbed eigenvalue $\lambda_{k}$
gives rise to a perturbed spectrum which, in the first order approximation,
equals the spectrum of the operator $\lambda_{k}P_{k}+\epsilon P_{k}K_{1}P_{k}$
restricted to the subspace $\mathop\mathrm{Ran}\nolimits P_{k}\subset\mathscr{K}$. If the degeneracy
of $\lambda_{k}$ is infinite then the character of the perturbed
spectrum may be arbitrary, depending on the properties of $P_{k}K_{1}P_{k}$.
The corresponding perturbed (generalized) eigenvectors span a subspace
which is the range of the orthogonal projection\begin{eqnarray*}
P_{k}(\epsilon) & := & e^{-W(\epsilon)}P_{k}e^{W(\epsilon)}\,=\, P_{k}-\epsilon\,[W_{1},P_{k}]+O(\epsilon^{2})\\
& = & P_{k}-\epsilon\,(\hat{S}_{k}K_{1}P_{k}+P_{k}K_{1}\hat{S}_{k})+O(\epsilon^{2})\end{eqnarray*}
where\[
\hat{S}_{k}=\sum_{\ell,\ell\neq k}(\lambda_{\ell}-\lambda_{k})^{-1}P_{\ell}\]
is the reduced resolvent of $K_{0}$ taken at the isolated eigenvalue
$\lambda_{k}$. Thus the first order averaging method is in fact nothing
but the standard quantum perturbation method in the first order but
accomplished on the extended Hilbert space simultaneously for all
eigenvalues of $K_{0}$ (compare to \cite[Chp.~II\S2]{Kato}).
Our strategy in the remainder of the paper is based on replacing the
true quasienergy $K(\epsilon)$ by its first order approximation \begin{equation}
K_{(1)}:=K_{0}+\epsilon\mathop\mathrm{diag}\nolimits K_{1}\label{eq:def_K_trunc1}\end{equation}
and, consequently, $U(T,0)$ is replaced by an approximate Floquet
operator $U_{(1)}$ associated with $K_{(1)}$. To determine the approximate
Floquet operator $U_{(1)}$ one has to solve the spectral problem
for $K_{(1)}$. To this end, as already pointed out above, one can
employ the orthonormal basis $\{\Psi_{k\ell}\}$ in order to identify
operators in $\mathscr{K}$ with infinite matrices indexed by $\mathbb{Z}\times\mathbb{Z}_{+}$.
Let $\{e_{k}^{1};k\in\mathbb{Z}\}$ denote the standard basis in $\ell^{2}(\mathbb{Z})$,
and $\{e_{\ell}^{2};\ell\in\mathbb{Z}_{+}\}$ denote the standard
basis in $\ell^{2}(\mathbb{Z}_{+})$. It is convenient to write $\ell^{2}(\mathbb{Z}\times\mathbb{Z}_{+})$
as the tensor product of Hilbert spaces $\ell^{2}(\mathbb{Z})\otimes\ell^{2}(\mathbb{Z}_{+})$
which also means identification of the standard basis in $\ell^{2}(\mathbb{Z}\times\mathbb{Z}_{+})$
with the set of vectors $\{e_{k}^{1}\otimes e_{\ell}^{2};\, k\in\mathbb{Z},\ell\in\mathbb{\mathbb{Z}}_{+}\}$.
Let $\boldsymbol{P}_{k}$ be the orthogonal projection onto the one-dimensional
subspace $\mathbb{C}e_{k}^{1}\subset\ell^{2}(\mathbb{Z})$. Recalling
(\ref{eq:def_K_trunc1}), (\ref{eq:def_K0}) and (\ref{eq:def_K1_K2}),
the matrix $\boldsymbol{K}_{(1)}$ of the operator $K_{(1)}$ expressed
in the basis (\ref{eq:basis_Psi_kl}) takes the form \begin{equation}
\boldsymbol{K}_{(1)}=\sum_{k\in\mathbb{Z}}\boldsymbol{P}_{k}\otimes(\lambda_{k}+\epsilon\boldsymbol{A}_{k})\label{eq:K1aver_matrix}\end{equation}
where $\boldsymbol{A}_{k}$ is the matrix operator in $\ell^{2}(\mathbb{Z}_{+})$
with the entries \begin{equation}
(\boldsymbol{A}_{k})_{\ell_{1},\ell_{2}}=\left\langle \,\Psi_{k,\ell_{1}},K_{1}\Psi_{k,\ell_{2}}\,\right\rangle _{\mathscr{K}}.\label{eq:Ak_matrix}\end{equation}
To compute the matrix entries of $\boldsymbol{A}_{k}$ one observes
that formally (see (\ref{eq:def_H_p}))\begin{equation}
K_{1}=f(\Omega t)\,\partial H(p)/\partial p\label{eq:K1_eq_f_derH}\end{equation}
and so\[
\left\langle \,\Psi_{k,\ell_{1}},K_{1}\Psi_{k,\ell_{2}}\,\right\rangle _{\mathscr{K}}=\mathscr{F}[f](\ell_{2}-\ell_{1})\,\left\langle \varphi_{n(k,\ell_{1})}(p),\left(\partial H(p)/\partial p\right)\varphi_{n(k,\ell_{2})}(p)\right\rangle \]
where \[
\mathscr{F}[f](j)=(2\pi)^{-1}\int_{0}^{2\pi}e^{-ij\cdot s}f(s)\,\mathrm{d}s\]
stands for the $j$th Fourier coefficient of $f$. Recall that, by
the assumption (\ref{eq:aver_f_0}), $\mathscr{F}[f](0)=0$. Moreover, for
$\ell_{1}\neq\ell_{2}$ one has $n(k,\ell_{1})\neq n(k,\ell_{2})$,
hence \begin{equation}
\left\langle \varphi_{n(k,\ell_{1})}(p),H(p)\varphi_{n(k,\ell_{2})}(p)\right\rangle =0.\label{eq:scalar_varphi_H_varphi_0}\end{equation}
In \cite{AschHradeckyStovicek} it is derived that, for $n_{1}\neq n_{2}$,\begin{equation}
\left\langle \varphi_{n_{1}}(p),\frac{\partial\varphi_{n_{2}}(p)}{\partial p}\right\rangle =\frac{1}{2(n_{2}-n_{1})}\,\min\!\left\{ \frac{\gamma(p;n_{2})}{\gamma(p;n_{1})}\,,\frac{\gamma(p;n_{1})}{\gamma(p;n_{2})}\right\} \label{eq:scalar_varphi_der_psi}\end{equation}
where \[
\gamma(p;n)=\big(\Gamma(n+p+1)/n!\big)^{1/2}.\]
Differentiating (\ref{eq:scalar_varphi_H_varphi_0}) with respect
to $p$ and using (\ref{eq:scalar_varphi_der_psi}) one finally obtains
the relation\begin{equation}
(\boldsymbol{A}_{k})_{\ell_{1},\ell_{2}}=\frac{\hbar\omega_{c}}{2}\,\mathscr{F}[f](\ell_{2}-\ell_{1})\,\min\!\left\{ \frac{\gamma(p;n(k,\ell_{2}))}{\gamma(p;n(k,\ell_{1}))}\,,\frac{\gamma(p;n(k,\ell_{1}))}{\gamma(p;n(k,\ell_{2}))}\right\} .\label{eq:Ak_l1l2}\end{equation}
Note that $n(k,\ell)$, as defined in (\ref{eq:indices_mn_kl}), is
$\mu$--periodic in the integer variable $k$, and so is the matrix
$\boldsymbol{A}_{k}$, i.e. $\boldsymbol{A}_{k+\mu}=\boldsymbol{A}_{k}$.
Moreover, since $\mu\omega_{c}=2\pi/T$ one also has $e^{-i\lambda_{k+\mu}T/\hbar}=e^{-i\lambda_{k}T/\hbar}$
(see (\ref{eq:def_K0})). For an integer $s$, $0\leq s<\mu$, let
$\mathscr{H}_{s}$ be the closed subspace in the original Hilbert space $\mathscr{H}=L^{2}(\mathbb{R}_{+},r\mbox{d}r)$
spanned by the vectors $\varphi_{s+j\mu}(r)$, $j=0,1,2,\ldots$.
Then $\mathscr{H}$ decomposes into the orthogonal sum \[
\mathscr{H}=\mathscr{H}_{0}\oplus\mathscr{H}_{1}\oplus\ldots\oplus\mathscr{H}_{\mu-1},\]
and from the relationship between $K_{(1)}$ and $U_{(1)}$, as recalled
in Section~\ref{sec:Floquet_quasienergy}, it follows that every
subspace $\mathscr{H}_{s}$ is invariant with respect to $U_{(1)}$.
In the example which we study in more detail in the following section
(for a sinusoidal function $f(t)$), the matrix operators $\boldsymbol{A}_{s}$
have purely absolutely continuous spectra. For the sake of simplicity
of the notation let us confine ourselves to this case. For a fixed
index $s$, $0\leq s<\mu$, suppose that all generalized eigenvectors
and eigenvalues of $\boldsymbol{A}_{s}$ are parametrized by a parameter
$\theta\in(a_{s},b_{s})$. Let us call them $\boldsymbol{x}_{s}(\theta)$
and $\eta_{s}(\theta)$, respectively, i.e. \[
\boldsymbol{A}_{s}\boldsymbol{x}_{s}(\theta)=\eta_{s}(\theta)\boldsymbol{x}_{s}(\theta),\]
and write \[
\boldsymbol{x}_{s}(\theta)=(\xi_{s;0}(\theta),\xi_{s;1}(\theta),\xi_{s;2}(\theta),\ldots).\]
The generalized eigenvectors $\boldsymbol{x}_{s}(\theta)$ are supposed
to be normalized to the $\delta$ function, i.e. \[
\langle\boldsymbol{x}_{s}(\theta_{1}),\boldsymbol{x}_{s}(\theta_{2})\rangle=\delta(\theta_{1}-\theta_{2}),\]
which in fact means that $\xi_{s;\ell}(\theta)$ as a function in
the variables $\ell\in\mathbb{Z}_{+}$ and $\theta\in(a_{s},b_{s})$
is a kernel of a unitary mapping between the Hilbert spaces $\ell^{2}(\mathbb{Z}_{+})$
and $L^{2}((a_{s},b_{s}),\mbox{d}\theta)$. Thus the spectral decomposition
of $\boldsymbol{A}_{s}$ reads:\[
\forall\boldsymbol{v}\in\ell^{2}(\mathbb{Z}_{+}),\ \boldsymbol{A}_{s}\boldsymbol{v}=\int_{a_{s}}^{b_{s}}\eta_{s}(\theta)\langle\boldsymbol{x}_{s}(\theta),\boldsymbol{v}\rangle\,\boldsymbol{x}_{s}(\theta)\,\mbox{d}\theta.\]
Put \begin{equation}
\Xi_{s}(\theta,r)=\sum_{j=0}^{\infty}\xi_{s;j}(\theta)\,\varphi_{s+j\mu}(p;r).\label{eq:Xi_th_r_def}\end{equation}
Then again,\[
\int_{0}^{\infty}\overline{\Xi_{s}(\theta_{1},r)}\,\Xi_{s}(\theta_{2},r)\, r\mbox{d}r=\delta(\theta_{1}-\theta_{2})\]
and, for all $\psi(r)\in\mathscr{H}_{s}$,\begin{equation}
U_{(1)}\psi(r)=e^{-2\pi i(s+p+1/2)/\mu}\int_{a_{s}}^{b_{s}}e^{-i\epsilon\,\eta_{s}(\theta)T/\hbar}\,\langle\,\Xi_{s}(\theta),\psi\rangle\,\Xi_{s}(\theta,r)\,\mbox{d}\theta.\label{eq:U_approx_def}\end{equation}
To get a correct approximation in the first order of the propagator
one further has to take into account the transformation which is inverse
to that generated by $W($$\epsilon)\approx\epsilon\, W_{1}$. First
observe that $W_{1}$ is a multiplication operator on the Hilbert
space $\mathscr{K}$ in the following sense. Let $S$ be the unitary operator
on $\mathscr{K}$ acting as\[
S\psi(t,r)=e^{i\Omega t}\psi(t,r),\ \forall\psi\in\mathscr{K}.\]
An operator $L$ on $\mathscr{K}$ commutes with $S$ if and only if there
exists a one-parameter $T$-periodic family of operators $\mathcal{L}(t)$
on $L^{2}(\mathbb{R}_{+},r\mbox{d}r)$ such that $L\psi(t,r)=\mathcal{L}(t)\psi(t,r)$.
Notice that \[
S^{-1}K_{0}S=K_{0}+\hbar\Omega.\]
With this equality, it is obvious from (\ref{eq:diagA_int}) that
if $A$ commutes with $S$ then the same is true for $\mathop\mathrm{diag}\nolimits A$. Furthermore,
as one can see from (\ref{eq:def_K1_K2}), $K_{1}$ commutes with
$S$, and from (\ref{eq:W1_int}) one infers that $W_{1}$ commutes
with $S$ as well. Hence there exists a one-parameter $T$-periodic
family of skew-Hermitian operators $\mathcal{W}_{1}(t)$ on $L^{2}(\mathbb{R}_{+},r\mbox{d}r)$
such that \[
W_{1}\psi(t,r)=\mathcal{W}_{1}(t)\psi(t,r),\ \forall\psi\in\mathscr{K}.\]
Next notice that a transformation of the quasienergy operator of the
form $\tilde{K}=e^{\mathcal{W}(t)}Ke^{-\mathcal{W}(t)}$, where again
$\mathcal{W}(t)$ is a $T$-periodic family of skew-Hermitian operators
on $L^{2}(\mathbb{R}_{+},r\mbox{d}r)$, implies a transformation of
the associated propagators according to the rule\[
\tilde{U}(t_{1},t_{2})=e^{\mathcal{W}(t_{1})}U(t_{1},t_{2})e^{-\mathcal{W}(t_{2})}.\]
Hence the correct approximation of the Floquet operator reads\begin{equation}
U(T,0)\approx U_{\text{approx}}=e^{-\epsilon\mathcal{W}_{1}(0)}U_{(1)}e^{\epsilon\mathcal{W}_{1}(0)}.\label{eq:Uapprox}\end{equation}
Let us note, however, that one has, for $N\in\mathbb{N}$ and $\psi\in\mathscr{K}$,\begin{equation}
\langle U_{\text{approx}}^{\ N}\psi,H(p)U_{\text{approx}}^{\ N}\psi\rangle=\langle U_{(1)}^{\ N}\psi_{1},\left(H(p)+\epsilon\,[\mathcal{W}_{1}(0),H(p)]\right)U_{(1)}^{\ N}\psi_{1}\rangle+O(\epsilon^{2})\label{eq:Emean_approx}\end{equation}
where $\psi_{1}=e^{\epsilon\mathcal{W}_{1}(0)}\psi$. If the commutator
$[\mathcal{W}_{1}(0),H(p)]$ happens to be bounded then it does not
contribute to the acceleration rate.
Finally let us indicate how to compute the operator-valued function
$\mathcal{W}_{1}(t)$. One has (here $\varphi_{n}=\varphi_{n}(p;r)$)\begin{equation}
\mathcal{W}_{1}(t)=\sum_{j=-\infty}^{\infty}\,\,\sum_{n_{1},n_{2}=0}^{\infty}e^{i\Omega jt}\, w(j,n_{1},n_{2})\,\langle\varphi_{n_{2}},\cdot\,\rangle\,\varphi_{n_{1}}\label{eq:W1_Fourier}\end{equation}
where\[
w(j,n_{1},n_{2})=\frac{1}{T}\int_{0}^{T}e^{-i\Omega jt}\,\langle\varphi_{n_{1}},\mathcal{W}_{1}(t)\varphi_{n_{2}}\rangle\,\mbox{d}t.\]
The commutator equation (\ref{eq:comm_W1_K0}) is equivalent to the
differential equation\begin{equation}
-i\hbar\,\mathcal{W}_{1}\,'(t)+[H(p),\mathcal{W}_{1}(t)]=\mathop\mathrm{offdiag}\nolimits K_{1}.\label{eq:W1_ODE}\end{equation}
Substituting (\ref{eq:W1_Fourier}) into (\ref{eq:W1_ODE}) and using
(\ref{eq:K1_eq_f_derH}) jointly with (\ref{eq:diagA_int}) one finds
that\begin{equation}
w(j,n_{1},n_{2})=\frac{\mathscr{F}[f](j)}{\hbar\omega_{c}(\mu j+n_{1}-n_{2})}\left\langle \!\varphi_{n_{1}},\frac{\partial H(p)}{\partial p}\,\varphi_{n_{2}}\!\right\rangle \ \,\text{if}\ \,\mu j+n_{1}-n_{2}\neq0\label{eq:w_jn1n2}\end{equation}
and $w(j,n_{1},n_{2})=0$ otherwise.
\section{A sinusoidally time-dependent AB flux}
In the remainder of the paper we discuss the example when $f(t)=\sin(t)$.
The goal of the current section is to provide more details on the
spectral decomposition of the averaged quasienergy $K_{(1)}$ derived
in (\ref{eq:def_K_trunc1}). Naturally, rather than directly with
the quasienergy we shall deal with its matrix, as given in (\ref{eq:K1aver_matrix})
and (\ref{eq:Ak_matrix}).
We still assume that $s\in\{0,1,\ldots,\mu-1\}$ is fixed. For this
choice of $f(t)$, an immediate evaluation of formula (\ref{eq:Ak_l1l2})
gives\[
(\boldsymbol{A}_{s})_{j_{1},j_{2}}=\frac{\hbar\omega_{c}}{4i}\,\delta_{|j_{2}-j_{1}|,1}\mathop\mathrm{sign}\nolimits(j_{2}-j_{1})\left(\prod_{\nu=1}^{\mu}\,\frac{\mu j_{<}+s+\nu}{\mu j_{<}+s+p+\nu}\right)^{\!1/2}\!.\]
where $j_{<}=\min\{j_{1},j_{2}\}$. Thus one has \[
\boldsymbol{A}_{s}=(\hbar\omega_{c}/4)\,\boldsymbol{D}\boldsymbol{J}\boldsymbol{D}^{-1}\]
where $\boldsymbol{J}$ is the Jacobi (tridiagonal) matrix with zero
diagonal,\begin{equation}
\boldsymbol{J}=\left(\begin{array}{ccccc}
0 & \alpha_{0} & 0 & 0 & \ldots\\
\alpha_{0} & 0 & \alpha_{1} & 0 & \ldots\\
0 & \alpha_{1} & 0 & \alpha_{2} & \ldots\\
0 & 0 & \alpha_{2} & 0 & \ldots\\
\vdots & \vdots & \vdots & \vdots & \ddots\end{array}\right)\!,\label{eq:J}\end{equation}
and with the positive entries \[
\alpha_{j}=\left(\prod_{\nu=1}^{\mu}\,\frac{\mu j+s+\nu}{\mu j+s+p+\nu}\right)^{\!1/2},\]
and $\boldsymbol{D}$ is the unitary diagonal matrix with the diagonal
$(1,i,i^{2},i^{3},\ldots)$.
This is an elementary fact that the spectrum of $\boldsymbol{J}$
is simple since any eigenvector or generalized eigenvector is unambiguously
determined by its first entry. Moreover, one readily observes that
the matrices $\boldsymbol{J}$ and $-\boldsymbol{J}$ are unitarily
equivalent, and so the spectrum of $\boldsymbol{J}$ is symmetric
with respect to the origin.
In our case, \[
\alpha_{j}=1-p/(2j)+O(j^{-2})\ \text{ as}\ j\to\infty.\]
Hence $\boldsymbol{J}$ is rather close to the {}``free'' Jacobi
matrix $\boldsymbol{J}_{0}$ for which $\alpha_{0,j}=1$ for all $j$.
The spectral problem for $\boldsymbol{J}_{0}$ is readily solvable
explicitly (see below). It turns out that the spectral properties
of $\boldsymbol{J}$ are close to those of $\boldsymbol{J}_{0}$ as
well \cite{JanasMoszynski}, see also \cite{Teschl}. In particular,
it is known that the singular continuous spectrum of $\boldsymbol{J}$
is empty, the essential spectrum coincides with the absolutely continuous
spectrum and equals the interval $[-2,2\,]$. Furthermore, there are
no embedded eigenvalues, i.e.\ if $\eta$ is an eigenvalue of $\boldsymbol{J}$
then $|\eta|\geq2$.
Splitting $\boldsymbol{J}$ into the sum of the upper triangular and
the lower triangular part, one notes that $\|\boldsymbol{J}\|\leq2\sup\{\alpha_{0},\alpha_{1},\alpha_{2},\ldots\}$.
In our example, $\alpha_{j}\leq1$ for all $j$ and so $\|\boldsymbol{J}\|\leq2$
and, consequently, the spectrum of $\boldsymbol{J}$ is contained
in the interval $[-2,2\,]$. This means that the only possible eigenvalues
of $\boldsymbol{J}$ are $\pm2$. But one can exclude even this possibility.
In fact, suppose that $\boldsymbol{J}\boldsymbol{u}=2\boldsymbol{u}$,
with $\boldsymbol{u}=(u_{0},u_{1},u_{2},\ldots)$ and $u_{0}=1$.
Then \[
\alpha_{j-1}u_{j-1}+\alpha_{j}u_{j+1}=2u_{j}\text{ for }j=0,1,2,\ldots\]
(while putting $u_{-1}=0$). Summing this equality for $j=0,1,\ldots,n$,
and using that $\alpha_{j}\leq1$, one finds that $u_{n+1}\geq u_{n}+1$
for $n=0,1,2,\ldots$. Hence $u_{j}\geq j+1$ for all $j$, and so
$\boldsymbol{u}$ is not square summable. Thus one can summarize that
the spectrum of $\boldsymbol{J}$ is simple, purely absolutely continuous
and equals $[-2,2\,]$.
Let us parametrize the spectrum of $\boldsymbol{A}_{s}=\boldsymbol{A}_{s}(p)$
by a continuous parameter $\theta$, $0<\theta<\pi$, so that \[
\eta(\theta):=(\hbar\omega_{c}/2)\cos(\theta)\]
is a point from the spectrum and $\boldsymbol{x}(p;\theta)$ is the
corresponding normalized generalized eigenvector with components $\xi_{j}(p;\theta)$,
$j=0,1,2,\ldots$ (here we drop the index $s$ at $\boldsymbol{x}$
and $\mathbb{\xi}$ in order to simplify the notation). The asymptotic
behavior of the components $\xi_{j}$ is known \cite{JanasNaboko,BelovRybkin};
one has\begin{equation}
\xi_{j}(p;\theta)\sim A(p;\theta)\, i^{j}\cos\!\big(j\theta-(p/2)\cot(\theta)\log(j+1)+\phi(p;\theta)\big)\label{eq:xi_j_asympt}\end{equation}
for $j\gg0$. Here $A(p;\theta)$ is a normalization constant and
$\phi(p;\theta)$ is a phase which depends on the initial conditions
imposed on the sequence $\{\xi_{j}\}$ (the initial condition is simply
$\xi_{-1}=0$) but the asymptotic methods employed in the cited articles
do not provide an explicit value for it. In the limit case $p=0$
the generalized eigenvectors are known explicitly, namely \[
\xi_{j}(0;\theta)=\sqrt{2/\pi}\, i^{j}\sin((j+1)\theta)\]
for all $j$. Hence $\phi(0;\theta)=\theta-\pi/2$.
The generalized eigenvectors are supposed to be normalized so that
\[
\langle\boldsymbol{x}(p;\theta_{1}),\boldsymbol{x}(p;\theta_{2})\rangle=\delta(\theta_{1}-\theta_{2}).\]
For $p=0$, one can use the equality \[
\sum_{n=1}^{\infty}e^{inx}=\pi\delta(x)-\mathcal{P}\frac{1}{1-e^{-ix}}\]
which is valid for $x=\theta_{1}-\theta_{2}\in(-\pi,\pi)$ and where
the symbol $\mathcal{P}$ indicates the regularization of a nonintegrable
singularity in the sense of the principal value. The normalization
is an immediate consequence of this identity.
For general $p$, the contribution to the $\delta$ function should
come from the most singular and, at the same time, the leading term
in the asymptotic expansion of $\xi_{j}(p;\theta)$, as given in (\ref{eq:xi_j_asympt}).
This time, when investigating the singularity near the diagonal $\theta_{1}=\theta_{2}$
in the scalar product of two generalized eigenvectors, one is lead
to considering the sum \[
\sum_{n=1}^{\infty}n^{iax}e^{inx}\]
where $a=p/(2\sin^{2}\theta_{1})$ is a real constant. Using the Lerch
function $\Phi(z,s,v)$ one has for $|z|<1$ (see \cite[\S~9.55]{GradshteynRyzhik}),\[
\sum_{n=1}^{\infty}n^{s}z^{n}=z\,\Phi(z,s,1)=\Gamma(1-s)\,\sum_{n=-\infty}^{\infty}\left(-\log(z)+2\pi ni\right)^{-1+s}\!.\]
From here one deduces that, for any real $a$,\begin{equation}
\sum_{n=1}^{\infty}n^{iax}e^{inx}=\pi\delta(x)+i\mathcal{P}\frac{1}{x}+g_{a}(x)\label{eq:sum_eq_pi_delta}\end{equation}
where $g_{a}(x)$ is a regular distribution, i.e.\ a locally integrable
function. Hence in the general case, too, the normalization constant
is given by \[
A(p;\theta)=\sqrt{2/\pi}\,.\]
As already mentioned, the phase $\phi(p;\theta)$ in the asymptotic
solution (\ref{eq:xi_j_asympt}) remains undetermined. But we remark
that a bit more can be said about the behavior of the phase near the
spectral point $0$ (the center of the spectrum) which corresponds
to the value of the parameter $\theta=\pi/2$. More precisely, one
can compute the derivative $\partial\phi(p;\pi/2)/\partial\theta$.
Though this result is not directly used in the sequel it represents
an additional information about generalized eigenfunctions of $\boldsymbol{J}$.
We briefly indicate basic steps of the computation in Appendix.
\section{The acceleration rate}
In the case when $f(t)=\sin(t)$ the commutator $[\mathcal{W}_{1}(0),H(p)]$
occurring in (\ref{eq:Emean_approx}) can be shown to be bounded.
This implies that instead of the approximate Floquet operator $U_{\text{approx}}$,
as given in (\ref{eq:Uapprox}), one can work directly with $U_{(1)}$
defined in (\ref{eq:U_approx_def}) when deriving a formula for the
acceleration rate. On the other hand, one should not forget about
the transformation of the initial state, i.e. $\psi_{0}$ has to be
replaced by $e^{\epsilon\mathcal{W}_{1}(0)}\psi_{0}$, see (\ref{eq:Emean_approx}).
First let us shortly discuss the boundedness of the commutator. From
(\ref{eq:W1_Fourier}) and (\ref{eq:w_jn1n2}) while using also (\ref{eq:scalar_varphi_der_psi})
one derives that for $n_{1},n_{2}\in\mathbb{Z}_{+}$, $n_{1}-n_{2}\neq\pm\mu$,\begin{align}
& \langle\varphi_{n_{1}}(p),[H(p),\mathcal{W}_{1}(0)]\varphi_{n_{2}}(p)\rangle\nonumber \\
\noalign{\medskip} & \quad=\,\frac{\hbar\omega_{c}}{4i}\,\frac{n_{1}-n_{2}}{(n_{1}-n_{2})^{2}-\mu^{2}}\,\min\!\left\{ \frac{\gamma(p;n_{2})}{\gamma(p;n_{1})}\,,\frac{\gamma(p;n_{1})}{\gamma(p;n_{2})}\right\} \!.\label{eq:commut_matrix}\end{align}
Of course, the parallels to the diagonal determined by $n_{1}-n_{2}=\pm\mu$
can be explicitly evaluated as well but for our purposes it is sufficient
to know that they are bounded. In \cite[Lemma~6]{AschHradeckyStovicek}
it is shown that the matrix operator \textbf{$\boldsymbol{Q}$} in
$\ell^{2}(\mathbb{Z}_{+})$ with the entries\begin{equation}
\boldsymbol{Q}_{n_{1},n_{2}}=\frac{\hbar\omega_{c}}{4i\,(n_{1}-n_{2})}\,\min\!\left\{ \frac{\gamma(p;n_{2})}{\gamma(p;n_{1})}\,,\frac{\gamma(p;n_{1})}{\gamma(p;n_{2})}\right\} \label{eq:Q_matrix}\end{equation}
for $n_{1}\neq n_{2}$ and $0$ otherwise is bounded. Thus to verify
the boundedness of the commutator it suffices to show that the difference
of matrices (\ref{eq:commut_matrix}) and (\ref{eq:Q_matrix}) has
a finite operator norm. This can be readily done, for example, with
the aid of the following estimate for the norm of a Hermitian matrix
operator $\boldsymbol{B}$ \cite[\S~I.4.3]{Kato},\[
\|\boldsymbol{B}\|\leq\sup_{n_{1}\in\mathbb{Z}_{+}}\,\sum_{n_{2}=0}^{\infty}|\boldsymbol{B}_{n_{1},n_{2}}|.\]
To proceed further, we again fix an integer $s$, $0\leq s<\mu$.
Suppose one is given a function $\varrho(\theta)\in C_{0}^{\infty}((0,\pi))$.
Recalling (\ref{eq:Xi_th_r_def}) we put \begin{equation}
\psi(r)=\int_{0}^{\pi}\Xi_{s}(\theta,r)\varrho(\theta)\,\mbox{d}\theta.\label{eq:varphi_int_varrho}\end{equation}
In what follows, we drop the index $s$ and, whenever convenient,
write simply $H$ instead of $H(p)$. Using (\ref{eq:U_approx_def}),
one has, for $N\in\mathbb{N}$,\begin{align*}
\langle U_{(1)}^{\,\, N}\psi,HU_{(1)}^{\,\, N}\psi\rangle & =\,\int_{0}^{\pi}\!\int_{0}^{\pi}e^{i\epsilon(\cos\theta_{1}-\cos\theta_{2})\omega_{c}TN/2}\\
\noalign{\smallskip} & \qquad\qquad\times\!\langle\,\Xi(\theta_{1}),H\Xi(\theta_{2})\rangle\,\overline{\varrho(\theta_{1})}\varrho(\theta_{2})\,\mbox{d}\theta_{1}\mbox{d}\theta_{2}\\
\noalign{\medskip} & =\,\sum_{j=0}^{\infty}E_{_{s+j\mu}}(p)\left|\int_{0}^{\pi}e^{-i\epsilon\cos(\theta)\omega_{c}TN/2}\xi_{j}(p;\theta)\varrho(\theta)\,\mbox{d}\theta\right|^{2}\!.\end{align*}
Note that $\{\xi_{j}(p;\theta)\}_{j=0}^{\infty}$ is an orthonormal
basis in $L^{2}((0,\pi),\mbox{d}\theta)$ and so \[
\sum_{j=0}^{\infty}\left|\int_{0}^{\pi}e^{-i\epsilon\cos(\theta)\omega_{c}TN/2}\xi_{j}(p;\theta)\varrho(\theta)\,\mbox{d}\theta\right|^{2}=\int_{0}^{\pi}|\varrho(\theta)|^{2}\,\mbox{d}\theta.\]
Hence, in view of (\ref{eq:E_n_p}), the leading contribution to the
acceleration rate comes from the expression\[
\mu\hbar\omega_{c}\sum_{j=0}^{\infty}(j+1)\left|\int_{0}^{\pi}e^{-i\epsilon\cos(\theta)\omega_{c}TN/2}\xi_{j}(p;\theta)\varrho(\theta)\,\mbox{d}\theta\right|^{2}.\]
Furthermore, restricting this sum to an arbitrarily large but finite
number of summands results in an expression which is uniformly bounded
in $N$. This justifies replacement of $\xi_{j}(p;\theta)$ by the
leading asymptotic term, as given in (\ref{eq:xi_j_asympt}) (with
$A(p;\theta)=\sqrt{2/\pi}$). Hence the leading contribution to the
acceleration rate is expressible as\[
\frac{2\hbar\Omega}{\pi}\,\int_{0}^{\pi}\!\int_{0}^{\pi}h(\theta_{1},\theta_{2})e^{i\epsilon(\cos\theta_{1}-\cos\theta_{2})\omega_{c}TN/2}\,\overline{\varrho(\theta_{1})}\varrho(\theta_{2})\,\mbox{d}\theta_{1}\mbox{d}\theta_{2}\]
where\begin{align}
h(\theta_{1},\theta_{2}) & =\sum_{j=0}^{\infty}\,(j+1)\cos\!\left(j\theta_{1}-\frac{p}{2}\,\cot(\theta_{1})\log(j+1)+\phi(p;\theta_{1})\right)\nonumber \\
& \qquad\quad\times\,\cos\!\left(j\theta_{2}-\frac{p}{2}\,\cot(\theta_{2})\log(j+1)+\phi(p;\theta_{2})\right).\label{eq:h_th1_th2}\end{align}
The singular part of the distribution $h(\theta_{1},\theta_{2})$
is supported on the diagonal $\theta_{1}=\theta_{2}$. The sum in
(\ref{eq:h_th1_th2}) can be evaluated analogously as that in (\ref{eq:sum_eq_pi_delta})
with the result\begin{align*}
h(\theta_{1},\theta_{2})=\, & -\frac{1}{2}\,\frac{\partial}{\partial\theta_{2}}\mathcal{P}\frac{1}{\theta_{1}-\theta_{2}}-\frac{\pi}{2}\!\left(\frac{\partial\phi(p;\theta_{1})}{\partial\theta}-1\right)\!\delta(\theta_{1}-\theta_{2})\\
\noalign{\smallskip} & +\,\text{a regular distribution}.\end{align*}
Estimating the acceleration rate we can restrict ourselves to a sufficiently
small but fixed neighborhood of the diagonal with a radius $d>0$.
Thus we arrive at the expression \[
\frac{\hbar\Omega}{\pi}\,\mathcal{P}\!\underset{\substack{\mbox{\ensuremath{\phantom{{s}}}}\\
|\theta_{1}-\theta_{2}|<d}
}{\int_{0}^{\pi}\!\int_{0}^{\pi}}\frac{1}{\theta_{1}-\theta_{2}}\,\frac{\partial}{\partial\theta_{2}}\!\left(e^{-i\epsilon\sin(\theta_{1})(\theta_{1}-\theta_{2})\omega_{c}TN/2}\,\,\overline{\varrho(\theta_{1})}\varrho(\theta_{2})\right)\!\mbox{d}\theta_{1}\mbox{d}\theta_{2}.\]
Further we carry out the differentiation, as indicated in the integrand,
and get rid of the terms which are not proportional to $N$ or which
are non-singular. Moreover, we use the substitution $\theta_{2}=\theta_{1}+u$.
Thus we obtain the expression\begin{align*}
& -\frac{i\epsilon\hbar\Omega\omega_{c}TN}{2\pi}\int_{0}^{\pi}\sin(\theta_{1})|\varrho(\theta_{1})|^{2}\left(\mathcal{P}\int_{-d}^{d}\frac{1}{u}\, e^{i\epsilon\sin(\theta_{1})\omega_{c}TNu/2}\,\mbox{d}u\right)\!\mbox{d}\theta_{1}\\
& =\frac{\epsilon\hbar\Omega\omega_{c}TN}{\pi}\int_{0}^{\pi}\sin(\theta)|\varrho(\theta)|^{2}\!\left(\,\int_{0}^{d}\frac{1}{u}\sin\!\Big(\,\frac{\epsilon}{2}\sin(\theta)\omega_{c}TNu\Big)\mbox{d}u\!\right)\!\mbox{d}\theta.\end{align*}
Finally note that, for any $a$ real, \[
\lim_{N\to\infty}\int_{0}^{d}\frac{1}{u}\,\sin\!\left(aNu\right)\mbox{d}u=\frac{\pi}{2}\,\mathop\mathrm{sign}\nolimits a.\]
Suppose that the initial state is chosen as $e^{-\epsilon\mathcal{W}_{1}(0)}\psi$.
Then we conclude that the formula for the acceleration rate in the
first-order approximation reads\begin{eqnarray}
\gamma_{\text{acc}} & := & \lim_{N\to\infty}\left\langle U_{(1)}^{\, N}\psi,H(p)U_{(1)}^{\, N}\psi\right\rangle \Big/(NT\|\psi\|^{2})\nonumber \\
\noalign{\medskip} & = & \frac{|\epsilon|\hbar\omega_{c}\Omega}{2}\int_{0}^{\pi}\sin(\theta)|\varrho(\theta)|^{2}\,\mbox{d}\theta\bigg/\!\int_{0}^{\pi}|\varrho(\theta)|^{2}\,\mbox{d}\theta.\label{eq:acc_rate_quant}\end{eqnarray}
Here we have used that \[
\|\psi\|^{2}=\int_{0}^{\pi}|\varrho(\theta)|^{2}\,\mbox{d}\theta.\]
Formula (\ref{eq:acc_rate_quant}) can be compared to formula (\ref{eq:acc_rate_class}),
as derived for a classical particle, in the case when $\Phi(t)$ is
given by (\ref{eq:Phi_t}) and $f(t)=\sin(t)$. Then (\ref{eq:acc_rate_class})
gives the acceleration rate \[
\gamma_{\text{acc}}=|\epsilon|\hbar\omega_{c}\Omega\sin(\xi)/2\]
where $\xi\in(0,\pi)$ depends on some data which can be learned from
the asymptotic behavior of the classical trajectory. Let us finally
note that, according to the analysis and discussion of the classical
case presented in \cite{AschKalvodaStovicek}, the first-order averaging
approximation may in fact yield the correct acceleration rate (valid
for the original system), and this is so even if the parameter $\epsilon$
is not necessarily assumed to be very small.
\section{A numerical test}
We conclude our discussion by a presentation of a numerical result
that agree quite nicely with the predicted acceleration rate (\ref{eq:acc_rate_quant}).
For the sake of simplicity we put $\Omega=\omega_{c}=1$, and so $\mu=1$
and $s=0$. We still assume that $f(t)=\sin(t)$. Concerning the physical
constants, we set $\hbar=1$, $e=1$ and $M=1$. Furthermore, we choose
$p=2.5$, $\epsilon=0.4$, and for the density $\varrho(\theta)$
determining an initial state according to (\ref{eq:varphi_int_varrho})
we take the Gaussian function\[
\varrho(\theta)=\left(\frac{20}{\pi}\right)^{\!1/4}\exp\!\big(-10(2-\theta)^{2}+8i\theta\,\big)\]
restricted to the interval $\theta\in(0,\pi)$. Its values near the
limit points of the interval are in fact numerically indistinguishable
from $0$. Particularly, $\varrho(\theta)$ is normalized to unity
with a negligible error, i.e.\[
\int_{0}^{\pi}|\varrho(\theta)|^{2}\,\mbox{d}\theta=1.\]
The numerical method we use is based on expanding a solution of the
time-dependent Schr\"odinger equation with respect to the time-dependent
basis $\{\varphi_{n}(a(t));\, n\in\mathbb{Z}_{+}\}$, with $\varphi_{n}(p)$
being defined in (\ref{eq:basis_varphi_n}). Below we call the solution
of the Schr\"odinger equation $\psi(t)$. Recalling (\ref{eq:Xi_th_r_def})
we put \[
\psi_{0}(r)=\int_{0}^{\pi}\Xi_{0}(\theta,r)\varrho(\theta)\,\mathrm{d}\theta,\]
and we have $\|\psi_{0}\|=1$. The task is to solve the Cauchy problem
for the time-dependent Schr\"odinger equation \[
i\partial_{t}\psi(t)=H(a(t))\psi(t),\ \psi(0)=\tilde{\psi}_{0}:=e^{-\epsilon\mathcal{W}_{1}(0)}\psi_{0}.\]
Let us note that in the case of $f(t)=\sin(t)$ the matrix entries
of $\mathcal{W}_{1}(0)$ are expressed as the finite sum\[
\langle\varphi_{n_{1}}(p),\mathcal{W}_{1}(0)\varphi_{n_{2}}(p)\rangle=w(1,n_{1},n_{2})+w(-1,n_{1},n_{2}),\]
with $w(j,n_{1},n_{2})$ being given in (\ref{eq:w_jn1n2}) (with
$\mu=1$). To carry out the computations we truncate the Fourier expansion
of $\psi(t)$, \[
\psi(t)=\sum_{n=0}^{\infty}x_{n}(t)\varphi_{n}(a(t)),\ x_{n}(t)=\langle\varphi_{n}(a(t)),\psi(t)\rangle,\ n=0,1,\ldots,\]
at some fixed order $n_{\mathrm{max}}$. In this way we obtain a system
of ordinary differential equations for the Fourier coefficients \begin{align*}
ix'_{n}(t) & =E_{n}(a(t))x_{n}(t)-ia'(t)\sum_{j=0}^{n_{\mathrm{max}}}\,\langle\varphi_{n}(a(t)),\varphi'_{j}(a(t))\rangle\, x_{j}(t),\\
x_{n}(0) & =\langle\varphi_{n}(a(0)),\tilde{\psi}_{0}\rangle,\ n=0,1,\ldots,n_{\mathrm{max}}.\end{align*}
Explicit formulas for the scalar products are known from \cite{AschHradeckyStovicek}
(see (\ref{eq:scalar_varphi_der_psi})). In order to approximately
solve this system we employ the explicit Runge-Kutta method of order
4 (RK4) with an adaptive step-size control, and we choose $n_{\mathrm{max}}=120$.
From the computational point of view it is convenient to introduce
the mean value of energy at time $t$ as \[
\mathcal{E}(t):=\langle\psi(t),H(a(t))\psi(t)\rangle.\]
$\mathcal{E}(t)$ is then approximated by the sum\[
\mathcal{E}(t)\approx\sum_{n=0}^{n_{\mathrm{max}}}E_{n}(a(t))|x_{n}(t)|^{2}.\]
The acceleration rate is computed according to formula (\ref{eq:acc_rate_quant})
in which one has to substitute $\psi_{0}$ for $\psi$. Let us point
out that this formula depends only on the time evolution over the
intervals which are integer multiples of the period $T$, and clearly,
$H(a(NT))=H(p)$ for $N=0,1,2,\ldots$. The predicted acceleration
rate for the above particular values of parameters is $\gamma_{\text{acc}}=0.1796$.
The numerically computed function $\mathcal{E}(t)/t$ is compared
to this value in Fig.~1.
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
\section*{Appendix. The phase $\phi(p;\theta)$ near the spectral point $0$}
Here we compute the derivative $\partial\phi(p;\pi/2)/\partial\theta$
of the phase $\phi(p;\theta)$ introduced in (\ref{eq:xi_j_asympt}).
We know that $0$ always belongs to the spectrum of the Jacobi matrix
$\boldsymbol{J}$ introduced in (\ref{eq:J}). Putting $\boldsymbol{u}=(u_{0},u_{1},u_{2},\ldots)$,
with $u_{2j+1}=0$ and\begin{equation}
u_{2j}=(-1)^{j}\,\prod_{k=0}^{j-1}\frac{\alpha_{2k}}{\alpha_{2k+1}}\label{eq:u_2j}\end{equation}
for $j=0,1,2,\ldots$, one has $\boldsymbol{J}\boldsymbol{u}=\boldsymbol{0}$
and $u_{0}=1$. Recalling that, in our example, $\alpha_{j}=1-p/(2j)+O(j^{-2})$
one derives that \[
u_{2j}=(-1)^{j}u_{\infty}\big(1+p/(8j)+O(j^{-2})\big)\quad\text{as\ }j\to\infty,\]
where \[
u_{\infty}=\lim_{j\to\infty}(-1)^{j}u_{2j}\]
is a finite constant (depending on $p$, however). Comparing to (\ref{eq:xi_j_asympt}),
with $A(p;\theta)=\sqrt{2/\pi}$ and $\theta=\pi/2$, one finds that
\[
\boldsymbol{x}(p;\pi/2)=(\sqrt{2/\pi}/u_{\infty})\,\boldsymbol{u}.\]
Moreover, $\phi(p;\pi/2)=0$.
Differentiating the equality \[
\boldsymbol{J}\boldsymbol{x}(p;\theta)=2\cos(\theta)\boldsymbol{x}(p;\theta)\]
with respect to $\theta$ at the point $\pi/2$ and using the substitution\[
\partial\boldsymbol{x}(p;\pi/2)/\partial\theta=-\left(2\sqrt{2/\pi}\big/u_{\infty}\right)\!\boldsymbol{v},\]
with $\boldsymbol{v}=(v_{0},v_{1},v_{2},\ldots)$, one arrives at
the equation $\boldsymbol{J}\boldsymbol{v}=\boldsymbol{u}$. From
(\ref{eq:xi_j_asympt}) one deduces that\begin{equation}
v_{j}\sim\frac{1}{2}\, u_{\infty}\sin\!\left(j\,\frac{\pi}{2}\right)\!\left(j+\frac{p}{2}\,\log(j+1)+\frac{\partial\phi(p;\pi/2)}{\partial\theta}\right)\label{eq:v_j_asympt}\end{equation}
for $j\gg0$. This suggests that one can seek a solution $\boldsymbol{v}$
such that $v_{2j}=0$ for all $j$. This assumption on $\boldsymbol{v}$
is in fact necessary and makes the solution unambiguous since otherwise
one could add to $\boldsymbol{v}$ any nonzero multiple of $\boldsymbol{u}$
thus violating the asymptotic behavior (\ref{eq:v_j_asympt}). Given
that all odd elements of the vector $\boldsymbol{u}$ and all even
elements of $\boldsymbol{v}$ vanish the equation $\boldsymbol{J}\boldsymbol{v}=\boldsymbol{u}$
effectively reduces to a linear system with a lower triangular matrix
which is explicitly solvable. Using (\ref{eq:u_2j}) one can express
the solution as\begin{equation}
v_{2j+1}=\frac{1}{\alpha_{2j}u_{2j}}\,\sum_{k=0}^{j}(u_{2k})^{2},\ \ j=0,1,2,\ldots.\label{eq:v_2j_plus_1}\end{equation}
Noting that\[
\sum_{k=0}^{j}\left(1+\frac{p}{4k+2}\right)=j+1+\frac{p}{4}\big(\log(4j+4)+\gamma_{E}\big)+O(j^{-1}),\]
where $\gamma_{E}$ is the Euler constant, and that \[
(u_{2k})^{2}=u_{\infty}^{\,2}(1+p/(4k)+O(k^{-2}))\]
one derives\begin{align*}
\sum_{k=0}^{j}\left(\frac{u_{2k}}{u_{\infty}}\right)^{\!2}=\,\, & j+1+\frac{p}{4}\,\big(\log(4j+4)+\gamma_{E}\big)\\
& +\,\sum_{k=0}^{\infty}\left(\!\left(\frac{u_{2k}}{u_{\infty}}\right)^{\!2}-1-\frac{p}{4k+2}\right)+O(j^{-1}).\end{align*}
Using (\ref{eq:v_2j_plus_1}) and comparing to (\ref{eq:v_j_asympt})
one finally arrives at the relation\[
\frac{\partial\phi(p;\pi/2)}{\partial\theta}=1+\frac{p}{2}\big(\log(2)+\gamma_{E}\big)+2\sum_{j=0}^{\infty}\!\left(\!\left(\frac{u_{2j}}{u_{\infty}}\right)^{\!2}-1-\frac{p}{4j+2}\right)\!.\]
\section*{Acknowledgments}
The authors wish to acknowledge gratefully partial support from the
following grants: Grant No.\ 201/09/0811 of the Czech Science Foundation
(P.\v{S}.), Grant No.\ LC06002 of the Ministry of Education of the
Czech Republic and Grant No.\ 202/08/H072 of the Czech Science Foundation
(T.K.).
|
1608.08578
|
\section*{Appendix}
\section{Omitted proofs}\label{app:proofs}
\rephrase{Lemma}{\ref{lem:lembitonicdrawing}}{\lembitonicdrawing}
\begin{proof}
The first and second statement hold for every $st$-ordering with $s$ and $t$ on the outer face.
For the third statement assume to the contrary, that for some $1 < k \leq |V|$
the neighbors of a vertex $v$ with $\pi(v) \leq k$
that are in $G - G_k$ do not appear consecutively in the embedding around $v$.
Then $v$ has two successors $w_a, w_c \in S(v)$
with $\pi(w_a) > k$ and $\pi(w_c) > k$. Assume that $w_a$ precedes $w_c$ in $S(v)$, that is $a < c$.
Since all vertices in $S(v)$ appear consecutively in the embedding, there exists
then a third successor $w_b$ between $w_a$ and $w_c$ in $S(v)$ that by our assumption is in $G_{k}$, that is, $\pi(w_b) \leq k$ holds.
Notice that $S(v)$ is of the form $S(v) = \{ \ldots, w_a, \ldots, w_b, \ldots, w_c, \ldots \}$ and
$\pi(w_a) > \pi(w_b) < \pi(w_c)$ holds, which contradicts that $S(v)$ is bitonic with respect to $\pi$.
\qed\end{proof}
\rephrase{Lemma}{\ref{lem:bitonic_alternative}}{\lembitonic}
\begin{proof}
Recall that $A$ is bitonic increasing if and only if
there exists $1 \leq h \leq n$ such that $a_1 < \cdots < a_h > \cdots > a_n$ holds.
We first prove~\lq\lq$\Rightarrow$\rq\rq, that is, if $A$ is bitonic increasing,
then there exists no pair $i,j$ with $1 \leq i < j < n$ and $a_i > a_{i+1} \wedge a_j < a_{j+1}$.
Assume to the contrary that there exists such a pair. Then from $a_i > a_{i+1}$, it follows
that $h \geq i$, and $a_j < a_{j+1}$ yields $j < h$, which contradicts $i < j$.
For~\lq\lq$\Leftarrow$\rq\rq~we choose, if it exists, $h = \min\{j \; | \; a_j > a_{j+1}\}$, otherwise
we set $h = n$. By our choice of $h$, $a_i < a_{i+1}$ holds for every $1 \leq i < h$.
Moreover, for every $h \leq j < n$, it must hold that $a_j > a_{j+1}$,
because otherwise, there exists $1 \leq h < j < n$ with $a_h > a_{h+1} \wedge a_j < a_{j+1}$.
\qed\end{proof}
\rephrase{Proposition}{\ref{pro:bitonic_h}}{\probitonicpaths}
\begin{proof}
We argue the same way as in the proof of Lemma~\ref{lem:bitonic_alternative}.
If there exists no path $v_{i+1} \rightsquigarrow v_{i}$ with $1 \leq i < m$, choose $h = m$.
Then $\forall\: 1 \leq i < m : v_{i+1} \centernot\rightsquigarrow v_{i}$ is satisfied in a trivial way.
If there exists at least one such path, we set ${ h = \min\{ i \mid v_{i+1} \rightsquigarrow v_{i} \} }$ which
satisfies $\forall\: 1 \leq i < h : v_{i+1} \centernot\rightsquigarrow v_{i}$ by construction.
Now assume to the contrary that there exists a path $v_{j} \rightsquigarrow v_{j+1}$ with $h \leq j < m$.
Then there exists $v_{h+1} \rightsquigarrow v_{h}$ and $h \leq j$ holds, which contradicts our assumption that
for every $1\leq i < j < m$, it holds that $ v_{i+1} \centernot\rightsquigarrow v_{i} \vee v_{j} \centernot\rightsquigarrow v_{j+1}$.
\qed\end{proof}
\rephrase{Lemma}{\ref{lem:path_after_split}}{\lempathsplit}
\begin{proof}
Notice that $w,x \in V$ implies $w \neq v'$ and $x \neq v'$. Every path in $G$ that contains $(u,v)$ can use $(u,v'),(v',v)$ in $G'$.
Assume there is a path $w \rightsquigarrow x$ in $G'$ that does not exist in $G$, thus, it contains $(u,v')$ or $(v',v)$.
From $w \neq v' \neq x$, it follows that the path contains both edges, $(u,v')$ and $(v',v)$, and that they appear consecutively.
Hence, $w \rightsquigarrow x$ can use the edge $(u,v)$ in $G$ instead.
\qed\end{proof}
\section{Description of the upward planar straight-line algorithm}\label{app:desc}
In the following, we describe how to adapt the canonical ordering-based planar straight-line algorithm to bitonic $st$-orderings
by borrowing some ideas from Harel and Sardas~\cite{harel98algorithm}.
They first describe a linear-time algorithm to compute a
biconnected canonical ordering.
Then a modification of the algorithm of de~Fraysseix et al.
is used to obtain a planar straight-line layout.
The key observation is that when installing a vertex $v_k$ that has at least two neighbors on the contour $C_{k-1}$,
one can proceed as in the original algorithm.
The only problematic case is the one in which
a vertex $v_k$ has only one neighbor on $C_{k-1}$, say $w_i$.
Harel and Sardas~\cite{harel98algorithm} introduce
the property of having \emph{left, right} and \emph{legal support} for these vertices.
Their solution to the problem is as follows:
If $v_k$ has left support at its only neighbor $w_i$,
then one may use $w_{i-1}$, the predecessor of $w_i$ on $C_{k-1}$,
as a second neighbor for $v_k$ and proceed as in the original
algorithm by pretending that the edge $(v_k, w_{i-1})$ exists.
However, this is only possible, because the property of having left support guarantees that
all edges that have to be attached to $w_i$ later, follow $(v_k, w_i)$ clockwise in the embedding.
Roughly speaking, all edges to be attached later appear to the right of $v_k$, so $v_k$ is
placed to the left of $w_i$ to keep $w_i$ accessible from above.
Similarly, when $v_k$ has right support, every edge incident to $w_i$ that is not yet present
will be attached from the left. Therefore, in case of right support, we may use $w_{i+1}$
as a second neighbor for $v_k$. An example for having right support is given in Fig.~\ref{fig:fpp_bitonic_support_1}.
It is not difficult to see that due to the third statement in Lemma~\ref{lem:lembitonicdrawing},
we can use the idea of Harel and Sardas
to deal with the case in which a vertex has only a single predecessor.
When placing such a vertex, say $v_k$, whose only predecessor is $u$, then
we can assume that $v_k$ is not preceded and followed in $S(u)$ by vertices with a label greater than $k$.
Therefore, the concept of having left and right support translates to bitonic $st$-orderings in the following sense:
$v_k$ has left support (at $u$) if no vertex preceding $v_k$ in $S(u)$ exists with a label greater than $k$.
And in a symmetric manner, $v_k$ has right support, if there is no vertex following $v_k$ in $S(u)$ with a label greater than $k$.
However, one problem arises: The approach by Harel and Sardas requires a vertex with only one neighbor on $C_{k-1}$ to have legal support,
not just left or right support. A quick look at their definition reveals that there is only a difference at the boundary of the contour.
More specifically, if the only predecessor of $v_k$ is $w_1$ (or $w_m$), then $v_k$ must have right support (or left support, respectively).
This is not necessarily the case in a bitonic $st$-ordering,
where it may happen for example that $v_k$ has right support at $w_m$.
Let us assume for a moment that we have to cope with this case in which $v_k$ has right support at $w_m$.
Hence, the edge $(v_k, w_m)$ must have a slope of $+1$,
thus, we are forced to choose $w_l = w_m$, whereas for $w_r$
we are then not able to find an appropriate vertex on $C_{k-1}$.
See Fig.~\ref{fig:fpp_steps_1} for an illustration of the problem of lacking legal support.
\begin{figure}[t]
\centering
\begin{minipage}[b]{0.49\textwidth}
\centering
\subfloat[\label{fig:fpp_steps_1}{}]
{ \includegraphics[page=1]{fpp_steps}}
\end{minipage}\hfill
\begin{minipage}[b]{0.49\textwidth}
\centering
\subfloat[\label{fig:fpp_steps_2}{}]
{\includegraphics[page=2]{fpp_steps}}
\end{minipage}
\caption{(a) The problem of having no legal support at the boundary of the contour $C_{k-1} = \{ w_1, \ldots, w_m \}$. The vertex to place has left support at $w_1$ or right support at $w_m$. (b) Two artificial vertices $v_L, v_R$, one at the beginning and one at the end of $C_{k-1} = \{ v_L = w_1, \ldots, w_m = v_R \}$ may serve as a second neighbor of $v_k$ in $G_{k-1}$.}
\end{figure}
To overcome this problem and without limiting the applicability of our bitonic $st$-ordering,
we make a small modification to the algorithm.
We add two dummy vertices $v_L$ and $v_R$ that take the roles of $v_1$ and $v_2$
in the original algorithm with the property that $v_L$ is always
the first, and $v_R$ always the last vertex in every contour,
that is, for every $1 \leq k \leq n$, $C_k = \{ v_L = w_1, \ldots, w_m = v_R\}$ holds.
Notice that $v_L$ and $v_R$ are isolated vertices,
thus, there exists no $v_k$ whose only predecessor is $v_L$ or $v_R$,
and that has left or right support.
Hence, we are always able to find a second neighbor on $C_{k-1}$ for $v_k$ as depicted in Fig.~\ref{fig:fpp_steps_2}.
\begin{algorithm}[p]
\small
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\SetKwInOut{Variables}{variables}
\SetKwData{decreasing}{decreasing}
\SetKw{KwTrue}{true}
\SetKw{KwFalse}{false}
\SetKw{UpTo}{to}
\SetKw{DownTo}{down to}
\Input{Embedded planar $st$-graph $G=(V,E)$ with successor lists $S(u)$ for every $u \in V$ and bitonic $st$-ordering $\pi$ for $G$.}
\Output{Grid-coordinates for an upward planar straight-line drawing.}
\Begin{
$x(v_L) \gets 0$; $y(v_L) \gets 0$\;
$x(v_1) \gets 1$; $y(v_1) \gets 1$\;
$x(v_R) \gets 2$; $y(v_R) \gets 0$\;
$C_1 \gets \{v_L, v_1, v_R\}$\;
\smallskip
\tcp{bottom-up pass}
\For{$k = 2$ \UpTo $ n$ }
{
$l \gets \min\{i~|~(w_i, v_k) \in E\}$\;
$r \gets \max\{i~|~(w_i, v_k) \in E\}$\;
\smallskip
\tcp{one predecessor case}
\If{l = r}{
$v_p \gets$ preceding vertex of $v_k$ in $S(w_r)$\;
$\textbf{if }v_p = nil \textbf{ or } \pi(v_p) \leq k\textbf{ then }l \gets l - 1$\;
$v_s \gets$ following vertex of $v_k$ in $S(w_r)$\;
$\textbf{if }v_s = nil \textbf{ or } \pi(v_s) \leq k\textbf{ then }r \gets r + 1$\;
}
\smallskip
\tcp{distance $w_l \leftrightarrow w_r$ after shift}
$d \gets 2 + \sum_{i = l+1}^r x(w_i)$\;
\smallskip
\tcp{place $v_k$}
$x(v_k) \gets (d + y(w_r) - y(w_l))/{2}$\;
$y(v_k) \gets (d + y(w_r) + y(w_l))/{2}$\;
\smallskip
\tcp{offset $w_{l+1}, \ldots, w_{r-1} \leftrightarrow v_k$}
$t \gets 1 - x(v_k)$\;
\For{$i = l+1$ \UpTo $r-1$ }{
$\textit{parent}(w_i) \gets v_k$\;
$t \gets t + x(w_i)$\;
$x(w_i) \gets t$\;
}
\smallskip
$x(w_r) \gets d - x(v_k)$\;
$C_{k} \gets$ replace $w_{l+1}, \ldots, w_{r-1}$ in $C_{k-1}$ with $v_k$
}
\smallskip
\For{$i = 2$ \UpTo $|C_n|$ }
{
$x(w_i) \gets x(w_i) + x(w_{i-1})$
}
\smallskip
\tcp{top-down pass}
\For{$k = n$ \DownTo $1$ }
{
$\textbf{if }\text{parent}(v_k) \neq nil\textbf{ then }x(v_k) = x(v_k) + x(parent(v_k))$\;
}
}
\caption{Shifting method for bitonic $st$-orderings}\label{alg:shifting_method}
\end{algorithm}
Now we put these ideas together by describing an algorithm (see Algorithm~\ref{alg:shifting_method}).
We start by placing $v_L, v_1$ and $v_R$ at $(0,0), (1,1)$ and $(2,0)$, respectively.
In every step $2 \leq k \leq n$, we proceed exactly as in the canonical ordering based variant
only the subroutine for determining $w_l$ and $w_r$ has to be adjusted according to the idea
of Harel and Sardas. However, notice that if $v_k$ has left and right support at $w_i$, then $w_l = w_{i-1}$ and $w_r = w_{i+1}$ is chosen.
A complete example is shown in Fig.~\ref{fig:bitonic_algo_example},
in which the drawing for a small graph with seven vertices is created step by step. The output of the algorithm for a larger example is given in Fig.~\ref{fig:bitonic_upward_polyline_output}.
\begin{figure}[t]
\centering
\begin{minipage}[b]{0.2\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_1}{}]
{ \includegraphics[page=1, scale = 0.95]{fpp_bitonic_example}
}
\end{minipage}\hfill
\begin{minipage}[b]{0.18\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_2}{}]
{ \includegraphics[page=2]{fpp_bitonic_example}}
\end{minipage}\hfill
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_3}{}]
{ \includegraphics[page=3]{fpp_bitonic_example}}
\end{minipage}\hfill
\begin{minipage}[b]{0.31\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_4}{}]
{ \includegraphics[page=4]{fpp_bitonic_example}}
\end{minipage}
\begin{minipage}[b]{0.45\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_5}{}]
{ \includegraphics[page=5]{fpp_bitonic_example}}
\end{minipage}\hfill
\begin{minipage}[b]{0.5\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_6}{}]
{ \includegraphics[page=6]{fpp_bitonic_example}}
\end{minipage}\\
\begin{minipage}[b]{0.468\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_7}{}]
{ \includegraphics[scale=0.9, page=7]{fpp_bitonic_example}}
\end{minipage}\hfill
\begin{minipage}[b]{0.53\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_example_8}{}]
{ \includegraphics[scale=0.9,page=8]{fpp_bitonic_example}}
\end{minipage}
\caption{(a) Example graph consisting of seven vertices with a bitonic $st$-ordering.
(b)-(h) Steps during the construction of the drawing. (b) $v_2$ is supported by $v_R$ and serves
in the next step (c) as supporting vertex for $v_3$. (f) $v_5$ uses $v_1$ as support.}\label{fig:bitonic_algo_example}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[angle=90,width=0.75\textwidth]{augmented_out_new}
\caption{Example of an upward planar poly-line drawing of a planar $st$-graph $G = (V,E)$ with $|V| = 16$ and $|E| = 30$.
Circles represent vertices of $G$, whereas squares indicate bends. The labels correspond to the rank in the bitonic $st$-ordering.}\label{fig:bitonic_upward_polyline_output}
\end{figure}
\section{Introduction}
Drawing directed graphs is a fundamental problem in graph drawing and
has therefore received a considerable amount of attention in the past.
Especially the so called \emph{upward planar drawings}, a planar drawing in which the
curve representing an edge has to be strictly $y$-monotone
from its source to target. The directed graphs that admit such a drawing
are called the \emph{upward planar} graphs. Deciding if a directed graph is upward planar
turned out to be {NP}-complete in the general case~\cite{Garg:1995fk}, but there exist special cases
for which the problem is polynomial-time solvable~\cite{Abbasi2010274,Bertolazzi:1994uq,DidimoGL09,hl-uptssad-96,p-uptod-95,DBLP:conf/walcom/SameeR07}. An important result in our context is from Di Battista and Tamassia~\cite{DiBattista1988175}. They show that every upward planar
graph is the spanning subgraph of a planar $st$-graph, that is, a planar directed acyclic graph with a single source
and a single sink. They also show that every such graph has an
upward planar straight-line drawing~\cite{DiBattista1988175}, but it may require exponential area
which for some graphs cannot be avoided~\cite{2014arXiv1410.1006D,Battista:1992fk}.
If one allows bends on the edges, then every upward planar graph can be drawn
within quadratic area.
Di~Battista and Tamassia~\cite{DiBattista1988175} describe an approach that is based
on the visibility representation of a planar $st$-graph. Every
edge has at most two bends, therefore, the resulting drawing has at most $6n-12$ bends with $n$ being the number of vertices.
With a more careful choice of the vertex positions and by employing a special visibility representation,
the authors manage to improve this bound to $(10n - 31)/3$. Moreover, the drawing requires only quadratic area and can be obtained in linear time.
Another approach by Di~Battista~et~al.~\cite{Battista:1992fk} uses an algorithm
that creates a straight-line dominance drawing as an intermediate step.
A dominance drawing, however, has much stronger requirements than an upward planar drawing.
Therefore, the presented algorithm in~\cite{Battista:1992fk} cannot handle planar $st$-graphs directly.
Instead it requires a \emph{reduced planar $st$-graph}, that is, a planar $st$-graph without \emph{transitive edges}.
In order to obtain such a graph, Di~Battista~et~al.~\cite{Battista:1992fk} split every transitive edge
by replacing it with a path of length two. The result is a reduced planar $st$-graph
for which a straight-line dominance drawing is obtained that requires only quadratic area
and can be computed in linear time. Then they reverse the procedure of splitting the edges
by using the coordinates of the inserted dummy vertices as bend points.
Since a planar $st$-graph has at most $2n-5$ transitive edges,
the resulting layout has not more than $2n-5$ bends and at most
one bend per edge. To our knowledge, this bound is the best achieved so far.
These techniques are very different to the ones used in the undirected case.
One major reason is the availability of \emph{canonical orderings} for undirected graphs,
introduced by de Fraysseix~et~al.~\cite{fpp-hdpgg-90} to draw every (maximal)
planar graph straight-line within quadratic area. From there on this concept has been further
improved and generalized~\cite{harel98algorithm,k-dpguc-96,KANT1997175}. Biedl and Derka~\cite{BiedlD15a} discuss various variants
and their relation. Another similar concept that extends to non-planar graphs is the Mondshein sequence~\cite{Schmidt2014}.
However, all these orderings have in common that they do not extend to directed graphs,
that is, for every edge $(u,v)$, it holds that $u$ precedes $v$ in the ordering. An exception
are $st$-orderings. While they are easy to compute for planar $st$-graphs, they lack
a certain property compared to canonical orderings. In~\cite{gronemann14} we introduced
for undirected biconnected planar graphs the \emph{bitonic $st$-ordering}, a special $st$-ordering which has
properties similar to canonical orderings. However, the algorithm in~\cite{gronemann14}
uses canonical orderings for the triconnected case as a subroutine. Since finding a canonical ordering
is in general not a trivial task, respecting the orientation of edges makes it even harder.
Nevertheless, such an ordering is desirable, since one would be able to use incremental drawing approaches
for directed graphs that are usually limited to the undirected case.
In this paper we extend the bitonic $st$-ordering to directed graphs, namely planar $st$-graphs.
We start by discussing the consequences of having such an ordering available.
Based on the observation that the algorithm of de~Fraysseix et al.~\cite{fpp-hdpgg-90} can easily be modified to obtain an
upward planar straight-line drawing, we show that for good reasons
not every planar $st$-graph admits such an ordering.
After deriving a full characterization of the planar $st$-graphs that do admit a bitonic $st$-ordering,
we provide a linear-time algorithm that recognizes these and computes a corresponding ordering.
For a planar $st$-graph that does not admit a bitonic $st$-ordering, we show that splitting at most $n-3$
edges is sufficient to transform it into one for which then an ordering can be found.
Furthermore, a linear-time algorithm is described that determines the smallest set of edges to split.
By combining these results, we are able to draw every planar $st$-graph with at most one bend per edge, $n-3$ bends in total
within quadratic area in linear time. This improves the upper bound on the total number of bends considerably.
\opt{lncs}{Some proofs have been omitted and can be found in the full version~\cite{bitonic-upward-arxiv} or in~\cite{Gronemann15}.}
\opt{arxiv}{Some proofs have been omitted and can be found in Appendix~\ref{app:proofs} or in~\cite{Gronemann15}.}
\section{Preliminaries}
In this work we are solely concerned with a special type of directed graph, the so-called \emph{planar st-graph},
that is, a planar acyclic directed graph $G= (V,E)$ with a single source $s \in V$, a single sink $t \in V$ and no parallel edges.
It should be noted that some definitions assume that $(s,t) \in E$,
we explicitly do not require this edge to be present. However, we assume a fixed embedding scenario
such that $s$ and $t$ are on the outer face. Under such constraints, planar $st$-graphs possess
the property of being \emph{bimodal}, that is, the incoming and outgoing edges
appear as a consecutive sequence around a vertex in the embedding.
Given an edge $(u,v) \in E$, we refer to $v$ as a \emph{successor} of $u$ and call $u$ a \emph{predecessor} of $v$.
Similar to~\cite{gronemann14}, we define for every vertex $u \in V$ a list of successors $S(u) = \{ v_1, \ldots, v_m\}$,
ordered by the outgoing edges $(u,v_1), \ldots, (u,v_m)$ of $u$ as they appear in the embedding clockwise around $u$.
For $S(s)$ we choose $v_1$ and $v_m$ such that $v_m, s, v_1$ appear clockwise on the outer face.
A central problem will be the existence of paths between vertices.
Therefore, we refer to a path from $u$ to $v$ and its existence with $u \rightsquigarrow v \in G$.
With a few exceptions, $G$ is clear from the context, thus, we omit it. If there exists no path
$u \rightsquigarrow v$, we may abbreviate it by writing $u \centernot\rightsquigarrow v$.
Let $G = (V,E)$ be a planar $st$-graph and $\pi : V \mapsto \{1, \ldots, |V| \}$
be the rank of the vertices in an ordering $s = v_1, \ldots, v_n = t$.
$\pi$ is said to be an \emph{$st$-ordering}, if for all edges $(u,v) \in E$, $\pi(u) < \pi(v)$ holds.
In case of a (planar) $st$-graph such an ordering can be obtained in linear time
by using a simple topological sorting algorithm~\cite{Cormen:2009}. We are interested in a special type
of $st$-ordering, the so called bitonic $st$-ordering introduced in~\cite{gronemann14}. We say an ordered sequence
$A = \{ a_1, \ldots, a_n \}$ is \emph{bitonic increasing},
if there exists $1 \leq h \leq n$ such that $a_1 \leq \cdots \leq a_h \geq \cdots \geq a_n$
and \emph{bitonic decreasing}, if $a_1 \geq \cdots \geq a_h \leq \cdots \leq a_n$.
Moreover, we say $A$ is bitonic increasing (decreasing) with respect to a function
$f$, if $A' = \{ f(a_1), \ldots, f(a_n) \}$ is bitonic increasing (decreasing).
In the following, we restrict ourselves to bitonic increasing sequences and abbreviate it by just referring to it as being bitonic.
An $st$-ordering $\pi$ for $G$ is a \emph{bitonic $st$-ordering} for $G$,
if at every vertex $u \in V$ the ordered sequence of successors $S(u) = \{ v_1, \ldots, v_m \}$
as implied by the embedding is bitonic with respect to $\pi$, that is, there exists
$1 \leq h \leq m$ with $\pi(v_1) < \cdots < \pi(v_h) > \cdots > \pi(v_m)$.
Notice that the successors of a vertex are distinct and so are their labels in an $st$-ordering.
\section{Upward planar straight-line drawings \& bitonic \textit{st}-orderings}\label{sec:bitonic_straight_line}
We start by assuming that we are given a planar $st$-graph $G = (V,E)$ together with a bitonic $st$-ordering $\pi$.
The idea is to use the straight-line algorithm from~\cite{gronemann14} which is based on the one
in~\cite{harel98algorithm} to produce an upward planar straight-line layout.
Due to space constraints, we omit details here and only sketch the two modifications that are
necessary. For a full pseudocode listing, an example and a detailed description,
\opt{lncs}{see the full version~\cite{bitonic-upward-arxiv} or~\cite{Gronemann15}.}%
\opt{arxiv}{see Appendix~\ref{app:desc} or~\cite{Gronemann15}.}
When using a bitonic $st$-ordering to drive the planar straight-line algorithm of de~Fraysseix~et~al.~\cite{fpp-hdpgg-90},
the only critical case is the one in which a vertex $v_k$ must be placed that has only one neighbor, say $w_i$, in the subgraph drawn so far.
In~\cite{gronemann14} we use the idea of Harel and Sardas~\cite{harel98algorithm} who guarantee with their ordering that
the edges preceding or following $(w_i, v_k)$ in the embedding around $w_i$ have already been drawn.
Hence one may just pretend that $v_k$ has a second neighbor either to the right or left of $w_i$.
The idea is illustrated in Fig.~\ref{fig:fpp_bitonic_support_1} where $v_k$ uses $w_{i+1}$, the successor of $w_i$ on the contour, as second neighbor.
The following lemma captures the required property and shows that a bitonic $st$-ordering complies with it.
\begin{figure}[t]
\centering
\begin{minipage}[b]{0.48\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_support_1}{}]
{ \includegraphics[page=1]{fpp_property}}
\end{minipage}\hfill
\begin{minipage}[b]{0.48\textwidth}
\centering
\subfloat[\label{fig:fpp_bitonic_support_2}{}]
{\includegraphics[page=2]{fpp_property}}
\end{minipage}
\caption{(a)~A vertex $v_k$ with only one predecessor $w_i$ using the vertex $w_{i+1}$ as second neighbor.
Vertices in grey have not been drawn yet. The two dummy vertices $v_L, v_R$ remain the left- and rightmost ones.
(b)~Example of an upward planar straight-line drawing on seven vertices.}
\label{fig:fpp_bitonic_support}
\end{figure}
\newcommand{\lembitonicdrawing}{
Let $G = (V,E)$ be an embedded planar $st$-graph with a corresponding bitonic $st$-ordering $\pi$.
Moreover, let $v_k$ be the {$k$-th} vertex in $\pi$ and $G_k=(V_k,E_k)$ the subgraph induced by $v_1, \ldots, v_k$.
For every $1 < k \leq |V|$ the following holds:
\begin{enumerate}
\item $G_{k}$ and $G-G_{k}$ are connected,
\item $v_k$ is in the outer face of $G_{k-1}$,
\item For every vertex $v \in V_k$, the neighbors of $v$ that are not in $G_k$
appear consecutively in the embedding around $v$.
\end{enumerate}}
\begin{lemma}\label{lem:lembitonicdrawing}
\lembitonicdrawing
\end{lemma}
\begin{sketchofproof}
The first two properties hold for all $st$-orderings. For the third, assume to the contrary, contradicting that
$S(v)$ is bitonic with respect to $\pi$.
\qed\end{sketchofproof}
Due to the third statement we can always choose a second neighbor either to the left or right, since otherwise the grey vertices in Fig.~\ref{fig:fpp_bitonic_support_1}
would not be consecutive in the embedding around $w_i$.
The second modification solves a problem that arises in the initialization phase
of the drawing algorithm. Recall that in~\cite{fpp-hdpgg-90} the first three vertices are drawn as a triangle.
This of course works in the case of a canonical ordering, but requires extra care when using a bitonic $st$-ordering.
In order to avoid subcases and keep things simple,
we add two isolated dummy vertices $v_L$ and $v_R$ that take the roles of the first two vertices
and pretend to form a triangle with $v_1 = s$. This has another side effect: It avoids distinguishing
between subcases when we have to find a second neighbor at the boundary of the contour, because
$v_L$ is always the first, and $v_R$ always the last vertex on every contour during the incremental construction. See the example in Fig.~\ref{fig:fpp_bitonic_support_2}.
\begin{theorem}\label{thm:bitonic_upward_straightline}
Given an embedded planar $st$-graph $G = (V,E)$ and a corresponding bitonic $st$-ordering $\pi$ for $G$.
An upward planar straight-line drawing for $G$ of size $(2|V|-2) \times (|V|-1)$
can be obtained from $\pi$ in linear time.
\end{theorem}
\begin{proof}
The upward property is obtained by the following observation: The original planar straight-line algorithm
installs every vertex $v_k$ with $k > 2$ above its predecessors. Since we start with $v_L, v_R, v_1$,
the drawing is upward. It remains to bound the area.
Notice that the input consists of the two additional vertices $v_L,v_R$.
The original algorithm, without any area improvements, produces a drawing with a size of
$2((|V|+2) - 4) \times (|V|+2) - 2$ = $2|V| \times |V|$.
However, $v_L$ and $v_R$ are dummy vertices and can be removed anyway.
Moreover, every other vertex is located above them. Hence,
their removal yields a smaller drawing of size $(2|V|-2) \times (|V|-1)$.
\qed\end{proof}
Now the first question that comes to mind is, if we can always find a bitonic $st$-ordering.
Although every planar $st$-graph admits an upward planar straight-line drawing~\cite{DiBattista1988175},
there exist some classes for which it is known that they require exponential area~\cite{2014arXiv1410.1006D,Battista:1992fk}.
Since Theorem~\ref{thm:bitonic_upward_straightline} clearly states that the drawing
requires only polynomial area, these graphs cannot admit a bitonic $st$-ordering.
\begin{corollary}
Not every planar $st$-graph admits a bitonic $st$-ordering.
\end{corollary}
While this had to be expected, we now have to solve an additional problem.
Before we think about how to compute a bitonic $st$-ordering,
we must first be able to recognize planar $st$-graphs that admit such an ordering.
\section{Characterization, recognition \& ordering}
We proceed as follows: As a first step, we identify a necessary condition
that a planar $st$-graph has to meet for admitting a bitonic $st$-ordering.
Then we exploit this condition to compute a bitonic $st$-ordering which proves sufficiency.
We start with an alternative characterization of bitonic sequences.
Since we will use the labels of an $st$-ordering, we can assume that the elements
are pairwise distinct.
\newcommand{\lembitonic}{
An ordered sequence $A = \{ a_1, \ldots, a_n \}$ of pairwise distinct elements is bitonic increasing
if and only if the following holds:
\[
\forall 1 \leq i < j < n \; : \; a_{i} < a_{i+1} \vee a_{j} > a_{j+1}.
\]}
\begin{lemma}\label{lem:bitonic_alternative}
\lembitonic
\end{lemma}
\begin{sketchofproof}
For~\lq\lq$\Rightarrow$\rq\rq, assume to the contrary which yields $i \geq j$.
For~\lq\lq$\Leftarrow$\rq\rq, we choose, if exists, $h = \min\{j \; | \; a_j > a_{j+1}\}$, otherwise
we set $h = n$.\qed
\end{sketchofproof}
In general a planar $st$-graph may have many $st$-orderings,
some of them being bitonic while others are not.
To deal with this in a more formal manner,
we introduce some additional notation.
Given an embedded planar $st$-graph $G = (V,E)$,
we refer with $\Pi(G)$ to all feasible $st$-orderings of $G$, that is,
\[
\Pi(G) = \{ \pi : V \mapsto \{1, \ldots, |V|\} \;| \;\pi \text{ is an $st$-ordering for } G \}.
\]
Furthermore, let $\Pi_b(G)$ be the subset of $\Pi(G)$ that contains
all bitonic $st$-orderings. By definition, we can describe $\Pi_b(G)$ by
\begin{equation*}
\Pi_b(G) = \{ \pi \in \Pi(G) \mid \forall u \in V \; : \; S(u) \text{ is bitonic with respect to } \pi\}.
\end{equation*}
\noindent Applying the alternative characterization of bitonicity from Lemma~\ref{lem:bitonic_alternative}
to the bitonic property of the successor lists $S(u)$ yields the following expression for
the existence of a bitonic $st$-ordering:
\begin{equation}\label{eq:bitonic_st_1}
\begin{split}
\exists \pi \in \Pi_b(G) \Leftrightarrow&\;\exists \pi \in \Pi(G) \quad \forall u \in V \text{ with } S(u)=\{v_1, \ldots, v_m \}\\
&\;\forall\: 1\leq i < j < m \;:\; \pi(v_{i}) < \pi(v_{i+1}) \vee \pi(v_{j}) > \pi(v_{j+1}).
\end{split}
\end{equation}
\begin{figure}[t]
\centering
\begin{minipage}[b]{.39\textwidth}
\centering
\subfloat[\label{fig:bitonic_forbidden_config}{}]
{\includegraphics[page=1, scale=0.9]{bitonic_st_condition_new}}
\end{minipage}
\begin{minipage}[b]{.19\textwidth}
\centering
\subfloat[\label{fig:bitonic_st_condition_1}{}]
{\includegraphics[page=2, scale=0.9]{bitonic_st_condition_new}}
\end{minipage}
\begin{minipage}[b]{.19\textwidth}
\centering
\subfloat[\label{fig:bitonic_st_condition_2}{}]
{\includegraphics[page=3, scale=0.9]{bitonic_st_condition_new}}
\end{minipage}
\begin{minipage}[b]{.19\textwidth}
\centering
\subfloat[\label{fig:bitonic_st_condition_3}{}]
{\includegraphics[page=4, scale=0.9]{bitonic_st_condition_new}}
\end{minipage}
\caption{(a)~A successor list $S(u) = \{ \ldots, v_i, v_{i+1}, \ldots, v_j, v_{j+1}, \ldots \}$ with $i < j$ and a
forbidden configuration of paths $v_{i+1} \rightsquigarrow v_{i}$ and $v_j \rightsquigarrow v_{j+1}$.
(b)-(d)~The three cases at a face between two successors $v_i$ and $v_{i+1}$ of the face-source $u$:
(b)~$v_{i+1}$ is the sink of the face indicating the existence of a path from $v_{i}$ to $v_{i+1}$.
(c)~A path from $v_{i+1}$ to $v_{i}$ results in a face having $v_i$ as sink.
(d)~There exists no path between $v_i$ and $v_{i+1}$, if and only if neither $v_i$ nor $v_{i+1}$ is the face-sink.}\label{fig:bitonic_st_condition_x}
\end{figure}
\noindent Next we translate this expression from $st$-orderings to the existence of paths.
Consider a path from some vertex $u$ to some other vertex $v$ in $G$,
then for every $\pi \in \Pi(G)$, by the definition of $st$-orderings, $\pi(u) < \pi(v)$ holds.
Now it is not hard to imagine that if there exists $\pi \in \Pi_b(G)$, then
there must exist configurations of paths that are forbidden.
To clarify this, let us rewrite the last part of the condition in Equation~\ref{eq:bitonic_st_1}, that is, $\pi(v_{i}) < \pi(v_{i+1}) \vee \pi(v_{j}) > \pi(v_{j+1})$,
using a simple boolean transformation, which yields \mbox{$\neg (\pi(v_{i}) > \pi(v_{i+1}) \wedge \pi(v_{j}) < \pi(v_{j+1}))$}.
So if there exists a path from $v_{i+1}$ to $v_i$ and one from $v_j$ to $v_{j+1}$ with $i < j$,
then this expression evaluates to false for every $\pi \in \Pi(G)$.
Therefore, we may refer to the pair of paths $v_{i+1} \rightsquigarrow v_i$ and $v_j \rightsquigarrow v_{j+1}$ with $i < j$
as a \emph{forbidden configuration} of paths.
See Fig.~\ref{fig:bitonic_forbidden_config} for an illustration.
We may state now that in case there exists a bitonic $st$-ordering, the aforementioned configuration of paths cannot exist:
\begin{equation*}
\begin{split}
\exists \pi \in \Pi_b(G) \Rightarrow&\; \forall u \in V \text{ with } S(u)=\{v_1, \ldots, v_m \}\\
&\;\forall\: 1\leq i < j < m \;:\; v_{i+1} \centernot\rightsquigarrow v_{i} \vee v_{j} \centernot\rightsquigarrow v_{j+1}.
\end{split}
\end{equation*}
Conversely, if we find an $u$ with $v_i$ and $v_j$ in a graph for which these paths exist,
then we can safely reject it as one that does not admit a bitonic $st$-ordering.
The following well-known property of planar $st$-graphs will prove itself useful when it comes to
testing for the existence of a path between two vertices.
\begin{lemma}\label{lem:bitonic_face_paths}
Let $F$ be the subgraph of an embedded planar $st$-graph $G = (V,E)$
induced by a face that is not the outer face\footnote{This restriction is necessary due to the possible absence of the $st$-edge
which is allowed by our definition of planar $st$-graphs.}, and $u,v$ two vertices of
$F$, that is, $u$ and $v$ are on the boundary of the face. Then there exists
a path from $u$ to $v$ in $G$, if and only if there exists such a path in $F$.
\end{lemma}
There are several ways to prove this result, one proof can be found in the
work of de~Fraysseix et al.~\cite{deFraysseix1995157}.
Notice that Lemma~\ref{lem:bitonic_face_paths} is concerned with every pair of vertices incident to the face.
But we are only interested in paths between two consecutive successors $v_{i}$ and $v_{i+1}$
of a vertex $u$. Notice that $v_{i}, v_{i+1}$ and $u$ share a common face which is not the
outer face and in which $u$ is the face-source. Fig.~\ref{fig:bitonic_st_condition_1}-d illustrates
all three possible cases: $v_i \rightsquigarrow v_{i+1}$~\subref{fig:bitonic_st_condition_1}, $v_{i+1} \rightsquigarrow v_{i}$~\subref{fig:bitonic_st_condition_2},
and no path at all~\subref{fig:bitonic_st_condition_3}. Hence, we can decide the existence of a path based on the sink of the common face.
To prove that the absence of forbidden configurations is sufficient for the existence of a bitonic $st$-ordering, we require
the following technical proposition.
\newcommand{\probitonicpaths}{
Given an embedded planar $st$-graph $G = (V,E)$ and a vertex $u \in V$ with successor list $S(u) = \{v_1, \ldots, v_m \}$.
If it holds that
\begin{equation*}
\forall\: 1\leq i < j < m \; : \; v_{i+1} \centernot\rightsquigarrow v_{i} \vee v_{j} \centernot\rightsquigarrow v_{j+1},
\end{equation*}
then there exists $1 \leq h \leq m$ such that
\begin{equation*}
(\forall\: 1 \leq i < h : v_{i+1} \centernot\rightsquigarrow v_{i}) \wedge (\forall\: h \leq i < m : v_{i} \centernot\rightsquigarrow v_{i+1})
\end{equation*}
holds. In other words, there exists at least one $v_h$ in $S(u)$ whose preceding vertices in $S(u)$ are only connected by paths in
clockwise direction, whereas paths between following vertices are directed counterclockwise.}
\begin{proposition}\label{pro:bitonic_h}
\probitonicpaths
\end{proposition}
\begin{sketchofproof}
If exists, set ${ h = \min\{ i \mid v_{i+1} \rightsquigarrow v_{i} \} }$, otherwise set $h = m$.
\qed\end{sketchofproof}
\begin{figure}[t]
\centering
\begin{minipage}[b]{.48\textwidth}
\centering
\subfloat[\label{fig:bitonic_h_example}{}]
{\includegraphics[page=1,scale=0.9]{proposition_lemma}}
\end{minipage}\hfill
\begin{minipage}[b]{.48\textwidth}
\centering
\subfloat[\label{fig:make_bitonic_edges}{}]
{\includegraphics[page=2,scale=0.9]{proposition_lemma}}
\end{minipage}
\caption{(a)~Paths orientations between consecutive successors of $u$.
All of them directed towards $v_h$ as described by Proposition~\ref{pro:bitonic_h}.
(b)~The augmented graph $G'$ in the proof of Lemma~\ref{lem:bitonic_planar_st_graph_ordering}
obtained by adding edges between consecutive successors of $u$ such that they are oriented towards $v_h$.}
\label{fig:bitonic_st_condition_0}
\end{figure}
The idea is now the following: If we have a graph that satisfies our necessary condition,
then we can find for every $u \in V$ with $u \neq t$ a successor $v_h$ with the property
as described in Proposition~\ref{pro:bitonic_h}. The intuition behind this property is that all paths
that exist between successors of $u$, are directed in some way towards $v_h$.
See Fig.~\ref{fig:bitonic_h_example} for an illustration.
The next lemma exploits this property to obtain a bitonic $st$-ordering,
which proves that this condition is indeed sufficient for the existence of a bitonic $st$-ordering.
\begin{lemma}\label{lem:bitonic_planar_st_graph_ordering}
Given a planar $st$-graph $G=(V,E)$ with a fixed embedding.
If at every vertex $u \in V$ with successor list $S(u)=\{v_1, \ldots, v_m \}$ the following holds:
\[
\forall\: 1\leq i < j < m \;:\; v_{i+1} \centernot\rightsquigarrow v_{i} \vee v_{j} \centernot\rightsquigarrow v_{j+1},
\]
then $G$ admits a bitonic $st$-ordering $\pi \in \Pi_b(G)$.
\end{lemma}
\begin{proof}
To show that there exists $\pi \in \Pi_b(G)$, we augment $G$ into a new graph $G'$
by inserting additional edges that we refer to as $E'$.
These edges ensure that between every pair of consecutive successors in $G$, there exists a path in $G' = (V, E \cup E')$.
Afterwards, we show that every $st$-ordering $\pi \in \Pi(G')$ for $G'$ is a bitonic $st$-ordering for $G$.
For every vertex $u$ with successor list $S(u) = \{v_1, \ldots, v_m\}$, we may assume
by Proposition~\ref{pro:bitonic_h} that there exists $1 \leq h \leq m$
such that for every $1 \leq i < h$ there exists no path from $v_{i+1}$ to $v_{i}$, and
for every $h \leq i < m$ no path from $v_{i}$ to $v_{i+1}$ in $G$.
Our goal is to add specific edges to fill the gaps such that there exist two paths in $G'$,
$v_1 \rightsquigarrow v_2 \rightsquigarrow \cdots \rightsquigarrow v_h \in G'$ and
$v_m \rightsquigarrow v_{m-1} \rightsquigarrow \cdots \rightsquigarrow v_h \in G'$.
Fig.~\ref{fig:make_bitonic_edges} illustrates the idea.
More specifically, for every $1 \leq i < m$, there are three cases to consider:
(i)~There already exists a path between $v_i$ and $v_{i+1}$ in $G$, that is,
$v_i \rightsquigarrow v_{i+1} \in G$ or $ v_{i+1} \rightsquigarrow v_{i} \in G$.
Proposition~\ref{pro:bitonic_h} ensures that the path is directed towards $v_h$, thus, we just skip the pair.
(ii)~If there exists no path between $v_i$ and $v_{i+1}$ in $G$ and $i < h$ holds,
we add an edge from $v_i$ to $v_{i+1}$. (iii)~When there also exists no path between
$v_i$ and $v_{i+1}$, but now $h \leq i < m$ holds, we add the reverse edge $(v_{i+1}, v_i)$ to $E'$.
Before we continue, we show that $G' = (V, E \cup E')$ is $st$-planar.
Consider a single edge in $E'$ which has been added either by case (ii) or (iii)
while traversing the successors $S(u)$ of some vertex $u \in V$.
This edge will be added to a face in which $u$ is the source, and since every face
has only one source, only one edge will be added to the corresponding face, hence, planarity is preserved.
Since case (ii) and (iii) only apply, when there exists no path between the two vertices,
adding this edge will not generate a cycle. Induction on the number of added edges yields then $st$-planarity for $G'$.
Consider now an $st$-ordering $\pi \in \Pi(G')$. Since clearly $E' \subseteq E \cup E'$ holds,
$\pi$ is also an $st$-ordering for $G$, that is, $\Pi(G') \subseteq \Pi(G)$ holds.
Recall that we constructed $G'$ such that
for every $u \in V$ with $S(u) = \{v_1, \ldots, v_m\}$,
there exists $v_1 \rightsquigarrow v_2 \rightsquigarrow \cdots \rightsquigarrow v_h \in G'$
and $v_m \rightsquigarrow v_{m-1} \rightsquigarrow \cdots \rightsquigarrow v_h \in G'$.
It follows that for every $\pi \in \Pi(G')$
\[
\forall\: 1 \leq i < h : \pi(v_{i}) < \pi(v_{i+1}) \; \wedge \; \forall\: h \leq i < m : \pi(v_{i}) > \pi(v_{i+1})
\] holds, which implies that $S(u)$ is bitonic with respect to $\pi$.
Since this holds for all $u\in V$, it follows that $\Pi(G') \subseteq \Pi_b(G)$.
Moreover, $G'$ has at least one $st$-ordering, that is, $\Pi(G') \neq \emptyset$, thus, there exists
$\pi \in \Pi_b(G)$.
\qed\end{proof}
Let us summarize the implications of the lemma. The only requirement is that the
graph complies with our necessary condition, that is, the absence of forbidden configurations.
If this is the case, then Lemma~\ref{lem:bitonic_planar_st_graph_ordering}
provides us with a bitonic $st$-ordering, which in turn proves that this condition is sufficient.
\begin{equation*}
\begin{split}
\exists \pi \in \Pi_b(G) \Leftrightarrow&\; \forall u \in V \text{ with } S(u)=\{v_1, \ldots, v_m \}\\
&\;\forall\: 1\leq i < j < m \;:\; v_{i+1} \centernot\rightsquigarrow v_{i} \vee v_{j} \centernot\rightsquigarrow v_{j+1} \\
\end{split}
\end{equation*}
\begin{algorithm}[t]
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\SetKwData{decreasing}{decreasing}
\SetKw{KwTrue}{true}
\SetKw{KwFalse}{false}
\SetKw{UpTo}{to}
\Input{Embedded planar $st$-graph $G = (V,E)$ with $S(u)$ for every $u \in V$.}
\Output{If exists, a bitonic $st$-ordering $\pi$ for $G$.}
\Begin{
$E' \gets \emptyset$\;
\For{$u \in V$ with $S(u) = \{v_1, \ldots, v_m\}$ }
{
$\textit{decreasing} \gets \KwFalse$\;
\For{$i = 1$ \UpTo $m-1$}{
$w \gets \textsc{faceSink}(u, v_i, v_{i+1})$\;
$\textbf{if }w = v_{i+1} \textbf{ and } \textit{decreasing} \text{ \bf then }$\Return \textsc{reject}\;
$\textbf{if }w = v_{i} \textbf{ then }\textit{decreasing}\gets \KwTrue$\;
\If{$v_{i} \neq w \neq v_{i+1}$}{
$\textbf{if }\textit{decreasing}\textbf{ then } E' \gets E' \cup (v_{i+1}, v_{i}) \textbf{ else } E' \gets E' \cup (v_{i}, v_{i+1})$\;
}
}
}
compute $\pi \in \Pi(V, E \cup E')$\;
\Return $\pi$
}
\caption{\small Recognition and ordering algorithm for planar-st graphs}\label{alg:bitonic_recognition}
\end{algorithm}
With a full characterization now at our disposal and in combination with Lemma~\ref{lem:bitonic_face_paths},
we are able to describe a simple linear-time algorithm
(Algorithm~\ref{alg:bitonic_recognition}) which tests a given graph and in case it admits
a bitonic $st$-ordering, computes one. We iterate over $S(u)$ and as long as there is no path $v_{i+1} \rightsquigarrow v_i$,
we assume $i < h$ and fill possible gaps. Once we encounter a path $v_{i+1} \rightsquigarrow v_i$ for the first time,
we implicitly set $h = i$ via the flag and continue to add edges, but now the reverse ones.
But in case we find a path $v_{i} \rightsquigarrow v_{i+1}$, then it forms with $v_{h+1} \rightsquigarrow v_h$ a forbidden configuration
and the graph can be rejected. If we succeed in all successor list, an $st$-ordering for $G'$ is computed,
which is a bitonic one for $G$. Since $G'$ is $st$-planar and has the same vertex set as $G$,
we can claim that the overall runtime is linear. Let us state this as the main result of this section.
\begin{theorem}\label{thm:bitonic_st_ordering_main_1}
Deciding whether an embedded planar $st$-graph $G$ admits a bitonic $st$-ordering $\pi$ or not is linear-time solvable.
Moreover, if $G$ admits such an ordering, $\pi$ can be found in linear time.
\end{theorem}
Next we will consider the case in which no bitonic $st$-ordering exists.
Although our initial motivation was to create upward planar straight-line drawings,
we now allow bends and shift our efforts to upward planar poly-line drawings.
\section{Upward planar poly-line drawings with few bends}
We start with a simple observation. Consider a forbidden configuration
consisting of two paths $v_{i+1} \rightsquigarrow v_{i}$ and $v_{j} \rightsquigarrow v_{j+1}$
with $i < j$ between successors of a vertex $u$ as shown in Fig.~\ref{fig:bitonic_forbidden_config}.
Notice that $(u,v_i)$ and $(u,v_{j+1})$ are transitive edges. Since a reduced planar $st$-graph has
no transitive edges, we can argue the following.
\begin{corollary}
Every reduced planar $st$-graph admits a bitonic $st$-ordering.
\end{corollary}
This leads to the idea to use the same transformation as Di~Battista~et~al.~\cite{Battista:1992fk} in their dominance-based
approach. We can split every transitive edge to obtain a reduced planar $st$-graph and draw it upward planar straight-line.
Replacing the dummy vertices with bends results in an upward planar poly-line drawing with at most
$2|V|-5$ bends, at most one bend per edge and quadratic area.
But we can do better using the following idea: If we have a single forbidden configuration, it suffices to split only one
of the two transitive edges. More specifically, if we split in Fig.~\ref{fig:bitonic_forbidden_config}
the edge $(u,v_i)$ into two new edges $(u,v'_i)$ and $(v'_i, v_i)$ with $v'_i$ being the dummy vertex,
then $v'_i$ replaces $v_i$ in $S(u)$. But now there exists no path from $v_{i+1}$ to $v'_i$, hence, the forbidden configuration
has been destroyed at the cost of one split. Moreover, a pair of transitive edges does not necessarily induce a forbidden configuration.
At this point the question arises how such a split affects other successor lists and if it may even create new forbidden configurations.
The following trivial observation is helpful in this regard.
\newcommand{\lempathsplit}{
Let $G' = (V',E')$ be the graph obtained from splitting an edge $(u,v)$ of a graph $G = (V,E)$ by inserting a dummy vertex $v'$.
More specifically, let $V' = V \cup \{ v' \}$ and $E' = (E - ( u,v )) \cup \{ (u,v'),(v',v) \}$. Then
for all $w, x \in V$ there exists a path $w \rightsquigarrow x \in G$, if and only if there exists a path $w \rightsquigarrow x \in G'$.
}
\begin{lemma}\label{lem:path_after_split}
\lempathsplit
\end{lemma}
Since a forbidden configuration is solely defined by the existence of paths, we can argue now
with Lemma~\ref{lem:path_after_split} that a split does not create nor resolves forbidden configurations
in other successor lists. However, one vertex that is not covered by the lemma is the dummy vertex itself,
but it has only one successor which is insufficient for a forbidden configuration. This locality
is of great value, because it enables us to focus on one successor list, instead of having to deal with a bigger picture.
Next we prove an upper bound on the number of edges to split in order to resolve all forbidden configurations.
\begin{lemma}\label{lem:bitonic_max_bends}
Every embedded planar $st$-graph $G = (V,E)$ can be transformed into a new one
that admits a bitonic $st$-ordering
by splitting at most \mbox{$|V|-3$ edges}.
\end{lemma}
\begin{proof}
Consider a vertex $u$ and its successor list $S(u) = \{ v_1, \ldots, v_m \}$
that contains multiple forbidden configurations of paths. Instead of arguing by means
of forbidden configurations, we use our second condition from Proposition~\ref{pro:bitonic_h},
that is, the existence of a vertex $v_h$ such that every path that exists between
two consecutive successors $v_i$ and $v_{i+1}$,
is directed from $v_i$ towards $v_{i+1}$ for $i < h$, or
from $v_{i+1}$ towards $v_{i}$ if $i \leq h$ holds.
Of course $h$ does not exist due to the forbidden configurations.
But we can enforce its existence by splitting some edges.
Assume that we want $v_h$ to be the first successor, that is, $h = 1$.
Then every path from $v_i$ to $v_{i+1}$ with $1 \leq i < m$ is in conflict with this choice.
We can resolve this by splitting every edge $(u,v_{i+1})$ for which a path $v_i \rightsquigarrow v_{i+1}$ exists.
Clearly, the maximum number of edges to split is at most $m-1$, that is the
case in which for every $1 \leq i < m$, there exists a path from $v_i$ to $v_{i+1}$.
However, there do not exist
paths $v_i \rightsquigarrow v_{i+1}$ and $v_{i+1} \rightsquigarrow v_{i}$ at the same time, because $G$ is acyclic.
So, if the number of edges to split is more than $\frac{m-1}{2}$, then there are less than
$\frac{m-1}{2}$ paths of the form $v_{i+1} \rightsquigarrow v_{i}$. In that case,
we may choose in a symmetric manner $v_h$ to be the last successor ($h = m$),
instead of being the first. Or in other words, we choose $v_h$ to be the first or the last successor,
depending on the direction of the majority of paths.
And as a result, at most $\frac{m-1}{2}$ edges have to be split.
Notice that the overall length of all successor lists is exactly the number of edges in the graph.
Hence, with $m = |S(u)|$ we get $\sum_{u \in V}{|S(u)|} = |E| \leq 3|V| - 6$, and the claimed upper bound can be derived by
\[
\sum_{u \in V}{\frac{|S(u)|-1}{2}} \leq \frac{3|V| - 6 - |V|}{2} = |V| - 3.
\] Moreover, the split procedure preserves $st$-planarity of $G$.
\qed\end{proof}
\begin{figure}[t]
\centering
\begin{minipage}[b]{.33\textwidth}
\centering
\subfloat[\label{fig:bitonic_sharpness_0}{}]
{\includegraphics[page=1, scale=0.85]{bitonic_sharpness}}
\end{minipage}
\begin{minipage}[b]{.66\textwidth}
\centering
\subfloat[\label{fig:bitonic_min_edges}{}]
{\includegraphics[page=1, scale=0.9]{edgesplits}}
\end{minipage}
\caption{(a)~Example of a graph with $|V|-3$ forbidden configurations, each requiring one split to be resolved.
(b)~Example for finding the smallest set of edges to split. The numbers indicate how many splits are
necessary when choosing the corresponding vertex to be $v_h$.
For $v_5, v_6, v_8$ and $v_9$ only two splits are necessary.
Choosing $h = 6$ results in $E_{\text{split}} = \{ (u,v_1), (u,v_8)\}$.
The squares indicate the result of the two splits, whereas the dotted edges represent $E'$ in Algorithm~\ref{alg:bitonic_recognition}.}\label{fig:bitonic_sharpness_1}
\end{figure}
One may wonder now if this bound can be improved. Unfortunately, the
graph shown in Fig.~\ref{fig:bitonic_sharpness_0} is an example that requires $|V| - 3$ splits, hence, the bound is tight.
It also shows that there exist graphs that can be drawn upward planar straight-line in polynomial area but do not admit a bitonic $st$-ordering.
But we will push the idea of splitting edges a bit further from a practical point of view, and
focus on the problem of finding a minimum set of edges to split.
In the following we describe an algorithm that solves this problem in linear time.
To do so, we introduce some more notation.
Let $u \in V$ be a vertex with successor list $S(u) = \{v_1, \ldots, v_m\}$.
We define
$L(u,h) = |\{ i < h : v_{i+1} \rightsquigarrow v_{i} \}|$ and $R(u,h) = |\{ i < h : v_{i} \rightsquigarrow v_{i+1} \}|.$
If we choose now a particular $1 \leq h \leq m$ at $u$, then we have to split
every edge $(u,v_{i+1})$ with $i < h$ for which there exists a path $v_{i+1} \rightsquigarrow v_{i}$, and
every edge $(u,v_{i})$ with $h \leq i$ for which $G$ contains a path $v_{i} \rightsquigarrow v_{i+1}$,
that is, we have to split $L(u,h) + R(u,m) - R(u,h)$ edges. See Fig.~\ref{fig:bitonic_min_edges} for an example.
When now considering all successor lists,
the minimum number of edge splits is
\[
\sum_{u \in V} \left(R(u,m) + \min_{1 \leq h \leq m }\left\{ L(u,h) - R(u,h)\right\} \right).
\]
Notice that the locality of a split allows us to minimize the number of edge splits for every successor list independently.
From an algorithmic point of view, we are interested in the value of $h$ and not in the number of splits, hence, we may drop $R(u,m)$
and consider the problem of finding $h$ for which $L(u,h) - R(u,h)$ is minimum.
Since this is now only a matter of counting paths for which we can again
exploit Lemma~\ref{lem:bitonic_face_paths}, a linear-time algorithm becomes straightforward (see Algorithm~\ref{alg:bitonic_min_split}).
And as a result, we may state the following lemma without proof.
\begin{algorithm}[t!]
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\SetKwData{decreasing}{decreasing}
\SetKw{KwTrue}{true}
\SetKw{KwFalse}{false}
\SetKw{UpTo}{to}
\Input{Embedded planar $st$-graph $G = (V,E)$ with $S(u)$ for every $u \in V$.}
\Output{Minimum set $E_{\text{split}} \subset E$ to split for admitting a bitonic $st$-ordering.}
\Begin{
$E_{\text{split}} \gets \emptyset$\;
\For{$u \in V$ with $S(u) = \{v_1, \ldots, v_m\}$ }
{
$h \gets 1$\;
$c_{\min} \gets c \gets 0$\;
\For{$i = 2$ \UpTo $m$}{
$w \gets \textsc{faceSink}(u, v_{i-1}, v_{i})$\;
$\textbf{if }w = v_{i-1} \textbf{ then } c \gets c + 1$\;
$\textbf{if }w = v_{i} \textbf{ then } c \gets c - 1$\;
\If{$c < c_{\min}$}{
$c_{\min} \gets c$\;
$h \gets i$\;
}
}
\For{$i = 1$ \UpTo $h-1$}{
$\textbf{if }v_{i} = \textsc{faceSink}(u, v_i, v_{i+1}) \textbf{ then } E_{\text{split}} \gets E_{\text{split}} \cup (u, v_{i}) $\;
}
\For{$i = h$ \UpTo $m-1$}{
$\textbf{if }v_{i+1} = \textsc{faceSink}(u, v_i, v_{i+1}) \textbf{ then } E_{\text{split}} \gets E_{\text{split}} \cup (u, v_{i+1}) $\;
}
}
\Return $E_{\text{split}}$
}
\caption{\small Algorithm for computing the minimum set of edges to split.}\label{alg:bitonic_min_split}
\end{algorithm}
\begin{lemma}\label{lem:bitonic_min_split}
Every embedded planar $st$-graph $G = (V,E)$ can be transformed into a planar $st$-graph that admits
a bitonic $st$-ordering by splitting every edge at most once.
Moreover, the minimum number of edges to split is at most $|V| - 3$ and they can be found
in linear time.
\end{lemma}
\noindent Now we may use this to create upward planar poly-line drawings with few bends.
\begin{theorem}\label{thm:bitonic_min_split}
Every embedded planar $st$-graph $G = (V,E)$ admits an upward planar poly-line drawing
within quadratic area having at most one bend per edge, at most $|V|-3$ bends in total, and
such a drawing can be obtained in linear time.
\end{theorem}
\begin{proof}
We use Lemma~\ref{lem:bitonic_min_split} to obtain a new planar $st$-graph $G'=(V',E')$ with
$|V'| \leq 2|V| -3$ and a corresponding bitonic $st$-ordering $\pi$ with Algorithm~\ref{alg:bitonic_recognition}.
With Theorem~\ref{thm:bitonic_upward_straightline},
an upward planar straight-line layout of size $(2|V'|-2) \times (|V'|-1)$ for $G'$ is computed. Replacement of the dummy vertices by bends,
yields an upward planar poly-line drawing for $G$ of size at most $(4|V|-8) \times (2|V|-4)$.
\qed\end{proof}
\noindent Recall that every upward planar graph is a spanning subgraph of a planar $st$-graph~\cite{DiBattista1988175}.
Therefore, the bound of $|V|-3$ translates to all upward planar graphs.
\begin{corollary}
Every upward planar graph $G=(V,E)$ admits an upward planar poly-line drawing
within quadratic area having at most one bend per edge and at most $|V|-3$ bends in total.
\end{corollary}
\section{Conclusion}\label{se:conclusions}
In this work we have introduced the bitonic $st$-ordering for planar $st$-graphs.
Although this technique has its limitations, it provides the properties of canonical orderings for the directed case.
We have shown that this concept is viable by using a classic undirected incremental drawing algorithm for creating
upward planar drawings with few bends.
\bibliographystyle{splncs03}
|
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